A Unified Framework for CVA, DVA and FVA Applied to

See the Disclaimer Appendix
A Unified Framework for CVA, DVA and
FVA Applied to Credit Portfolios
Youssef Elouerkhaoui
Managing Director, Markets Quantitative Analysis
[email protected]
19 March 2014
Citigroup Global Markets Limited
Outline
• Motivation: CVA, DVA and FVA Risk Mitigation for Credit
• Master Funding Equation with CVA, FVA and FVO
• Fundamental Invariance Principle for Funding and CVA
• FVA vs FVO
• CVA in the Enlarged Filtration
• Credit Options Revisited: Impact of the Choice of Filtration
• Wrong-Way Risk vs Gap Risk
2
Introduction
There are a lot of discussions within the industry around best
practices for marking, managing and mitigating counterparty risk
charges and funding costs.
The inclusion of FVA adjustements for unsecured derivatives has
been heavily debated between practitioners and academics; and the
marking methodology for DVA is still raising many questions.
According to the E&Y 2012 CVA/FVA Survey, all banks record
CVA and OCA (Own Credit Adjustment) on liabilities under
FVO accounting; and most report DVA. (DVA reporting will be
mandatory with the introduction of IFRS 13).
The majority of dealers have moved to CSA discounting. But for
uncollateralized derivatives, only a handful record an FVA. Funding
is usually priced in at trade inception, but there is no adjustment
3
made to the FVA during the life of the trade for financial reporting
purposes.
Unsecured CVA will be less of a problem going forward as we move
to mandatory clearing for derivatives, and cash-rich corporates are
incentivized to reduce their CVA charges and earn CSA interest on
cash reserves.
For credit, even Secured CVA can still be sizeable because of the
Gap risk induced by the enlarged filtrations. In that case, The Base
PV will be funded by the CSA and discounted at the CSA rate. The
CVA and DVA are funded via Treasury and will incur a funding
charge (FVA).
The main rationale for the inclusion of FVA in books and records
reporting is based on the concept of exit price, i.e, the price that
would be paid between market participants to novate a transaction.
There is no consensus on how the FVA should be calculated. To
mark the OCA, more banks are moving away from CDS curves to
using either primary issuance data or target funding curves.
The FVA Debate
In a nutshell, the FVA debate is about whether we should include
FVA in derivatives pricing or not. Academics, Hull and White
(2012a), argue that we shouldn’t:
• “It might be argued that the use of a risk-free discount rate indicates the
valuation is only appropriate when the bank can fund the derivative at the riskfree rate. This is not true. The risk-free rate is used for discounting because
this is required by the risk-neutral valuation principle. Risk-neutral valuation is
an artificial – but fantastically useful – tool that gives the correct economic
valuation for a derivative, taking into account all its market risks.
• Another argument against FVA is a well-established principle in corporate finance theory that pricing should be kept separate from funding. The discount
rate used to value a project should depend on the risk of the project rather
than the riskiness of the firm that undertakes it.
• FVA is closely related to debit value adjustment (DVA), but it is important
to avoid confusing the two different types of DVA. One is the DVA arising
because a dealer may default on its derivatives portfolio (we will refer to this
as DVA1). The other (DVA2) is the DVA arising because a dealer may default
on its other liabilities – long-term debt, short-term debt, and so on.”
4
Pro-FVA
Traders on the other hand reject the“academic” arguments and
state that FVA is a market reality that drives the profitability of the
business. The response from Laughton and Vaisbrot (2012):
• “They use the Black-Scholes-Merton (BSM) theory to argue derivatives should
be valued on a risk-neutral basis, independent of the cost to the trader of
funding the position, but the theory rests on the ability of market participants to
fully hedge all risk factors. In reality, they are not able to do so because markets
are incomplete. As a result, risk preference is reintroduced into valuations, and
the law of one price no longer holds.
• In practice, a bank borrows at the rate it can – usually unsecured. The bank
may try to convince counterparts that its borrowing costs should go down as
Hull and White argue. But most of its debt will already have been issued at
a fixed rate, which is unlikely to be renegotiated, even assuming the creditors
believe the incremental debt is actually risk-free. And it doesn’t matter to the
trader at which rate the bank should be able to borrow, only the rate at which
it can.
• models need to be amended to be useful to traders. They should remove the
assumption of the ability to borrow at the risk-free rate to finance an apparently
risk-free basket, and instead assume that: a) the cost of borrowing for an
institution is exogenous and unaffected by a single trade; b) market-makers
give no value to their expected profit or loss upon their own default.”
5
The FVA Debate Continues
Hull and White (2012b) maintain in their follow-up article:
• “the Merton argument is not the only justification of BSM – economic arguments, such as Fisher Black and Myron Scholes’ original contention based on
the capital asset pricing model, give the same result and do not assume any
risk-free borrowers. All they require is that, in equilibrium, a risk-free portfolio
should earn the risk-free rate. We may not know exactly what that rate is, but
the concept is clear and there are excellent proxies for it.
• The key question in this debate appears to be whether the debit valuation
adjustment (DVA) that accounts for the dealer’s own default is a real benefit
or an accounting quirk. DVA2 arises from the possibility that the bank may
default on its borrowings, and seems to be more controversial. Accountants signalled acceptance of DVA2 in the US Financial Accounting Standards Board’s
directive 159 in 2007.
• In the end, the reporting of DVA2 is just a move to market accounting. For
those who believe DVA2 is an unreal accounting abstraction that carries this
too far, FVA makes sense: it is equal but opposite to DVA2, so can be regarded
as a way of removing it from pricing. But we believe the DVA2 benefits are
real and should accrue to the funding desk. This means FVA should not be
included in prices.”
6
CVA and FVA Literature
Some of the key papers that tackled CVA and FVA consistently,
include:
• D. Brigo, A. Pallavicini, D. Perini (2011): “Funding Valuation Adjustment:
A Consistent Framework Including CVA, DVA, Collateral, Netting Rules and
Re-hypothecation”.
• D. Brigo, A. Pallavicini, D. Perini (2012), “Funding, Collateral and Hedging:
Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustemnts”.
• C. Burgard, M. Kjaer (2010): “PDE Representations of Options with Bilateral
Counterparty Risk and Funding Costs”.
• C. Burgard, M. Kjaer (2012), “A Generalised CVA with Funding and Collateral
via Semi-Replication”.
• S. Crepey (2011): “A BSDE Approach to Counterpary Risk Under Funding
Constraints”.
• S. Crepey (2012), “Counterparty Risk and Funding: The Four Wings of the
TVA”.
7
Our Approach: Key Principles
There is no need for a new arbitrage-free pricing theory.
Use the same theory, with the same risk-neutral measure and the
same money-market account (risk-free) numeraire.
The only difference is that we price a more complex payoff which
includes all the default contingent legs and all funding legs. There is
no need to make any ad-hoc or a-priori assumptions on what the
final result should be.
The fundamental result is that by pricing all the funding legs,
since we are by construction funding flat (or funding neutral), the
(risk-free) money-market accruals drops off and the dependence on
the theoretical risk-free rate vanishes.
See Crepey (2012a) for an attempt to re-build the arbitrage-free
pricing theory from first principles.
8
Our Approach: Hedging Portfolio
There is no need to include the hedging instruments and the hedge
portfolio. The hedging instruments just define which risk-neutral
probability measure should be used. This defines the drift of the
diffusions, and in the case of credit, defines the intensity processes
to be used.
If we hedge with bonds, we use the bond measure, and hence the
intensity would be the one infered by bond pricing which accounts
for the bond-CDS basis. If we hedge with CDS, then the intensity
would be derived by calibrating on the CDS prices.
See Brigo et al. (2012) where they include the hedge portfolio then
simplify the equations and remove the dependence on the hedge by
choosing the appropriate ”hedge” risk measure.
See Burgard and Kjaer (2012) where all the intensities that they
use are bond-implied intensities since they use bonds to hedge the
default risk and neutralize the JTD of the hedged portfolio.
9
Bank Funding Structure
Source: Citi. Bank funding structure for a derivative contract.
10
In this Presentation
• We shall:
– Derive a Master Funding Equation with CVA, FVA and FVO
– Develop a new technique based on the Fundamental Invariance
Principle for Funding and CVA
– Use the Invariance Principle to solve the Master equation for
the Desk CVA and Treasury FVO
– Derive the relationship between FVA and FVO and show that
FVA doesn’t disappear
– Apply to Credit CVA and link Gap risk and CDS Option Skews
through Filtration enlargement techniques. Analyze the relationship between Gap risk and Wrong-Way risk for CDSs.
11
Set-up
We work on a probability space (Ω, G, P), where we have a set of
default times (τ1, ..., τn), representing the defaults of a reference
portfolio on n obigors (i.e., the names in the netting set).
The other names in our credit universe are represented as
τn+1, ..., τn+m .
We denote by τc, the default time of the counterparty, and we
denote by τb, the default time of the bank. Their recovery rates are
R1, .., Rn+m , Rc and Rb respectively. And their default indicators
are denoted by Dti = 1{τi≤t}, Dtc = 1{τc≤t}, Dtb = 1{τb≤t} respectively.
The enlarged filtration {Gt} that we work with contains both the
n+m
n
i ∨ Hb ∨ Hc and the
i ∨
H
defaults filtration {Ht} =
H
t
t
i=1 t
i=n+1 t
background filtration {Ft}.
12
Arbitrage-Free Pricing Theory
We define a generic derivative contract by its cumulative dividend
process Ct. In general, the dividend process is considered to be of
the form Ct = At − Bt, where A and B are bounded increasing
adapted right-continuous left-limit (cadlag) processes.
Risk-Neutral Pricing. The value of the derivative security C
without default risk and funding is given by:
(r)
V t = Et
T
t
(r)
pt,s dCs ,
(r)
where pt,s is the risk-free discount factor with maturity s, at time t,
(r)
pt,s
exp (− ts rudu).
Note. The risk-free rate rt is the (theoretical) money-market
account that we use in the classic arbitrage-free pricing theory.
13
Market Discount Factor Rates
We will also have two (market) discount factors:
(r c )
• pt,s
exp (− ts ruc du), where rtc is the CSA rate paid or received
on the collateral.
rF
• pt,s
exp − ts ruF du , where rtF is the (unsecured) funding rate
we get from treasury.
Example. For a Standard CDS contract, with payment schedule
(T0 = 0, T1, ...TN ), δi is the accrual fraction between Ti and Ti−1, and
S is the running spread. The dividend process is given by
Cs = Dsk −
Ti
1{Ti≤s}Sδi 1 − DTki .
14
Close-Out Value
Bilateral CVA. We consider the bank’s own default probability
and the possibility of the bank defaulting before the counterparty.
The loss incurred when the counterparty defaults after the bank
is referred to as Asset CVA. The benefit gained when the bank
defaults before the counterparty is called Liability CVA.
Let τf denote the first-to-default time of the counterparty and the
bank τf = min (τb, τc).
Assumption. We assume that we do not have simultaneous joint
defaults.
Close-Out Value. Upon termination of the contract, the Close-Out
Value will be denoted by χ(τc). In general, the close-out value of
the contract is considered to be the value of the contract without
counterparty risk at the time of default χ(τc) = Vτc .
15
Close Out Amount. The ISDA Market Review of OTC Derivative
Bilateral Collateralization Practices (2010) summarizes:
“Upon default close-out, valuations will in many circumstances
reflect the replacement cost of transactions calculated at the
terminating party’s bid or offer side of the market, and will often
take into account the credit-worthiness of the terminating party.
However, it should be noted that exposure is calculated at
mid-market levels so as not to penalize one party or the other.”
ISDA Documentation (2009)
The relevant sections in the ISDA Close-out Amount Protocol
(2009) are highlighted:
“Close-out Amount” means, with respect to each Terminated Transaction and a
Determining Party, the amount of the losses or costs of the Determining Party...
in replacing, or in providing for the Determining Party the economic equivalent
of,... When considering information described in clause (i), (ii) or (iii), the
Determining Party may include costs of funding, to the extent costs of funding
are not and would not be a component of the other information being utilised...
the Determining Party may in addition consider in calculating a Close-out Amount
any loss or cost incurred in connection with its terminating, liquidating or
re-establishing any hedge related to a Terminated Transaction ... for the purpose
of determining a Close-out Amount, the Determining Party will: if obtaining
quotations from one or more third parties, ask each third party (A) not to take
account of the current creditworthiness of the Determining Party or any existing
Credit Support Document and (B) to provide mid-market quotations”
16
ISDA Documentation (2002)
The relevant sections in the ISDA 2002 Master Agreement are
highlighted:
“the Calculation Agent will determine the Cash Settlement Amount on the basis
of quotations (either firm or indicative) for a replacement transaction supplied by
Cash Settlement Reference Banks (but the Calculation Agent may not take into
account any loss or cost incurred by a party in connection with its terminating,
liquidating or re-establishing any hedge related to the Relevant Swap Transaction
(or any gain resulting from any of them)).”
17
Collateral and Funding
We denote by Mt the value of the margin (or collateral) posted at
time t:
• Mt ≥ 0 means that we are receiving margin,
• Mt ≤ 0 means that we are posting margin.
We denote by Ft the value of the funding from treasury at time t:
• Ft ≥ 0 means that we are receiving funding, i.e., borrowing from
treasury and paying interest (this corresponds to Vt ≥ 0 where we
pay the premium),
• Ft ≤ 0 means that we are providing funding, i.e., lending to treasury and receiving interest (this corresponds to Vt ≤ 0 where we
receive the premium).
18
FVA, DVA and FVO
We derive a general formula that combines the effect of (internal)
treasury funding, (exogenous) CSA funding, and counterparty
default risk. The formula is generic and does not depend on the
type of CSA that we have in place or the details of the margining
agreement.
Funding Equation. We start with the default-free case where we
consider all the cash-flows of the transaction including the CSA and
the Funding swap.
This is the case considered in Piterbarg (2010) in a Black-Scholes
PDE framework, which is limited to diffusions and Brownian motion
filtrations.
We do this in a general (probabilistic) set-up that includes jumps
and larger filtrations needed for credit products.
We include both the CSA Funding and Treasury Funding.
19
Collateral and Funding Accounts
CSA Funding. When we are funded via the CSA agreement we
should include the interest paid or received on the collateral account,
which will change the value of the contract without default risk and
will have an impact on the CVA/DVA formulas.
The value of the collateral account between the valuation date t and
the maturity T , is given by the sum of the variation margins and the
interest paid on the account:
AC
t = Mt + Et
T
t
(r)
pt,s dαC
s ,
where we have
c
dαC
t = dMt − rt Mt dt,
MT = 0, at trade maturity.
By integration by parts, we have
Mt +
T
t
T
T
(r)
(r)
(r)
(r)
pt,s dMs = pt,T MT −
rsMspt,s ds.
Msdpt,s =
t
t
20
Hence
AC
t = Et
T
t
(r)
(rs − rsc) Mspt,s ds .
Treasury Funding. When we are funded via Treasury we need to
account for the interest charge or benefit on the (cash) funding
account.
Using a similar reasonning for the funding account, we have
F
dαF
t = dFt − rt Ft dt,
FT = 0, at trade maturity.
Ft = Vt − Mt, for all t < T .
which leads to:
AF
t = Et
T
t
(r)
rs − rsF Fspt,s ds .
The Funding Equation
We need to value the package which contains the derivative
contract, the CSA agreement and the Funding (swap) with Treasury.
Proposition 1 (The Funding Equation). The value of the defaultfree trade with funding from the CSA and Treasury will be:
V t = Et
T
t
(r)
pt,s dCs +
T
t
(r)
pt,s · (rs − rsc) Msds +
T
t
(r)
pt,s · rs − rsF Fsds ,
where the funding account is given by Ft = Vt − Mt.
21
Equivalent Representations
The funding equation can also be re-written in other useful forms:
• either using CSA discounting:
T
V t = Et
t
(r c )
pt,s dCs +
T
t
(r c )
pt,s · rsc − rsF
Fsds ,
• or using (unsecured) funding discounting:
V t = Et
T
t
rF
pt,s dCs +
T
t
rF
pt,s · rsF − rsc Msds .
Both representations have been used in Piterbarg (2010).
22
The Invariance Principle for Funding
Funding equation invariance principle states that we can discount
the cash-flows of the trade (inclusive of CSA and Treasury funding)
with any rate that we choose.
Proposition 2 (Funding Invariance Principle). Let rt∗ t≥0 be any
(positive) interest rate process, then the funding equation can be
re-written equivalently using the discounting with rt∗ -process, i.e.,
the value of the (default-free) trade with CSA and Treasury funding is:
V t = Et
T
t
(r ∗ )
pt,s dCs +
T
t
(r ∗ )
pt,s · rs∗ − rsc Msds +
T
t
(r ∗ )
pt,s · rs∗ − rsF
where the funding account is given by Ft = Vt − Mt.
Proof. Define the net cash-flow stream (of the three contracts)
(r)
d Cs
dCs + (rs − rsc) Msds + rs − rsF Fsds.
23
Fsds ,
Then, we can write the Funding equation as
T
V t = Et
t
(r)
(r)
pt,s dCs
.
(r)
Define the process At :
(r)
p0,t Vt +
(r)
At
t (r)
(r)
p0,s dCs .
0
(r)
Using the Funding equation, we can show that At
(r)
(r)
At
= p0,t Vt +
(r)
= p0,t Et
= Et
T
0
t
t (r)
(r)
p0,s dCs
0
T (r)
(r)
pt,s dCs
+
(r)
(r)
p0,s dCs
is a martingale:
t (r)
(r)
p0,s dCs
0
(r)
= Et A T
.
On the other hand, differentiating we get the following SDE:
(r)
dAt
(r)
(r)
= p0,t −rtVtdt + dVt + dCt
.
(r ∗ )
Similarly, if we introduce the process At , we have
(r ∗)
(r ∗ )
(r ∗ )
∗
dAt
= p0,t −rt Vtdt + dVt + dCt
.
Observing that
(r ∗ )
d Ct
= dCs + rt∗ − rtc Mtdt + rt∗ − rtF Ftdt
(r)
= dCt + rt∗ − rt Mtdt + rt∗ − rt Ftdt,
we obtain, for all t < T ,
(r ∗ )
(r ∗ )
(r ∗ )
∗
dAt
= p0,t −rt Vtdt + dVt + dCt
(r ∗ )
(r)
= p0,t −rtVtdt + rt − rt∗ Vt + dVt + dCt + rt∗ − rt (Mt + Ft) dt
(r ∗ )
= p0,t
(r)

(r ∗ ) 
rt − rt∗ [Vt − (Mt + Ft)] dt − rtVtdt + dVt + dCt


= p0,t  rt − rt∗ Vt − Mt − Ft dt +
=
(r ∗ )
p0,t
(r)
dAt .
(r)
p0,t
=0
(r)
dAt
(r)
p0,t



(r ∗ )
Hence, the process At
is also a martingale, which gives the result
(recall that VT is the post-dividend price):
(r ∗ )
(r ∗ )
At
= Et A T
= Et
T
0
(r ∗ )
(r ∗)
p0,s dCs
.
Note. In a deterministic setting, we can get the intuition behind
the martingale based proof by solving the deterministic differential
equation instead.
Corollary. Setting rt∗ = rtc or rt∗ = rtF gives the equivalent funding
equations with CSA discounting or (unsecured) funding discounting.
The Master Equation with Funding and CVA
Now, we add default risk to the equation and we include all the
cashflows from the trade, CSA funding, Treasury funding, and the
default terms.
Proposition 3 (The Master Equation). The value of the risky
trade will cover all the cashflows from the trade, the financing
from the CSA and Treasury, and recovery payments at the time
of default:
V t = Et
+Et
T
t
1{τ
T
t
(r)
f
>s} pt,s dCs
(r)
1{τ >s}pt,s
· (rs − rsc) Msds +
f
(r)
T
t
1{τ
(r)
F F ds
p
·
r
−
r
s
s
s
f >s} t,s
(r)
(r)
+Et 1{τ ≤T }pt,τ Mτf + 1{τ ≤T }pt,τ FτRf − Fτ − + 1{τ ≤T }pt,τ VτR
,
f
f
f
f
f
f
f
f
where the funding account is given by Ft = Vt − Mt, for all t < τf .
24
• The recovery payoff (of the swap) will be after netting with
the margin from the collateral account




=
VτR

f

+
1{τ
f =τc }
+1{τ =τ }
f
b
Rc V τ f − M τ −
f
+ V τf − Mτ −
f
+
V τf − Mτ −
f
−
+ Rb V τ f − M τ −
f
−



,


where we use the notations x+ = max (x, 0) and x− =
min (x, 0).
• The recovery payoff of the funding swap from treasury (if reallocated back to the desk)
FτRf = F −− + F +− 1{τ =τc} + RbF F +− 1{τ =τ }.
f
f
b
τ
τ
τ
f
f
f
Note. The contract Vt pays the cashflows dCs; the CSA agreement
pays financing and exchanges the margin Mτf at the time of default;
the treasury funding pays financing and exchanges the funding
account balance Fτf at the time of default.
If the DVA of the Treasury funding position is re-allocated back to
the desk, the benefit from the funding leg loss is
F,b
ξ τ = FτRf − Fτ − = 1 − RbF F +− 1{τ =τ }.
( f)
f
b
τf
f
The recovery leg on the funding swap is the DVA of the funding leg.
It is accounted for in the FVO for Debt CVA. This is referred to in
Hull and White (2012) as DVA2.
The DVA of the funding leg should be on the aggregate positions
across all netting sets and all counterparties:
Ft =
FτRf
Vtk − Mtk .
Ftk =
k
= F −− +
τf
k
k
F +− 1{τ =τ } + RbF F +− 1{τ =τ }.
f
k
f
b
τ
τ
f
f
Collateral and Funding Accounts with Default
Discrete Set-up. On a discrete time grid {ti}, the change in the
funding account (or the value of each funding position) is given by
(r)
(r)
F ∆t F
F
=
p
F
1
−
1
+
r
−
A
AF
ti−1 pt,ti 1{τf >ti }
t
i
ti−1
ti−1
ti
i−1 t,ti−1 {τf >ti−1 }
(r)
−pt,τ FτRf 1{t <τ ≤t }.
f
i−1
i
f
Summing up all the individual funding positions, we get
AF
tN


(r)
n
n
(r)
(r)
 pt,ti−1

F
= −
rti−1 Fti−1 ∆tipt,ti 1{τ >t } +
Fti−1 pt,ti 1{τ >t } 
− 1
(r)
i
i
f
f
pt,ti
i=1
i=1


(r)
n
pt,ti−1  (r)
 R
−
 pt,τ 1{t <τ ≤t }.
Fτf − Fti−1
f
(r)
i−1
i
f
pt,τ
i=1
f
25
Similarly, we have for the collateral account
AC
tN


(r)
n
n
(r)
(r)
 pt,ti−1

c
= −
rti−1 Mti−1 ∆tipt,ti 1{τ >t } +
Mti−1 pt,ti 1{τ >t } 
− 1
(r)
i
i
f
f
pt,ti
i=1
i=1


(r)
n
pt,ti−1  (r)

−
 pt,τ 1{t <τ ≤t }.
Mτf − Mti−1
f
(r)
i−1
i
f
pt,τ
i=1
f
The Invariance Principle for Funding and CVA
The invariance principle holds as well for the master equation with
CSA funding, Treasury funding and CVA.
Proposition 4 (Funding Invariance Principle). Let rt∗ t≥0 be any
(positive) interest rate process, then the master funding equation
with CVA can be re-written equivalently using the discounting with
the rt∗ -process, i.e., the value of the risky trade with CSA and
Treasury funding is:
V t = Et
+Et
T
t
(r ∗ )
1{τ >s}pt,s dCs
f
T
t
(r ∗ )
1{τ >s}pt,s · rs∗ − rsc Msds +
f
(r ∗ )
(r ∗ )
+Et 1{τ ≤T }pt,τ Mτf + 1{τ ≤T }pt,τ
f
f
f
f
T
t
(r ∗ )
1{τ >s}pt,s · rs∗ − rsF
f
FτRf − Fτ −
f
Fsds
(r ∗ ) R
+ 1{τ ≤T }pt,τ Vτf ,
f
f
where the funding account is given by Ft = Vt − Mt, for all t < τf .
26
Proof. Define the net payoff at default (of the three contracts)
,
Mτf + FτRf − Fτ − + VτR
f
Θτ f
f
and the net cash-flow stream (of the three contracts)
(r)
dCs + (rs − rsc) Msds + rs − rsF Fsds.
d Cs
Then, we have
V t = Et
T
t
T
(r)
(r)
1{τ >s}pt,s
d Cs
+ Et
f
t
(τ )
(r)
pt,s ΘsdDs f
(r)
Define the process At :
t
t (r)
(τ )
p0,s ΘsdDs f ds
0
0
(r)
Using the Master funding equation, we can show that At is a
(r)
At
(r)
p0,t Vt +
(r)
(r)
1{τ >s}p0,s
d Cs +
f
martingale:
(r)
A t = Et
T
0
(r)
(r)
1{τ >s}p0,s
d Cs +
f
T
0
(τ )
(r)
p0,s ΘsdDs f .
(r ∗ )
Similarly, we have for the process At :
(τ )
(r ∗)
(r ∗ )
(r ∗)
(r ∗ )
(r ∗ )
(r ∗ )
∗
+ p0,t ΘtdDt f ,
= −rt p0,t Vt + p0,t dVt + 1{τ >t}p0,t dCt
dAt
f
(r ∗ )
(r)
where we substitute dCt
with dCt , and we obtain, for all t < τf ,





(r)
d
A
(r ∗ )
(r ∗ ) 
t 
dAt
= p0,t  rt − rt∗ 1{τ >t} Vt − Mt − Ft dt +

(r)
f
p0,t
=0
(r ∗)
p0,t
(r)
=
dAt .
(r)
p0,t
(r ∗ )
Hence, the process At
is also a martingale, which gives the result.
The Bank’s Balance Sheet
We need to separate the bank’s balance sheet into the Desk balance
sheet and Treasury’s balance sheet, and allocate each term in the
(bank) total economic value to the appropriate cost centre.
Bank Balance sheet = Desk Balance sheet + Treasury Balance sheet.
The full economic value of the derivative and the funding position
can be written as:
T reasury
Vt = VtDesk + Vt
.
27
The Desk Balance Sheet
On the desk balance sheet, we only consider the funding ticket from
Treasury. The external DVA of the funding leg would show up on
Treasury balance sheet in the FVO accounting of the issued debt.
VtDesk
= Et
T
t

1{τ
 T

+Et 
 t

 T

+Et 
 t
(r)
(r)
(r)
R
p
dC
+
E
1
p
M
+
1
p
V
s
τ
t
f
{τf ≤T } t,τf
{τf ≤T } t,τf τf
f >s} t,s


(r)
c
1{τ >s}pt,s · (rs − rs) Msds

f

CSA Funding
F
1{τ >s}p(r)
·
r
−
r
s
s
t,s
f
Treasury Funding



Fsds

Ft = VtDesk − Mt.
Treasury funds the base PV (or the unsecured portion of the base
PV), the CVA P&L and the DVA P&L.
28
The Treasury Balance Sheet
On the Treasury balance sheet, we will have: the back-to-back
funding ticket with the desk, the external (market) funding, and the
FVO Debt CVA from the issued debt.
Treasury’s funding position with FVO DVA is
T reasury
Vt

 T

= −Et 
 t
+Et
(r)
· rs − rsF
1{τ >s}pt,s
f
Internal Funding Ticket
T
t
1{τ
(r)



Fsds

(r)
F F ds + 1
R−F
·
r
−
r
F
p
p
s
s
s
τf
{τf ≤T } t,τf
τ−
f >s} t,s
f
External Debt Issuance
29
.
CVA with Funding and Margining
• To solve the CVA equation with Funding, we proceed in two steps:
– First, we solve the pricing equation with CSA and Treasury
funding without default-risk, i.e., we find the funded PV of the
contract,
– Second, we plug the funded PV into the CVA equation to derive
the additional CVA terms.
• We can separate the CVA and funding problem because:
– The value of the collateral account Mt used for margining and
the value of the swap recovery term VτR
are a function of the
f
(funded) default-free PV.
– The Funding rates rtF that we pay and receive from treasury
are symmetric.
30
Note. Brigo et al. (2012) and Burgard and Kjaer (2012) consider
asymmetric funding rates, which introduce an implicit dependence
of the (base) PV funding charge on Vt, and the CVA funding charge
on Vt.
This level of generality is not necessary given the way banks raise
long-term funding.
Banks need to pre-fund their balance sheets. They lock-in their
funding rates and warehouse the cash needed to finance the balance
sheet. For a given balance-sheet size and maturity profile, cash is
raised in the debt markets at a fixed cost determined at the time of
the debt issuance.
The trading desks will be allocated a portion of that balance sheet
to use for their trading needs. By netting assets and liabilities, for
each trading desk, we obtain the net balance sheet usage that then
gets charged at the locked-in issuance level.
The only case where asymmetric funding rates may make sense is
for short-term (rolling) funding rates.
Solving the CVA Funding Equation
Using our funding invariance principle for both base PV and CVA,
we can simplify the equations by discounting with rtF , and dropping
the funding adjustment terms:
1. Funded (base) PV:
V t = Et
T
t
rF
pt,s dCs +
T
t
rF
pt,s · rsF − rsc Msds ,
2. Funded (margined) CVA:
V t = Et
T
t
rF
1{τ >s}pt,s dCs +
f
T
t
rF
1{τ >s}pt,s · rsF − rsc Msds
f
rF
rF
.
+Et 1{τ ≤T }pt,τ Mτf + 1{τ ≤T }pt,τ VτR
f
f
f
f
f
Note. This works in the most general case. But to proceed further,
we use the fact that the funding rates (although they can be
stochastic) do not depend on the PV of trade.
31
Funded Margined CVA
Proposition 5 (Funded (margined) CVA) The value of the trade
with default risk, margining, CSA funding and (unsecured) Treasury funding, is given by
VtDesk = Vt − CV At − DV At,
+
rF
,
CV At = Et 1{τ ≤T }1{τ =τc}pt,τ (1 − Rc) Vτf − Mτf
f
f
f
−
rF
,
DV At = Et 1{τ ≤T }1{τ =τ }pt,τ (1 − Rb) Vτf − Mτf
f
f
f
b
where the Default-Free Funded PV of the trade Vt is the solution
of the funding equation
V t = Et
T
t
rF
pt,s dCs +
T
t
rF
pt,s · rsF − rsc Msds .
32
r F −r c
rF
the default-free PV with
and Vt
Proof. We denote by Vt
no CSA funding and the PV of the CSA funding cash-flow stream
respectively
Vt
Vt
rF
r F −r c
Et
Et
T
rF
pt,s dCs ,
T
rF
pt,s · rsF − rsc Msds .
t
t
The value of the funded default-free PV will be the sum of the two
terms:
Vt = Vt
rF
+ Vt
r F −r c
.
By introducing the survival indicator, we can re-write each one of the
extinguishing terms as a function of the equivalent risk-free term:
T
Et
T
Et
t
t
rF
1{τ >s}pt,s dCs
f
rF
1{τ >s}pt,s · rsF − rsc Msds
f
= Vt
= Vt
rF
rF
rF
− Et pt,τ Vτf
f
r F −r c
1{τ
f ≤T }
r F −r c
rF
− Et pt,τ Vτf
f
1{τ
,
f ≤T }
.
Summing up the two extinguishing terms, we get
T
Et
t
rF
1{τ >s}pt,s dCs + Et
f
T
t
rF
1{τ >s}pt,s · rsF − rsc Msds
f
rF
= Vt − Et pt,τ Vτf 1{τ ≤T }
f
f
Plugging into the expression of the risky PV with CSA funding gives
the CVA and DVA formulas.
Note. To compute the default-free PV, for a fully collateralized
trade, we need to discount the cash-flows with the CSA rate, to
compute the CVA value we need to discount the CVA payoff with
the unsecured funding rate. The trade P&L is funded via the CSA,
but the CVA P&L is not and should therefore be discounted at the
unsecured rate.
Close-Out with Funding
Using the Close-out amount based on the 2009 ISDA
documentation, the CVA and DVA formulas should be based on Vτbf
and Vτcf , respectively, i.e., the funded PV using either the bank’s
F,b
F,c
cost of funding rs , or the counterparty’s cost of funding rs .
With asymmetric Close-out, we get
+
r F,b
,
CV At = Et 1{τ ≤T }1{τ =τc}pt,τ (1 − Rc) Vτbf − Mτf
f
f
f
−
r F,b
DV At = Et 1{τ ≤T }1{τ =τ }pt,τ (1 − Rb) Vτcf − Mτf
f
f
f
b
r F,b
+Et 1{τ ≤T }1{τ =τ }pt,τ
f
f
f
b
Vτbf − Vτcf
,
where the funded PVs are given by the solution of the funding
33
equations for each counterparty with its own funding costs
Vtb = Et
Vtc = Et
T
t
T
t
r F,b
pt,s dCs +
r F,c
pt,s dCs +
T
r F,b
· rsF,b − rsc Msds ,
pt,s
T
r F,c
· rsF,c − rsc Msds .
pt,s
t
t
Note. This will be more relevant for dealers and financial institutions
than for corporate end-users.
FVO Debt CVA
Typically, FVO Debt CVA is computed for the Bank’s Structured
Debt and its Primary Debt Issuance, but Funding is charged on an
Accrual Accounting basis.
Gross DVA is defined as the PV of the bond cashflows with today’s
credit spreads (including liquidity basis, i.e., Bond-CDS basis). The
Net DVA is the difference between the Gross DVA at today’s spreads
and the Gross DVA at issuance spreads. FVO is based on the Net
DVA.
For bonds (debt issuance), we use a Recovery of Par assumption.
This is also consistent with the recovery assumption used for CDSs.
Other recovery assumptions are also possible: e.g., Recovery of
Treasury or Recovery of Market Value.
34
Debt Issuance Spreads
We use the following notations for short rates and forward rates:
(r)
pt,s = exp −
(b)
Qt,s = exp −
s
t
t
s
s
ft,udu = Et exp −
λbt,udu
t
= Et exp −
s
t
rudu
,
λbudu
.
We issue debt with maturity T ≥ 0, at par, with a coupon
rtF0,t
. We have at the time of debt issuance t0
t0 ≤t≤T
T
T (r) (b)
(r) (b)
(r)
(b)
F
F
1=
pt0,sQt0,s · rt0,sds + pt ,T Qt ,T + Rb
pt0,sQt0,s · λbt0,sds.
0
0
t0
t0
Integrating by parts, we get:
T
1=
t0
(r)
(b)
(r)
(b)
pt0,sQt0,s · ft0,s + λbt0,s ds + pt ,T Qt ,T ,
0
0
which gives
0=
T
t0
(r)
(b)
pt0,sQt0,s · rtF0,s − ft0,s − 1 − RbF λbt0,s ds.
35
Hence, the par (break-even) issuance rate is given by
rtF0,s = ft0,s + 1 − RbF λbt0,s.
Gross DVA
The Gross DVA is the bond value with today’s spreads:
T
T (r) (b)
(r) (b)
(r) (b)
F
F
pt,s Qt,s · rt0,sds + pt,T Qt,T + Rb
Bt =
pt,s Qt,s · λbt,sds
t
t
T (r) (b)
pt,s Qt,s · rtF0,s − ft,s − 1 − RbF λbt,s ds
= 1+
t
T (r) (b)
pt,s Qt,s · ft0,s − ft,s + 1 − RbF λbt0,s − λbt,s ds.
= 1+
t
It can also be re-written as a function of the new (break-even)
funding rate:
Bt = 1 +
T
t
(r)
(b)
F ds,
pt,s Qt,s · rtF0,s − rt,s
F = f − 1 − R F λb .
rt,s
t,s
t,s
b
36
Net DVA
Similarly, we have today’s bond value with the issuance spreads
T
T (r) (b)
(r) (b)
(r) (b)
F
F
pt,s Qt0,s · rt0,sds + pt,T Qt ,T + Rb
pt,s Qt0,s · λbt0,sds
0
t
t
T (r) (b)
pt,s Qt0,s · rtF0,s − ft,s − 1 − RbF λbt0,s ds
= 1+
t
T (r) (b)
pt,s Qt0,s · ft0,s − ft,s ds.
= 1+
t
Bt∗ =
Thus, the FVO Debt CVA is given by
F V Ot = Bt − Bt∗
=
T
t
T (r) (b)
(r) (b)
F
F
pt,s Qt,s · rt0,s − rt,s ds −
pt,s Qt0,s · ft0,s − ft,s ds.
t
37
FVA vs FVO
The funding leg (for a unit notional) of my balance sheet is
F V At =
=
T
t
T
t
= −
(b) (r)
Qt,s pt,s · ft,s − rtF0,s ds
(b) (r)
F ds +
Qt,s pt,s · ft,s − rt,s
T
t
T
t
(b) (r)
Qt,s pt,s · sF
t,s ds − F V Ot +
(b) (r)
F − rF
Qt,s pt,s · rt,s
t0,s ds
T
t
(r)
(b)
pt,s Qt0,s · ft0,s − ft,s ds.
FVO Debt CVA does not exclude the credit risk from the valuation,
it merely marks the bond to market. If we issue the bond at par,
traditional accounting pre-FAS 157, will keep the bond on the
balance sheet marked at Par. With the FVO adjustement, it marks
the bond back to market. The change in the bond price is not solely
due to credit spreads, it can also move because of interest rates and
liquidity spread changes.
38
FVO Bond Maths
Using traditional bond maths and a continuous (implied) yield to
convert the bond cash flows to price, we can write
1 =
T
(y)
(y)
pt0,s rtF0,s ds + pt ,T , where
0
t0
(y)
pt0,s = exp
−
s
t0
yt0,udu .
We also have by integration
1=
T
(y)
(y)
pt0,syt0,sds + pt ,T ,
0
t0
hence, the yield will be equal to the break-even (par) issuance rate
yt0,s = rtF0,s.
If the new bond price is Bt, then the new implied yield will be
Bt =
T
t
(y)
pt,s
(y)
rtF0,s ds + pt,T ,
39
and the FVO adjustment is
F V Ot =
T
t
(y)
(y)
pt,s rtF0,s ds + pt,T −
T
t
(y)
(y)
pt0,s rtF0,s ds + pt ,T
0
= Bt − Bt∗ = Bt − 1.
If we are marking an FVO adjustment, we need to change the
funding rate from the lock-in funding rate to the current funding
rate, and charge the desk accordingly.
In essence, this is equivalent to buying back the issued date and
locking-in the gain or the loss, then re-issuing the same outstanding
debt notional at the new prevailing issue levels.
The new levels can move because of: a) base interest rates, b)
credit spreads, or c) liquidity basis.
The current methodology for computing FVO includes all three
changes in the bond credit spreads. But Treasury also hedges the
interest rate risk on the issued debt, which would offset the change
in the bond price due to interest rates.
The choice of interest rate curve to use is arbitrary, it could be
driven by the intrest rate hedges. If we hedge with swaps, then it’s
Libor. If we hedge with Treasuries, then it’s treasury rates.
Another alternative is to use Asset Swaps instead. The asset swap
would be collateralised, hence discounted at the CSA rate, but the
cashflows would be Libor + spread, in exchange for the bond
cashflows.
CVA with Gap Risk and Funding
In this section, we focus on pricing the unilateral margined CVA with
Gap risk and Funding. We consider a netting set with one CDS
position to highlight some of key features relevant for credit. A large
netting set of CDSs can be compressed into a CDO with a large
number of names, and a similar approach can then be applied to
price the CVA on the CDO.
The margined CVA is given by
rF
CV A0 = (1 − Rc) E p0,τc (Vτc − Mτc )+ 1{τc≤T } ,
where Vt is the default-free value of the trade with funding.
If we have a fully collateralized trade, the cash-flows will be
discounted at the CSA funding rate
V t = Et
T
t
(r c )
pt,s dCs ,
40
and the margin will be equal to the value of the derivative prior to
default
Mτc = Vτc−∆,
where ∆ is the time lag between time t and the last margin date
before time t.
The CVA will be given by the value of the jump in MTM at the time
of default
rF
CV A0 = (1 − Rc) E p0,τc
= (1 − R c )
T
0
Vτc − Vτc−∆
rF
p0,τc E
+
Vτc − Vτc−∆
1{τc≤T }
+
|τc = t P (τc ∈ dt) .
This is a series of cliquet (or ratchet) option conditional on default.
Their value is mainly driven by Gap risk. If we have a cure period,
then there will be an additional impact of the forward volatility on
the value of this option, but the gap risk will still be the dominant
factor. To evaluate this option, we need to compute the conditional
forwards first, then overlay the vol-induced diffusion on top.
Forwards. Starting with the forwards, we have
Vτc − Vτc−∆
+
= 1{τi>τc} Vτc − Vτc−∆
+
= 1{τi>τc} Vτc − Vτc−∆ .
The pricing boils down to computing the jumps in the expected loss
DTi
Vt
= E DTi |Gt , for each time horizon
DTi
∆Vτc
DTi
DTi
= Vτc − Vτc−∆
= Eτc DTi − Eτc−∆ DTi , for all T ≥ τc,
CVA in Own Natural Filtration
Single-Name Case. We start with the single-name case and show
how the jump upon default is computed. We use the Dellacherie
formula to derive the conditional expectations before default and
after default.
Own Filtration. The filtration here contains the default of the
single name τi and the counterparty τc: Gti = Ft ∨ Hti ∨ Htc.
Working in own filtration, we have
Et 1 − DTi
= P τi > T Gti
P (τi > T, τc > t |Ft )
= 1{τi>t}1{τc>t}
P (τi > t, τc > t |Ft )
P (τi > T |Ft ∨ σ (τc))
+1{τi>t}1{τc≤t}
.
P (τi > t |Ft ∨ σ (τc))
41
CVA in the Enlarged Filtration
Enlarged Filtration. We use a top-down approach and assume that
the {Gt}-filtration can be approximated as





n+m

j

H
Gt = Ft ∨ Htc ∨ 
Hti  = Ft ∨ Htc ∨ Hti ∨ 

t

 j=1
i=1
j=i
n+m
(−i)
= Gti ∨ σ Lt
= Gti ∨ σ L∗t ,
i.e., we only need to condition on realizations of the (macro) loss
variable L∗t (of all the other names in our credit universe) and we
do not need the (micro) information on the individual single-name
defaults.
42
In this case, we will have
Et 1 − DTi
= P (τi > T |Gt )
P τi > T, τc > t |Ft ∨ σ L∗t
= 1{τi>t}1{τc>t}
P τi > t, τc > t Ft ∨ σ L∗t
P τi > T |Ft ∨ σ L∗t ∨ σ (τc)
+1{τi>t}1{τc≤t}
.
∗
P τ i > t F t ∨ σ L t ∨ σ (τ c )
Note. In the general case, to compute this for the whole portfolio,
we work on the enlarged filtration and we use the top-down
decomposition above.
The choice of the (macro) filtration conditioning should be general
and should not depend on the names within the netting set. The
pricing for each trade (or each name) in the netting set would
depend on its own natural filtration and the (macro) conditioning
filtration.
The (macro) conditioning filtration defines the gap risk. If we are in
a distressed state of the economy, then gap risk will be small as
most names would have already widened before default. If we are in
a benign state of the economy, then gap risk will be large.
Pre-Default and Post-Default Value
We have the following result.
Lemma 6 (Pre and Post Default Value Process) The value process is of the form
Et 1 − DTi = 1{τc>t}F 1 t, T, L∗t + 1{τc≤t}F 2 t, T, L∗t , τc ,
where F 1 t, T, L∗t and F 2 t, T, L∗t , τc are the value functions predefault and post-default respectively
F 1 t, T, L∗t
F 2 t, T, L∗t , τc
P
= 1{τi>t}
P
P
= 1{τi>t}
P
τi > T, τc > t |Ft ∨ σ L∗t
,
∗
τi > t, τc > t Ft ∨ σ Lt
τi > T |Ft ∨ σ L∗t ∨ σ (τc)
.
τi > t Ft ∨ σ L∗t ∨ σ (τc)
Note. In Elouerkhaoui (2012), we have used the filtration of the
netting set, and a top-down approximation of the filtration with the
loss variable of the netting set.
43
The new approach is a hybrid that accounts for the micro default
information of each name (or each trade) in the netting set, by
conditioning on its own natural filtration, then it is augmented with
a top-down approximated loss variable that would drive the gap risk.
Hence, the PVs at the time of default and prior to default will be
given by
Eτc 1 − DTi 1{τc=t} = F 2 t, T, L∗t , τc 1{τc=t},
Eτc−∆ 1 − DTi 1{τc=t} = F 1 t − ∆, T, L∗t−∆ 1{τc=t}.
Expectation of the Jump at Default
Proposition 7 The expectation of the jump at default is
1−DTi
E ∆Vτc
|τc = t
= E 1{τi>τc}
1−DTi
1−DTi
V τc
− Vτc−∆
|τc = t ,
where the post-default and pre-default expectations are given by
1. Post-default
1−DTi
E 1{τi>τc}Vτc
|τc = t
= P (τi > T |τc = t ) ,
2. Pre-default
1−DTi
E 1{τi>τc}Vτc−∆ |τc = t
=
∗
L
P
τ
>
T
i
t−∆ = x, τc > t − ∆
∗
.
P Lt−∆ = x, τi > t |τc = t
∗
P τi > t − ∆ Lt−∆ = x, τc > t − ∆
x
44
We need to compute the following objects:
1. The CDS survival probability conditional on default (with WWR)
P (τi > T |τc = t ) , for all T ≥ t,
2. The CDS survival probability conditional on realization of L∗t−∆
and survival
P τi > T L∗t−∆ = x, τc > t − ∆ , for all T ≥ t − ∆,
3. The joint probability of CDS survival and L∗t−∆ conditional on
default
P L∗t−∆ = x, τi > t |τc = t .
To compute the marginal distributions of the loss variable
conditional on default or conditional on survival, we use the
CDO-Squared correlation model (see Elouerkhaoui (2012)).
Credit Options Revisited
• The choice of filtration that we work with has a fundamental
impact on the value of CDS options as well.
• Working in the CDS own natural filtration, we get the standard
Black formula used in the market.
• By enlarging the filtration, we have more information from the
other names in the credit universe, which in turn skews the forward
distribution of the CDS spread.
• This leads to fatter tailed distributions and an implied volatility
skew effect.
• We see a similar effect with the margined CVA payoff, where the
forward vol (distribution) is driven by the conditioning macro loss
variable.
45
CDS Options in Own Natural Filtration
Pricing Single Name CDS Options. The Filtration in this case is
Gt = Ft ∨ Hti .
We define the survival probability and the conditional (forward)
survival probability of the reference entity:
Qi0,T
Qit,U,T
P (τ i > T ) ,
P (τ > T |Ft )
,
P (τ > U |Ft )
which is used to define a strictly positive a.s. (and {Ft}-adapted)
forward annuity and its associated forward annuity measure.
The Break-even spread is a martingale under this measure, and we
can use the standard Black formula to compute this expectation
(e.g., see Brigo and Morini (2005) for technical details).
46
Proposition 8 The price of a CDS option in its own filtration
Gt = Ft ∨ Hti is given by



O0,t,T = 
Tj >t

p0,Tj Qi0,Tj δj  Black S0,t,T , S 0, σBS
√
T −t ,
where the break-even spread and the annuity are defined as
− (1 − Ri) tT p0,sdQi0,s
S0,t,T =
,
i
Tj >t p0,Tj Q0,T δj
j
A0,t,T =
Tj >t
p0,Tj Qi0,Tj δj .
CDS Options in the Enlarged Filtration
Enlarged Filtration. When we work on the enlarged filtration we
have: Gt = Ft ∨ Hti ∨ σ L∗t .
We use a top-down approach to approximate the default filtration
with the macro loss variable. Then, we get a mixture of Black
formulas by conditioning on realizations of the macro loss variable.
We define the Spot and Forward survival probabilities of the
reference entity conditional on the (macro) loss variable:
i,x
Qt,T
i,x
Qt,U,T
P τi > T |L∗t = x ,
P τi > T Ft ∨ L∗U = x
P τi > U Ft ∨ L∗U = x
.
Conditioning on {L∗t = x}, the forward survival probabilities define a
(strictly positive) conditional forward annuity, which is used to derive
a Black-Scholes price under the associated forward annuity measure.
47
Proposition 9 The price of a CDS option in the enlarged filtration
Gt = Ft ∨ Hti ∨ σ L∗t is given by a mixture of Black-Scholes prices:
O0,t,T =
x
x
x
, S 0, σBS
P L∗t ∈ dx Ax0,t,T Black S0,t,T
√
T −t ,
where the break-even spread and the annuity are defined as
x
=
S0,t,T
Ax0,t,T
i,x
− (1 − Ri) tT p0,sdQt,s
,
i,x
Tj >t p0,Tj Qt,Tj δj
i,x
p0,Tj δj E
p0,Tj δj Qt,T =
=
j
Tj >t
Tj >t
1 − DTi j |L∗t = x .
CDS Options Filtration Conditioning
We analyze the impact of the Macro filtration conditioning on the
implied Black volatilities. We set the Macro market spread to
SM = 500 bps, and the Market correlation to ρM = 0.2.
Source: Citi. Distribution of the Market conditioning variable L∗T .
48
CDS Options Filtration Induced Volatility Skew
The Option maturity is 1 year. The CDS maturity is 5 years. The
CDS spread is 100 bps. The Base volatility is 0.2.
Source: Citi. Filtration conditioning implied volatility skew.
49
Applications
We analyze the impact of post-default and pre-default correlations
on Credit CVA. Post-default correlation is driven by ρc. Pre-default
correlation is driven by ρc, ρM and SM .
The primary driver of pre and post spread widening is correlation.
The conditioning Market spreads have a milder impact.
Valuation date is 18-Mar-13. The CDS trade maturity is 20-Mar-16.
Counterparty correlation is set to ρc = 0.5. Market correlation is
ρM = 0.5. Market spread is SM = 500 bps. The counterparty spread
is 200 bps. The reference CDS spread is 200 bps. SNAC Coupon is
100 bps.
We have the following pre and post value profiles.
50
Gap Risk vs Wrong-Way Risk
Market correlation is set to ρM = 0.5. Market spread is set to
SM = 500 bps. We vary the counterparty correlation ρc.
Source: Citi. Pre and Post-Default PV Profile for a counterparty
correlation of ρc = 0.5.
51
Wrong-Way Risk Correlation
As we change the counterparty correlation ρc both the pre and post
PVs move.
Source: Citi. Pre and Post-Default PV Profile for a counterparty
correlation of ρc = 0.9.
52
Gap Risk Correlation
Counterparty correlation is set to ρc = 0.5. Market spread is set to
SM = 500 bps. We vary the Market correlation ρM .
Source: Citi. Pre and Post-Default PV Profile as we change the
Market correlation of ρM .
53
Gap Risk Market Spread
Counterparty correlation is set to ρc = 0.5. Market correlation is set
to ρM = 0.5.We vary the Market spread SM .
Source: Citi. Pre and Post-Default PV Profile as we change the
Market spread SM .
54
CVA Pricing
We have the following results.
1. Market correlation is set to ρM = 0.5. Market spread is set to
SM = 500 bps. We vary the counterparty correlation ρc.
ρc
PV
NonMarginedPV
MarginedPV
0
26,625
800
0
0.1
26,625
2,968
1,090
0.5
26,625
7,158
3,737
0.9
26,625
11,395
7,319
2. Counterparty correlation is set to ρc = 0.5. Market spread is set
to SM = 500 bps. We vary the Market correlation ρM .
ρM
PV
NonMarginedPV
MarginedPV
0
26,625
7,158
6,646
0.1
26,625
7,158
5,788
0.5
26,625
7,158
3,737
0.9
26,625
7,158
1,004
55
3. Counterparty correlation is set to ρc = 0.5. Market correlation is
set to ρM = 0.5. We vary the Market spread SM .
SM
PV
NonMarginedPV
MarginedPV
5
26,625
7,158
1,823
100
26,625
7,158
2,905
500
26,625
7,158
3,737
1000
26,625
7,158
4,124
Conclusion
• We have addressed the problem of computing CVA, FVA and FVO
from a credit perspective.
• We have developed a new technique to solve the Master Funding
Equation based on a Fundamental (Funding) Invariance Principle.
• We have analyzed the relationship between FVA and FVO, and
we have shown that the FVA doesn’t disappear when you account
for FVO as it is usually claimed in the literature.
• We have derived the CVA for a CDS by computing the Gap risk
in the Enlarged Filtration. And we have shown that this is similar
to a Filtration-Induced Volatility Skew for CDS Options.
• We have analyzed the impact of pre and post default correlations
and market conditioning on the Gap risk and Wrong-Way risk.
56
References
• D. Brigo, M. Morini (2005), “CDS Market Formulas and Models”, Working
Paper.
• D. Brigo, A. Pallavicini, D. Perini (2012), “Funding, Collateral and Hedging:
Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustemnts”, Working Paper.
• C. Burgard, M. Kjaer (2012), “A Generalised CVA with Funding and Collateral
via Semi-Replication”, Working Paper.
• S. Crepey (2012a), “Bilateral Counterparty Risk Under Funding Constraints,
Part I: Pricing”, Working Paper.
• S. Crepey (2012b), “Bilateral Counterparty Risk Under Funding Constraints,
Part II: CVA”, Working Paper.
• S. Crepey (2012c), “Counterparty Risk and Funding: The Four Wings of the
TVA”, Working Paper.
• C. Dellacherie (1972), “Capacites et Processus Stochastiques”, Springer-Verlag,
Berlin, 1972.
• Deloitte, Solum Partners (2013), “Counterparty Risk and CVA Survey: Current
Market Practice Around Counterparty Risk Regulation, CVA Management and
Funding”, February 2013.
57
• Y. Elouerkhaoui (2012), “From Funding to Gap Risk: A Consistent Methodology for Credit CVA”, ICBI Conference.
• Ernst & Young (2012), “Reflecting Credit and Funding Adjustments in Fair
Value”, Spring 2012.
• M. Fujii, A. Takahashi (2010), “Derivatives Pricing under Asymmetric and
Imperfect Collateralization and CVA”, CARF Working Paper Series F-240.
• J. Hull, A. White (2012a), “The FVA Debate”, Risk, 25th Anniversary edition,
July 2012.
• J. Hull, A. White (2012b), “The FVA Debate Continues”, Risk, October 2012.
• S. Laughton, A. Vaisbrot (2012), “In Defence of FVA: a Response to Hull and
White”, Risk, September 2012.
• V. Piterbarg (2010), “Funding Beyond Discounting: Collateral Agreements and
Derivatives Pricing”, Risk, February 2010.
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