Ma sa r yk Un iv e rs i t y
Faculty of Economics and Administration
Major: Finance
ARBITRAGE OPPORTUNITIES IN SELECTED
FINANCIAL MARKETS OF EAST ASIA
Arbitrážní příležitosti na vybraných finančních trzích
Východní Asie
Diploma thesis
Advisor:
Author:
Ing. Boris Šturc, CSc.
Pavel WERL
Brno, 2014
Masarykova univerzita
Ekonomicko-správní fakulta
Katedra financí
Akademický rok 2013/2014
ZADÁNÍ DIPLOMOVÉ PRÁCE
Pro:
WERL Pavel
Obor:
Finance
Název tématu:
ARBITRÁŽNÍ PŘÍLEŽITOSTI NA VYBRANÝCH
FINANČNÍCH TRZÍCH VÝCHODNÍ ASIE
Arbitrage opportunities in selected financial markets of East
Asia
Zásady pro vypracování:
Cíl práce:
Hlavním cílem práce je retrospektivně posoudit arbitrážní příležitosti vybraných investičních
instrumentů, obchodovaných na finančních trzích ve Východní Asii. Dílčím cílem je pak
navrhnout vhodný algoritmus pro identifikaci a měření arbitrážních příležitostí z burzovních
dat. Dalším dílčím cílem je aplikovat zvolený algoritmus na reálná data z finančních trhů
Východní Asie a vyhodnotit zjištěné souvislosti.
Postup a použité metody:
 sběr dat,
 analýza dat,
 navržení algoritmu pro identifikaci dat,
 syntéza a dedukce.
Rozsah grafických prací:
dle pokynů vedoucího práce
Rozsah práce bez příloh:
60 – 80 stran
Seznam odborné literatury:
REJNUŠ, Oldřich. Finanční trhy. Třetí rozšířené. Ostrava: Key Publishing, 2011. 690 s.
Ekonomie. ISBN 978-80-7418-128-3.
JÍLEK, Josef. Finanční trhy a investování. 1. vyd. Praha: Grada, 2009. 648 s. ISBN 978-80247-1653.
SHARPE, William F. a Gordon J. ALEXANDER. Investice. Translated by Zdeněk Šlehofr. 4.
vyd. Praha: Victoria Publishing, 1994. 810 s. ISBN 80-85605-47-3.
POLOUČEK, Stanislav. Peníze, banky, finanční trhy. Vyd. 1. Praha: C.H. Beck, 2009. xvii,
415. ISBN 978-80-7400-152.
PLUMMER, Tony. Prognóza finančních trhů :psychologie úspěšného investování. Vyd. 1.
Brno: Computer Press, 2008. xvi, 373 s. ISBN 978-80-251-1592.
Vedoucí diplomové práce:
Ing. Boris Šturc, CSc.
Datum zadání diplomové práce:
5. 3. 2013
Termín odevzdání diplomové práce a vložení do IS je uveden v platném harmonogramu
akademického roku.
……………………………………
vedoucí katedry
V Brně dne 5. 3. 2013
…………………………………………
děkan
Aut hor’s ful l nam e :
Pavel Werl
Topi c :
Arbitrage opportunities in selected financial markets of
East Asia
Náz ev di pl om ové práce :
Arbitrážní příležitosti na vybraných finančních trzích
Východní Asie
Depart m ent :
Finance
Advi s or :
Ing. Boris Šturc, CSc.
Year o f t hes i s def en se :
2014
Annotation
The goal of this paper is to develop algorithms to detect and evaluate arbitrage opportunities on
financial data from East Asia. The first part is concentrated on thorough literature research on
most types of arbitrage strategies, including the development of methods to exploit the arbitrage
opportunities. Among those strategies are triangular arbitrage, cash-and-carry arbitrage,
volatility arbitrage, convertible arbitrage, fixed-income arbitrage, and others. In the second part,
after describing acquired data from East Asian forex markets, I propose a software algorithm
to detect multilateral arbitrage opportunities on foreign exchange markets, and another
algorithm to generate all combinations of foreign exchange arbitrage loops. Subsequently, I
develop a custom computer program to analyze the market data in automated fashion, including
the computerized generation of Excel reports and charts. Finally, I evaluate and interpret the
results of the analysis, and conclude.
Anotace
Cílem této práce je vyvinout algoritmy na detekci a vyhodnocení arbitrážních příležitostí na
finančních datech z Východní Asie. První část se zaměřuje na podrobný výzkum literatury o
většině typů arbitrážních strategií, včetně navrhnutí metod pro využití arbitrážních příležitostí.
Mezi zmíněnými strategiemi je triangulární arbitráž, cash-and-carry arbitráž, arbitráž volatility,
konvertibilní arbitráž, arbitráž instrumentů s fixními příjmy, a další. V druhé části, po popisu
získaných dat z východoasijských měnových trhů, navrhnu softwarový algoritmus na detekci
multilaterálních arbitrážních příležitostí na měnových trzích, a další algoritmus na generování
všech kombinací arbitrážních měnových smyček. Poté vyvinu vlastní počítačový program k
automatizované analýze tržních dat, a to včetně následného komputerizovaného generování
výstupu v Excelu včetně grafů. Nakonec vyhodnotím a interpretuji výsledky analýzy a vyvodím
závěr.
Keywords
Arbitrage, triangular arbitrage, multilateral arbitrage, East Asia, statistical arbitrage, pricing
model, risk, algorithm, programming, implementation, forex
Klíčová slova
Arbitráž, triangulární arbitráž, multilaterální arbitráž, Východní Asie, statistická arbitráž,
oceňovací model, riziko, algoritmus, programování, implementace, forex
Prohlášení
Prohlašuji, že jsem diplomovou práci Arbitrážní příležitosti na vybraných finančních trzích
Východní Asie vypracoval samostatně pod vedením Ing. Boris Šturc, CSc. a uvedl v ní všechny
použité literární a jiné odborné zdroje v souladu s právními předpisy, vnitřními předpisy
Masarykovy univerzity a vnitřními akty řízení Masarykovy univerzity a Ekonomicko-správní
fakulty MU.
Proclamation
I declare that the diploma thesis titled Arbitrage opportunities in selected financial markets of
East Asia was written by me and only me, under the supervision of Ing. Boris Šturc, CSc.
Furthermore, I declare that I have referenced and properly cited all sources used in this paper,
in accordance to the contemporary law, Masaryk University regulations, and internal acts of
both Masaryk University and Faculty of Business and Administration MU.
V Brně dne 16. května 2014
In Brno on May 16th, 2014
vlastnoruční podpis autora | signature
TABLE OF CONTENTS
INTRODUCTION ................................................................................................................................................. 1
GOALS.................................................................................................................................................................. 1
METHODOLOGY ................................................................................................................................................... 1
NOTATIONS AND PRELIMINARIES .............................................................................................................. 2
TRADE PAIRS NOTATION ....................................................................................................................................... 2
FOREIGN EXCHANGE (CURRENCY) NOTATIONS..................................................................................................... 3
Currency codes ................................................................................................................................................ 3
Quotations ....................................................................................................................................................... 3
1
DEFINITION OF ARBITRAGE ................................................................................................................. 5
2
GENERAL IDEA BEHIND ARBITRAGE ................................................................................................ 6
2.1
2.2
3
INTEREST RATES ....................................................................................................................................... 7
OTHER ASPECTS ....................................................................................................................................... 7
RISKS OF ARBITRAGE ............................................................................................................................. 8
3.1
LIQUIDITY RISK ........................................................................................................................................ 8
3.2
COUNTERPARTY RISK ............................................................................................................................... 9
3.3
TECHNICAL AND EXECUTION RISK ............................................................................................................ 9
3.3.1
Processor speed ............................................................................................................................. 10
3.3.2
Network latency ............................................................................................................................. 10
3.3.3
Software or hardware malfunction ................................................................................................ 11
3.3.4
Other hardware limitations ........................................................................................................... 11
3.4
IMPLICIT RISK ......................................................................................................................................... 12
4
DETERMINISTIC ARBITRAGE ............................................................................................................. 13
4.1
BILATERAL ARBITRAGE .......................................................................................................................... 13
4.2
TRILATERAL ARBITRAGE ........................................................................................................................ 14
4.3
MULTILATERAL ARBITRAGE................................................................................................................... 14
4.4
CASH-AND-CARRY ARBITRAGE .............................................................................................................. 16
4.4.1
Generic futures (reverse) cash-and-carry arbitrage ..................................................................... 16
4.4.2
Foreign exchange futures cash-and-carry arbitrage..................................................................... 20
4.4.3
Options cash-and-carry arbitrage ................................................................................................. 23
4.4.4
Futures-options cash-and-carry arbitrage .................................................................................... 26
4.4.5
Fixed-income cash-and-carry arbitrage........................................................................................ 27
4.5
TAX ARBITRAGE ..................................................................................................................................... 27
4.6
REGULATORY ARBITRAGE ...................................................................................................................... 27
5
STATISTICAL ARBITRAGE ................................................................................................................... 28
5.1
SCALPING (MARKET MAKING) ................................................................................................................ 29
5.2
RELATIVE VALUE ARBITRAGE ................................................................................................................ 32
5.2.1
Convertible arbitrage .................................................................................................................... 33
5.2.2
Fixed-income arbitrage ................................................................................................................. 34
5.2.3
Capital structure arbitrage ............................................................................................................ 35
5.2.4
Pairs trading.................................................................................................................................. 36
5.2.5
Merger arbitrage ........................................................................................................................... 44
5.3
VOLATILITY ARBITRAGE ........................................................................................................................ 45
5.3.1
Directional volatility bet................................................................................................................ 46
5.3.2
Relative value volatility arbitrage ................................................................................................. 48
5.4
LIQUIDATION ARBITRAGE....................................................................................................................... 51
6
MULTILATERAL ARBITRAGE ALGORITHM STUDY .................................................................... 53
6.1
DATA ..................................................................................................................................................... 53
6.2
METHODOLOGY ..................................................................................................................................... 54
6.2.1
Detecting arbitrage for single set of quotes .................................................................................. 54
6.2.2
Detecting arbitrage for all sets of quotes ....................................................................................... 55
6.2.3
Detecting arbitrage for all variants of arbitrage loops ................................................................. 56
6.2.4
Putting it all together ..................................................................................................................... 56
6.3
RESULTS ................................................................................................................................................. 58
6.3.1
Preliminary datasets analysis ........................................................................................................ 58
6.3.2
Number of variations analysis ....................................................................................................... 60
6.3.3
Arbitrage analysis .......................................................................................................................... 60
CONCLUSION .................................................................................................................................................... 65
BIBLIOGRAPHY ................................................................................................................................................ 67
LIST OF TABLES ............................................................................................................................................... 76
LIST OF FIGURES ............................................................................................................................................. 76
LIST OF EQUATIONS ....................................................................................................................................... 76
APPENDIX A: EAST ASIAN EXCHANGES ................................................................................................... 78
Introduction
1
INTRODUCTION
Arbitrage is an exciting field of study in finance, because the premise of riskless profit is so
tempting. Apart from that, arbitrage is a paramount principle in pricing financial instruments,
and some believe that arbitrage is the omnipresent principle that drives markets into
equilibrium.
My personal interest in both finance and programming made me choose this exciting topic,
which gave me the opportunity to delve deeper into this thrilling area, expand my knowledge
greatly, and learn a great deal about automated analysis.
Among others, this paper also puts most types of arbitrages into a clear hierarchy, so anyone
interested in the field can quickly get into it.
1st. chapter deals with the definition of arbitrage for the purposes of this paper.
2nd. chapter explains the general idea behind arbitrage and some not-so-known nuances
3rd. chapter examines risks in arbitrage, because arbitrage is actually a risky endeavor
4th. chapter takes a look at deterministic (a.k.a. riskless) arbitrage and its kinds
5th. chapter explores the area of statistical arbitrage, a different approach to trading
6th. chapter presents arbitrage algorithms, conducted analysis and interpreted results
Goals
The main goal of this paper is to evaluate arbitrage opportunities of selected financial
instruments on historical data from East Asian financial markets. This main goal can be
decomposed into these partial goals:

Develop appropriate algorithm to identify and measure arbitrage opportunities from
market data

Apply that algorithm to real market data from East Asian markets and evaluate the
results
Methodology
The exact methodology of how the actual algorithm works and the nuances of my approach to
arbitrage measurement are all part of the chapter “6 Multilateral arbitrage algorithm ”. It
involves:

Thorough literature research

Collection of historical data from reliable source, which was deemed to be Bloomberg
terminal

Design of required algorithms to detect arbitrage opportunities and to generate foreign
exchange arbitrage loops

Development of custom software to make use of the algorithms and to automate data
analysis

Final evaluation and interpretation of the results
2
Notations and preliminaries
NOTATIONS AND PRELIMINARIES
For the purposes of this paper, the following notations will be used. These are meant to improve
clarity and correctness of the text.
Trade pairs notation
Each trade requires at least 2 assets. When more than 2 assets are involved, then such trade can
be substituted to a mathematical equivalent by using a set of trades each using exactly 2 assets
(called trade pair). This can be best explained by an example:
If someone wants to buy a car (asset 1 – car), he pays with money (asset 2 – currency).
Sometimes, the trade can be more complex and may involve many more assets – such as buying
bicycle (asset 1), helmet (asset 2) and clothes (asset 3) and then paying for the whole basket
once with cash (asset 4). This trade can be decomposed into 3 elementary trades – buying
bicycle with cash, then buying helmet with cash, and lastly buying clothes with cash.
Trade pair will be denoted as “A/B”, where “A” stands for one asset, and “B” stands for the
other. The first asset “A” is always called base asset, and “B” is the counter asset.
When it is said that the trade pair “A/B” is bought, it means that “B” is exchanged for “A”, “B”
is called input asset, and “A” is called output asset. When the trade pair “A/B” is sold, “A” is
exchanged for “B”. In this case, “A” is the input asset, and “B” is the output asset.
Price and sometimes volume are associated with each trade pair. Price is quoted as the number
of counter assets required to buy single base asset. When selling the base asset, it represents the
number of counter assets received for each unit of base asset sold. Volume is quoted as the
number of base assets available at that price (for purchase or sale).
To give you an idea of all these notations in action, let’s take a look at the following example.
Suppose you are given the following quotation:
KRW/CZK Price: 60 Volume: 1000
In this case, KRW is the base currency, CZK is the counter currency. 1 KRW costs 60 CZK,
and there are 1 000 KRW available at that price. When you buy this currency pair, CZK is
called the input currency, and KRW the output currency. When the currency pair is sold, CZK
is called the output currency, and KRW the input currency.
Sometimes, trades can be chained one after another. Such as buying KRW/CZK, then buying
JPY/KRW, and then selling JPY/CZK. These are called trade chains. When the first input assets
matches the last output asset, we can call it a trade loop.
Also, please be very aware whether the text talks about KRW/CZK, or CZK/KRW. Although
both involve the same assets, it’s important to keep in mind which asset is the base asset, counter
asset, input asset, output asset, and also what do both the price and volume relate to.
And last but not least, both base and counter assets do not have to be currencies. There can be
Car/CZK as well as Car/Bicycle trade pairs, although the latter is not very common in the real
world.
Notations and preliminaries
3
Foreign exchange (currency) notations
Currency codes
When talking about currencies, a currency code might be used instead. These currency codes
follow the ISO 4217 standard, as defined by the International Organization for Standardization,
and is available at the organization’s website. Here are some examples:
Table 1: Currency codes
USD
United States Dollar
CZK
Czech Koruna
KRW Korean Won
Source: ISO 42171
Quotations
Industry practice is that currency pairs are quoted either directly (1 unit of foreign currency for
domestic currency) or indirectly (1 unit of domestic currency for foreign currency) 2. Direct
quotation of CZK/USD in Czech Republic means 1 USD = x CZK, but direct quotation of
CZK/USD in United States means 1 CZK = x USD. So, there is also a difference of whether to
write CZK/USD or USD/CZK. A style of writing where domestic currency is in the first
position and foreign currency in the second position is called European2. In case of Czech
Republic and European style, that would be CZK/USD. American style is to write foreign
currency in the first position and domestic currency in second position. So in case of Czech
Republic and American style, that would be USD/CZK. So you can see, without explicit
statements regarding the direct/indirect quotation and European/American style, one cannot
really determine what an exchange rate of CZK/USD means. On top of that, these are just
common industry practices and not all exchanges, companies and traders might decide to use
them.
For this reason and for the matter of consistency, these rules will be used in this paper:

In case of European style (𝐷/𝐹), use indirect quotation (1 𝐷 = 𝑥 𝐹) and vice versa

In case of American style (𝐹/𝐷), use direct quotation (1 𝐹 = 𝑥 𝐷) and vice versa
Where 𝐷 refers to domestic currency and 𝐹 refers to foreign currency. This approach is the
same as for the general trade pairs mentioned in the previous section. The other reasoning
behind this is that once the arbitrage algorithms are designed to work with the more general
type of quotation, it’s very simple to download quotes from data providers and fetch those into
the algorithm, which will continue to work without modifications even when the data provider
decides to use different type of quotation later.
1
ISO 4217. In: International Organization for Standardization [online]. [Accessed 2014-04-28]. Available from:
http://www.iso.org/iso/home/standards/currency_codes.htm
2
POLOUČEK, Stanislav. Peníze, banky, finanční trhy. Vyd. 1. Praha: C.H. Beck, 2009, xviii, 414 s. Beckovy
ekonomické učebnice. ISBN 978-80-7400-152-9. p. 374 – 375
4
Notations and preliminaries
After abstracting from the direct/indirect quotation issue, the notation is interchangeable with
the rules used in practice3,4. In practice, foreign exchange spot market is traded in “lots”
(contracts of certain size)4. However, this paper will abstract from those details as those can be
dependent on the broker.
FOREXZONE. FOREX - jak zbohatnout a nekrást: obchodování na měnových trzích. 1. vyd. Praha: Grada,
2011, 185 s. Finanční trhy a instituce. ISBN 978-80-247-3739-3. p. 20, 25
4
HARTMAN, Ondřej. Jak se stát forexovým obchodníkem: naučte se vydělávat na měnových trzích. 1. vyd.
Praha: FXstreet, 2011, 230 s. Finanční trhy a instituce. ISBN 978-80-904418-0-4. p. 21 – 22, 27
3
Definition of arbitrage
5
1 DEFINITION OF ARBITRAGE
There are many different definitions of the term arbitrage. Some define it as a trade operation
exploiting different prices in different markets at a certain moment 5. Others might say that
arbitrage is an exploitation of different real values of the same or similar financial instruments
at the very same moment by the arbitrager at different markets, where the difference of real
values might originate from inefficient transfer of information or because of the difference in
taxation or regulation6.
However, these definitions do not really cover all cases, e.g. “volatility arbitrage”. As will be
shown later, some types of volatility arbitrage are just directional bets, yet it is considered
arbitrage because the bet is not on the market direction. Also, the requirement of those
definitions to execute transactions simultaneously is too strict. In reality, it is not practically
possible to execute 2 or more trades at the very same moment because of technical limitations,
which are further covered in separate chapter. So it’s clear that there must be some tolerance to
the “same moment” criteria. The other points is that many of these definitions often require the
whole operation to be risk-free. As shown in chapter 3 Risks of arbitrage, it is not possible to
achieve risk-free operations in the real world because of the current state of technology. For
these reasons, I propose the following definition:
Arbitrage is a set of trade operations based on a certain model that yield
profit at negligible risk.
By model, it is meant an algorithm using a set of equations (algebraic, statistical, or other). It’s
worth noting that the “negligible risk” also inherently implies that the time span it takes to
execute the whole set of trade operations is short enough that price fluctuations do not
significantly affect the riskiness of the whole operation. The term “negligible” is very subjective
and might differ for various arbitrageurs. Some might prefer virtually no risk at all, while others
might be absolutely fine with statistical arbitrage, which is sometimes so risky it’s a bit
controversial to call it an arbitrage in the first place.
REJNUŠ, Oldřich. Finanční trhy. 3., rozš. vyd. Ostrava: Key Publishing, 2011, 689 s. Ekonomie (Key
Publishing). ISBN 978-80-7418-128-3. p. 64
6
JÍLEK, Josef. Finanční trhy a investování. 1. vyd. Praha: Grada, 2009, 648 s. ISBN 978-80-247-1653-4. p. 85
5
6
General idea behind arbitrage
2 GENERAL IDEA BEHIND ARBITRAGE
As mentioned before, arbitrage is based on the exploitation of pricing discrepancies in financial
assets. According to law of one price, assets with the same cash-flow should trade for the same
price. In case the prices are different, it’s possible to earn almost riskless profit. Financial
derivatives prices are based on a certain model, and in case it doesn’t match, it’s possible to
carry out the cash-and-carry arbitrage to trade both the derivative and underlying asset in such
a specific way that leads to almost riskless profit.
Any practical algorithm trying to exploit arbitrage opportunities must account for transaction
costs, execution risks, and other details that are often overlooked in theoretical papers.
When there are mispriced assets, it does not necessarily mean there is also a riskless profit to
be earned. This is because prices of financial instruments often move in so-called no-arbitrage
zones7. This zones are price intervals in which the prices are not necessarily in accordance with
the model, but because of transaction costs and other barriers it’s not possible to arbitrage the
prices to match the theoretical value. The higher the transaction costs, the larger is the noarbitrage zone. So, arbitrage strategies can be only executed when the prices move away from
the zone, either through the upper bound or lower bound.
Sometimes the difference between arbitrage and speculation is not that large in real world7.
Because of globalization and high competition, arbitrage is profitable only when doing huge
volumes of transactions. But that increases the risks, because single point of failure can lead to
huge losses. In practice, arbitrage is usually conducted by few professional, highly specialized
investors who combine their knowledge with resources of outside investors8.
A real-world study revealed that markets with most transparency and liquidity are characterized
by highest levels of arbitrage activity9. This means that these markets are the most competitive
and for any retail investor that wants to profit in arbitrage, he/she should probably focus on the
less liquid markets, where the profit might still be sufficient for an individual rather than
institution. Especially on young emerging markets (i.e. those right after economic
transformation) is the arbitrage activity very limited9.
Although there are many ways how to categorize arbitrages, these are the 2 main categories:

Deterministic arbitrage – profits can be precisely calculated ex ante

Statistical arbitrage – overall profit is expected (under a model), but losses can happen
Other approach to categorize arbitrages is to think about when the profit cash-flow occurs:

Arbitrages with immediate profit cash flow

Arbitrages with future profit cash flow (cash-and-carry arbitrage)
Yet another way is to think about how many asset classes are involved in the operation:
ZÁŠKODNÝ, Přemysl, Vladislav PAVLÁT and Josef BUDÍK. Finanční deriváty a jejich oceňování. 1. vyd.
Praha: Vysoká škola finanční a správní, 2007, 161 s. ISBN 978-80-86754-73-4. p. 28 – 29
8
SHLEIFER, Andrei and Robert W. VISHNY. The Limits of Arbitrage. In: The Journal of Finance [online].
1997 [Accessed 2014-04-27]. 52. Available from: http://onlinelibrary.wiley.com/doi/10.1111/j.15406261.1997.tb03807.x/pdf. p. 36
9
HANOUSEK, Jan and Libor NEMECEK. Mispricing and lasting arbitrage between parallel markets in the
Czech Republic. Prague: CERGE-EI, 1998, 33 s. Working paper series (CERGE-EI). ISBN 80-862-8606-1. p. 5
and 24
7
General idea behind arbitrage

Arbitrages with single asset class (bilateral arbitrage)

Arbitrages with multiple asset classes (cash-and-carry arbitrage)
7
Or we can categorize even by the location of the trades:

Intra-exchange arbitrage (an exchange can have multiple markets)

Inter-exchange arbitrage (also called geographical arbitrage)
And many others.
2.1 Interest rates
Many of the textbook models assume single riskless rate for valuating assets. There is nothing
wrong about that from theoretical standpoint, but such model is close to useless in practice. In
real life, the assumption of being able to borrow or lend at risk-free rates including the absence
of bid-ask spread is just unreasonable. For almost all investors, the rate at which they borrow is
different to the rate at which they can lend.
Thus, practical arbitrage models should incorporate bid-ask spread for interest rates, where bid
rate is the rate at which the investor can actually (in the real life) lend money, and ask rate is
the rate at which he can actually borrow. Risk-free rates can be used to analyze equilibriums
and other theoretical aspects of modern financial markets, and are probably also a good estimate
of the rates at which large and powerful institutional investors can borrow/lend, and for that can
be used to model feared competition.
2.2 Other aspects
For actual real trading, full order book should be always analyzed in full. Only then it’s possible
to discover how deep the arbitrage opportunity is, and subsequently generate market orders that
arbitrage away the precise amount available, and not more. However, full order book data are
not easy to obtain at high frequencies, and that is one of the reasons why arbitrage seems to
have large barriers of entry.
8
Risks of arbitrage
3 RISKS OF ARBITRAGE
In academic world, arbitrage is often defined as a risk-free operation. However, nothing is riskfree in the real world, and the goal of this chapter is to explain why.
First risk is the liquidity risk – some arbitrage strategies require to take a position against the
market movements, and even though the profit might be likely in the end, arbitrageur might run
out of money in the meantime and margin calls may force the arbitrageur to close his positions
and end up in loss.
Second risk is the counterparty risk – to trade in the modern financial system requires the use
of banks, brokers, and other institutions. Default of any of those institutions might lead to a loss
and certain headaches.
Third risk is the execution risk – which is inherent to the contemporary technological
development. Computers are discrete machines and operate in “jumps” rather than continuously
and speed of light is also limited. For both of those reasons, there might be delays and lags in
communication which might lead to an undesired situations, possibly resulting in a loss for the
arbitrageur.
And finally the fourth risk is the risk inherent to the arbitrage model itself (implicit risk). Such
as in the case of statistical arbitrage, the risk is implied by the use of statistics (statistical errors).
3.1 Liquidity risk
Contrary to the common belief (which states that arbitrage does not require any capital), most
real-life arbitrage opportunities actually require a lot of capital, such as in the case of futures –
daily price changes might move against the arbitrageur (mispricing may deepen), resulting in a
margin call and further capital requirements10. In other words, if the market mispricing deepens,
the arbitrageur is required to add capital to his margin account, and in case he doesn’t have any
more capital, his position might get closed and he will incur a loss11.
When the arbitrageur is a fund (institution) and uses money of its investors to conduct arbitrage,
mark-to-market value of the fund might have high volatility even though there is actually very
little risk in the fund operations. Those investors might withdraw their cash from the fund in
fear of loss, which might subsequently lead to the fund not being able to meet the margin
requirements and be forced to close the positions in loss10. This is known as a liquidity risk, or
in some literature as noise trader risk – which comes from the idea that irrational noise traders
widen the spreads which leads to the lack of liquidity12.
For the reasons mentioned in previous paragraph, highly volatile markets might not be as
attractive to arbitrageurs, because even though the arbitrage is often more profitable, the implied
liquidity risk might prevent them from exploiting those opportunities in the fear of running out
of money to cover margin accounts10. Another aspect to liquidity arbitrage is that some arbitrage
10
SHLEIFER, Andrei and Robert W. VISHNY. The Limits of Arbitrage. In: The Journal of Finance [online].
1997 [Accessed 2014-04-27]. 52. Available from: http://onlinelibrary.wiley.com/doi/10.1111/j.15406261.1997.tb03807.x/pdf. p. 36 – 37, 54
11
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 18
12
ZHANG, Jie. The Limits to Arbitrage and the Fundamental Value-to-Price Trading Strategies [online]. Hong
Kong, 2006 [Accessed 2014-04-29]. Available from: http://www.proquest.com/. Research thesis. The Hong
Kong University of Science and Technology. p. 56
Risks of arbitrage
9
opportunities might require significant capital to even enter the positions, not the mention the
subsequent maintenance of margin accounts
When creating an arbitrage strategy, it’s important to pay attention to the profit/loss cash flow.
For example, in a cash-and-carry type of arbitrage between stock and futures of that stock, the
profit is gained at the maturity of the futures. However, in the meantime, the futures gets marked
to market (daily settlement), and thus incurs positive and negative cash flow throughout its
lifetime. If there is not enough cash in the margin account, the investor might receive margin
call and subsequently the futures contract might get closed, locking the position in an almost
certain loss. So, even though the strategy might be riskless on paper, there is always a risk of
not having sufficient capital to cover the daily marking-to-market. So, it’s always a good idea
to design an arbitrage strategies such that the cash flows from each assets cancel out, and the
only real cash flow is the profit. This can be illustrated in a put/call parity arbitrage – if the
underlying asset is just a common stock, then there is the margin call risk again. However, if
the underlying asset is a futures to a common stock, then the daily mark-to-market from futures
and options more or less cancel out, leaving us only with the profit.
However, some exchanges do not necessarily cancel-out the cash-flows from each individual
open position. That is, they do not calculate the net position (such as 100 long and 100 short
equals 0 net), but the gross position (100 long and 100 short equals 200 contracts) 13. If that is
the case, one must maintain a considerably large margin, which might limit the arbitrage
strategy a lot.
Historical example of a cash-flow mismatch was the case of the company MGRM14. When oil
prices fell, long oil futures positions required additional capital because of margin calls.
However, the short offsetting forward positions did not generate any cash-flow, which resulted
in massive losses to the company because of forced closing of the futures positions.
3.2 Counterparty risk
Arbitrageurs must use financial system to conduct arbitrage. They must use dealers, brokers,
banks, exchanges, and other institutions and intermediates to trade, deposit their capital at, etc.
Depositing capital at any of the aforementioned institutions imply the counterparty risk – risk
of default of that institution. For that reason, almost all arbitrage opportunities are not risk-free
at all – there is always at least the risk of default of the exchange, bank, or other subject that the
arbitrageur deals with. A recent example of that is the bankruptcy of “Mt. Gox”, a BitCoin
(crypto-currency) exchange15, which will probably result in a loss for many clients of that
exchange.
3.3 Technical and execution risk
Technical risks are risks associated with the contemporary state of technology, its limitations,
imperfections, and characteristics. These risks can turn arbitrage into a risky operations – e.g.
if there is a sufficient delay between issuance and execution of a market order. Electronic
13
POWERS, Mark J and Mark G CASTELINO. Inside the financial futures markets. 3rd ed. New York: Wiley,
c1991, x, 390 p. ISBN 04-715-3674-1. p. 20
14
JÍLEK, Josef. Deriváty, hedžové fondy, offshorové společnosti. 1. vyd. Praha: Grada, 2006, 260 s. ISBN 80247-1826-X. p. 116
15
KANOKOGI, Yasushi, Hideki KANAZAWA and Masaki HIGUCHI. Bankrupcy procedure decision. In:
MtGox.com [online]. 2014 [Accessed 2014-04-27]. Available from:
https://www.mtgox.com/img/pdf/20140424_order.pdf
10
Risks of arbitrage
trading is done over a computer network, and each networking element introduces additional
delays. These delays often stack on each other and create a cascade effect, so it’s very important
to be aware of how all these devices work and optimize the network such that the delay is
minimized. As mentioned earlier, these delays cause that it’s often not possible to execute 2 or
more market orders at the very same time. In practice, this means that a position can remain
open for too long so that before the next market order arrives at the exchange, the price
fluctuations that occurred in the meantime lead to a net loss.
3.3.1 Processor speed
Processor, or CPU (central processing unit), is a sophisticated electronic circuit with many
inherent limitations, either caused by design (architecture), or the state of technology at the
time. CPUs operate at a certain frequency (e.g. 2.4 GHz/s), execute instructions sequentially,
and each instruction takes a certain number of cycles.
E.g. if the sum of two numbers takes 1 cycle (but can also take more, depending on the
architecture), and the CPU operates at 2.4 GHz/s, then the processor could theoretically execute
2 400 000 000 sum operations per second. Practically speaking, the processor would be able to
execute a lot less than that, because it must execute other instructions (where each might take
different amount of cycles to complete) that fetch the data from memory into registers, then
sum the registers, and then store the result in either a third register or into the memory. These
operations incur further delays caused by limits and speed constraints of processor cache, bus,
and memory. These operations are also absolutely crucial and inevitable for the algorithm to
operate correctly, so they must be accounted for.
CPUs may contain multiple logical processors (i.e. multicore processors), which can execute
multiple instructions at the same time. Although there are some limits to this type of architecture
(such as when these logical processors use single shared memory), it is possible to execute
operations in parallel using this technology. Unfortunately, that does not necessarily mean that
algorithms that calculate arbitrages can be effectively parallelized. In addition, market orders
are processed on remote servers (exchanges), and to make effective use of the CPU
parallelization, we would have to have parallelized networks (a separate cable for each
exchange, connecting our computer directly to the exchange servers). Because exchanges are
often geographically far apart, this is simply not feasible in most cases.
For these reasons, the arbitrage trading algorithm should be optimized for speed, to limit the
amount of risk incurred by the delay caused by the calculation itself.
3.3.2 Network latency
Network latency is the time it takes for a request to leave the computer, transfer through the
whole computer network consisting of network cards, cables, routers, bridges, hubs, and other
networking devices (which often use CPUs, introducing further delays), and reach the
destination. In practice, network latencies are often in the order of milliseconds, so it’s not
possible to execute any series of trades at the very same moment. In a network cable, the market
orders are transferred as a set of 0s and 1s, one after another, at a certain speed (MB/s, GB/s)
and thus there is an inherent delay in executing these orders. Not to mention additional network
latency caused by the limited capacity of network cables (the network might be overloaded),
limited speed of CPUs, etc.
Risks of arbitrage
11
Apart from these delays caused by the hardware, latency can also come from the software
(network protocol). The most used protocol TCP/IP does guarantee integrity of data, correct
order of data packets, and data confirmations (addressee confirms that he received data from
the sender). However, this protocol does not guarantee when the data is received. It can take up
to several seconds or more. The TCP/IP protocol works with a timeout principle in mind – if
the receiving machine does not confirm that it received data packet in a certain time span, the
sender will resend the packet. The sender can try several times and only after several attempts
it may yield the control back to the calling program. This can take many seconds and for this
reason, it’s important to optimize the network communication module such that it allows for
high-frequency trading, otherwise the risk of price fluctuation in the meantime might become
too high.
In practice, network latencies are mostly reduced by colocation, i.e. putting the trading robot
inside the exchange’s datacenter.
3.3.3 Software or hardware malfunction
When calculating an arbitrage opportunity, software or hardware malfunction might happen.
In case of hardware malfunction, there might be a logical error in CPU, data corruption in
memory, hard drive, etc. These can be caused by faulty design, or by external events, such as
some high-speed particles colliding with the atoms in memory, changing the charge of a bit and
thus corrupting the data. When it comes to networking, a data packet might be lost during its
transfer to destination, or might be corrupted, etc. And last but not least, a power outage would
bring the whole system down, even in the middle of transaction broadcasting.
In case of software malfunction, a device driver, operating system kernel, or other code running
in ring-0 execution mode might execute an invalid operation, causing the whole system to crash,
and thus requiring a computer reboot. Also, the arbitrage trading algorithm might contain design
errors that are not inherent to the algorithm itself, but rather to the implementation (such as
threading synchronization problems, bad error checking, etc.).
It’s worth mentioning that any of these malfunctions can occur between market orders, i.e. after
first market order is sent and executed, and before the second market order is sent. This means
that the trader might end up with an undesired open position in a certain asset until the system
is rebooted or the malfunction is otherwise fixed and the set of orders is either finished, or
cancelled.
3.3.4 Other hardware limitations
Virtually all commercial computing devices are based on electricity and/or electromagnetic
waves. These phenomena are limited by the maximum speed of light, which is also dependent
on the medium in which the electric current or electromagnetic wave is travelling through. Thus,
there is an inherent speed limitation at which the information can travel, incurred by the laws
of the physics.
Apart from that, other hardware components of the computer do have other speed limitations,
such as RAM frequency, RAM latency, bus size, length of cables and circuits, etc. All these
contribute to delays and speed limits of contemporary computing devices.
12
Risks of arbitrage
3.4 Implicit risk
Implicit risks are all risks that are inherent to the arbitrage model. In other words, these risks
come from the design of arbitrage model itself. For example, in statistical arbitrage, the risk
comes from the use of statistical methods. This can be characterized not only by the statistical
error, but also by the fact that if it’s not possible to repeat the trade many times, the strategy
might end up with a net loss, even though the profit was statistically expected.
Deterministic arbitrage
13
4 DETERMINISTIC ARBITRAGE
Deterministic arbitrage is any arbitrage strategy where the profit can be calculated ex ante with
absolute certainty, under the assumption that all trades can be executed simultaneously (so there
is no risk involved)16. This means the profit is known even before the trades are executed.
However, after the trades are executed initially, the profit can be:

Earned immediately

Locked-in, but the profit cash-flow happens later (such as cash-and-carry arbitrage)
Thorp17 describes the arbitrage between the same or similar assets as buying the relatively
underpriced security while simultaneously selling the overpriced security. Because the assets
should be equivalent (not necessarily same), this trade would achieve a perfect -1 correlation,
hedge against beta and alpha risk and yield a riskless profit.
Kuepper18 explains that deterministic arbitrage opportunity exists only when one or more of the
following conditions are violated:

The same security must trade at the same price on all markets

Two securities with identical cash-flows must trade at the same price

Security with a known future price must trade today at that price discounted by the riskfree rate
Because many investors are looking for a guaranteed riskless profit, deterministic arbitrage is
a highly competitive area and requires access to real-time prices, instantaneous execution and
ability to borrow at low interest rates19. Kuepper18 further adds that retail investors cannot be
competitive when it comes to deterministic arbitrage, as this area is completely dominated by
market makers. For example, value of most financial instruments is affected when central bank
announces to change interest rates. In order to profit from the imminent market movements, the
arbitrageur must be as fast as possible to receive that information, then process it and finally
react to it accordingly.
4.1 Bilateral arbitrage
Bilateral (geographical, spatial) arbitrage is the most well-known and simplest type of arbitrage,
in which single asset (such as stock or financial derivative) is traded at 2 or more markets at
different prices, so an arbitrageur can buy on the market where the asset is cheaper and sell on
market where the price is higher20. Obviously there are actually 2 assets involved – the stock or
derivative, and the currency.
16
BRETZLER, Marcus and Dirk RUDOLPH. Hedge Fonds und Alternative Investments. First edition. Frankfurt
am Main: Bankakademie-Verlag GmbH, 2004. ISBN 3-937519-08-4. p. 22
17
THORP, Edward. THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK
MARKET. In: BJMath [online]. 1997 [Accessed 2014-04-13]. Available from:
http://www.bjmath.com/bjmath/thorp/paper.htm. p. 22
18
KUEPPER, Justin. Trading The Odds With Arbitrage. In: Investopedia [online]. 2012 [Accessed 2014-04-26].
Available from: http://www.investopedia.com/articles/trading/04/111004.asp
19
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 33
20
JÍLEK, Josef. Finanční trhy a investování. 1. vyd. Praha: Grada, 2009, 648 s. ISBN 978-80-247-1653-4. p. 85
14
Deterministic arbitrage
I.e. in case of forwards/futures, the profit is locked-in at the time arbitrageur enters both long
and short positions, but the profit is earned at expiration date, when the actual delivery of the
underlying asset happens21.
An example of bilateral futures arbitrage can be the case of Baring Futures Singapore Pte
Limited22. They had the idea was to trade on SIMEX (in Singapore), Osaka Securities Exchange
(Japan), Tokyo Stock Exchange, and Tokyo Financial Exchange. SIMEX tried to compete with
other Japanese exchanges by also listing Nikkei 225 futures, 10-year Japanese government bond
futures, EUR/YEN 3-month futures, and also options on Nikkei 225 futures. The idea was to
take simultaneous long position in one country and short position in the other as soon as the
prices misaligned.
4.2 Trilateral arbitrage
Trilateral (triangular) arbitrage is an extension to classical arbitrage, only that it requires 3 assets
instead of just two. Arbitrage opportunity exists if the cross-asset exchange rates are not
properly balanced, meaning that an arbitrageur can start with one asset and through the series
of trades can end up with more than what he began with.
The most well-known asset for triangular arbitrage are currencies – simply because the crossrates for stocks are not used in practice (no exchange is trading IBM stocks for Microsoft stocks
directly, at least to my knowledge). However, this type of arbitrage is not limited only to spot
exchange rates, but can be used with forward/futures rates as well, where the mechanics for
detecting the arbitrage opportunity is actually identical21. The only difference is that the profit
is earned at expiration date rather than immediately.
Even though theoretically triangular arbitrage is not limited to any asset class, in reality only
foreign exchange (currency) markets are suitable for this kind of arbitrage, as forex markets do
offer the ability to trade many assets in triangle. In case of foreign exchange, the basic triangular
arbitrage equation is23:
𝐴/𝐵 = 𝐴/𝐶 × 𝐶/𝐵
(1)
4.3 Multilateral arbitrage
The previous two types of arbitrage are special cases of a more generalized variant. The first
type is basically an arbitrage between two markets, while in second variant, 3 markets are
involved. Multilateral (polygonal) arbitrage is when 𝑛 markets are involved.
The triangular arbitrage for foreign exchange can be can be expanded to use more than just 1
intermediate currency. In case of using 2 intermediate currencies, we get quadrangular
arbitrage23:
𝐴/𝐵 = 𝐴/𝐶 × 𝐶/𝐷 × 𝐷/𝐵
(2)
JÍLEK, Josef. Finanční a komoditní deriváty v praxi. 2. upr. vyd. Praha: Grada, 2010, 630 s. Finance (Grada).
ISBN 978-80-247-3696-9. p. 200 – 201
22
JÍLEK, Josef. Deriváty, hedžové fondy, offshorové společnosti. 1. vyd. Praha: Grada, 2006, 260 s. ISBN 80247-1826-X. p. 134
23
MAVRIDES, Marios John. The efficiency of triangular arbitrage in the foreign exchange market [online].
Chicago, 1991 [Accessed 2014-05-12]. Available from: http://www.proquest.com/. Dissertation. University of
Illinois at Chicago. p. 53, 61 – 63
21
Deterministic arbitrage
15
And when using 3 intermediate currencies, we get quintangular arbitrage23:
𝐴/𝐵 = 𝐴/𝐶 × 𝐶/𝐷 × 𝐷/𝐸 × 𝐸/𝐵
(3)
The total amount of all possible multilateral parity conditions is determined by the following
formula23:
𝑃=
𝑁!
(𝑁 − 𝑅)!
(4)
Where 𝑁 is the number of available currencies and 𝑅 is the order of parity (3 for triangular
arbitrage). The generalized arbitrage condition is24:
𝑛
∏ 𝑟𝑖 > 1
(5)
𝑖=1
Which basically means that the product of all exchange rates should be higher than 1. For any
currency chain, there are actually two 2 “directions” one needs to monitor to check for arbitrage
opportunities, which is because of the presence of bid-ask spread. In case of currencies USD,
CZK, and EUR, those 2 directions would be:
(𝑈𝑆𝐷/𝐶𝑍𝐾)𝑎𝑠𝑘 → (𝑈𝑆𝐷/𝐸𝑈𝑅)𝑏𝑖𝑑 → (𝐸𝑈𝑅/𝐶𝑍𝐾)𝑏𝑖𝑑
(6)
(𝐸𝑈𝑅/𝐶𝑍𝐾)𝑎𝑠𝑘 → (𝑈𝑆𝐷/𝐸𝑈𝑅)𝑎𝑠𝑘 → (𝑈𝑆𝐷/𝐶𝑍𝐾)𝑏𝑖𝑑
(7)
It can be proven with example that multilateral arbitrages of higher order than 3 can
theoretically exist, and not all of them can be substituted by a set of arbitrages of lower orders
(i.e. it’s not possible to decompose quadrangular arbitrage into any number of equivalent
triangular arbitrages, and there are some higher-order chains that also cannot be decomposed):
Table 2: Quadrangular arbitrage example
USD/EUR EUR/CZK CZK/JPY JPY/USD Arbitrage?
Ask
Bid
1
1
1
USD/EUR EUR/CZK USD/CZK
Ask
Bid
2
1
1
(0.5)
EUR/CZK
CZK/JPY
JPY/EUR
1
1
0.5
CZK/JPY
JPY/USD
USD/CZK
Ask
Bid
Ask
Bid
24
(2)
1
1.5
0.5
1,5
Yes: 1.5 > 1
Arbitrage?
No: 0.5 < 1
Arbitrage?
No: 0.5 < 1
Arbitrage?
No: 0.75 < 1
FENN, Daniel J., Sam D. HOWISON, Mark MCDONALD, Stacy WILLIAMS and Neil F. JOHNSON. The
Mirage of Triangular Arbitrage in the Spot Foreign Exchange Market [online]. 2008 [Accessed 2014-05-12].
Available from: http://arxiv.org/abs/0812.0913. Research paper. p. 4
16
Deterministic arbitrage
JPY/USD
USD/EUR
JPY/EUR
Ask
2
Bid
1.5
1
(0.5)
Source: Author’s own construction
Arbitrage?
No: 0.75 < 1
It should be mentioned that high-order arbitrages are a lot riskier than arbitrages with lower
order. All additional operations (buy, sell) that must be taken introduce additional execution
risk. The more operations, the more likely it is that the market may move before we are able
to close the loop.
4.4 Cash-and-carry arbitrage
Cash-and-carry arbitrage refers to the arbitrage between cash (spot, prompt) and derivatives
(forwards, futures, swaps, options) markets25, which means there are almost unlimited types of
this kind of arbitrage. Because the profit is locked-in at the beginning, but earned at expiration
of the derivative, it might as well be called temporal arbitrage. The underlying asset (which
refers to the cash market) can be anything – from currencies, stocks, interest rates, to
commodities and indices. Each have their own specifics - i.e. some stocks cannot be short sold,
which means the arbitrage is possible only in one of the two types of imbalances. Furthermore,
it might be quite challenging to perform index arbitrage – there are significant transaction costs
in buying all components of an index.
There are two types of cash-and-carry arbitrage26,27:

Cash-and-carry arbitrage – Sell overvalued futures, buy underlying

Reverse cash-and-carry arbitrage – Buy undervalued futures, sell underlying
4.4.1 Generic futures (reverse) cash-and-carry arbitrage
In case of futures, we can think of the following two strategies28,29:
1. Buy futures contract and deposit cash, wait until maturity, then sell the contract
2. Buy the underlying today, then wait until maturity of the futures and sell it, while paying
the storage fees (or interest) in the meantime
Both strategies should result in identical situation. An arbitrage opportunity exists in case they
do not. Depending on the type of imbalance, arbitrageur would either both sell the futures and
buy the underlying (cash-and-carry), or buy the futures and short the underlying (reverse cashand-carry).
25
BRETZLER, Marcus and Dirk RUDOLPH. Hedge Fonds und Alternative Investments. First edition. Frankfurt
am Main: Bankakademie-Verlag GmbH, 2004. ISBN 3-937519-08-4. p. 22
26
ZÁŠKODNÝ, Přemysl, Vladislav PAVLÁT and Josef BUDÍK. Finanční deriváty a jejich oceňování. 1. vyd.
Praha: Vysoká škola finanční a správní, 2007, 161 s. ISBN 978-80-86754-73-4. p. 55
27
DVOŘÁK, Petr. Deriváty. Praha: Oeconomica, 2010. ISBN 978-80-245-1435-2. p. 155 – 156
28
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 4
29
SHARPE, William F. Investice. Praha: Victoria Publishing, 1994, 810 s. ISBN 80-856-0547-3. p. 527
Deterministic arbitrage
17
The arbitrage-free continuous-time pricing formula for a forward with no inter-temporal cashflow (no income, no costs) is30,31:
𝐹0 = 𝑆0 𝑒 𝑟𝑇
(8)
Where 𝐹0 is the price of futures today, 𝑆0 is the current spot price, 𝑟 is the risk-free annual
interest rate, and 𝑇 is the time until expiration in years. The forward and futures prices are equal
when interest rates are constant31, so the pricing formula can be used for futures too. In practice,
the forward and futures prices may differ for the reasons of stochastic interest rates (which
affect borrowing/lending), taxes, transaction costs, treatment of margins, and different risks of
default of the counterparty31. For the purposes of marking-to-market, the value of the contract
at some point of time after the contract is entered is31:
𝑓 = (𝐹0 − 𝐾)𝑒 −𝑟𝑇
(9)
Where 𝐾 is the delivery price set in the contract, 𝐹0 is the current forward price, and 𝑓 is the
value of the contract. In case there is an arbitrage opportunity (forward price differs to the
theoretical price calculated by the aforementioned pricing formula), the profit can be made with
the following decision table31:
Table 3: Generic cash-and-carry arbitrage decision table
Market situation
𝐹0 > 𝑆0 𝑒 𝑟𝑇
(cash-and-carry arbitrage)
How to profit
1. Borrow 𝑆0 at interest rate 𝑟 for 𝑇 years, which
becomes debt of 𝑆0 𝑒 𝑟𝑇 at maturity
2. Buy underlying asset for 𝑆0
3. Sell the futures for 𝐹0
4. Profit at maturity = 𝐹0 − 𝑆0 𝑒 𝑟𝑇
1. Short-sell underlying asset for 𝑆0
2. Deposit 𝑆0 at interest rate 𝑟 for 𝑇 years, which
𝐹0 < 𝑆0 𝑒
becomes claim of 𝑆0 𝑒 𝑟𝑇 at maturity
(reverse cash-and-carry
arbitrage)
3. Buy the futures contract for 𝐹0
4. Profit = 𝑆0 𝑒 𝑟𝑇 − 𝐹0
Source: Constructed from information provided by Pelletier30
𝑟𝑇
The 𝐹0 < 𝑆0 𝑒 𝑟𝑇 type of arbitrage requires the ability to short-sell the underlying asset, which
is not always possible. Especially in the case where underlying asset is an index, it’s quite likely
30
PELLETIER, Dennis. ECG590I Asset Pricing. Lecture 5: Forward and Futures Prices. In: University of
Illinois at Chicago [online]. 2006 [Accessed 2014-04-27]. Available from:
http://www.uic.edu/cuppa/pa/academics/Duplicate7/Lectures,%20Outlines%20and%20Handouts/Public%20Fina
nce/Asset%20Pricing%20Dennis%20Pelletier%20of%20North%20Carolina%20State%20University/Forward%20and%20Future%20
Prices.pdf. p. 1
31
HULL, John. Options, futures. 5th ed. Upper Saddle River, NJ: Prentice Hall, c2003, xxi, 744 p. ISBN 0-13009056-5. p. 46, 50 – 51, 68
18
Deterministic arbitrage
that some of the index components won’t have the option of short-selling. Not to mention the
inevitable transaction costs32.
The profit is locked-in immediately once the positions are taken33. At the expiration date, all
positions have to be unwound to finally earn the profit34. Arbitrageur might unwind his position
prematurely in case the marked-to-market situation got the arbitrageur in a profit which is
higher than the discounted arbitrage profit33. Sharpe further adds that34:

Both strategies are self-financing – no capital is required

Depositing money can be also thought of as buying riskless bonds

Bought riskless bonds can act as a collateral for long futures margin account

Bought underlying asset can act as a collateral for short futures margin account

Arbitrage models for stocks should account for dividends

Daily marking-to-market can lead to extra capital requirements to cover the margin
account, and in such case the positions would no longer be self-financing
An important property of futures contracts is the process called marking-to-market, which is
something arbitrageurs must be aware of, as that process might liquidate their position before
the arbitrage profit can be reached35,36,37,38 – i.e. beware liquidity risk.
To sum things up, the market imperfections involved in futures arbitrage (which should be all
taken into consideration in practical implementations) are33:
32

Transaction costs – which lead to bounds where the pricing formula does not match, yet
the arbitrage is not possible39

Taxes

Storage fees

Lease (convenience from owning the underlying asset)

Different borrowing and lending rates

Seasonal dividend payout
YU, Wei. The dynamics of cash and futures markets for stock [online]. South Carolina, 1992 [Accessed 201405-04]. Available from: http://www.proquest.com/. Dissertation. Clemson University. p. 20
33
LEE, Jae Ha. Stock index futures arbitrage and intraday tests of market efficiency [online]. Indiana, 1988
[Accessed 2014-05-06]. Available from: http://www.proquest.com/. Dissertation. Indiana University. p. 5, xiv
34
SHARPE, William F. Investice. Praha: Victoria Publishing, 1994, 810 s. ISBN 80-856-0547-3. p. 527 – 529
35
JÍLEK, Josef. Finanční trhy. 1.vyd. Praha: Grada Publishing, 1997, 527 s. ISBN 80-716-9453-3. p. 395
36
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 18
37
SHLEIFER, Andrei and Robert W. VISHNY. The Limits of Arbitrage. In: The Journal of Finance [online].
1997 [Accessed 2014-04-27]. 52. Available from: http://onlinelibrary.wiley.com/doi/10.1111/j.15406261.1997.tb03807.x/pdf. p. 36
38
POLOUČEK, Stanislav. Peníze, banky, finanční trhy. Vyd. 1. Praha: C.H. Beck, 2009, xviii, 414 s. Beckovy
ekonomické učebnice. ISBN 978-80-7400-152-9. p. 187
39
CHENG, Kevin H.K., Joseph K.W. FUNG and Yiuman TSE. How Electronic Trading Affects Bid-ask
Spreads and Arbitrage Efficiency between Index Futures and Options [online]. China, 2005 [Accessed 2014-0506]. Available from: http://ied.hkbu.edu.hk/publications/fdp/FDP200403.pdf. Research paper. Hong Kong
Baptist University and University of Texas at San Antonio. p. 7 – 8
Deterministic arbitrage

19
Daily marking-to-market effects – which might result in further borrowing/lending of
the capital to maintain the positions
Example
Imagine a BitCoin spot and futures exchange (such as https://btc-e.com/ and https://icbit.se/).
BitCoin futures expiring in 6 months for the delivery of 1 BitCoin is quoted at 500 USD, while
the current spot price is 450 USD. Because the arbitrageur is a small retail client, he can borrow
at local bank for 20 % p.a. (far from risk-free rate). If he borrows 450 USD now, he will have
6
to give 450(1 + 0.2)12 = 492.95 𝑈𝑆𝐷 back to the bank in 6 months. However, after 6 months,
he can deliver 1 BitCoin for 500 USD (abstracting from marking-to-market) thanks to the
futures contract, which means there is a riskless profit of 500 − 492.95 = 7.05 𝑈𝑆𝐷 (actually
as mentioned before, the profit is not riskless as there is the counterparty risk among others).
Of course, such arbitrageur can’t do this forever because the bank will not lend him unlimited
money, but the more he earns, the more he can use of his own capital in the arbitrage without
borrowing.
This example also shows that institutional investors with access to better loans can do much
more profitable arbitrage trades, even such trades that are not available to retail clients at all.
On the other hand, retail investor might use his own capital instead of borrowing money at high
rates, as long as the return he expects is lower than the loan interest rate. Such investor would
follow this algorithm:

Loan interest rate allows arbitrage – take a loan and use bank’s money, or use own
capital

Loan interest rate does not allow arbitrage, but the investor’s expected return does – use
own capital

Investor’s expected return does not allow arbitrage (i.e. he can use deposit account at
bank with higher return) – do nothing
Let’s dig deeper into the arbitrage mechanics and see how marking-to-market (abstracting from
the margin account lock of money) would affect the trade:
Table 4: Detailed example of cash-and-carry arbitrage
Today
In 2 months
In 4 months
Maturity
Spot = 450
Spot = 460
Spot = 300
Spot = 100
Futures = 500
Futures = 520
Futures = 290
Futures = 100
Short futures
0
- 20
210
400
Long spot
450
460
300
100
Bank loan
- 450
- 463.88
- 478.20
- 492.95
31.80
7.05
0
- 23.88
Total
Source: Author’s own construction
It’s not as easy to calculate the return per annum, because the amount of locked money in the
investment depends on the margin account, which is updated daily based on the actual price
20
Deterministic arbitrage
movements. And characteristic feature of futures trading is that when entering a long or short
position, the buyer/seller does not immediately pay the price for the instrument40. In any case,
the fair arbitrage-free price of futures from the view of the investor would have been the exact
amount that he paid to the bank, which is 492.95 USD. Please note that in case of BitCoin, it’s
not yet possible to short-sell the underlying asset. So the arbitrage is possible only in one
direction.
4.4.2 Foreign exchange futures cash-and-carry arbitrage
In this type of arbitrage (sometimes also called covered interest arbitrage), the underlying asset
is a currency. Foreign exchange (currency) futures is based on future exchange of fixed amount
of one currency for another currency, at an exchange rate determined at the moment the contract
is bought/sold for the first time40.
For a retail investor, it might be very complicated to borrow (foreign or domestic) currency at
risk-free rate. Using foreign exchange brokers, there is a way how to somewhat borrow/lend at
a rate close to riskless, using a mechanism called carry trade. When you sell one currency for
another through a broker, you pay the interest rate of the short currency and receive interest rate
of the long currency41. This trade is fairly popular and is the reason for large speculative capital
movements when a change in interest rates occur. However, these interest rates change daily
(and the arbitrage model requires fixed interest rates), because they are subject to trading as
well. Many brokers also use their own rates as a hidden fee41. In any case, because the rates
change daily and thus there is a temporal mismatch between the forex loan and the futures,
carry trade is not suitable for arbitrage, only for speculative trading. Some possible solutions
might include:

Interest rate swaps, to convert the floating interest rate into fixed rate – however, interest
rate swaps are traded only OTC40,42 and as such not likely to be available for retail
investors

Interest rate futures – however the additional spread at that instrument would probably
lead to not being able to execute any arbitrage in real world

Futures pricing model with stochastic interest rates
Carry trading also refers to the practice of borrowing for a short-term rate (e.g. 2 %) and
subsequent long-term lending (e.g. rate of 6 %), under the assumption that yield curve has an
upward sloping shape43. In other words, it’s a bet that short-term interest rates won’t grow.
The general principle behind arbitrage-free pricing models for currency forwards is that the
following two investments result in the same value44:

Buy foreign currency for one unit of domestic currency now, deposit at foreign
exchange rate till contract expiration
REJNUŠ, Oldřich. Finanční trhy. 3., rozš. vyd. Ostrava: Key Publishing, 2011, 689 s. Ekonomie (Key
Publishing). ISBN 978-80-7418-128-3. p. 462, 465 – 466, 469
41
FOREXZONE. FOREX - jak zbohatnout a nekrást: obchodování na měnových trzích. 1. vyd. Praha: Grada,
2011, 185 s. Finanční trhy a instituce. ISBN 978-80-247-3739-3. p. 109 – 110
42
JÍLEK, Josef. Finanční trhy. 1.vyd. Praha: Grada Publishing, 1997, 527 s. ISBN 80-716-9453-3. p. 400
43
JÍLEK, Josef. Deriváty, hedžové fondy, offshorové společnosti. 1. vyd. Praha: Grada, 2006, 260 s. ISBN 80247-1826-X. p. 123
44
DVOŘÁK, Petr. Deriváty. Praha: Oeconomica, 2010. ISBN 978-80-245-1435-2. p. 186
40
Deterministic arbitrage

21
Deposit one unit of domestic currency at domestic exchange rate till contract expiration,
then exchange domestic currency for foreign currency at the contract (forward) rate
Regardless, there are many different types of currency forward pricing models mentioned in
literature. Some are based on simple interest, others on compounded interest or even
continuously compounded interest, and some use various types of approximations rather than
precise expressions.
Discrete simple interest approximation model is45 (edited to match used notation):
𝑡
𝑎𝑠𝑘
𝑏𝑖𝑑
𝐹 𝑎𝑠𝑘 = 𝑆 𝑎𝑠𝑘 (1 + (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
− 𝑟𝑏𝑎𝑠𝑒
)
)
360
( 10 )
Where 𝐹 𝑎𝑠𝑘 is a forward rate to buy base currency for a counter currency (i.e. in case of
USD/CZK, it’s a contract to buy USD for CZK in the future), 𝑆 𝑎𝑠𝑘 is the spot rate to buy base
𝑎𝑠𝑘
currency for counter currency, 𝑟𝑏𝑎𝑠𝑒
is the interest rate of counter currency that investor pays
𝑏𝑖𝑑
(investor borrows money), and 𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
is the interest rate of base currency that investor
receives (investor lends money). The use of interest rate difference (e.g. 5% − 3% ≅ 2%) is a
form of approximation, which is considered to be precise enough for small (i.e. lower than 10)
numbers.
𝑡
𝑏𝑖𝑑
𝑎𝑠𝑘
( 11 )
𝐹 𝑏𝑖𝑑 = 𝑆 𝑏𝑖𝑑 (1 + (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
− 𝑟𝑏𝑎𝑠𝑒
)
)
360
Where 𝐹 𝑏𝑖𝑑 is a forward rate to sell base currency for counter currency, 𝑆 𝑏𝑖𝑑 is the spot rate to
𝑏𝑖𝑑
sell base currency for counter currency, 𝑟𝑏𝑎𝑠𝑒
is the interest rate of counter currency that investor
𝑎𝑠𝑘
receives (investor lends money), and 𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 is the interest rate of base currency that investor
pays (investor borrows money).
There is also a question why divide 𝑡 by 360. This actually depends on the day count convention
(𝑎𝑐𝑡 represents the actual number of days in the calendar)46:
Table 5: Day counting convention
Convention
Act/360
Act/365
30/360
Act/Act
Description
𝐴𝑐𝑡𝑢𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑖𝑛 𝑚𝑜𝑛𝑡ℎ
360
𝐴𝑐𝑡𝑢𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑖𝑛 𝑚𝑜𝑛𝑡ℎ
365
30
360
𝐴𝑐𝑡𝑢𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑖𝑛 𝑚𝑜𝑛𝑡ℎ
𝐴𝑐𝑡𝑢𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑖𝑛 𝑦𝑒𝑎𝑟
Usage
Money market
British money market
Bonds trading
T-notes and T-bonds
Source: Jílek47
45
JÍLEK, Josef. Finanční trhy. 1.vyd. Praha: Grada Publishing, 1997, 527 s. ISBN 80-716-9453-3. p. 391
JÍLEK, Josef. Finanční trhy a investování. 1. vyd. Praha: Grada, 2009, 648 s. ISBN 978-80-247-1653-4. p. 214
47
JÍLEK, Josef. Finanční a komoditní deriváty v praxi. 2. upr. vyd. Praha: Grada, 2010, 630 s. Finance (Grada).
ISBN 978-80-247-3696-9. p. 189, 195, 198 – 199
46
22
Deterministic arbitrage
Discrete simple interest model is based on simple interest, but without interest rate difference
approximation48,49:
𝐹=𝑆
1 + 𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 × 𝑛
1 + 𝑟𝑏𝑎𝑠𝑒 × 𝑛
( 12 )
Discrete compound interest model is49:
𝐹 𝑎𝑠𝑘 = 𝑆
𝐹
𝑏𝑖𝑑
=𝑆
𝑏𝑖𝑑
(1 + 𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
)
𝑎𝑠𝑘
(1 +
𝑏𝑖𝑑
𝑡
( 13 )
𝑎𝑠𝑘 𝑡
𝑟𝑏𝑎𝑠𝑒
)
𝑎𝑠𝑘
(1 + 𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
)
𝑏𝑖𝑑
(1 + 𝑟𝑏𝑎𝑠𝑒
)
𝑡
( 14 )
𝑡
And Continuous compounding model49,50:
𝑏𝑖𝑑
𝑎𝑠𝑘
𝑎𝑠𝑘
𝑏𝑖𝑑
𝐹 𝑏𝑖𝑑 = 𝑆 𝑏𝑖𝑑 𝑒 (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 −𝑟𝑏𝑎𝑠𝑒)𝑇
( 15 )
𝐹 𝑎𝑠𝑘 = 𝑆 𝑎𝑠𝑘 𝑒 (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 −𝑟𝑏𝑎𝑠𝑒)𝑇
( 16 )
Question might come as to which model should be used. This ultimately depends on the
borrowing/lending facility that the investor uses – i.e. in case of bank load/deposit, compound
interest model might be adequate. Continuous compounding model seems to be the most
flexible, because it can be easily further analytically enhanced to account for stochastic interest
rates, stochastic taxes, etc.
The following decision table is a result of modifying the example provided by Jílek49 to follow
used notation and continuous compounding model, and then also generalized for arbitrary
values:
Table 6: Forex futures cash-and-carry arbitrage decision table
Market situation
How to profit
𝑏𝑖𝑑
𝐾𝐹𝑎𝑠𝑘 𝑒 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡
1. Borrow
of base currency till
𝑆 𝑏𝑖𝑑
expiration, where 𝐾 is the contract size denominated
in base currency. At expiration, the debt becomes
𝑎𝑠𝑘
𝑏𝑖𝑑
(𝑟
−𝑟
)𝑡
𝐾𝐹𝑎𝑠𝑘 𝑒 𝑏𝑎𝑠𝑒 𝑐𝑜𝑢𝑛𝑡𝑒𝑟
𝐹 𝑎𝑠𝑘 < 𝑆
𝑎𝑠𝑘
𝑏𝑖𝑑
𝑏𝑖𝑑 (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 −𝑟𝑏𝑎𝑠𝑒 )𝑇
𝑒
𝑆 𝑏𝑖𝑑
of base currency
𝑏𝑖𝑑
2. Exchange
𝐾𝐹𝑎𝑠𝑘 𝑒 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡
𝑆 𝑏𝑖𝑑
of base currency to
𝑏𝑖𝑑
−𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
𝑡
𝐾𝐹 𝑎𝑠𝑘 𝑒
of counter currency
𝑏𝑖𝑑
𝑎𝑠𝑘 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
𝑡
3. Lend 𝐾𝐹 𝑒
of counter currency till
expiration, at which point the claim becomes
𝐾𝐹 𝑎𝑠𝑘 of counter currency
DVOŘÁK, Petr. Deriváty. Praha: Oeconomica, 2010. ISBN 978-80-245-1435-2. p. 187
JÍLEK, Josef. Finanční a komoditní deriváty v praxi. 2. upr. vyd. Praha: Grada, 2010, 630 s. Finance (Grada).
ISBN 978-80-247-3696-9. p. 189, 195, 198 – 199
50
HULL, John. Options, futures. 5th ed. Upper Saddle River, NJ: Prentice Hall, c2003, xxi, 744 p. ISBN 0-13009056-5. p. 56
48
49
Deterministic arbitrage
23
4. Buy futures contract for 𝐹 𝑎𝑠𝑘 to buy 𝐾 base
currency for 𝐾𝐹 𝑎𝑠𝑘 counter currency
𝑎𝑠𝑘
5. Profit at maturity = 𝐾 (1 −
𝑏𝑖𝑑
(𝑟
−𝑟
)𝑡
𝐹𝑎𝑠𝑘 𝑒 𝑏𝑎𝑠𝑒 𝑐𝑜𝑢𝑛𝑡𝑒𝑟
𝑆 𝑏𝑖𝑑
) of
base currency
𝑎𝑠𝑘
1. Borrow 𝐾𝐹 𝑏𝑖𝑑 𝑒 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡 of counter currency till
expiration, at which point the debt becomes 𝐾𝐹 𝑏𝑖𝑑
𝑎𝑠𝑘
2. Exchange 𝐾𝐹 𝑏𝑖𝑑 𝑒 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡 of counter currency to
𝑎𝑠𝑘
𝐾𝐹𝑏𝑖𝑑 𝑒 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡
𝑆 𝑎𝑠𝑘
of base currency
𝑎𝑠𝑘
𝐾𝐹𝑏𝑖𝑑 𝑒 −𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑡
𝐹 𝑏𝑖𝑑 > 𝑆 𝑎𝑠𝑘 𝑒
𝑏𝑖𝑑
𝑎𝑠𝑘
(𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟
−𝑟𝑏𝑎𝑠𝑒
)𝑇
3. Lend
of base currency till expiration,
𝑆 𝑎𝑠𝑘
at
which
point
the
claim
becomes
𝑏𝑖𝑑
𝑎𝑠𝑘
(𝑟
−𝑟
)𝑡
𝐾𝐹𝑏𝑖𝑑 𝑒 𝑏𝑎𝑠𝑒 𝑐𝑜𝑢𝑛𝑡𝑒𝑟
𝑆 𝑎𝑠𝑘
4. Sell futures contract for 𝐹 𝑏𝑖𝑑 to sell 𝐾 base currency
for 𝐾𝐹 𝑏𝑖𝑑 counter currency
𝑏𝑖𝑑
5. Profit at maturity = 𝐾 (
𝑎𝑠𝑘
(𝑟
−𝑟
)𝑡
𝐹𝑏𝑖𝑑 𝑒 𝑏𝑎𝑠𝑒 𝑐𝑜𝑢𝑛𝑡𝑒𝑟
𝑆 𝑎𝑠𝑘
− 1) of
counter currency
Source: Constructed and generalized from information provided by Jílek49
𝑏𝑖𝑑
𝑎𝑠𝑘
𝑎𝑠𝑘
𝑏𝑖𝑑
The interval 〈𝑆 𝑏𝑖𝑑 𝑒 (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 −𝑟𝑏𝑎𝑠𝑒)𝑇 ; 𝑆 𝑎𝑠𝑘 𝑒 (𝑟𝑐𝑜𝑢𝑛𝑡𝑒𝑟 −𝑟𝑏𝑎𝑠𝑒)𝑇 〉 is called no-arbitrage band, where
arbitrage is not possible because of the bid-ask spread. As mentioned before, the profit is earned
at expiration, but is locked-in in the beginning.
Note: A long USD/EUR futures is a contract to buy USD for EUR at maturity, and the contract
size is denominated in base currency51.
4.4.3 Options cash-and-carry arbitrage
It is possible to construct a (replication) portfolio of underlying asset and the riskless asset such
that it provides the same cash flows as a call or put option52. Since both have the same cash
flows, the portfolio and the option have to trade at the same price. Such position might look
like this:

51
5
Buy 7 shares and borrow $22.50
LABUSZEWSKI, John W., Sandra RO and David GIBBS. Understanding FX Futures. In: CME Group
[online]. 2013 [Accessed 2014-05-11]. Available from:
http://www.cmegroup.com/education/files/understanding-fx-futures.pdf. p. 7
52
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 22 – 28
24
Deterministic arbitrage
5

In case that spot price is $50, the cost of this portfolio is: 7 50 − 22.5 = 13.21

In case the option is traded at less than 13.21, arbitrageur would buy the option and sell
the portfolio and claim the difference
Generalization of aforementioned arbitrage to an arbitrage-free model is the famous BlackSholes option pricing formula53,54 – based on stochastic (Wiener) process. The model itself
assumes European-type option on stocks with no dividends55:
𝐶 = 𝑆 × 𝑁(𝑑1 ) − 𝐾 × 𝑒 −𝑟𝑇 × 𝑁(𝑑2 )
𝑑1 =
𝑆
𝜎2
ln 𝐾 + (𝑟 + 2 ) 𝑇
( 17 )
( 18 )
𝜎√𝑇
𝑑2 = 𝑑1 − 𝜎√𝑇
( 19 )
Where 𝑆 is the spot price of the underlying asset, 𝐾 is the strike price, 𝑇 is the time to maturity
in years, 𝜎 is the annualized standard deviation of logarithmic returns, 𝑟 is the risk-free rate and
𝑁 is the cumulative distribution function of the standard normal distribution.
Other type of arbitrage is possible when the put-call parity does not hold. The advantage of this
type of arbitrage is that it does not depend on the option pricing model and does not have any
assumptions about volatility. It is based on the idea that when arbitrageur sells a call, buys a
put, and buys the underlying asset, he will end up with the value of strike at maturity56:
Table 7: Put-call parity derivation
Position
Payoff in future if spot > strike Payoff in future if spot < strike
Short call
- (Spot – Strike)
0
Long put
0
Strike – Spot
Long stock
Spot
Spot
Total
Strike
Source: Damodaran56
Strike
The cost of creating this position must be equal to the present value of strike. Under the
assumption that the options in question are of European-type and the underlying asset does not
yield any dividends, the parity equation is55:
𝑆 + 𝑃 − 𝐶 = 𝐾𝑒 −𝑟𝑡
( 20 )
REJNUŠ, Oldřich. Finanční trhy. 3., rozš. vyd. Ostrava: Key Publishing, 2011, 689 s. Ekonomie (Key
Publishing). ISBN 978-80-7418-128-3. p. 486
54
JÍLEK, Josef. Finanční trhy. 1.vyd. Praha: Grada Publishing, 1997, 527 s. ISBN 80-716-9453-3. p. 413 – 414
55
SHARPE, William F. Investice. Praha: Victoria Publishing, 1994, 810 s. ISBN 80-856-0547-3. p. 488
56
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 22 – 28
53
Deterministic arbitrage
25
Where 𝑆 is spot price, 𝑃 is price of put option, 𝐶 is price of call option, 𝐾 is strike, 𝑟 is riskfree rate and 𝑡 is time to maturity. When the equation does not hold, strategies (called
conversion and reverse conversion) in the following table can be used to realize a profit57:
Table 8: Options cash-and-carry arbitrage decision table
Market situation
How to profit
Sell call 𝐶
Buy put 𝑃
Buy underlying asset 𝑆
Borrow 𝑆 + 𝑃 − 𝐶 at 𝑟 to finance the investment, which
becomes debt of (𝑆 + 𝑃 − 𝐶)𝑒 𝑟𝑡 at maturity
5. At maturity, exercise both options, which also covers the long
position in 𝑆, and results in net cash-flow of 𝐾
6. Realize a profit at maturity of 𝐾 − (𝑆 + 𝑃 − 𝐶)𝑒 𝑟𝑡 in future
prices
Realize a discounted profit at maturity of 𝐾𝑒 −𝑟𝑡 − 𝑆 − 𝑃 + 𝐶
in current prices
1.
2.
3.
4.
𝑆 + 𝑃 − 𝐶 < 𝐾𝑒 −𝑟𝑡
(conversion)
Buy call 𝐶
Sell put 𝑃
Sell underlying asset 𝑆
Deposit 𝑆 + 𝑃 − 𝐶 at 𝑟 until maturity, at which point the
claim becomes (𝑆 + 𝑃 − 𝐶)𝑒 𝑟𝑡
𝑆 + 𝑃 − 𝐶 > 𝐾𝑒 −𝑟𝑡
5. At expiration, exercise both options, which also covers the
(reverse conversion)
short position in 𝑆, and results in net cash-flow of 𝐾
6. Realize a profit at maturity of (𝑆 + 𝑃 − 𝐶)𝑒 𝑟𝑡 − 𝐾 in future
prices
Realize a discounted profit at maturity of 𝑆 + 𝑃 − 𝐶 − 𝐾𝑒 −𝑟𝑡
in current prices
Source: Constructed from information provided by Hull57
1.
2.
3.
4.
Both the put and call options must have the same exercise price and expiration date. The amount
of options must match the amount of underlying assets. Furthermore, the price to buy call option
is different to the price to sell call option, because of the bid-ask spread. This applies to all
assets mentioned. For that reason, there will also be the no-arbitrage band.
57
HULL, John. Options, futures. 5th ed. Upper Saddle River, NJ: Prentice Hall, c2003, xxi, 744 p. ISBN 0-13009056-5. p. 174 – 175
26
Deterministic arbitrage
4.4.4 Futures-options cash-and-carry arbitrage
When both futures and options have the same underlying asset and mature at the same
expiration date, the futures can be replicated by a combination of put option, call option and
riskless asset58. This results into a futures-put-call parity of58:
−𝐶𝑡 + 𝑃𝑡 +
𝐹𝑡 − 𝑋
=0
(1 + 𝑟)(𝑇−𝑡)
( 21 )
𝐹𝑡 = (𝐶𝑡 − 𝑃𝑡 )(1 + 𝑟)(𝑇−𝑡) + 𝑋
( 22 )
Where 𝐹𝑡 is the futures price at the time 𝑡, 𝑇 is the time of expiration, 𝐶𝑡 is the price of call at
time 𝑡, 𝑃𝑡 is the price of put at time 𝑡, 𝑋 is the exercise price of the options, and 𝑟 is the riskfree rate. In case the parity is violated, use the following decision table58:
Table 9: Futures-options cash-and-carry arbitrage decision table
Market situation
How to profit
𝐹𝑡𝑎𝑠𝑘 > 𝑋
1. Buy futures
2. Sell call option
3. Buy put option
4. Lend
𝐹𝑡 −𝑋
(1+𝑟)(𝑇−𝑡)
at
interest rate 𝑟
𝐹𝑡𝑎𝑠𝑘 < (𝐶𝑡𝑏𝑖𝑑 − 𝑃𝑡𝑎𝑠𝑘 )(1 + 𝑟)(𝑇−𝑡) + 𝑋
𝐹𝑡𝑎𝑠𝑘 < 𝑋
1. Buy futures
2. Sell call option
3. Buy put option
4. Borrow
𝑋−𝐹𝑡
(1+𝑟)(𝑇−𝑡)
at
interest rate 𝑟
𝐹𝑡𝑏𝑖𝑑 > 𝑋
1. Sell futures
2. Buy call option
3. Sell put option
4. Borrow
𝐹𝑡 −𝑋
(1+𝑟)(𝑇−𝑡)
interest rate 𝑟
𝐹𝑡𝑏𝑖𝑑 > (𝐶𝑡𝑎𝑠𝑘 − 𝑃𝑡𝑏𝑖𝑑 )(1 + 𝑟)(𝑇−𝑡) + 𝑋
𝐹𝑡𝑏𝑖𝑑 < 𝑋
1. Sell futures
2. Buy call option
3. Sell put option
4. Lend
𝑋−𝐹𝑡
(1+𝑟)(𝑇−𝑡)
interest rate 𝑟
58
Source: Constructed from the information by Bae et al.
58
at
BAE, Kee-Hong, Kalok CHAN and Yan-Leung CHEUNG. The Profitability of Index Futures Arbitrage:
Evidence from Bid-Ask Quotes [online]. Hong Kong, 1998 [Accessed 2014-05-05]. Available from:
http://www.proquest.com/. Research paper. City University of Hong Kong and Hong Kong University of
Science & Technology. p. 4, 12
at
Deterministic arbitrage
27
4.4.5 Fixed-income cash-and-carry arbitrage
It’s relatively simpler to arbitrage fixed-income securities thanks to the fixed cash flows59. For
example, in case of government 10-year bond, one could replicate the cash flows using zerocoupon bonds with expirations matching those of coupon payment dates on the 10-year bond.
The present value of such portfolio should match the 10-year bond. In case it doesn’t, there is
an arbitrage opportunity. For corporate bonds, such arbitrage is a bit more complicated because
corporate bonds carry the risk of default, so arbitrageurs might have to hedge using CDS.
Fixed-income securities (such as bonds) can be examined from 2 perspectives – from the
perspective of investors and from the perspective of traders. Investor just buys the bond and
holds it till maturity, periodically receiving coupons and then face value at last. For a trader,
bonds change value constantly because the interest rates change constantly as well60.
4.5 Tax arbitrage
A special type of arbitrage is the so-called tax arbitrage. This can be also thought of as tax
optimizations, although in this case the rather common use of complex financial instruments –
such as financial derivatives – is not encompassed by the common notion of tax optimizations.
An example of such arbitrage61:
German tax system used to moderately support German companies in holding German shares.
So, Salomon Brothers made a system in which German companies bought local shares, and
then sold the stock risk to foreign investors using stock swaps. Thus, foreign investors were
basically investing in German stocks as if they were German investors, with all the related tax
benefits.
Using tax heavens can be thought of as tax arbitrage as well61. For example, manufacturing
company can sell goods to its offshore company for a low price, which then subsequently resells
the goods to consumers for market price. Thus, there is almost no profit generated in the country
where manufacturing company resides. The transfer of profits back to the country of origin is
a bit more complicated, but most companies are not interested in that anyway. Another example
of tax arbitrage is what Sholes used in LTCM fund61:
In USA, the income tax is significantly higher than capital gains tax. So, Sholes had to think of
a way how to get his money out of LTCM such that he wouldn’t have to pay the income tax. He
partnered with UBS. UBS bought LTCM shares and sold to Sholes newly emitted European
options on those shares, which would expire in 7 years. After 7 years, Sholes would pay only
capital gains tax and not income tax.
4.6 Regulatory arbitrage
Regulatory arbitrage is the practice of taking advantage of regulatory differences between two
or more markets. An example of regulatory arbitrage is the fact that banks are much more
regulated than pension funds or insurance companies. Banks can use financial derivatives (such
as CDS) to get rid of toxic instruments and sell it to less-regulated institutions61.
59
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 31
60
JÍLEK, Josef. Finanční trhy a investování. 1. vyd. Praha: Grada, 2009, 648 s. ISBN 978-80-247-1653-4. p. 93
61
JÍLEK, Josef. Deriváty, hedžové fondy, offshorové společnosti. 1. vyd. Praha: Grada, 2006, 260 s. ISBN 80247-1826-X. p. 32, 74, 171, 247
28
Statistical arbitrage
5 STATISTICAL ARBITRAGE
Statistical arbitrage is a strategy that is market-neutral (no correlation with market returns), has
systematic (rules-based) trading signals, and the general mechanism for generating profit is
statistical, i.e. based on a mathematical model62,63. Holding period for individual positions may
range from seconds or days to even longer. Contemporary statistical arbitrageurs mostly focus
on high-frequency trading using neural networks and/or statistical models64.
Obviously there is some risk in statistical arbitrage. Statistical arbitrage is called arbitrage
because the investment risks are expected to be largely offsetting (such as in simultaneous
long/short position), with a likely profit65. It’s usually dependent on statistical behavior of large
number of favorable bets to eventually deliver profit. In other words, the profit is expected to
manifest thanks to the law of large numbers66. Thus, it’s expected to have some losses as long
as the overall strategy is profitable. This means that statistical arbitrage is mostly limited to
those that control large capital.
It has been proven in practice that statistical arbitrage opportunities based on volatility errors
and option mispricing do exist - strategies that exploit them yield profit with a probability higher
than 50%67.
Thorp’s statistical arbitrage hedge fund was based on the principle of mean-reversion (big
gainers in recent history will probably fall and big losers will probably grow), and very large
number of many diversified short-term long positions with offsetting short positions to hedge
against market movements (large number of short-term positions led to a new position being
open every few seconds)65. This supposedly resulted in a market-neutral portfolio with
significant profit. For an example of a simple mean-reverting statistical arbitrage strategy based
on Ornstein-Uhlenbeck model, refer to paper written by Avellaneda68.
Statistical arbitrage is a strategy used by investors who believe that behavior of securities can
be described mathematically, such as using the time series analysis, neural networks or
62
AVELLANEDA, Marco and Jeong-Hyun LEE. Statistical Arbitrage in the U.S. Equities Market [online]. New
York, 2008 [Accessed 2014-04-22]. Available from:
http://www.math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb071108.pdf. Research paper. New York
University: Department of Mathematics. p. 1 – 2
63
TASHMAN, Adam Paul. Modeling Risk in Arbitrage Strategies Using Finite Mixtures [online]. New York,
2006 [Accessed 2014-04-29]. Available from: http://www.proquest.com/. Dissertation. Stony Brook University.
p. 4
64
KUEPPER, Justin. Arbitrage Squeezes Profit From Market Inefficiency. In: Investopedia [online]. 2013
[Accessed 2014-04-26]. Available from: http://www.investopedia.com/articles/trading/07/statisticalarbitrage.asp
65
THORP, Edward. Statistical Arbitrage – Part II. In: WILMOTT, Paul. Wilmott [online]. 2004 [Accessed
2014-04-26]. Available from: http://www.wilmott.com/pdfs/080630_thorp.pdf. p. 48 – 49
66
AHMED, Murat, Anwei CHAI, Xiaowei DING, Yunjiang JIANG and Yunting SUN. Statistical Arbitrage in
High Frequency Trading Based on Limit Order Book Dynamics [online]. Stanford, 2009 [Accessed 2014-04-09].
Available from: http://www.stanford.edu/class/msande444/2009/2009Projects/2009-2/MSE444.pdf. Research
paper. Stanford University. p. 1
67
WILL, Matthew. Statistical arbitrage opportunities resulting from volatility errors and option mispricing
[online]. Anderson, 2007 [Accessed 2014-04-29]. Available from: http://www.proquest.com/. Dissertation.
Anderson University. p. 87
68
AVELLANEDA, Marco. Risk and Portfolio Management [online]. New York, 2011 [Accessed 2014-04-26].
Available from: http://www.math.nyu.edu/faculty/avellane/Lecture8Risk2011.pdf. New York University:
Department of Mathematics.
Statistical arbitrage
29
econometric models69. Neural networks have the advantage of being able to find non-linear
patterns in time series64.
Thus, characteristics of statistical arbitrage are:

Market (beta) neutral

Trading mechanism based on statistics, usually mean-reversion
Technical analysis is in some aspects similar to statistical arbitrage. It can use the same tools.
However, technical analysis is highly speculative trading based on trends, whereas statistical
arbitrage model should generate significant alpha (the return above market return) and have low
exposure to market risk (such as through hedging)70. Thus, I find the biggest difference between
statistical arbitrage and technical analysis to be the market neutral approach.
However, statistical arbitrage is susceptible to the practice of over-fitting a trading strategy to
a particular sample period during back-testing. One solution might be to use multiple sets of
data, some for back-testing and others for subsequent validation71.
5.1 Scalping (market making)
Scalpers are those that always stand ready to buy at one tick below or sell at one tick above the
last price, rarely hold positions overnight and trade at very high frequency72. Thus, scalpers
often play the role of liquidity-providing market makers who profit from their own bid-ask
spreads72. The scalper’s profits are positively related to the bid-ask spread and negatively
related to the length of time a position is held73. The idea behind scalping is not to make profit
on every trade, but to be in profit on average – make more correct guesses than incorrect ones,
based on gauging the short term imbalances in supply and demand74.
Market maker is exposed to losses from the positions in the security (a.k.a. inventory holding
cost), order processing costs (operating expenses) and adverse selection costs (when trading
with informed investors), which all have to be offset by the bit-ask difference75,76. In other
69
BRETZLER, Marcus and Dirk RUDOLPH. Hedge Fonds und Alternative Investments. First edition. Frankfurt
am Main: Bankakademie-Verlag GmbH, 2004. ISBN 3-937519-08-4. p. 31 – 32
70
ALONSO, Miquel Noguer. Statistical Arbitrage and Algorithmic Trading: Overview and Applications
[online]. Madrid, 2010 [Accessed 2014-04-09]. Available from: http://espacio.uned.es:8080/fedora/get/tesisuned:CiencEcoEmp-Mnoguer/Documento.pdf. Dissertation. National
University of Distance Education. p. 130
71
THOMAIDIS, Nikos S. Statistical arbitrage and pairs trading. In: University of the Aegean [online]. 2013
[Accessed 2014-04-21]. Available from: http://casl.ucd.ie/hosted/Thomaidis.pdf
72
LENG, Hsiaohua. The intraday price dynamics of foreign exchange futures market [online]. Madison, 1996
[Accessed 2014-04-29]. Available from: http://www.proquest.com/. Dissertation. University of Wisconsin –
Madison. p. 37 – 38
73
LIU, Shi-Miin. Market efficiency and the microstructure of grain futures markets implied by return series of
various time intervals [online]. Illinois, 1990 [Accessed 2014-05-07]. Available from:
http://www.proquest.com/. Dissertation. University of Illinois at Urbana-Champaign. p. 16
74
LIU, Dongqing. Market-Making Behavior in Futures Markets [online]. California, 2002 [Accessed 2014-0507]. Available from: http://www.proquest.com/. Dissertation. University of California. p. 9 – 10
75
ASCIOGLU, N. 2000. Market Maker’s Use of Depth in Price Experimentation [online]. Dissertation.
Memphis: The University of Memphis. Available from: http://www.proquest.com/ [Accessed 2014-05-08] p. 1
76
HEIDLE, H. 2001. Information, trading, and market making [online]. Dissertation. Nashville: Vanderbilt
University. Available from: http://www.proquest.com/ [Accessed 2014-05-08] p. 5
30
Statistical arbitrage
words, the profits of market makers from bid-ask differences offset the risk faced by market
makers from holding of undiversified unwanted portfolio77.
This can be summarized to the following characteristics of a scalper:

High-frequency trading

Relatively small transactions, which add up to a huge volume

Short-lived positions

Almost no overnight positions (which leads to narrow spreads in the market at the end
of trading day77)

Trading based on statistical model of short-term supply and demand

Main factor behind the profits of market makers is their ability to forecast future trends78
For example, the following simple price-reversal scalping strategy generated significant profits
and performed best out of the proposed strategies in the paper by Leng79:
“Close out all outstanding positions and buy one more contract if the current price is lower
than the previous price which is also lower than immediate prior price (𝑃𝑡 < 𝑃𝑡−1 < 𝑃𝑡−2 ).
Offset all outstanding long positions and sell one more contract if the opposite is true (𝑃𝑡 >
𝑃𝑡−1 > 𝑃𝑡−2 ). All outstanding positions are closed at the last transaction price.”
It has been shown that high-frequency data have the property of bid-ask bounce, which means
that some buy orders are bought at ask price and some sell orders are sold at the bid prices,
which leads to large first order autocorrelation in price changes, which leaves a potential room
for exploitation80.
For a theoretically-proved profitable market-making algorithm requiring a mean-reverting price
development (profitable even for the weakest mean-reversion processes without any specific
form), refer to the paper by Chakraborty et al.81.
Some argue that market making is not a type of statistical arbitrage, because statistical arbitrage
often takes directional bet (i.e. in case of co-integration arbitrage, there is one long and one
short position), while the goal of market making (scalping) is minimizing directional risk and
profit from non-directional volatility81. However, market maker’s algorithms are still based on
the assumption of mean-reversal (e.g. Chakraborty et al. assumed mean-reverting stochastic
77
ALAMPIESKI, K. and Lepone, A.. 2006. Intraday Patterns in High Frequency Trading: Evidence from the
UK Equity Markets [online]. Research paper. Sydney: University of Sydney. Available from:
http://www.ecmcrc.org/Articles/Intraday_Patterns_in_High_Frequency_TradingEvidence_from_the_UK_Equity_Markets.pdf [Accessed 2014-05-08] p. 4 – 5
78
PANAYIDES, Marios Andrea. The Market Making system of the NYSE and other Markets: Implementation
in Emerging Markets [online]. New Haven, 2004 [Accessed 2014-05-07]. Available from:
http://www.proquest.com/. Dissertation. Yale University. p. 56
79
LENG, Hsiaohua. The intraday price dynamics of foreign exchange futures market [online]. Madison, 1996
[Accessed 2014-04-29]. Available from: http://www.proquest.com/. Dissertation. University of Wisconsin Madison. p. 39 – 40)
80
MURRAY, Jennifer Wells. A study of intraday volatility trading utilizing high frequency data and the
microstructure effects of implementation [online]. New York, 2010 [Accessed 2014-04-29]. Available from:
http://www.proquest.com/. Dissertation. Fordham University. p. 32
81
CHAKRABORTY, T. and Kearns, M.. 2011. Market Making and Mean Reversion [online]. Research paper.
Philadelphia: University of Pennsylvania. Available from:
http://www.cis.upenn.edu/~mkearns/papers/marketmaking.pdf [Accessed 2014-05-07]
Statistical arbitrage
31
Ornstein-Uhlenbeck process in their paper81) and statistical models, and for that reason I have
categorized it as a kind of statistical arbitrage. In my opinion, the feature of statistical arbitrage
is not about presence/absence of directional bets, but rather the statistical approach and the goal
of eventual market-neutral (average) profits rather than making the correct bet every time.
Research have shown a distinctive feature of monopolistic market makers called price
experimentation. In the very first (about 30) minutes after the market opens, he/she tries to
encourage informed trading (because the information asymmetry is high when the market
opens) by widening spread and increasing depth. In the later stage, when the market maker
discovers the true value of the security, he/she subsequently narrows the spreads and reduce
depth, to profit on uninformed (noise) and liquidity traders82. Market makers are willing to take
short-term losses incurred by price discovery, because those losses are then recouped later
throughout the day82.
A different study (that no longer assumed monopolistic market maker) showed that actually the
price experimentation mechanism is actually quite different. It was showed that the first
discovery stage consists of large spreads but shallow depth, while after-discovery quotes are
characterized by narrower spread and significant depth83.
For a practical algorithmic approach to market making (which is being used in Deutsche Bank),
refer to the paper by Nuti84. Othman85 presents another practical approach to high-frequency
market making.
Ahmed et al.86 describe in their paper a trading strategy based on high-frequency trading of the
bid-ask spread in order book of a single asset. This system analyzes the order book, and if the
structure is appropriate, it places an order at the best ask price and waits for the best bid to raise
to the purchase level and above. It uses statistical model to predict how the placement of the
order itself affects the market and whether the market will react to that event favorably.
High-frequency trading is done by powerful computers located as close to the computers that
drive the exchange as possible, and accounts for a significant portion of profits made by hedge
funds or institutions like Goldman Sachs87. The trading is done by automated algorithms
executing millions of orders a second. Some algorithms may issue and simultaneously cancel
many orders simultaneously, in a hope to influence the market to go in a desired direction.
However some argue that high-frequency trading is not very fair – to the extent that a loophole
ASCIOGLU, N. 2000. Market Maker’s Use of Depth in Price Experimentation [online]. Dissertation.
Memphis: The University of Memphis. Available from: http://www.proquest.com/ [Accessed 2014-05-08] p. 1 –
5, 96 – 97, v – vi
83
ALAMPIESKI, K. and Lepone, A.. 2006. Intraday Patterns in High Frequency Trading: Evidence from the
UK Equity Markets [online]. Research paper. Sydney: University of Sydney. Available from:
http://www.ecmcrc.org/Articles/Intraday_Patterns_in_High_Frequency_TradingEvidence_from_the_UK_Equity_Markets.pdf [Accessed 2014-05-08] p. 1 – 3
84
NUTI, G. 2008. An Electronic Market-Making Algorithm [online]. Dissertation. London: University of
London. Available from: http://www.proquest.com/ [Accessed 2014-05-08]
85
OTHMAN, A. 2012. Automated Market Making: Theory and Practice [online]. Dissertation. Pittsburgh:
Carnegie Mellon University. Available from:
http://repository.cmu.edu/cgi/viewcontent.cgi?article=1160&context=dissertations [Accessed 2014-05-08]
86
AHMED, Murat, Anwei CHAI, Xiaowei DING, Yunjiang JIANG and Yunting SUN. Statistical Arbitrage in
High Frequency Trading Based on Limit Order Book Dynamics [online]. Stanford, 2009 [Accessed 2014-04-09].
Available from: http://www.stanford.edu/class/msande444/2009/2009Projects/2009-2/MSE444.pdf. Research
paper. Stanford University.
87
DUHIGG, Charles. Stock Traders Find Speed Pays, in Milliseconds. In: The New York Times [online]. 2009
[Accessed 2014-04-10]. Available from: http://www.nytimes.com/2009/07/24/business/24trading.html?_r=0
82
32
Statistical arbitrage
in regulations allows exchanges to charge a fee to traders who wish to receive the market data
ahead of everyone else87. In any case, high-frequency traders generated about
21 000 000 000 USD of profits in the year 2008.
A very simple high-frequency order book arbitrage algorithm (which is very simple to trade
against by smarter algorithms) works as follows. Suppose the following table describes actual
order book for some stock:
Table 10: High-frequency order book example
Amount
Ask
Price
300
10
Bid
Source: Author’s own construction
1 002
1 000
Because the bid amount is much lower than ask amount, it’s somewhat reasonable to expect
that the price will fall88. One explanation for that is that if the imminent orders coming to the
exchange are random, those are much more likely to trade through the 10 bids rather than 300
asks. So, trader might want to place a small ask at the price 1 001 for example. That order is
somewhat likely to get filled, and soon afterwards it’s reasonable to expect the price to drop
because the bid quotes at 1 000 get all filled. Once the bid amount is high enough, we can do
the opposite and place a small bid at a tick higher.
More complicated and practical strategies might require very good statistical model of the highfrequency market behavior.
5.2 Relative value arbitrage
Relative value arbitrage is based on trading the distortions between individual securities –
distortions in the relative value of one security to another. The basic strategy is to take a long
position in undervalued security (that is expected to appreciate) and short position in overvalued
security (that is expected to depreciate), which leads to an overall market-neutral position89. For
example, if there are two bonds and one of them is undervalued, then the undervalued bond is
bought but at the same time the other bond is short-sold as a hedge. In other words, relative
value arbitrage is all about trading similar or related assets (that are historically or
mathematically interconnected), rather than identical assets90.
Some examples of related assets are stock and bond of a company, stocks of two different
companies in the same sector, or two bonds issued by the same company but with different
parameters91. Other examples might be dual-listed or multiple-listed (e.g. companies traded at
88
CHEN, Rocko. Orderbook High Frequency Trading Tactics. In: Ingenuity Space [online]. 2011 [Accessed
2014-04-22]. Available from: http://matdays.blogspot.co.nz/2011/09/orderbook-strategies.html
89
BRETZLER, Marcus and Dirk RUDOLPH. Hedge Fonds und Alternative Investments. First edition. Frankfurt
am Main: Bankakademie-Verlag GmbH, 2004. ISBN 3-937519-08-4. p. 31
90
FOLGER, Jean. Arbitrage and Pairs Trading. In: Investopedia [online]. © 2014b [Accessed 2014-04-21].
Available from: http://www.investopedia.com/university/guide-pairs-trading/arbitrage-and-pairs-trading.asp
91
BARUFALDI, Dan. Hedge Funds: Strategies. In: Investopedia [online]. © 2014 [Accessed 2014-04-26].
Available from: http://www.investopedia.com/university/hedge-fund/strategies.asp
Statistical arbitrage
33
multiple exchanges in different countries) stocks and depository receipts92,94, stocks and plainvanilla certificate on that stock93, or any other combination of certificate and the underlying
asset. However, certificates are emitted by financial institution, such as investment banks93,95,
and for that reason it’s quite unlikely that price of those instruments would allow for any kind
of arbitrage.
Depository receipts are claims equivalent to the one investor would have in case he would
purchase shares in the local market92, and also typically entitle their owner to convert depository
receipts into the original shares, as well as give the owner the right to receive dividends94. In
other words, depository receipts are securities that are traded at domestic financial market, but
represent a stock emitted by foreign company95. They allow investors to purchase shares of a
company that is otherwise not available to foreign investors.
Research have shown that arbitrage between Taiwanese depository receipts and their
underlying stocks is very profitable, with annualized returns exceeding 100% in most cases96.
The strategy had a high variance, yet the Sharpe ratio was much better than that of the market
anyway.
There are other arbitrage strategies when it comes to depository receipts – such as converting
shares to ADRs and vice versa, or use mean-reversion arbitrage96. Especially in case of
American depository (ADR) receipts emitted on Taiwanese shares, the arbitrage strategies
showed great profitability96, but require the arbitrageur to be able to buy and short sell
Taiwanese shares as well as to buy the ADRs in the U.S. market96.
The general principle is that in case that depository receipt is trading at a discount relatively to
the original share, arbitrageur may wish to purchase the depository receipt and at the same time
short the original share to hedge against company risk. The resulting position is a bet on
convergence of depository receipt with the original share. It may also be possible to convert the
depository receipt into the underlying stock and use it to cover the short position.
5.2.1 Convertible arbitrage
On the matter of convertible arbitrage, Thorp et al. says: “Any security that may be exchanged
for common stock is a convertible security. Besides warrants, there are convertible bonds,
convertible preferred stocks, calls, * stock rights, and stock options.”97
Convertible bond is a bond that can be converted into a certain number of shares98. So in fact it
has 2 components – a bond and a stock option. Convertible arbitrage is then based on purchasing
92
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012 [Accessed
2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf. New York
University. p. 35, 38
93
REJNUŠ, Oldřich. Finanční trhy. 3., rozš. vyd. Ostrava: Key Publishing, 2011, 689 s. Ekonomie (Key
Publishing). ISBN 978-80-7418-128-3. p. 599, 602
94
JÍLEK, Josef. Finanční trhy. 1.vyd. Praha: Grada Publishing, 1997, 527 s. ISBN 80-716-9453-3. p. 275 – 278
95
POLOUČEK, Stanislav. Peníze, banky, finanční trhy. Vyd. 1. Praha: C.H. Beck, 2009, xviii, 414 s. Beckovy
ekonomické učebnice. ISBN 978-80-7400-152-9. p. 163, 179
96
CHEN, Andrew Y. Three Essays in International Finance [online]. New York, 2002 [Accessed 2014-04-29].
Available from: http://www.proquest.com/. Dissertation. Columbia University. p. 107 – 108, 171 – 172, 174
97
THORP, Edward O. and Sheen T. KASSOUF. Beat the Market: A Scientific Stock Market System [online].
New York: Random House, Inc., 1967 [Accessed 2014-04-26]. Available from:
http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/beatthemarket.pdf. p. 148)
98
BARUFALDI, Dan. Hedge Funds: Strategies. In: Investopedia [online]. © 2014 [Accessed 2014-04-26].
Available from: http://www.investopedia.com/university/hedge-fund/strategies.asp
34
Statistical arbitrage
convertible bond and simultaneously short-selling shares of the issuer in order to hedge against
the company-specific risk98,99,102. In case that stock declines, the value of bond might decline
too (although a little less) but that loss will be covered by the short equity position. Similarly,
when the stock rallies the bond can be converted into stocks to cover the losses from short
equity position. This leads to a delta-neutral portfolio. The motivation for convertible arbitrage
can vary:

Anticipation of bond’s price increasing and/or stock price decreasing

Anticipated reduction in interest rates, which is expected to increase the value of bond
more than the value of stock

Implied volatility of the embedded option is too low
For example, when the implied volatility seems to be too low, the arbitrageur can perform a
directional volatility bet – buy the convertible bond, and hedge with short equity position. Profit
is generated by dynamically hedging against delta exposure, provided that the realized volatility
is higher than implied volatility at the time of purchase. Such position should be hedged against
change in interest rate as well, though.
Convertible bond arbitrage requires that companies have convertible bonds outstanding in
conjunction with common stock, warrants, and conventional bonds100. In case of convertible
preferred bond, preferred stocks should be outstanding as well. If one of the securities is not
offered, arbitrageur might use security of the same type of very similar company.
A famous historical example of convertible arbitrage was the Meriwether’s arbitrage group,
which focused on Japanese convertible bonds101. The market inefficiency was caused by an
effort of Japanese government to keep stock prices high (by limiting supply through regulation),
but they did not limit the supply of convertible bonds. Demand for convertible bonds was low
(because investors did not know how to price the embedded stock option), which led to the
bonds being underpriced. The strategy was to buy the convertible bond and sell the bond risks
using interest rate swaps, which resulted in pure stock option position. Because the option in
convertible bond was undervalued, they could sell it on its own for a bonus.
5.2.2 Fixed-income arbitrage
Fixed-income arbitrage is about finding pricing anomalies in various types of bonds (mortgagebacked securities, government bonds, etc.)102.
Hasanhodzic et al. add that: “fixed income arbitrageur aims to profit from price anomalies
between related interest rate securities. Most managers trade globally with a goal of generating
steady returns with low volatility. This category includes interest rate swap arbitrage, US and
non-US government bond arbitrage, forward yield curve arbitrage, and mortgage-backed
99
HASANHODZIC, Jasmina and Andrew W. LO. Can hedge-fund returns be replicated?: The linear case. In:
Journal Of Investment Management [online]. 2007 [Accessed 2014-04-26]. Vol. 5, No. 2. Available from:
http://web.mit.edu/alo/www/Papers/replicate.pdf. p. 43
100
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012
[Accessed 2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf.
New York University. p. 46
101
JÍLEK, Josef. Deriváty, hedžové fondy, offshorové společnosti. 1. vyd. Praha: Grada, 2006, 260 s. ISBN 80247-1826-X. p. 30 – 31, 120
102
BRETZLER, Marcus and Dirk RUDOLPH. Hedge Fonds und Alternative Investments. First edition.
Frankfurt am Main: Bankakademie-Verlag GmbH, 2004. ISBN 3-937519-08-4. p. 31 – 32
Statistical arbitrage
35
securities arbitrage. The mortgage-backed market is primarily US-based, over-the-counter,
and particularly complex.”103
One way to arbitrage yield curve is to exploit deviations of bond yields from smoothed yield
curve101. The bond with relatively higher yield would be bought and bonds with close maturity
but significantly lower yields would be sold.
Inverse floating rate bonds are instruments which can be decomposed into 2 basic instruments
– fixed rate bond and interest rate swap101. In case the inverse floating rate bond is not priced
as a sum of the two underlying instruments, it’s possible to earn a risk-free profit.
Interest rate swaps can be used for example by companies with different credit scores to exploit
the interest rate spread offered by banks to both companies104. It’s a popular choice for lowering
debt interest rates using comparative advantage of swap partners105. For example, if one partner
has the choice of taking debt with fixed interest rate of 5.5% or variable rate of LIBOR, and the
other partner has the choice of fixed rate of 6.5% or variable rate LIBOR + 0.5%, then with the
use of interest rate swap, both can achieve a discount of 0.25% on their debts. The arbitrage
condition is:
𝐹𝐴 − 𝐹𝐵 < 𝑉𝐴 − 𝑉𝐵
( 23 )
And arbitrage potential can be expressed as:
𝑈𝐴𝐵 = (𝑉𝐴 − 𝑉𝐵 ) − (𝐹𝐴 − 𝐹𝐵 )
( 24 )
Where 𝐴 is the partner who has a fixed rate debt and ends up with variable rate at the end, and
𝐵 is the partner who has variable rate debt and ends up with fixed rate. 𝑉𝑥 is the variable rate of
the respective partner, and 𝐹𝑥 is the fixed rate.
The thing to note here is that even though both partners take the loan from banks, they are not
necessarily immune to the default risk of the other partner. In case one of the partners defaults,
the other partner will not receive any discount (arbitrage profit), but will also end up with a debt
with different interest than he intended to (i.e. fixed rate instead of variable rate).
5.2.3 Capital structure arbitrage
Capital structure arbitrage opportunity exists if market price of debt cannot be justified by the
company’s capital structure106. In case that arbitrageur believes that the debt of a company in
underpriced, he/she should106:
103

Buy company’s cheap bonds

Hedge against default by purchasing puts on the stock
HASANHODZIC, Jasmina and Andrew W. LO. Can hedge-fund returns be replicated?: The linear case. In:
Journal Of Investment Management [online]. 2007 [Accessed 2014-04-26]. Vol. 5, No. 2. Available from:
http://web.mit.edu/alo/www/Papers/replicate.pdf. p. 43
104
ELTON, Edwin J. Interest rate swaps [online]. New York, 1999 [Accessed 2014-04-27]. Available from:
http://pages.stern.nyu.edu/~eelton/debt_inst_class/Swap.pdf. New York University. p. 17
105
ZÁŠKODNÝ, Přemysl, Vladislav PAVLÁT and Josef BUDÍK. Finanční deriváty a jejich oceňování. 1. vyd.
Praha: Vysoká škola finanční a správní, 2007, 161 s. ISBN 978-80-86754-73-4. p. 91 – 95
106
CHIRAYATHUMADOM, Ramakrishnan, Ronnie Zachariah GEORGE, Venkateshwarlu BALLA, Dhiraj
BHAGCHANDKA, Neeraj SHAH a Kunal SHAH. Capital Structure Arbitrage: using non-Gaussian approach
[online]. Illinois, 2007 [Accessed 2014-04-26]. Available from:
http://users.iems.northwestern.edu/~armbruster/2007msande444/2004CSAslides.pdf. Northwestern University.
p. 6 – 7
36
Statistical arbitrage

Profit is the yield of the bond in excess of what the investor paid for the option
Capital structure arbitrage is basically a bet on the convergence of equity and credit markets,
and the best results are achieved when the company in financial distress recovers 107. Popular
capital structure arbitrage is arbitrage between a put options and CDS (Credit Default Swap)107.
5.2.4 Pairs trading
The fundamental idea behind pairs trading is mean-reversion108. Empirically proven examples
of mean-reverting assets are oil and foreign exchange rates109. Pairs trading is a type of
statistical arbitrage based on the following ideas110:

Identify two securities with similar historical price trajectories (correlated, cointegrated, etc.)

When the spread exceeds a threshold, simultaneously long the undervalued security and
short the overvalued one (which also results in reduced systematic risk)

This joint bet will generate profit if the spread closes again in the near future
In the long run, the price ratio between two stocks fluctuates less than the stocks themselves –
providing an opportunity for an arbitrage when one of the prices deviate from the mean too
much, because the price should revert to mean eventually111. Pairs trading is a strategy aiming
to find mispricing in the financial market not by finding true absolute values of securities (also
known as fundamental values), but rather rely on the (relative) value of securities in relationship
with each other111.
The idea behind pairs trading is in fact quite simple – find two stocks that historically moved
together. Then, when the spread between them widens, short the winner and buy the loser (also
known as pairs trading based on co-integration)112. A study found pairs trading to be quite
profitable and the returns to be uncorrelated with the market112.
There are many different pairs trading strategies, and the main ones are111:
107

Distance arbitrage

Co-integration arbitrage

Stochastic spread arbitrage
BENNETT, Colin and Miguel A. GIL. Volatility Trading: Trading Volatility, Correlation, Term Structure and
Skew. In: Chicago Board Options Exchange [online]. 2012 [Accessed 2014-04-26]. Available from:
http://cfe.cboe.com/education/TradingVolatility.pdf. p. 202 and 210
108
FOLGER, Jean. Arbitrage and Pairs Trading. In: Investopedia [online]. © 2014b [Accessed 2014-04-21].
Available from: http://www.investopedia.com/university/guide-pairs-trading/arbitrage-and-pairs-trading.asp
109
CHAKRABORTY, T. and Kearns, M.. 2011. Market Making and Mean Reversion [online]. Research paper.
Philadelphia: University of Pennsylvania. Available from:
http://www.cis.upenn.edu/~mkearns/papers/marketmaking.pdf [Accessed 2014-05-07] p. 2
110
THOMAIDIS, Nikos S. Statistical arbitrage and pairs trading. In: University of the Aegean [online]. 2013
[Accessed 2014-04-21]. Available from: http://casl.ucd.ie/hosted/Thomaidis.pdf
111
ZHANG, Mengyun. Research on Modern Implications of Pairs Trading [online]. Berkeley, 2012 [Accessed
2014-04-22]. Available from: http://www.stat.berkeley.edu/~aldous/Research/Ugrad/Amy_Zhang.pdf. Research
paper. University of California, Berkeley. p. 2 – 3
112
GATEV, Evan. Pairs Trading: Performance of a Relative Value Arbitrage Rule [online]. Connecticut, 2001
[Accessed 2014-04-29]. Available from: http://www.proquest.com/. Dissertation. Yale University. p. 89, 114,
797, 801 – 804, 826
Statistical arbitrage
37
A common approach when performing high-frequency statistical arbitrage is to construct a
stationary, mean-reverting synthetic asset as a linear combination of many securities113. Speed
is vital for such arbitrage strategies, and for that reason many practical systems use analytic or
approximate solutions rather than slower numerical methods113. Individual stock prices are
hardly stationary (they drift from their initial value) and exhibit geometric random walk114.
However, it’s often possible to find a pair of stocks that if you long one and short the other, the
market value of such position is stationary and thus a good candidate to co-integration arbitrage.
Pairs trading is a popular statistical arbitrage strategy used by hedge funds and investment
banks. Even very simple co-integration arbitrage algorithm can make substantial market-neutral
profits (uncorrelated to S&P 500), as was shown by a study on daily historical data from 1962
to 2002 in the U.S. equity market115. The algorithm was115:

Based on trading co-integrated pairs

Positions were open when the spread of assets diverge by more than 2 standard
deviations

Positions were closed at the next crossing of prices
As was the case of relative value arbitrage, the best pairs for trading are those where there is a
fundamental reason to believe that the spread is mean-reverting, such as common and priority
stocks, stocks of very similar companies, stocks and their depository receipts sector ETF and
sector stocks, etc.
Volatility arbitrage
Volatility arbitrage will be covered further down the paper, but at this place for the sake of
completeness it’s worth mentioning that there are more sides to volatility arbitrage, and one of
them can be categorized under pairs trading as well. Further details are provided in separate
chapter.
Correlation arbitrage
There are many different approaches to designing an arbitrage strategy based on Pearson
correlation coefficient. The coefficient is calculated as116:
0,𝑇
𝜌𝑖,𝑗
113
=
∑𝑇𝑡=0(𝑟𝑖,𝑡 − 𝑟̅𝑖 )(𝑟𝑗,𝑡 − 𝑟̅𝑗 )
√∑𝑇𝑡=0(𝑟𝑖,𝑡
−
𝑟̅𝑖 )2 √∑𝑇𝑡=0(𝑟𝑗,𝑡
− 𝑟̅𝑗
=
)2
𝑐𝑜𝑣(𝑟𝑖 ; 𝑟𝑗 )0,𝑇
𝜎𝑖0,𝑇 𝜎𝑗0,𝑇
( 25 )
BERTRAM, William Karel. Analytic Solutions for Optimal Statistical Arbitrage Trading. In: Social Science
Electronic Publishing [online]. 2009 [Accessed 2014-04-22]. Available from: http://ssrn.com/abstract=1505073.
p. 2
114
ALONSO, Miquel Noguer. Statistical Arbitrage and Algorithmic Trading: Overview and Applications
[online]. Madrid, 2010 [Accessed 2014-04-09]. Available from: http://espacio.uned.es:8080/fedora/get/tesisuned:CiencEcoEmp-Mnoguer/Documento.pdf. Dissertation. National
University of Distance Education. p. 27
115
GATEV, Evan, William N. GOETZMANN and K. Geert ROUWENHORST. Pairs Trading: Performance of
a Relative-Value Arbitrage Rule [online]. 2006 [Accessed 2014-04-22]. Available from:
http://stat.wharton.upenn.edu/~steele/Courses/434/434Context/PairsTrading/PairsTradingGGR.pdf. p.797, 801 –
804, 826
116
CHOI, Yujin. Analysis and Development Of Correlation Arbitrage Strategies on Equities. University of Paris
1, 2008. Available from: http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/5007_dissertation.pdf. Report
on internship. University of Paris 1. p. 12
38
Statistical arbitrage
This coefficient can have a value in the interval 〈−1; 1〉 where value 1 means the variables have
perfect linear relationship (variables move in tandem), value −1 implies perfect inverted linear
relationship (variables move in opposite directions), and finally value 0 implies no relationship
at all117.
When it comes to asset correlations, it’s important to note that these coefficients are not set in
stone. For example, currency correlations are to a large extent result of geopolitics, economic
integration and trade. And as those change throughout the time, the correlations will also likely
change. For example, Canada is a major exporter of oil, while Japan is a major importer118. So
there is a causal relationship between the CAD/JPY and the price of oil. This means that the
Canadian dollar (CAD) is positively correlated with the price of oil, while Japanese Yen (JPY)
is negatively correlated. However, because oil is usually priced in U.S. dollars (USD), the
fluctuations in USD value affect Canadian export and Japanese import. For that reason, the
correlation coefficient is not as high as one might expect, and the prices can sometimes diverge
and seemingly not correlate at all. For finding correlated currency pairs with causal relationship,
it’s reasonable to first find major exporters and importers of certain commodity and continue
from there118. However, one must keep in mind that correlations do change over time,
sometimes they might even reverse, and divergences can last for a long period of time.
Correlations can easily change when one of the assets in the pair is affected by events that do
not affect the second asset119.
One of the possible way how to trade correlations commonly mentioned in literature is the socalled range trading117. “Range” refers to the size of the movement - such as the height of a
candle in candle-stick chart, rather than its direction:
𝑅𝑎𝑛𝑔𝑒 = |𝑂𝑝𝑒𝑛 − 𝐶𝑙𝑜𝑠𝑒|
( 26 )
For example, EUR/USD is strongly correlated with GBP/USD, yet EUR/USD range is smaller
than GBP/USD (which means the movements have same direction most of the time, but are
different in absolute size). In other words, GBP/USD should have “lower valleys” and “higher
peaks”120. When the movement of EUR/USD is higher than GBP/USD, it means GBP/USD is
lagging and provides a trade opportunity:
FOREXZONE. FOREX - jak zbohatnout a nekrást: obchodování na měnových trzích. 1. vyd. Praha: Grada,
2011, 185 s. Finanční trhy a instituce. ISBN 978-80-247-3739-3. p. 106 – 108
118
MITCHELL, Cory. How To Trade Currency And Commodity Correlations. In: Investopedia [online]. 2010
[Accessed 2014-04-08]. Available from: http://www.investopedia.com/articles/forex/10/trading-currencycommodity-correlations.asp
119
FOLGER, Jean. Pairs Trading: Correlation. In: Investopedia [online]. © 2014a [Accessed 2014-04-21].
Available from: http://www.investopedia.com/university/guide-pairs-trading/pairs-trading-correlation.asp
120
FIELDER, Jason. How to Trade a Correlation Strategy. In: MoneyShow.com [online]. 2009 [Accessed 201404-21]. Available from: http://www.moneyshow.com/articles.asp?aid=TradingIdea-18101
117
Statistical arbitrage
39
Figure 1: Range trading example
Source: Fielder121
Once this lag is beyond certain threshold, we can take the opposite position. In the picture, the
GBP/USD is behind EUR/USD by 24 pips, yet the range of GBP/USD should be actually higher
than that of EUR/USD.
In my opinion, this strategy is no different to trading just GBP/EUR and basically betting on
mean-reversion of that single pair. And that strategy can be used for any assets, such as stocks.
Also, this strategy seems to completely miss the point of correlation – which quantifies whether
there is any linear relationship between two time series. This means only the direction is of the
movements matter rather than the size. The spread between two time series with perfect
correlation can be absolutely random and in that case this strategy would prove pointless. This
strategy might be viable with co-integrated pairs, and particularly with pairs that are both
correlated and co-integrated.
Different approach for trading highly correlated pairs is when the pair is highly positively
correlated, yet it happens that the prices move in opposite direction. In that case, investor can
quickly exploit this divergence by shorting the over-performing asset and going long with the
under-performing asset122. In this case, the investor is again just betting that the pair is meanreverting, although the use of correlation seems questionable for the reasons already mentioned
before.
This strategy can be described in following steps:
1. Get prices for asset A and B (𝑃𝑖,𝑥 , where 𝑖 is the period number and 𝑥 is the asset)
121
FIELDER, Jason. How to Trade a Correlation Strategy. In: MoneyShow.com [online]. 2009 [Accessed 201404-21]. Available from: http://www.moneyshow.com/articles.asp?aid=TradingIdea-18101
122
FOLGER, Jean. Pairs Trading: Correlation. In: Investopedia [online]. © 2014a [Accessed 2014-04-21].
Available from: http://www.investopedia.com/university/guide-pairs-trading/pairs-trading-correlation.asp
40
Statistical arbitrage
2. Calculate price ratios (𝑃𝑅𝑖 ) for all periods
𝑃𝑅𝑖 =
𝑃𝑖,𝐴
𝑃𝑖,𝐵
( 27 )
3. Calculate average price ratio
∑𝑛𝑖=1 𝑃𝑅𝑖
𝑛
̅̅̅̅
4. Trade when the price ratio diverges from 𝑃𝑅 by 2 standard deviations
̅̅̅̅
𝑃𝑅 =
( 28 )
Volatility pumping
Volatility pumping (also known as optimal growth portfolios, constantly rebalanced portfolios,
universal portfolios), is based on an idea of being invested in a portfolio of assets with constant
weights, and maintaining these fixed weights continuously through rebalancing, based on socalled Kelly formula123. For elaborate study of mathematics behind Kelly formula and its
practical application in finance please refer to the paper by Edward O. Thorp124.
It has been proven that for a portfolio of assets with given volatility 𝜎, if we model the asset
prices as processes (geometric Brownian motion), investor can earn extra return on top of buyand-hold returns using continuous rebalancing (rebalancing gains - RG) to maintain fixed
weights125:
𝑛
1
𝑅𝐺 = (∑ 𝑤𝑖 𝜎𝑖2 − 𝜎𝑝2 )
2
( 29 )
𝑖=1
Where 𝑤𝑖 is fixed weight of the particular asset in portfolio, 𝜎𝑖 is the volatility (standard
deviation) of the asset, and 𝜎𝑝 is the volatility of the whole portfolio. That is, for a continuously
rebalanced portfolio, the overall return is125:
𝑛
𝑟~𝑁 (∑ 𝑤𝑖 𝑟𝑖 + 𝑅𝐺 , 𝜎𝑝2 )
( 30 )
𝑖=1
Portfolio volatility can be calculated with the formula from modern portfolio theory126:
𝑛
𝑛
𝜎𝑝2 = ∑ ∑ 𝜎𝑖,𝑗 𝑤𝑖 𝑤𝑗
( 31 )
𝑖=1 𝑗=1
123
GABAY, Daniel and Daniel HERLEMONT. Volatility Pumping: optimal growth portfolios revisited. In:
YATS [online]. 2004 [Accessed 2014-04-13]. Available from: http://www.yats.com/doc/VolatilityPumping.pdf
124
THORP, Edward. THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK
MARKET. In: BJMath [online]. 1997 [Accessed 2014-04-13]. Available from:
http://www.bjmath.com/bjmath/thorp/paper.htm
125
MULVEY, John M. and Woo Chang KIM. Constantly Rebalanced Portfolio: Is Mean-Reverting Necessary?
[online]. Princeton, 2008 [Accessed 2014-04-13]. Available from:
http://idlab.kaist.ac.kr/papers/full/%5Bmain%20text%5D%20Constantly%20Rebalanced%20Portfolio.pdf.
Research paper. Princeton University. p. 4
126
SHARPE, William F. Investice. Praha: Victoria Publishing, 1994, 810 s. ISBN 80-856-0547-3. p. 122
Statistical arbitrage
41
Where 𝑤𝑖 is the weight of the asset 𝑖 in the portfolio, and 𝜎𝑖,𝑗 is the covariance of assets 𝑖 and
𝑗. Covariance can be also expressed with the use of correlation and standard deviation127:
𝜎𝑖,𝑗
𝜌𝑖,𝑗 =
( 32 )
𝜎𝑖 𝜎𝑗
Where 𝜌𝑖,𝑗 is the correlation of assets 𝑖 and 𝑗, 𝜎𝑖,𝑗 is the covariance of assets 𝑖 and 𝑗, and finally
𝜎𝑖 is the standard deviation of asset 𝑖. Putting these two equations together gets us:
𝑛
𝜎𝑝2
𝑛
= ∑ ∑ 𝜌𝑖,𝑗 𝜎𝑖 𝜎𝑗 𝑤𝑖 𝑤𝑗
( 33 )
𝑖=1 𝑗=1
Which means the rebalancing gains are equal to128:
𝑛
𝑛
𝑛
1
𝑅𝐺 = (∑ 𝑤𝑖 𝜎𝑖2 − ∑ ∑ 𝜌𝑖,𝑗 𝜎𝑖 𝜎𝑗 𝑤𝑖 𝑤𝑗 )
2
𝑖=1
( 34 )
𝑖=1 𝑗=1
1
Assume that the weights are the same for all assets (𝑤𝑖 = 𝑛), the equation can be further
simplified to:
𝑛
𝑛
𝑛
1 𝑛−1
𝑅𝐺 = ( 2 ∑ 𝜎𝑖2 − ∑ ∑ 𝜌𝑖,𝑗 𝜎𝑖 𝜎𝑗 )
2
𝑛
𝑖=1
( 35 )
𝑖=1 𝑗=1,𝑗≠𝑖
And in case correlation 𝜌 and volatility 𝜎 are same for all assets in the portfolio129:
(𝑛 − 1)𝜎 2 (1 − 𝜌)
( 36 )
2𝑛
By plotting the rebalancing gains on a chart with various values of correlation and volatility, it
comes clear that the closer the correlation is to 0 and the higher the volatility, the higher are
rebalancing gains (for 𝑛 = 5)129:
𝑅𝐺 =
127
CHEREWYK, Peter. Calculating Covariance For Stocks. In: Investopedia [online]. 2014 [Accessed 2014-0414]. Available from: http://www.investopedia.com/articles/financial-theory/11/calculating-covariance.asp
128
KWOK, Yue Kuen. Optimal investment strategy. In: The Hong Kong University of Science & Technology
[online]. Hong Kong, 2005 [Accessed 2014-04-14]. Available from:
http://www.math.ust.hk/~maykwok/courses/Fin_econ_05/Fin_econ_05_1.pdf
129
MULVEY, John M. and Woo Chang KIM. Constantly Rebalanced Portfolio: Is Mean-Reverting Necessary?
[online]. Princeton, 2008 [Accessed 2014-04-13]. Available from:
http://idlab.kaist.ac.kr/papers/full/%5Bmain%20text%5D%20Constantly%20Rebalanced%20Portfolio.pdf.
Research paper. Princeton University. p. 5, 10
42
Statistical arbitrage
Figure 2: Rebalancing gains for n = 5
4.0%
ρ = 0.0
Rebalancing gains
3.5%
3.0%
ρ = 0.3
2.5%
2.0%
1.5%
ρ = 0.6
1.0%
0.5%
ρ = 0.9
0.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30%
Volatility
Source: Mulvey et al.130
Because of the existence of rebalancing gains, Sharpe ratio might not be the best indicator when
comparing assets – asset with no return, high volatility and no correlation with other assets of
the same properties in single portfolio might be a great source of rebalancing gains131 as well
as diversification, and rebalancing can even lower the overall risk of portfolio132. It has been
shown that in a world where all assets have zero return and 30 % annual volatility, a
continuously rebalanced portfolio consisting of 10 independent assets yield 4.275 % normally
distributed expected return with 9.5 % volatility, yet the Sharpe ratio is 0131. This example can
be taken to the extreme and assume that all assets have the same negative return. Even in such
world, the return of continuously rebalanced portfolio can be still positive in the case that
rebalancing gain is higher than absolute value of the asset return. Thus, volatility pumping can
be reminiscent to Parrondo’s paradox in certain situations.
In any case, investor must be very aware of the transaction costs and capital gain taxes131, as
those can absolutely negate any return gained through rebalancing. For practical applications,
investors should modify the model with so-called no-trade zones, which is the idea that
rebalancing should be done only once the balance moves out of certain boundary132.
130
MULVEY, John M. and Woo Chang KIM. Constantly Rebalanced Portfolio: Is Mean-Reverting Necessary?
[online]. Princeton, 2008 [Accessed 2014-04-13]. Available from:
http://idlab.kaist.ac.kr/papers/full/%5Bmain%20text%5D%20Constantly%20Rebalanced%20Portfolio.pdf.
Research paper. Princeton University. p. 10
131
MULVEY, John M., Woo Chang KIM and Mehmet BILGILI. Dynamic investment strategies and
rebalancing gains [online]. Princeton, 2008 [Accessed 2014-04-14]. Available from:
http://idlab.kaist.ac.kr/papers/full/%5bmain%20text%5d%20Dynamic%20investment%20strategies%20and%20
rebalancing%20gains.pdf. Research paper. Princeton University.
132
MULVEY, John M. and Koray D. SIMSEK. Rebalancing Strategies for Long-Term Investors [online].
EDHEC, 2010 [Accessed 2014-04-14]. Available from: http://facultyresearch.edhec.com/servlet/com.univ.collaboratif.utils.LectureFichiergw?ID_FICHIER=1328885973687.
Research paper. EDHEC. p. 13
Statistical arbitrage
43
It seems that emerging countries might provide a great opportunity for volatility pumping, as
their returns have high volatility and low correlations133. Depending on an investment strategy,
the investor who wishes to gain rebalancing profits might want to choose a portfolio of assets
with similar returns. Because in case the returns are extensively different, the continuous
rebalancing will maintain fix weights and thus sell more and more of the higher-yield asset over
time, which might or might not be desirable.
Distance arbitrage
This strategy keeps track of the sum of squared differences between two normalized stock
prices, and the trading trigger is usually set at two standard deviations from the mean134. Study
tested this strategy on historical prices of selected stocks and found it to be profitable134, where
the profit was highest when the correlation between the pair was high rather than low134.
Co-integration arbitrage
Co-integration arbitrage is based on the idea that when the spread of co-integrated pair of assets
diverge from mean by a certain number of standard deviations (z-score), both a long and short
position should be taken with the idea that the spread will converge to mean again135,136. Both
positions are taken also for the reason to hedge against market risk. The difference between
distance arbitrage and co-integration arbitrage is that distance arbitrage is based on squared
difference rather than simple difference.
Key characteristics of co-integration is the assumption of mean-reverting tracking error
(stationary residuals)137.
Intra-day pairs trading based on co-integration was shown to be profitable138. The common
signal to open a position is when the spread moves two standard deviations from the mean, and
also the pair is already in the process of reversion. Stationarity is tested with augmented DickeyFuller test138, and co-integration with Engle-Granger or Johansen test (latter being superior)138.
Difference between correlation and co-integration
As mentioned before, sometimes literature is confusing the terms correlation and co-integration.
133
STEIN, David M., Vassilii NEMTCHINOV and Sam PITTMAN. Diversifying and rebalancing emerging
market countries. In: Parametric: Engineered Portfolio Solutions [online]. 2008 [Accessed 2014-04-14].
Available from: http://www.parametricportfolio.com/wp-content/uploads/2010/09/EM-Rebalancing-WhitePaper-JWM.CA_.pdf. p. 14
134
ZHANG, Mengyun. Research on Modern Implications of Pairs Trading [online]. Berkeley, 2012 [Accessed
2014-04-22]. Available from: http://www.stat.berkeley.edu/~aldous/Research/Ugrad/Amy_Zhang.pdf. Research
paper. University of California, Berkeley. p. 3, 7, 19
135
CHAN, Ernest P. An updated analysis of the arbitrage between gold and gold-miners. In: Quantitative
Trading: Quantitative investment and trading ideas, research, and analysis. [online]. 2006 [Accessed 2014-0415]. Available from: http://epchan.blogspot.cz/2006/11/updated-analysis-of-arbitrage-between.html
136
CHAN, Ernest P. An arbitrage trade between energy stocks and futures. In: Quantitative Trading:
Quantitative investment and trading ideas, research, and analysis. [online]. 2007 [Accessed 2014-04-15].
Available from: http://epchan.blogspot.cz/2006/10/arbitrage-trade-between-energy-stocks.html
137
DUNIS, Christian L. and Richard HO. Cointegration Portfolios of European Equities for Index Tracking and
Market Neutral Strategies [online]. Liverpool, 2005 [Accessed 2014-04-21]. Available from:
http://www.ljmu.ac.uk/AFE/AFE_docs/ARTCDRH_01051.PDF. Research paper. Liverpool John Moores
University and CIBEF. p. 14
138
MURRAY, Jennifer Wells. A study of intraday volatility trading utilizing high frequency data and the
microstructure effects of implementation [online]. New York, 2010 [Accessed 2014-04-29]. Available from:
http://www.proquest.com/. Dissertation. Fordham University. p. 24 – 25, 27, 102
44
Statistical arbitrage
Highly correlated assets move in the same direction, regardless of the actual size of the
movement. This means the prices move in tandem up or down most of the time, but the spread
can increase or decrease without limitation.
Highly co-integrated assets maintain mean-reverting spread. This means the prices can move
in any direction without any seeming relationship, but the spread oscillates around some value.
Both correlation and co-integration are used by investors in a hope to find assets whose price
development is somehow related. One must be very aware of the fundamental differences of
both statistics when used in trading strategies. This is not to say that one statistics is better than
other, as both have their own role and different trading strategies can be build using correlation,
co-integration, or even both at the same time, as assets that are both correlated and co-integrated
move very closely together.
5.2.5 Merger arbitrage
Merger arbitrage (also called risk arbitrage) is a strategy based on simultaneously taking long
and short positions in shares that will be affected by merger, acquisition or hostile takeover139.
The long position is typically in shares of a company that is subject to takeover, and the short
position is in the shares of purchasing company. When the acquisition is announced, the
acquiring company offers a price for the other company’s shares. Because of the risk that the
acquisition will fail, the shares trade lower than what was offered140. Speed is crucial because
markets react to announced merger/takeover in mere seconds141.
The risk is that the merger or acquisition gets cancelled, in which case the share prices will
return to the pre-announcement levels, or even a bit lower. Because of that risk, the shares of
the company that is to be acquired are traded cheaper than what the purchasing company offered
for them. Profit is realized as a difference between the buyout price and the market price142.
Another risk in merger arbitrage is the time value of money – mergers can take a long time and
that time will degrade the investor’s profit142.
A study has shown on Canadian equities that merger arbitrage is not profitable when the
position is entered one day after the announcement, proving how crucial the speed is143. This is
because the companies experience very high abnormal return on the day before – the
announcement day143.
139
BRETZLER, Marcus and Dirk RUDOLPH. Hedge Fonds und Alternative Investments. First edition.
Frankfurt am Main: Bankakademie-Verlag GmbH, 2004. ISBN 3-937519-08-4. p. 28
140
TASHMAN, Adam Paul. Modeling Risk in Arbitrage Strategies Using Finite Mixtures [online]. New York,
2006 [Accessed 2014-04-29]. Available from: http://www.proquest.com/. Dissertation. Stony Brook University.
p. 5
141
KUEPPER, Justin. Trading The Odds With Arbitrage. In: Investopedia [online]. 2012 [Accessed 2014-0426]. Available from: http://www.investopedia.com/articles/trading/04/111004.asp
142
KUEPPER, Justin. Arbitrage Squeezes Profit From Market Inefficiency. In: Investopedia [online]. 2013
[Accessed 2014-04-26]. Available from: http://www.investopedia.com/articles/trading/07/statisticalarbitrage.asp
143
BELANGER, Marcel. Risk arbitrage trading and the characteristics of arbitrage spreads: The Canadian
evidence [online]. Montreal, 2000 [Accessed 2014-04-29]. Available from: http://www.proquest.com/. Master's
thesis. Concordia University. p. 47 – 48
Statistical arbitrage
45
5.3 Volatility arbitrage
Important realization for anyone wanting to participate in volatility arbitrage is that volatility
can be traded as an asset144. That is, one can be long or short in volatility, and does not need to
worry about the actual direction of the market. Because of that latter property, volatility
arbitrage can yield returns uncorrelated with the performance of the market in virtually any
environment145. Volatility arbitrage can be divided into these main categories145:

Directional volatility bet (song, short)

Relative value volatility arbitrage (such as song/short - being long in one volatility, and
at the same time short in other volatility
There are virtually unlimited possible volatility arbitrage strategies, and that is mainly due to
the fact that because volatility arbitrage is a kind of statistical arbitrage, there are as many
strategies as there are statistical models that one can come up with. Not to mention that many
of the relative value arbitrage strategies mentioned before can be applied to volatility as well145.
Arbitrageur could for example find co-integrated volatilities between 2 baskets of assets (or
individual assets as well of course), and use some simple mean-reversion strategy to trade the
volatility when the spread of volatilities deviates from mean by 2 standard deviations.
For any practical implementations of volatility arbitrage, one should keep in mind how to
properly annualize volatility. A common notation to annualize monthly volatility is to multiply
the volatility by √12. It was shown that this is only valid when the monthly volatility has been
calculated using logarithmic returns146, which is fortunately a prerequisite when dealing with
Black-Sholes model anyway.
An interesting feature of options is that it’s possible to use them quantify the market view of
probabilities of underlying asset movements147. Delta not only represents the change in value
of option when the underlying asset’s value change by 1, but also the probability that the
underlying asset price will be above/below the strike value of call/put option at expiration date.
In other words, the probability that the underlying asset finishes above (below) the strike of call
option at expiration date is delta (1 - delta). For information about how to calculate the
probability of option finishing above (below) strike before expiration date, refer to paper by
Su148. This allows to trade options according to implied probabilities rather than somewhat
ambiguous price. Also, this explains why deep-in-the-money call options have delta very close
to 1.
144
RICE, Bob. Bob's Daily Buzzword: `Volatility Arbitrage' [online video]. Bloomberg News, 2012 [Accessed
2014-04-06]. Available from: https://www.youtube.com/watch?v=UZY1ZonQT5c
145
CUSWORTH, Emma. Evolved volatility arbitrage strategy performs well in most market environments. In:
Risk.net [online]. 2012 [Accessed 2014-04-10]. Available from: http://www.risk.net/hedge-fundsreview/feature/2248594/evolved-volatility-arbitrage-strategy-performs-market-environments
146
KAPLAN, Paul D. What’s Wrong with Multiplying by the Square Root of Twelve. In: Morningstar [online].
2013 [Accessed 2014-04-23]. Available from:
http://corporate.morningstar.com/US/documents/MethodologyDocuments/MethodologyPapers/SquareRootofTw
elve.pdf
147
LIEN, Kathy. Using Options Tools To Trade Foreign-Exchange Spot. In: Investopedia [online]. 2010
[Accessed 2014-04-08]. Available from: http://www.investopedia.com/articles/optioninvestor/05/022305.asp
148
SU, Lujing. Theoretical and Empirical Estimation: How Likely Is It to Hit a Barrier? [online]. Bielefeld, 2009
[Accessed 2014-04-25]. Available from: http://erasmus-mundus.univparis1.fr/fichiers_etudiants/1459_dissertation.pdf. Master thesis. Bielefeld University.
46
Statistical arbitrage
For a detailed discussion of volatility, variance and correlation derivatives, as well as volatility
trading itself, refer to paper by Bennett et al.149
5.3.1 Directional volatility bet
Even though in reality this is just a normal directional (long or short) bet, it’s often called a
volatility arbitrage in practice144, because of the nuances of trading volatility – such as the
indifference to the direction of underlying asset market movement. It is also called gamma
position150, and can be even traded across different strike prices and option maturities151. This
strategy relies upon the ability of the investor to predict the future (realized) volatility better
than the market152.
Historically implied volatility was usually higher than realized volatility153. The spread was
shown to be 4% on average in case of one-month S&P 500 index154 (which might be however
because of implied correlation – more on that later), but this may not be the case for all strikes
and maturities (can vary widely). During spikes (crisis), realized volatility is actually higher
than implied volatility154.
The simplest way to take a directional bet in volatility is to have a long (short) position in option
and opposite position in underlying asset in a ratio as to form a delta-neutral portfolio155. To list
the variants, these strategies provide the opportunity to take a directional bet in volatility, and
in case those are delta-neutral then the movement of the underlying market does not matter:
149

Long call and short in underlying asset

Short call and long in underlying asset

Long put and long in underlying asset

Short put and short in underlying asset

Long (short) straddle – put and call options with the same strike; the strike and
expiration date should be chosen such that the portfolio is delta-neutral151

And other variants (such as butterfly, condor, etc.)
BENNETT, Colin and Miguel A. GIL. Volatility Trading: Trading Volatility, Correlation, Term Structure and
Skew. In: Chicago Board Options Exchange [online]. 2012 [Accessed 2014-04-26]. Available from:
http://cfe.cboe.com/education/TradingVolatility.pdf
150
WARNER, Adam. Long Gamma – How to Make a Long Gamma Position Work for You. In: InvestorPlace
[online]. 2010 [Accessed 2014-04-24]. Available from: http://investorplace.com/2010/01/long-gamma-position/
151
SKEGGS, James. Welcome turbulence – how to make money from volatility arbitrage. In: Hedge Funds
Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 5 – 7
152
LOGGIE, Keith. Volatility arbitrage indices – a primer. In: The Markit Magazine [online]. 2008 [Accessed
2014-04-23]. Available from: http://www.markit.com/assets/en/docs/markit-magazine/issue-2/volatilityarbitrage.pdf. p. 61 – 2
153
CHOI, Yujin. Analysis and Development Of Correlation Arbitrage Strategies on Equities. University of Paris
1, 2008. Available from: http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/5007_dissertation.pdf. Report
on internship. University of Paris 1. p. 9, 11
154
CUSWORTH, Emma. Evolved volatility arbitrage strategy performs well in most market environments. In:
Risk.net [online]. 2012 [Accessed 2014-04-10]. Available from: http://www.risk.net/hedge-fundsreview/feature/2248594/evolved-volatility-arbitrage-strategy-performs-market-environments
155
RICE, Bob. Bob's Daily Buzzword: `Volatility Arbitrage' [online video]. Bloomberg News, 2012 [Accessed
2014-04-06]. Available from: https://www.youtube.com/watch?v=UZY1ZonQT5c
Statistical arbitrage
47
Of those variants, those that are long in options are also long in volatility, and in that case the
profit is made if the realized volatility turns out to be higher than the implied volatility at the
time of trade152. Opposite is true as well – when being short in volatility, the profit is made if
the realized volatility turns out to be lower than the implied volatility at the time of trade.
To maintain delta-neutral portfolio over time, it’s required to continuously rebalance portfolio
(gamma hedging)156. However this comes at a price – continuous portfolio rebalancing is not
only impossible because of the discrete nature of computers, but more importantly very costly
due to transaction fees. And rebalancing done periodically leads to imperfect hedge and thus
exposure to delta risk. Thus, one must find an optimal balance between minimizing transaction
fees and minimizing delta exposure. In practice, many hedge funds use daily rebalancing, but
others might prefer longer or shorter intervals.
To mitigate the transaction costs incurred by rebalancing, one might use variance swaps, which
are derivatives specifically tailored to trade volatility, unlike options156,153 ,151. There are also
other methods to take bets on volatility direction, such as trading futures or options contracts
on VIX, or trading volatility swaps157. Variance and volatility swaps are however OTC
contracts, and as such might not be available to all investors.
In case of long volatility position, the profit is locked-in by continuous portfolio
rebalancing158,159, and the cost is theta157 – meaning that if the market doesn’t move sufficiently,
the income gained through portfolio rebalancing will not cover the loss incurred from option
time value decay. For short volatility positions, the true is opposite – investor profits through
time decay (because he is short options), but can potentially lose money through portfolio
rebalancing (gamma).
There are actually two ways to profit from directional volatility bet160. Either the market realizes
that the implied volatility is wrong and corrects itself, in which case the profit can be harnessed
by liquidating the position, or the position is held to maturity and the profit is harnessed by
daily gamma management161.
When it comes to delta-hedging, there is also a question of whether to use the implied volatility
or the actual volatility that we bet against. The advantage of hedging with actual volatility is
that the investor knows exactly the profit he will get at expiration, and the disadvantage is that
the profit & loss daily fluctuations are huge and do not offer much of a hint as to whether the
guess of the actual volatility was correct161. In case of hedging with implied volatility, the daily
156
WOODARD, Jared. The Lazy Guide to Delta Hedging. In: Condor Options [online]. 2009 [Accessed 201404-06]. Available from: http://condoroptions.com/2009/07/10/the-lazy-guide-to-delta-hedging/
157
SKEGGS, James. Welcome turbulence – how to make money from volatility arbitrage. In: Hedge Funds
Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 5 – 7
158
KUEPPER, Justin. Arbitrage Squeezes Profit From Market Inefficiency. In: Investopedia [online]. 2013
[Accessed 2014-04-26]. Available from: http://www.investopedia.com/articles/trading/07/statisticalarbitrage.asp
159
GLEADALL, Simon. The difference between long gamma and short gamma. In: Volcube [online]. © 2014
[Accessed 2014-04-24]. Available from: http://www.volcube.com/resources/options-articles/the-differencebetween-long-gamma-and-short-gamma/
160
BUENOS, Alain, Peng TANG and David WALKER. Sigma Square – making money on both sides of
volatility’s golden coin. In: Hedge Funds Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 12 – 13
161
AHMAD, Riaz and Paul WILMOTT. Which Free Lunch Would You Like Today, Sir?: Delta Hedging,
Volatility Arbitrage and Optimal Portfolios. In: WILMOTT, Paul. Wilmott [online]. 2008 [Accessed 2014-0424]. Available from: http://www.wilmott.com/pdfs/110309_riaz_wilmott.pdf. p. 66 and 72 – 73
48
Statistical arbitrage
profit & loss does not fluctuate at all and provide a steady income (as long as the initial guess
was correct of course). On the other hand, the actual size of the profit is not known until
expiration. It is absolutely possible to also use other values of volatility for hedging, to balance
the pros and cons.
5.3.2 Relative value volatility arbitrage
It’s possible to trade implied volatilities of options with the same underlying asset (known as
“volatility surface arbitrage”)162. The other option is to trade implied volatilities of 2 assets
relative to one another:

Long/short positions, also known as volatility-spread positions162

Dispersion trades
As an example of volatility-spreads, one could trade the implied volatility of Japanese Yen
versus the volatility of Nikkei, or the volatility of gold versus volatility of gold producer
stock162.
Volatility surface arbitrage
Volatility surface is a shape obtained by plotting implied volatility as a function of strike157,
and is called a vertical skew (“smile” is a common shape). Horizontal skew is a situation when
options further from expiry are cheaper than those near expiry, and such situation can be easily
noticed when the implied volatility is plotted as a function of strike for options of various
expirations on the same asset. One possible explanation for volatility surface shape is the
skewness of return distribution of the underlying asset163.
The implied volatility surface (implied volatility as y-axis, expiration date as z-axis, and strike
prices as x-axis) might not be a smooth shape, but there might be certain distortions such as
“bumps” and “holes”, which might be caused for example by investors trading according to buy
and sell recommendations164. Thus, arbitrageur can trade against these distortions using a deltaneutral portfolio with the idea that the realized volatility will be somewhere in-between.
Vertical skew can be traded by buying the cheaper options in volatility terms, and selling the
more expensive options in volatility terms. To trade horizontal skew, one can buy the cheaper
options (further from expiry) and sell options closer to expiry. This type of trade is called
SKEGGS, James. Welcome turbulence – how to make money from volatility arbitrage. In: Hedge Funds
Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 5 – 7
163
MULVEY, John M. and Woo Chang Kim. What Causes the Volatility Asymmetry? - A Clue to the Volatility
Puzzle [online]. 2007 Proceedings of Korean Security Association Conference 2007. Available from:
http://idlab.kaist.ac.kr/papers/full/%5bmain%20text%5d%20What%20causes%20the%20Volatility%20Asymme
try.pdf
164
CHOI, Yujin. Analysis and Development Of Correlation Arbitrage Strategies on Equities. University of Paris
1, 2008. Available from: http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/5007_dissertation.pdf. Report
on internship. University of Paris 1. p. 9
162
Statistical arbitrage
49
calendar spread165. These trades should be done using delta-neutral portfolios if the investor
does not wish to bet on the direction of the underlying market166.
Some hedge funds trade on the assumption that volatilities of main indices are short-term
correlated, which presents another opportunity for arbitrage between implied volatility forward
skew and implied volatility smile of those indices164.
For more details on implied volatility surface arbitrage, refer to the paper by Fengler167.
Trading return distribution skewness
It’s also possible to trade the properties of the return distribution of the underlying asset. For
example, if the arbitrageur finds out the skewness of the distribution is undervalued, he can buy
OTM calls and sell OTM puts168, while maintaining the position delta-neutral and vega-neutral.
To arbitrage overvalued kurtosis, he can sell ATM and deep OTM options, and buy near OTM
options168. For more information about skewness arbitrage refer to the paper by Bali et al.169.
Please be aware that skewness of return distribution and implied volatility skew are not the
same thing. But some studies suggest that skewness of option-implied return distribution is the
source of volatility skew170.
Dispersion and correlation trading
Volatility dispersion trade is based on trading the volatility of index against the volatilities of
index components166, using at-the-money options, but requires constant rebalancing of the
portfolio to keep the options at-the-money171. It can also be thought of as correlation
trading172,173.
165
DAMODARAN, Aswath. Too good to be true?: The Dream of Arbitrage [online]. New York, 2012
[Accessed 2014-04-26]. Available from: http://people.stern.nyu.edu/adamodar/pdfiles/invphiloh/arbitrage.pdf.
New York University. p. 30
166
SKEGGS, James. Welcome turbulence – how to make money from volatility arbitrage. In: Hedge Funds
Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 7
167
FENGLER, Matthias R. Arbitrage-free smoothing of the implied volatility surface [online]. Frankfurt am
Main, 2005 [Accessed 2014-04-25]. Available from: http://edoc.hu-berlin.de/series/sfb-649-papers/200519/PDF/19.pdf
168
ALONSO, Miquel Noguer. Statistical Arbitrage and Algorithmic Trading: Overview and Applications
[online]. Madrid, 2010 [Accessed 2014-04-09]. Available from: http://espacio.uned.es:8080/fedora/get/tesisuned:CiencEcoEmp-Mnoguer/Documento.pdf. Dissertation. National
University of Distance Education. p. 34 – 35
169
BALI, Turan G. and Scott MURRAY. Does Risk-Neutral Skewness Predict the Cross-Section of Equity
Option Portfolio Returns?. In: [online]. 2013 [Accessed 2014-04-22]. Available from:
http://ssrn.com/abstract=1572827
170
CORRADO, C. J. and Tie SU. Implied volatility skews and stock return skewness and kurtosis implied by
stock option prices. In: The European Journal of Finance [online]. 1997 [Accessed 2014-04-25]. 3. Available
from: http://moya.bus.miami.edu/~tsu/ejf1997.pdf
171
Murray, 2010, p. 19
172
BUENOS, Alain, Peng TANG and David WALKER. Sigma Square – making money on both sides of
volatility’s golden coin. In: Hedge Funds Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 12
173
CBOE. CBOE S&P 500 Implied Correlation Index. In: Chicago Board Options Exchange [online]. 2009
[Accessed 2014-04-23]. Available from:
http://www.cboe.com/micro/impliedcorrelation/ImpliedCorrelationIndicator.pdf. p. 1 – 2
50
Statistical arbitrage
Long dispersion trade is a long position on volatility of individual components and short
position on index volatility166, and provides profit when components are uncorrelated172 – so in
effect long dispersion is almost the same as short correlation174. Dispersion trading can be
thought of as trying to reconstruct the index option by trading options on index components175.
Furthermore, dispersion/correlation trading is historically popular with volatility arbitrage
funds176. However, liquidity of the index components is a serious obstacle.
There is actually a difference between trading dispersion and correlation - correlation swaps
have pure exposure to correlation, and dispersion trades have exposure to both the realized
volatility and also the correlation174. To quantify, dispersion is a difference of index volatility
and weighted average of component volatilities174:
n
Dispersion = ∑ wi σi − σp
( 37 )
i=1
Unlike stock volatility, index volatility is driven by not only component volatilities, but also by
correlations of component price returns173. Thus, with some algebra it’s possible to calculate
implied correlation of index components, because index variance is a weighted average of
individual variances and covariance terms177. The following equation maps the relationship
between index volatility and the volatility of individual components173,178,179:
𝑛
𝜎𝑝2
𝑛
= ∑ ∑ 𝜌𝑖,𝑗 𝜎𝑖 𝜎𝑗 𝑤𝑖 𝑤𝑗
( 38 )
𝑖=1 𝑗=1
From the equation, it’s clear that in case we know the values of implied volatilities for both the
index and its components, we can calculate the implied correlation, which might be different to
the historical correlation – depending on the option prices. In other words, index volatility may
be driven not only by the volatility of components, but also by the changes in correlation, also
known as correlation risk177. The following implied correlation equation was derived from a
stochastic model177, but here written in its equivalent model-free equation form180:
𝜌̅̂ =
174
𝜎̂𝑝2 − ∑𝑛𝑖=1 𝜎̂𝑖2 𝑤𝑖2
𝜎̂𝑝2 − ∑𝑛𝑖=1 𝑤𝑖2 𝜎̂𝑖2
=
𝑛
∑𝑛𝑖=1 ∑𝑛𝑗≠𝑖 𝜎̂𝑖 𝜎̂𝑗 𝑤𝑖 𝑤𝑗 2 ∑𝑛−1
𝑖=1 ∑𝑗>𝑖 𝑤𝑖 𝑤𝑗 𝜎𝑖 𝜎𝑗
( 39 )
CHOI, Yujin. Analysis and Development Of Correlation Arbitrage Strategies on Equities. University of Paris
1, 2008. Available from: http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/5007_dissertation.pdf. Report
on internship. University of Paris 1. p. 3, 12
175
AVELLANEDA, Marco. Dispersion Trading. In: New York University: Department of Mathematics [online].
2009 [Accessed 2014-04-08]. Available from: https://www.math.nyu.edu/faculty/avellane/Lecture10Quant.pdf.
p. 4
176
CUSWORTH, Emma. Evolved volatility arbitrage strategy performs well in most market environments. In:
Risk.net [online]. 2012 [Accessed 2014-04-10]. Available from: http://www.risk.net/hedge-fundsreview/feature/2248594/evolved-volatility-arbitrage-strategy-performs-market-environments
177
DRIESSEN, Joost, Pascal MAENHOUT and Grigory VILKOV. Option-Implied Correlations and the Price of
Correlation Risk [online]. 2005 [Accessed 2014-04-23]. Available from:
http://www.gsb.stanford.edu/sites/default/files/documents/dmv_sept2005.pdf. p. 1, 3, 4, 14, 16, 19, 32
178
SHARPE, William F. Investice. Praha: Victoria Publishing, 1994, 810 s. ISBN 80-856-0547-3. p. 122
179
CHEREWYK, Peter. Calculating Covariance For Stocks. In: Investopedia [online]. 2014 [Accessed 2014-0414]. Available from: http://www.investopedia.com/articles/financial-theory/11/calculating-covariance.asp
180
CBOE. CBOE S&P 500 Implied Correlation Index. In: Chicago Board Options Exchange [online]. 2009
[Accessed 2014-04-23]. Available from:
http://www.cboe.com/micro/impliedcorrelation/ImpliedCorrelationIndicator.pdf. p. 1 – 2
Statistical arbitrage
51
Where 𝜌̅̂ is the implied average correlation, 𝜎̂𝑝 is the implied volatility of index, 𝜎̂𝑖 is the implied
volatility of index component 𝑖, and 𝑤𝑖 is the weight of component 𝑖 in the index. This leads us
to the ability to trade the difference between implied and realized correlations177.
To be long in volatility dispersion, we sell at-the-money index straddles and purchase at-themoney straddles of index components177,180, but other ways are possible such as using just calls
or puts181.
Option-implied correlations are substantially higher than realized correlations on average,
which presents almost a risk-free statistical profit177,181,182,183. Other equivalent ways to look at
the same problem is to say that risk of certain stocks is undervalued with respect to their market,
or that market risk is overvalued181. However, in case of index components, there is not a
statistically significant difference between the implied and realized volatility of those
components. To make use the overpriced correlation, one should take short position in index
options and long position in index component options (short correlation, long dispersion). In
case of opposite scenario – when implied correlations are predicted to be lower than realized
correlations, arbitrageur should take long position in index options and short positions in
component options. The logic behind that is:
↑ 𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 → ↑ 𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝑖𝑛𝑑𝑒𝑥 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 → ↑ 𝐼𝑛𝑑𝑒𝑥 𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒
In case the implied correlation is the same as realized correlation on average, one can use the
implied correlation as a signal to sell index options when the correlation is high (in that case
index options are relatively expensive). Similar to volatility, correlation can be also thought of
as an asset. For that reason, arbitrageur can use mean-reverting arbitrage strategies (such as cointegration arbitrage) to profit from the movements in correlation.
This also provides an intuitive reason why index puts should be relatively more expensive than
stock puts. It’s because index puts allow the investor to hedge against correlation increases – in
other words, investor can hedge against the loss of diversification177.
Correlation swap is a financial derivative engineered specifically to trade the difference
between implied and realized correlations, but recent years brought other innovative ways to
trade correlation and dispersion, such as gamma swaps and barrier variance swaps181. For
valuation models of correlation and dispersion derivatives, refer to the paper by Choi181. CBOE
even used to provide a volatility arbitrage strategy benchmark based on rolling short position
of S&P 500 variance futures184. The Sharpe ratio for that benchmark was more than double than
that of the S&P 500 for the measured period.
5.4 Liquidation arbitrage
When investor expects that the company will come to liquidation, and the book value per share
is higher than what is the share currently trading at, arbitrageur can purchase the stock in an
181
CHOI, Yujin. Analysis and Development Of Correlation Arbitrage Strategies on Equities. University of Paris
1, 2008. Available from: http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/5007_dissertation.pdf. Report
on internship. University of Paris 1. p. 9, 11, 13
182
Loggie (2008, p. 63),
183
MELVIN, William O., Andrew GREELEY and David WALKER. From little acorns... a $900m business
grows. In: Hedge Funds Review [online]. 2006 [Accessed 2014-04-24]. Available from:
http://www.incisivemedia.com/hfr/specialreport/VolArbSupp.pdf. p. 11
184
CBOE. CBOE S&P 500 VARB-X Strategy Benchmark. In: Chicago Board Options Exchange [online]. 2007
[Accessed 2014-04-23]. Available from: http://www.cboe.com/micro/vty/cboevarbxbenchmark.pdf
52
Statistical arbitrage
expectation of liquidation profit185. The assessment whether the company is likely to liquidate
can be done using a statistical model analyzing various financial ratios.
185
KUEPPER, Justin. Trading The Odds With Arbitrage. In: Investopedia [online]. 2012 [Accessed 2014-0426]. Available from: http://www.investopedia.com/articles/trading/04/111004.asp
Multilateral arbitrage algorithm study
53
6 MULTILATERAL ARBITRAGE ALGORITHM STUDY
As mentioned before, multilateral arbitrage algorithm is such that takes arbitrary amount of
trade pairs and detects whether there is an arbitrage opportunity.
First, let’s begin by descripting the data sample that will be used to look for arbitrage
opportunities, datasets of currency exchange rates from East Asia.
Then, I will derive an appropriate algorithm to check whether there is an arbitrage (given single
set of bid-ask quotes).
After that, I will derive another appropriate algorithm the generate all possible variants of
arbitrage loops (chains) – this means an algorithm that takes a list of currencies and outputs all
possible triangular arbitrages, all possible quadrangular arbitrages, and so on.
And finally, let’s use both algorithms to analyze all datasets and interpret the results.
6.1 Data
To test the arbitrage algorithms, high-quality high-frequency market data is required, because
arbitrage is highly competitive area where only the fastest wins.
The data was acquired as an intraday tick data exported from Bloomberg terminal. The period
monitored is from January 1st till May 9th 2014.
Bloomberg allows to export high-frequency bid-ask price and volume quotations. However,
they do not provide full order book, which somewhat limits the possible analysis. Because there
is no full order book, I have decided to analyze only prices and abstract from volume, simply
because only one level of volume quotations is not enough for thorough analysis. Using only
prices does not allow to check how deep the arbitrage opportunity is, but having only one level
of volume quotes does not allow for that either.
As mentioned in the Appendix A, these are the currencies in East Asian region:
𝐾𝑅𝑊, 𝐶𝑁𝑌, 𝑀𝑁𝑇, 𝐽𝑃𝑌, 𝑀𝑂𝑃, 𝐻𝐾𝐷, 𝑇𝑊𝐷, 𝐾𝑃𝑊
𝐾𝑃𝑊 is North Korean currency is not traded publicly. Out of the remaining currencies, these
currency pairs are not available in Bloomberg:
𝑀𝑂𝑃/𝑀𝑁𝑇, 𝑀𝑁𝑇/𝑇𝑊𝐷
And these currency pairs had just too much data for the analyzed period, so it was not possible
to download them:
𝐻𝐾𝐷/𝐽𝑃𝑌, 𝐽𝑃𝑌/𝐾𝑅𝑊
So I was able to acquire data for the following 17 currency pairs:
Table 11: Analyzed currency pairs
CNY/HKD
CNY/JPY
CNY/TWD
HKD/KRW
MOP/JPY
MOP/KRW MOP/TWD
KRW/MNT KRW/TWD
CNY/KRW CNY/MOP
CNY/MNT
HKD/MOP HKD/MNT HKD/TWD
JPY/MNT
JPY/TWD
54
Multilateral arbitrage algorithm study
Bloomberg allows to export the data into Excel, which turned out to be quite insufficient
software for this kind of analysis. Millions of records and hundreds of megabytes of data seem
to be too much for that application, which resulted in its frequent crashes and unresponsiveness.
For that reason, I had to create a VBA script which exported all data from Excel files into CSV
files – which are just plain text files with values separated by comma (to make columns) and
new lines (to make rows). Then, I had to develop my own analysis software that would go
through the data line by line and do the necessary analysis.
6.2 Methodology
6.2.1 Detecting arbitrage for single set of quotes
As per equation ( 5 ):
𝑛
∏ 𝑟𝑖 > 1
𝑖=1
The multilateral arbitrage algorithm is disarmingly simple. However, one must pay particular
attention as to what values he puts in place of 𝑟𝑖 . An arbitrage loop
𝐶𝑍𝐾/𝑈𝑆𝐷 → 𝐶𝑍𝐾/𝐸𝑈𝑅 → 𝑈𝑆𝐷/𝐸𝑈𝑅
is an ordered set of “buy, sell, buy” operations, because only then we can end up with a currency
we also begun with (input currency of first operation equals output currency of last operation).
As explained in the notation, when we are buying, the rate says how much to pay in counter
currency for one unit of base currency. Thus, the amount of base currency we get is equal to:
−1
𝐴𝑏𝑎𝑠𝑒 = 𝑃𝑎𝑠𝑘
𝐴𝑐𝑜𝑢𝑛𝑡𝑒𝑟
( 40 )
−1
Where 𝐴𝑏𝑎𝑠𝑒 is the amount of base currency, 𝐴𝑐𝑜𝑢𝑛𝑡𝑒𝑟 the amount of counter currency, and 𝑃𝑎𝑠𝑘
is the exchange rate – notice the use of ask quote and also the value inversion. And for selling:
𝐴𝑐𝑜𝑢𝑛𝑡𝑒𝑟 = 𝑃𝑏𝑖𝑑 𝐴𝑏𝑎𝑠𝑒
( 41 )
Also note the lack of value inversion in this case. For an ordered set of objects
(IEnumerable<ArbItem>), where each object contains information about one currency pair (bid
quote, ask quote, and whether the pair is to be bought or sold), the algorithm to calculate
arbitrage in C# language is:
protected decimal Arbitrage(IEnumerable<ArbItem> input)
{
var pairs = input.ToArray();
if (pairs.Length < 2)
return -1;
// Arbitrage opportunity equation
var profitMultiplier = pairs[0].Buy ? 1m / pairs[0].BidAsk.Ask :
pairs[0].BidAsk.Bid;
for (var i = 1; i < pairs.Length; i++)
profitMultiplier *= pairs[i].Buy ? 1m / pairs[i].BidAsk.Ask :
pairs[i].BidAsk.Bid;
return profitMultiplier;
}
( 42 )
Multilateral arbitrage algorithm study
55
Where arbitrage opportunity exists in case the return value is higher than 1.
6.2.2 Detecting arbitrage for all sets of quotes
Because the dataset for each currency pair is not synchronized, most of the quotes might differ
in their timestamps. For example, assume an arbitrage loop of
𝐻𝐾𝐷/𝐾𝑅𝑊 → 𝐾𝑅𝑊/𝑇𝑊𝐷 → 𝐻𝐾𝐷/𝑇𝑊𝐷
( 43 )
Then the quotes for HKD/KRW and KRW/TWN might be:
Table 12: Quotes example
HKD/KRW
Time
Bid
1:00:50
Ask
KRW/TWD
Bid
Ask
135.4304 0.0282
1:00:52 135.3443
0.0283
1:00:55
0.0284
1:00:57 135.3442 135.4303
Source: Author’s own construction
As you can see, the quotes arrive to the algorithm with some delays, so it’s not really feasible
to test only those cases when all timestamps match exactly.
For that reason, the choice was made to build in a tolerance for 10 seconds of difference between
youngest and oldest quotes. This is checked by first generating all combinations (here referring
to combinatorics) of 2 quote timestamps, calculating a difference for each combination, and
then checking if the values fall within 10 seconds. If the time difference is higher than 10
seconds, then the oldest quote is discarded and a new quote is fetched in place of the discarded
one. This process continues until all quotes fall within 10 seconds apart from each other.
Once an arbitrage is found, then all those quotes are discarded (for the assumption that they
would be arbitraged away), and then the algorithm repeats – find another set of quotes that all
fall within 10 seconds, test for arbitrage, and so on. The reported timestamp for detected
arbitrage opportunity is the average of timestamps of all quotes that are part of that arbitrage.
There are 2 reasons for the choice of 10 seconds maximum difference:
1. Foreign exchange market is the market with highest volume in the world. Because of
that, it seems quite unlikely that if there was a longer gap in the dataset, that it would
actually mean there was no trading at all. For that reason, 10 seconds were chosen to
account for possible missing entries in the dataset. Some datasets provided by
Bloomberg – even though claimed to be high-frequency – do in fact have large gaps
between quotes, as well as some missing quotes here and there (dataset defects).
2. It is generally assumed that arbitrage opportunities last for a short time because of how
tempting it is to harvest “free money” from the market. It is a highly competitive area
and speed is crucial. For that reason, it seems that most arbitrage opportunities should
56
Multilateral arbitrage algorithm study
last only few seconds. On highly competitive markets – such as London or New York –
arbitrage opportunities last for mere milliseconds or microseconds186.
Because all datasets begin on January 1st 2014 and end somewhere in the middle of May 9th
2014, I decided to artificially stop checking for arbitrage opportunities once some quote is older
than 0:00:00 May 9th 2014. This is to standardize all datasets and to make sure some arbitrage
loops do not get a bit more quotes than others. So, the analyzed period has exactly 128 calendar
days, in which are 18 Saturdays and 18 Sundays, and 92 business days.
6.2.3 Detecting arbitrage for all variants of arbitrage loops
Now given the knowledge of arbitrage algorithm, I had to develop another algorithm that would
generate all the arbitrage loop variants.
The algorithm works like this:
1. For each arbitrage loop length (3, 4, 5), do:
1. Enumerate all possible variations (term from combinatorics) of all currency
pairs, where the size of each variation set is equal to arbitrage loop length (e.g.
17!
for 17 currencies and loop length of 5, there are (17−5)! = 742 560 variations)
2. Validate all variations that they can in fact form a loop, and discard those that
cannot
3. Because certain variations can be considered equal (use the same currency pairs
and bid-ask quotes, only the first input currency is different), filter out all
equivalent variations and leave only one
To clarify the term equivalent variations, think of the following two loops:
𝑠𝑒𝑙𝑙 𝐶𝑍𝐾/𝑈𝑆𝐷 → 𝑠𝑒𝑙𝑙 𝑈𝑆𝐷/𝐸𝑈𝑅 → 𝑠𝑒𝑙𝑙 𝐸𝑈𝑅/𝐶𝑍𝐾
𝑠𝑒𝑙𝑙 𝑈𝑆𝐷/𝐸𝑈𝑅 → 𝑠𝑒𝑙𝑙 𝐸𝑈𝑅/𝐶𝑍𝐾 → 𝑠𝑒𝑙𝑙 𝐶𝑍𝐾/𝑈𝑆𝐷
As you can see, both variants have the same currency pairs which all use bid quotes, yet the
input/output currency of first loop differs with that of the second loop. To make sure arbitrages
are not double or triple counted (in case of triangular loops), the equivalent variations filtering
algorithm makes sure only single loop gets analyzed.
In regards to the overall variation generation algorithm, it might not be the fastest and most
effective algorithm to generate arbitrage loops, because there are many variations generated
that also get filtered out. A more effective algorithm might be to generate the correct variations
directly. However, such algorithm seems somewhat more complex and because the variations
are generated only once before the analysis begins, it doesn’t really matter.
With the ability to generate all arbitrage loop variants as well as to check for arbitrage for each
variant given a bid-ask quotes, it’s now possible to proceed to final analysis.
6.2.4 Putting it all together
For all individual arbitrage loops, the following information is analyzed:
186
MCCABE, Thomas. When The Speed Of Light Is Too Slow: Trading at the Edge. In: Kurzweil Accelerating
Intelligence [online]. 2010 [Accessed 2014-05-15]. Available from: http://www.kurzweilai.net/when-the-speedof-light-is-too-slow
Multilateral arbitrage algorithm study
57

All arbitrage opportunities and individual returns per each arbitrage

Annualized continuous time return in case whole portfolio would be used to exploit an
arbitrage opportunity each time it is detected

Average, mode and median return per single arbitrage

Average, mode and median time difference between detected arbitrage opportunities

Number of detected arbitrage opportunities
Then, for each variant of arbitrage loop length (e.g. aggregating all triangular arbitrages
together, then all quadrangular arbitrages, then all quintangular arbitrages, and so on):

All arbitrage opportunities and individual returns per each arbitrage combined across
all aggregated arbitrage loops of certain length

Average, mode and median annualized continuous time returns

Average, mode and median number of detected arbitrage opportunities

Average, mode and median return per single arbitrage

Average, mode and median time difference between detected arbitrage opportunities

Total number of detected arbitrage opportunities

List of aggregated arbitrage loops ordered by annualized continuous time returns

List of aggregated arbitrage loops ordered by number of detected arbitrage opportunities
And finally analysis for all arbitrage loops regardless of loop length (i.e. aggregating all
variants):

All arbitrage opportunities and individual returns per each arbitrage combined across
all aggregated arbitrage loops

Average, mode and median return per single arbitrage

Average, mode and median time difference between detected arbitrage opportunities

Average, mode and median annualized continuous time returns

Average, mode and median the number of detected arbitrage opportunities

Total number of detected arbitrage opportunities

List of each variant of arbitrage loop length ordered by average annualized continuous
time return

List of each variant of arbitrage loop length ordered by the number of arbitrage
opportunities

List of all variants of arbitrage loops ordered by average annualized continuous time
return

List of all variants of arbitrage loops ordered by the number of arbitrage opportunities
So, there are total of 268 analyses of individual arbitrage loops, 3 analyses of aggregated
arbitrage loops per the arbitrage loop length, and single overall analysis of all information
combined.
58
Multilateral arbitrage algorithm study
The program has 2 192 lines of code, excluding the open-source combinatorics component. Full
source code is available on the attached medium. The output of the program is in the forms of
CSV files and Excel files (which include automatically generated charts). The whole analysis
took about one full day to calculate on my laptop (for a maximum length of arbitrage loop set
to 5).
6.3 Results
6.3.1 Preliminary datasets analysis
First analysis was to find out how many quotes there actually are in all datasets.
Figure 3: Amount of quote entries per each currency pair
2 500 000
2 000 000
1 500 000
1 000 000
500 000
0
Amount of entries
Figure 4: Overall statistics of the amount of quote entries
12000 000
10 707 121
10000 000
8000 000
6000 000
4000 000
2000 000
629 831
0
Average
Total
Amount of entries
Multilateral arbitrage algorithm study
59
It can be seen that most of the datasets actually contain relatively few entries. The four currency
pairs with most entries account for about 70 % of all entries. This does not necessarily indicate
anything wrong with the datasets. It might be the case that currency pairs with few entries are
just too exotic and are not traded as much. Or it might be the case that the datasets for those
currency pairs are actually not high-frequency. Because of that, we need to take a look at quote
time gap analysis.
Time gap is a time difference between two subsequent quotes. If one quote has a timestamp of
14:35:22 and second quote has timestamp of 14:35:28, then time gap is 6 seconds. Because the
time gap analysis was based on calendar days and did not take into account non-business days,
it seems best to look at median time gaps rather than average time gaps, as median has the
feature to better filter out extreme values:
Figure 5: Time gap medians
60
50
40
30
20
10
0
There seems to be some kind of indirect relationship between the number of entries in the
dataset and the median time gap, which however does not apply to all pairs, namely:
𝐶𝑁𝑌/𝐾𝑅𝑊, 𝐾𝑅𝑊/𝑇𝑊𝐷, 𝐶𝑁𝑌/𝑇𝑊𝐷
To conclude, the following currency pairs are high-frequency, because they have relatively
large number of entries and small time gaps:
𝐶𝑁𝑌/𝐽𝑃𝑌, 𝐻𝐾𝐷/𝐾𝑅𝑊, 𝐶𝑁𝑌/𝐻𝐾𝐷, 𝐻𝐾𝐷/𝑇𝑊𝐷, 𝑀𝑂𝑃/𝐽𝑃𝑌, 𝐽𝑃𝑌/𝑀𝑁𝑇, 𝐽𝑃𝑌/𝑇𝑊𝐷
While these currency pairs are also considered high-frequency, because despite having fewer
entries, they still have small time gaps:
𝐶𝑁𝑌/𝐾𝑅𝑊, 𝐾𝑅𝑊/𝑇𝑊𝐷, 𝐶𝑁𝑌/𝑇𝑊𝐷
And the following pairs are deemed not high-frequency:
𝑀𝑂𝑃/𝐾𝑅𝑊, 𝐶𝑁𝑌/𝑀𝑂𝑃, 𝐶𝑁𝑌/𝑀𝑁𝑇, 𝐻𝐾𝐷/𝑀𝑂𝑃, 𝑀𝑂𝑃/𝑇𝑊𝐷, 𝐾𝑅𝑊/𝑀𝑁𝑇, 𝐻𝐾𝐷/𝑀𝑁𝑇
This matters because if it is found out that the most profitable arbitrage loop has some of the
non-high-frequency pairs, it might indicate the lack of proper data rather than the arbitrage loop
being actually so profitable. So, analysis of such loops might require special attention.
60
Multilateral arbitrage algorithm study
6.3.2 Number of variations analysis
It was also analyzed how many possible arbitrage loops there are for each loop length. The
methodology of generating all variants was discussed before.
Figure 6: Arbitrage loop variations
152
160
140
120
100
82
80
60
40
34
20
0
3
4
5
Amount
On the figure, x-axis represents the loop length – 3 means triangular arbitrages, 4 means
quadrangular arbitrages, and so on. It’s clear that the more currencies are involved in the loop,
the more possibilities there are, and there seems to be an exponential relationship – a loop length
of one order higher has roughly twice the amount of loop variants.
For real-time trading, it might be required to actually sort all the loops by preferences, and first
check the most preferred/profitable loops, and then later if there is time check the other loops
as well.
6.3.3 Arbitrage analysis
In overall, for 17 currency pairs, 34 triangular arbitrage loops, 82 quadrangular arbitrage loops
and 152 quintangular arbitrage loops, there were 350 292 arbitrage opportunities available – 1
opportunity every 30 seconds. Out of 268 arbitrage loops, 127 did not have any arbitrage
opportunities at all (47 %). For the rest of 141 loops, the average amount of arbitrage
opportunities was 2 484 in the analyzed period.
Amount of arbitrage opportunities
The highest amount of arbitrage opportunities were found in quadrangular arbitrages, with a
total of 158 399 opportunities (45 %). 137 358 arbitrage opportunities in total were found
among triangular arbitrage loops (39 %), and only 54 535 opportunities among quintangular
arbitrages (16 %).
Out of 34 triangular arbitrage loops, only 5 (15 %) did not have any arbitrage opportunities.
Out of 82 quadrangular loops, 31 (38 %) did not have any opportunities, and finally out of 152
quintangular loops, 91 (60 %) did not have any opportunities.
Multilateral arbitrage algorithm study
61
The following chart shows average, mode and median of the amount of arbitrage opportunities
per chains that had arbitrage opportunities.
Figure 7: Statistics regarding number of arbitrages in individual loops
3 000
2 484
2 500
2 000
1 500
1 000
444
500
3
0
Average count
Mode count
Median count
Number of arbitrages
The high average and low median suggests that very few arbitrage loops had most of the
arbitrage opportunities, and the rest had very few. This can be confirmed by the following
distribution chart, which shows that few leftmost arbitrage loops had the most opportunities:
Figure 8: Distribution of amount of opportunities in individual loops
60 000
50 000
40 000
30 000
20 000
10 000
0
Profitability of arbitrage opportunities
The most profitable arbitrage opportunities were found among triangular arbitrages (54 %),
then among quadrangular arbitrages (33 %) and finally among quintangular arbitrages (13 %).
Following chart displays average, mode and median of profitability of individual loops:
62
Multilateral arbitrage algorithm study
Figure 9: Statistics regarding profitability of individual arbitrage loops
4.50000000
4.13840718
4.00000000
3.50000000
3.00000000
2.50000000
2.00000000
1.50000000
1.00000000
0.59949838
0.50000000
0.22526202
0.00000000
Average return
Mode return
Median return
Annual return (%/100)
Again, the high average and low median seems to suggest large inequality amongst arbitrage
loops, which is confirmed by the following distribution chart:
Figure 10: Distribution of returns of individual arbitrage loops
100.00
90.00
80.00
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0.00
Arbitrage loops with most amounts of arbitrage opportunities
The most amount of arbitrage opportunities were found among these loops (ordered from
highest to lowest; “S” denotes that the pair is being sold, while “B” denotes the pair is being
bought):
1.
S:KRW/TWD  B:HKD/TWD  S:HKD/KRW
2.
S:HKD/TWD  B:KRW/TWD  B:HKD/KRW
3.
S:CNY/KRW  S:KRW/TWD  B:HKD/TWD  B:CNY/HKD
Multilateral arbitrage algorithm study
4.
S:HKD/TWD  B:KRW/TWD  B:CNY/KRW  S:CNY/HKD
5.
S:HKD/TWD  B:JPY/TWD  B:CNY/JPY  S:CNY/HKD
6.
S:HKD/KRW  S:KRW/TWD  B:CNY/TWD  S:CNY/HKD
7.
S:KRW/TWD  B:CNY/TWD  S:CNY/KRW
8.
S:CNY/TWD  B:KRW/TWD  B:HKD/KRW  B:CNY/HKD
9.
S:CNY/TWD  B:KRW/TWD  B:CNY/KRW
63
10. S:CNY/JPY  S:JPY/TWD  B:HKD/TWD  B:CNY/HKD
11. S:CNY/KRW  S:KRW/TWD  B:JPY/TWD  B:CNY/JPY
12. S:HKD/KRW  S:KRW/TWD  B:JPY/TWD  B:CNY/JPY  S:CNY/HKD
These 12 loops (5 %) combined account for 72 % of all arbitrage opportunities found. The loops
of number 11 and 12 are those that did not get among the most profitable arbitrage loops.
The currencies in these loops are KRW (in 10 loops), TWD (in all 12 loops), HKD (in 9 loops),
CNY (in 10 loops), and JPY (in 4 loops), and the total amount of currencies analyzed is 7. It
seems that the first 4 currencies (often considered somewhat exotic) are particularly not
effective, because they offer so much arbitrage opportunities. This might indicate a potential
business opportunity for those in the area.
It’s worth noting though that the 4 currency pairs with most data available are composed of
these 4 currencies. This might indicate potential skew in results due to incomplete databases
for the other currency pairs, but that also implies the idea that Bloomberg is not reliable source
for high-frequency financial data. The other side of the coin is that 4 other currency pairs – that
are also composed only out of these 4 currencies – do not have as much data.
There are a total of 8 currency pairs involved in those loops combined, where the most common
is KRW/TWD (10 times), then CNY/HKD (7 times), HKD/TWD (6 times), HKD/KRW and
CNY/KRW (both 5 times), and finally JPY/TWD, CNY/JPY and CNY/TWD (each 4 times).
This makes me come to the conclusion that TWD (Taiwan) and KRW (South Korea) are
probably the most important sources of arbitrage opportunities, meaning that these currencies
show particular lack of market efficiency.
Most profitable arbitrage loops
The most profitable loops were:
1.
S:KRW/TWD  B:HKD/TWD  S:HKD/KRW
2.
S:HKD/TWD  B:KRW/TWD  B:HKD/KRW
3.
S:CNY/KRW  S:KRW/TWD  B:HKD/TWD  B:CNY/HKD
4.
S:HKD/TWD  B:JPY/TWD  B:CNY/JPY  S:CNY/HKD
5.
S:HKD/TWD  B:KRW/TWD  B:CNY/KRW  S:CNY/HKD
6.
S:CNY/JPY  B:MOP/JPY  S:MOP/KRW  B:HKD/KRW  B:CNY/HKD
7.
S:KRW/TWD  B:CNY/TWD  S:CNY/KRW
8.
S:HKD/KRW  S:KRW/TWD  B:CNY/TWD  S:CNY/HKD
9.
S:CNY/JPY  S:JPY/TWD  B:HKD/TWD  B:CNY/HKD
64
Multilateral arbitrage algorithm study
10. S:HKD/KRW  B:MOP/KRW  S:MOP/JPY  B:CNY/JPY  S:CNY/HKD
11. S:CNY/TWD  B:KRW/TWD  B:HKD/KRW  B:CNY/HKD
12. S:CNY/TWD  B:KRW/TWD  B:CNY/KRW
The loops of number 6 and 10 are those that did not get among the loops with highest amounts
of arbitrage opportunities.
The currencies in these loops are KRW (in 10 loops), TWD (in 10 loops), HKD (in 10 loops),
CNY (in 10 loops), JPY (in 4 loops), and MOP (in 2 loops).
There are a total of 10 currency pairs involved in these loops combined, where the most
common is KRW/TWD and CNY/HKD (both 8 times), HKD/KRW and HKD/TWD (both 6
times), CNY/KRW, CNY/JPY, and CNY/TWD (each 4 times), and finally JPY/TWD,
MOP/KRW, and MOP/JPY (each 2 times). The profit analysis also confirms that indeed TWD
and KRW seem to be a cause of most profits. HKD seems to be also a significant source of
profits.
Combined arbitrage returns
The following chart displays returns of all arbitrage opportunities combined:
Figure 11: Timeline of combined arbitrage returns
1.01500000
1.01000000
1.00500000
1.00000000
0.99500000
1.1.2014 0:59:56
6.1.2014 3:49:55
8.1.2014 4:20:12
10.1.2014 5:38:55
14.1.2014 5:47:00
16.1.2014 7:54:06
22.1.2014 4:53:05
24.1.2014 8:17:19
29.1.2014 4:03:03
4.2.2014 5:36:58
6.2.2014 5:03:27
10.2.2014 5:40:46
13.2.2014 3:02:20
17.2.2014 6:21:41
19.2.2014 7:38:18
21.2.2014 7:19:45
26.2.2014 2:57:21
28.2.2014 3:50:50
4.3.2014 5:42:11
6.3.2014 5:41:24
11.3.2014 3:44:32
14.3.2014 1:09:07
26.3.2014 3:59:21
16.4.2014 4:26:24
0.99000000
Multiplier
It can be seen that most of the time, the returns are relatively small, but from time to time there
is a large opportunity available, likely due to large orders affecting the market relatively sharply.
There doesn’t seem to be any pattern in the data that would explain the large opportunities. This
might imply that large arbitrage opportunities are stochastic and do not follow any real trend.
Conclusion
65
CONCLUSION
Analysis of 268 types of multilateral arbitrage on 10 700 000 high-frequency market quotes in
the period from 1st January 2014 till 9th May 2014 has shown that KRW (Korean Won) and
TWD (New Taiwan Dollar) currencies exhibit the most arbitrage opportunities, particularly in
conjunction with HKD (Hong Kong Dollar). In other words, these currency pairs are the cause
behind most multilateral arbitrage opportunities and can be considered significantly inefficient:

KRW/TWD

HKD/TWD

HKD/KRW
Furthermore, it has been shown that triangular (39 %) and quadrangular arbitrages (45 %) are
largely similar when it comes to profitability or the amount of arbitrage opportunities, but
quintangular arbitrages are quite rare (only 16 %). It does seem that the longer is the arbitrage
loop, the fewer opportunities there are, because the more unique market situation is required.
However, the longer is the arbitrage loop, the higher is the amount of loop variants, and so even
though there are fewer opportunities per variant, there are more variants. However, both
triangular and quadrangular arbitrages seem to be able to capture most of the profits, and profits
are also bound to only few currency pairs, so arbitraging over longer loops at all costs does not
seem to make much sense.
Additionally, only very few arbitrage loop variants (5 %) account for the vast majority of found
arbitrage opportunities (72 %). This means some loops are extremely prone to market
ineffectiveness, and proves that multilateral arbitrage is not uniformly distributed across all
loop variants and currency pairs.
Thus, for institutions interested in leveraging available arbitrage opportunities, it is most
efficient to focus on few selected currency pairs and few selected arbitrage loops rather than
going overboard to try to arbitrage on all available markets and very long arbitrage loops.
The contribution of this paper is the design and implementation of arbitrage detection algorithm,
design and implementation of algorithm that generates all possible variants of multilateral
arbitrage loops, and the application of the algorithms to real-world market data, which led to
the discovery of the source of arbitrage opportunities in East Asian foreign exchange markets.
This also fulfilled the main goals of this thesis. Furthermore, the practical conclusions drawn
from the results of automated analysis and recommendations for potential real-world
arbitrageurs all form the output of this research.
Future research
In future research, the full order book might be analyzed to actually determine the real depth of
arbitrage opportunities. However, the biggest obstacle in that is that full order book highfrequency databases are very hard to obtain with limited budget, because these databases are
considered an asset in the commercial world and thus priced accordingly.
Additional paper might focus on analyzing longer arbitrage loops, such as 6-order, 7-order or
even higher, and include additional currencies. But such analysis might require expensive
hardware because of the incredible amount of calculations required.
Other direction of further research might be to look for more complex arbitrage opportunities,
such as spot-futures cash-and-carry arbitrages, co-integration arbitrages, and others.
66
Conclusion
A somewhat different, but related direction might be to research what it actually takes to enter
the real-world arbitrage scene, what are the barriers to entry, and how to practically proceed.
Yet another research might be to compare arbitrage opportunities in Europe, USA and Asia. It
might be very interesting to get a comparison to gauge how much tempting the Asian markets
are from the perspective of arbitrage.
It might be also quite interesting to take the last figure showing timeline of combined arbitrage
opportunities and try to match them with real-world events, such as market development and
news. A conclusion would be drawn on whether important events and surprising news are the
main source of arbitrage opportunities, or whether those are rather caused by liquidity
management and thus are somewhat stochastic.
Bibliography
67
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76
List of tables
LIST OF TABLES
TABLE 1: CURRENCY CODES......................................................................................................................... 3
TABLE 2: QUADRANGULAR ARBITRAGE EXAMPLE ........................................................................... 15
TABLE 3: GENERIC CASH-AND-CARRY ARBITRAGE DECISION TABLE ........................................ 17
TABLE 4: DETAILED EXAMPLE OF CASH-AND-CARRY ARBITRAGE ............................................. 19
TABLE 5: DAY COUNTING CONVENTION ................................................................................................ 21
TABLE 6: FOREX FUTURES CASH-AND-CARRY ARBITRAGE DECISION TABLE ......................... 22
TABLE 7: PUT-CALL PARITY DERIVATION............................................................................................. 24
TABLE 8: OPTIONS CASH-AND-CARRY ARBITRAGE DECISION TABLE ........................................ 25
TABLE 9: FUTURES-OPTIONS CASH-AND-CARRY ARBITRAGE DECISION TABLE .................... 26
TABLE 10: HIGH-FREQUENCY ORDER BOOK EXAMPLE.................................................................... 32
TABLE 11: ANALYZED CURRENCY PAIRS ............................................................................................... 53
TABLE 12: QUOTES EXAMPLE..................................................................................................................... 55
LIST OF FIGURES
FIGURE 1: RANGE TRADING EXAMPLE ................................................................................................... 39
FIGURE 2: REBALANCING GAINS FOR N = 5 ........................................................................................... 42
FIGURE 3: AMOUNT OF QUOTE ENTRIES PER EACH CURRENCY PAIR ........................................ 58
FIGURE 4: OVERALL STATISTICS OF THE AMOUNT OF QUOTE ENTRIES ................................... 58
FIGURE 5: TIME GAP MEDIANS .................................................................................................................. 59
FIGURE 6: ARBITRAGE LOOP VARIATIONS ........................................................................................... 60
FIGURE 7: STATISTICS REGARDING NUMBER OF ARBITRAGES IN INDIVIDUAL LOOPS ....... 61
FIGURE 8: DISTRIBUTION OF AMOUNT OF OPPORTUNITIES IN INDIVIDUAL LOOPS ............. 61
FIGURE 9: STATISTICS REGARDING PROFITABILITY OF INDIVIDUAL ARBITRAGE LOOPS . 62
FIGURE 10: DISTRIBUTION OF RETURNS OF INDIVIDUAL ARBITRAGE LOOPS ......................... 62
FIGURE 11: TIMELINE OF COMBINED ARBITRAGE RETURNS ......................................................... 64
LIST OF EQUATIONS
( 1 ) ........................................................................................................................................................................ 14
( 2 ) ........................................................................................................................................................................ 14
( 3 ) ........................................................................................................................................................................ 15
( 4 ) ........................................................................................................................................................................ 15
( 5 ) ........................................................................................................................................................................ 15
( 6 ) ........................................................................................................................................................................ 15
( 7 ) ........................................................................................................................................................................ 15
List of equations
77
( 8 )......................................................................................................................................................................... 17
( 9 )......................................................................................................................................................................... 17
( 10 )....................................................................................................................................................................... 21
( 11 )....................................................................................................................................................................... 21
( 12 )....................................................................................................................................................................... 22
( 13 )....................................................................................................................................................................... 22
( 14 )....................................................................................................................................................................... 22
( 15 )....................................................................................................................................................................... 22
( 16 )....................................................................................................................................................................... 22
( 17 )....................................................................................................................................................................... 24
( 18 )....................................................................................................................................................................... 24
( 19 )....................................................................................................................................................................... 24
( 20 )....................................................................................................................................................................... 24
( 21 )....................................................................................................................................................................... 26
( 22 )....................................................................................................................................................................... 26
( 23 )....................................................................................................................................................................... 35
( 24 )....................................................................................................................................................................... 35
( 25 )....................................................................................................................................................................... 37
( 26 )....................................................................................................................................................................... 38
( 27 )....................................................................................................................................................................... 40
( 28 )....................................................................................................................................................................... 40
( 29 )....................................................................................................................................................................... 40
( 30 )....................................................................................................................................................................... 40
( 31 )....................................................................................................................................................................... 40
( 32 )....................................................................................................................................................................... 41
( 33 )....................................................................................................................................................................... 41
( 34 )....................................................................................................................................................................... 41
( 35 )....................................................................................................................................................................... 41
( 36 )....................................................................................................................................................................... 41
( 37 )....................................................................................................................................................................... 50
( 38 )....................................................................................................................................................................... 50
( 39 )....................................................................................................................................................................... 50
( 40 )....................................................................................................................................................................... 54
( 41 )....................................................................................................................................................................... 54
( 42 )....................................................................................................................................................................... 54
( 43 )....................................................................................................................................................................... 55
APPENDIX A: EAST ASIAN EXCHANGES
The member countries of the region of Eastern Asia187, and their respective local stock
exchanges188 and currencies (as per ISO 4217189) are:

China (currency CNY, sometimes also known as RMB)
o China Financial Futures Exchange (CFFEX) - http://www.cffex.com.cn/
o Dalian Commodity Exchange (DCE) - http://www.dce.com.cn/
o Shanghai Futures Exchange (SHFE) - http://www.shfe.com.cn/
o Shanghai Steel Electronic Exchange - http://www.ssec-steel.com/
o Shanghai Stock Exchange (SSE) - http://www.sse.com.cn/
o Shenzhen Stock Exchange - http://www.szse.cn/
o Zhengzhou Commodity Exchange (ZSE) - http://www.czce.com.cn/

China, Hong Kong Special Administrative Region (currency HKD)
o Chinese Gold & Silver Exchange Society (CGSE) - http://www.cgse.com.hk/cn/
o Growth Enterprise Market - http://www.hkgem.com/
o Hong Kong Exchanges and Clearing (HKEx) - http://www.hkex.com.hk/
o Hong Kong Stock Exchange (SEHK)
o Hong Kong Futures Exchange

China, Macao Special Administrative Region (currency MOP)
o None

Democratic People's Republic of Korea (currency KPW)
o None

Japan (currency JPY)
o Osaka Dojima Commodity Exchange - http://ode.or.jp/
o Fukuoka Stock Exchange - http://www.fse.or.jp/
o Nagoya Stock Exchange - http://www.nse.or.jp/
o Sapporo Securities Exchange - http://www.sse.or.jp/
o Tokyo Commodity Exchange (TOCOM) - http://www.tocom.or.jp/
o Tokyo Financial Exchange (TFX) - http://www.tfx.co.jp/
o Japan Exchange Group - http://www.jpx.co.jp/en/
o Tokyo Stock Exchange (TSE) - http://www.tse.or.jp/
187
UNITED NATIONS. Composition of macro geographical (continental) regions, geographical sub-regions,
and selected economic and other groupings. In: United Nations [online]. 2013 [Accessed 2014-04-28]. Available
from: https://unstats.un.org/unsd/methods/m49/m49regin.htm
188
WORLD-STOCK-EXCHANGES.NET. Asian Stock Exchanges. In: World-Stock-Exchanges.net [online]. ©
2008-2012 [Accessed 2014-04-28]. Available from: http://www.world-stock-exchanges.net/asia.html
189
ISO 4217. In: International Organization for Standardization [online]. [Accessed 2014-04-28]. Available
from: http://www.iso.org/iso/home/standards/currency_codes.htm
o Osaka Securities Exchange (OSE) - http://www.ose.or.jp/

Mongolia (currency MNT)
o Mongolian Stock Exchange - http://www.mse.mn/

Republic of Korea (currency KRW)
o Korea Exchange (KRX) - http://www.krx.co.kr/
Although not recognized by United Nation, there countries are also in the region:

Republic of China, Taiwan (currency TWD)
o GreTai Securities Market - http://www.gretai.org.tw/
o Taiwan Stock Exchange (TSEC) - http://www.tse.com.tw/
o Taiwan Futures Exchange (TAIFEX) - http://www.taifex.com.tw/

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