ESG
Calibration
Report
.
Example ESG Calibration Report
Model IT
Economic
Conditions
Calibration
.
Market-Consistent Economic Scenarios
Q1/2014
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
Model IT Ltd
Bibliography
10.4.2014
www.modelit.fi
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Notice
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
This document is proprietary and confidential.
For Model IT and Model IT client use only.
c 2014 Model IT Ltd
⃝
Disclaimer
Contact
Bibliography
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Calibration Report
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
This report provides documentation regarding the risk-neutral
(market-consistent, MC) economic scenarios, including
Input data
Models and assumptions
Calibration
Simulation and outputs
Disclaimer
Contact
Quality tests
Bibliography
.
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Market-Consistent Framework
ESG
Calibration
Report
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
...
The scenarios produced are market-consistent, or risk-neutral
Rationale: “Market prices of financial instruments reflect all available
information regarding the expected future development of the risk
factors to which the instruments are exposed to.”
Try to back out these “expectations” from the observed market
prices (by postulating a model to calibrate).
In practice: market-consistent ⇔ traded instruments are priced correctly.
Market-consistent scenarios
should be used for valuation (of traded assets)
Quality Tests
are used for valuation of (non-traded) options & guarantees
embedded in technical reserves
Disclaimer
Contact
Bibliography
should not be used for any other purpose (such as risk management
or planning, for which real-world scenarios are needed)
Models are calibrated (i.e., model parameters are estimated) by minimizing
. . . . . . . . . . . . . . .
.
. .
the (mean-squared) error between market prices
and model-implied prices.
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Economic Scenario Generation
Modeling Stages
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Goodness-of-Fit testing
Calibration
Interest Rates
Equity
Property
Inflation
Simulations from the
Market. prices
Calibration
Model of the
Market
Simulation
Model of the
Market
Simulation
and Outputs
Quality Tests
Calibration error
Disclaimer
Simulation error
Contact
Bibliography
.
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
.
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www.modelit.fi
For marketing purposes only
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Economic Conditions
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Swap Rates
ESG
Calibration
Report
...
Model IT
LIBOR rates as of 31.3.2014
3.5
Economic
Conditions
3
Calibration
2.5
Interest Rate (%)
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
2
Forward
Spot
1.5
1
Quality Tests
0.5
Disclaimer
0
Contact
0
10
20
30
40
Time (years from Valuation Date)
50
60
Bibliography
Source: Based on market data from SuperDerivatives
.
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Economic Conditions
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Smith-Wilson Extrapolation
Used initial curves
ESG
Calibration
Report
...
Smith−Wilson Extrapolated Yield Curves as of 31.3.2014
4.5
Model IT
4
Economic
Conditions
3.5
Interest Rate (%)
3
Calibration
Interest Rates
Equity
Property
Inflation
2.5
S−W Forward
2
S−W Spot
Market Forward
1.5
Simulation
and Outputs
Market Spot
1
0.5
Quality Tests
0
Disclaimer
0
10
20
30
40
Time (years from Valuation Date)
50
60
Contact
Bibliography
.
Assumptions
.
“Risk-free” base curve: Swap Curve − 10 bps
Smith-Wilson (S-W) parameters: UFR = 4.2 %, LLP = 20 y, α = 0.1
.
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www.modelit.fi
For marketing
purposesEIOPA
only (Smith-Wilson approach)
Economic Conditions
Source: Based on market data from
SuperDerivatives,
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Swaption Volatility Surface
ESG
Calibration
Report
...
ATM Swaption Volatility Surface as of 31.3.2014
Model IT
Economic
Conditions
Calibration
100
Interest Rates
Equity
Property
Inflation
Volatility (%)
80
Simulation
and Outputs
Quality Tests
60
40
20
0
Disclaimer
10
0
0
Contact
20
10
30
20
Bibliography
40
30
40
50
50
Option Maturity (y)
Swap Tenor (y)
.
.. ..
Source: SuperDerivatives
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Economic Conditions
.
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Swaption Volatilities
A Sample of Swap Tenors
ESG
Calibration
Report
...
Model IT
Swaption Volatilities for certain Swap Tenors as of 31.3.2014
90
Economic
Conditions
80
Calibration
70
Volatility (%)
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
1Y
5Y
10Y
20Y
30Y
60
50
40
30
Quality Tests
Disclaimer
20
Contact
10
0
Bibliography
5
10
15
20
25
30
Option Expiry (years)
35
40
45
50
Source: SuperDerivatives
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Economic Conditions
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Equity Index Volatilities
ESG
Calibration
Report
...
Model IT
ATM Implied Volatilities for Equity Indices as of 31.3.2014
50
Economic
Conditions
SPX
FTSE
N225
OMXS30
OMXH25
MIASJ0000PUS
BVSPUSD
IRTS
NIFTY
HSCEI
45
40
Calibration
Simulation
and Outputs
35
Volatility (%)
Interest Rates
Equity
Property
Inflation
30
25
20
15
Quality Tests
10
Disclaimer
5
Contact
0
0
1
2
Bibliography
3
4
5
Time (years)
6
7
8
9
10
Source: SuperDerivatives
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Economic Conditions
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
.
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www.modelit.fi
For marketing purposes only
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Calibration
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Component Models
ESG
Calibration
Report
...
Model IT
The following models are used for generating the scenarios:
Economic
Conditions
Interest rates: Libor Market Model (LMM)
Calibration
Interest Rates
Equity
Property
Inflation
Calibrated to swaption volatilities (and initial forward curve)
Index assets: geometric Brownian motion (GBM)
Equities: Calibrated to ATM equity option prices
Property: Historical volatility assumption is used (no liquid
option markets for real-estate)
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
Inflation: Cox-Ingersoll-Ross square-root diffusion (CIR)
Calibrated to historical inflation (Finland)
Bibliography
.
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Calibration
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Dependency
ESG
Calibration
Report
Model IT
...
The dependencies between random processes (risk factors) are modeled
using Gaussian copula with correlation matrix ρ:
Economic
Conditions
Correlations between driving processes / risk factors
1
IR_level 1.00 0.00 0.00 0.20 0.20 0.20 0.20 0.20 0.10 0.10 0.10 0.10 0.10 0.00 0.00
IR_twist 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Calibration
0.9
IR_butterfly 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Interest Rates
Equity
Property
Inflation
0.8
SPX 0.20 0.00 0.00 1.00 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.20 0.00
FTSE 0.20 0.00 0.00 0.80 1.00 0.80 0.90 0.90 0.80 0.80 0.80 0.80 0.80 0.30 0.00
0.7
N225 0.20 0.00 0.00 0.80 0.80 1.00 0.80 0.90 0.80 0.80 0.80 0.80 0.80 0.20 0.00
0.6
Simulation
and Outputs
OMXS30 0.20 0.00 0.00 0.80 0.90 0.80 1.00 0.90 0.80 0.80 0.80 0.80 0.80 0.30 0.00
OMXH25 0.20 0.00 0.00 0.80 0.90 0.90 0.90 1.00 0.80 0.80 0.80 0.80 0.80 0.30 0.00
0.5
MIASJ0000PUS 0.10 0.00 0.00 0.80 0.80 0.80 0.80 0.80 1.00 0.80 0.80 0.80 0.80 0.10 0.00
Quality Tests
0.4
BVSPUSD 0.10 0.00 0.00 0.80 0.80 0.80 0.80 0.80 0.80 1.00 0.80 0.80 0.80 0.10 0.00
Disclaimer
IRTS 0.10 0.00 0.00 0.80 0.80 0.80 0.80 0.80 0.80 0.80 1.00 0.80 0.80 0.10 0.00
0.3
NIFTY 0.10 0.00 0.00 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 1.00 0.80 0.10 0.00
Contact
0.2
HSCEI 0.10 0.00 0.00 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 1.00 0.10 0.00
Bibliography
Property 0.00 0.00 0.00 0.20 0.30 0.20 0.30 0.30 0.10 0.10 0.10 0.10 0.10 1.00 0.00
0.1
Inflation 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00
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For marketing purposes only
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n
EI
In
fla
tio
rty
pe
SC
H
Pr
o
TS
TY
IF
IR
N
25
PU
S
BV
SP
U
SD
XH
00
M
IA
SJ
00
5
XS
30
O
M
O
M
SE
22
N
SP
X
FT
is
t
rfl
y
w
te
_t
ut
_b
IR
IR
IR
_l
ev
el
0
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Calibration
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
.
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Calibration
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Libor Market Model
ESG
Calibration
Report
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
...
Libor Market Model is used for modeling interest rates
Instead of hypothetical continuous interest rates, model discrete
forward rates actually observed in the market
⇒ Intuitive, possible to capture (more) accurately the prices of
market instruments
Libor Market Model takes the initial forward rates (fwd curve) as
input, and therefore calibrates to the initial term structure
automatically
Calibration consists of estimating the forward rate volatility and
correlation (function) parameters from prices of actively traded
instruments
Bibliography
In this case, ATM Swaptions are used
For more information and details, see e.g. [BME06].
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Calibration
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Libor Market Model
Model Specification
ESG
Calibration
Report
Model IT
...
.
Basic quantities
.
Tenor dates: 0 = T0 < T1 < · · · < TM < TM+1
Economic
Conditions
Interval lengths: δj = Tj+1 − Tj
Calibration
Forward rates: Lj (t) := L(t, Tj )
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
Bibliography
Lj (t) is the forward rate, at time t, for accrual period
[Tj , Tj+1 ]
.
The development of forward rates is describe by the stochastic differential
equation
dLj (t)
= µj (t) dt + σj (t) dWj (t),
(1)
Lj (t)
0 ≤ t ≤ Tj , j = 1, . . . , M, where W = (Wj ) is (M-dimensional) Brownian
motion with correlation matrix ρ(t) (i.e., E[ dWi (t) dWj (t)] = ρi,j (t) dt
with the common notation).
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Calibration
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LMM Calibration
ESG
Calibration
Report
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
...
.
Calibrating correlations between forward rates
.
1. Estimate historical correlation matrix
2. Fit a smooth functional form to historical correlations
3. Perform a principal components analysis on the fitted correlation
matrix
.
.
Calibrating volatilities of forward rates
.
1. Specify a functional form for the volatilities of forward rates
Disclaimer
. Here piece-wise linear, continuous
2. “Pivot” points as decision variables
1
Contact
Bibliography
2.
Minimize the weighted sum-of-squared differences (SSE) between
market and model swaption volatilities
.
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Calibration
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LMM Calibration
Calibrating Volatilities
ESG
Calibration
Report
Model IT
Economic
Conditions
...
The swap option volatility – which functions as the calibration target
– can be approximated as a function of the forward rate volatilities
(see [RJA03]):
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
2
να,β
∫
β
∑
wi (0)wj (0)Li (0)Lj (0) Tα
=
σi (t)σj (t)ρi,j (t) dt,
Sα,β (0)2 Tα
0
i,j=α+1
wi (0) = ∑β
δi P(0, Ti )
j=α+1 δj P(0, Tj )
,
Disclaimer
Contact
Bibliography
where Sα,β (t) is the forward swap-rate at time t for a swap with
payment times Tα+1 , . . . , Tβ , and P(t, Ti ) is the time-t price of a
zero-coupon bond maturing at Ti .
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Calibration
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LMM Calibration
Input Data
ESG
Calibration
Report
...
ATM Swaption Volatility Surface as of 31.3.2014
50
Model IT
45
Economic
Conditions
40
Calibration
35
Swap Tenor (y)
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
30
25
20
15
Disclaimer
10
Contact
5
Bibliography
0
0
5
10
15
20
25
30
Option Maturity (y)
.
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(See also p.9.)
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35
For marketing purposes only
40
45
50
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Calibration
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LMM Calibration
Correlations & Principal Components
ESG
Calibration
Report
...
Model IT
Correlation between 1−year fwd rates
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Principal component weights for 1−year fwd rates
1
0.8
1
0.4
Simulation
and Outputs
0.6
0.2
Quality Tests
0.4
40
0
40
Disclaimer
20
20
Contact
Bibliography
PC1
PC2
PC3
0.6
0.8
Time−to−maturity
for 1y fwd rate
0
0
−0.2
−0.4
0
.
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For marketing purposes only
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20
30
40
Time to maturity (y)
50
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Calibration
60
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LMM Calibration
Forward Rate Volatility – Functional Form
ESG
Calibration
Report
...
Model IT
Forward Rate Volatility (function)
0.4
Economic
Conditions
time
maturity
residual maturity
0.35
Calibration
Interest Rates
Equity
Property
Inflation
Volatility multiplier
0.3
Simulation
and Outputs
Quality Tests
0.25
0.2
0.15
Disclaimer
0.1
Contact
Bibliography
0.05
0
5
10
15
20
25
Time (years)
30
.
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35
40
45
50
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Calibration
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LMM Calibration
Swaption Volatility Surfaces
ESG
Calibration
Report
...
Model volatilities
Market volatilities
Relative difference
Model IT
1
1
0.1
0.5
0.5
0
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
50
0
0
0 Swap
Option 50
Maturity
Expiry
Model volatilities ABS
Simulation
and Outputs
50
0
0
50
−0.1
0
0 Swap
Option 50
Maturity
Expiry
Market volatilities ABS
0 Swap
Option 50
Maturity
Expiry
Absolute difference
−4
x 10
Quality Tests
Disclaimer
Contact
0.01
0.01
5
0.005
0.005
0
Bibliography
50
0
0
0
Option 50
Expiry
Swap
Maturity
50
0
0
0
Option 50
Expiry
Swap
Maturity
.
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For marketing purposes only
50
−5
0
0
Option 50
Expiry
Swap
Maturity
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Calibration
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LMM Calibration
Volatilities of Forward Rates
ESG
Calibration
Report
...
Absolute Volatilities
60
Model IT
55
Economic
Conditions
50
Calibration
45
Interest Rates
Equity
Property
Inflation
0.00616
0.00616
0.00696
0.00696
0.00535
0.00535
0.00776
0.00776
0.00776
0.00776
0.00776
0.00776
0.00616
0.011
0.011
0.00455
0.00455
0.00455
0.00455
Forward Rate Maturity
Quality Tests
25
Disclaimer
20
Contact
15
Bibliography
0.0174
0.0174
0.011
0.011
0.01660.0102
0.0166
0.0102
0.0102
0.0102
0.0158 0.011
0.0158
0.011
0.015
0.015
0.0126
0.0126
0.01420.00857
0.0142
0.00857
0.0134 0.00696
0.0134
0.00696
0.0118
0.0118
0.00134
0.00134
35
30
10
5
0.00616
0.00937
0.0118
0.0118
40
Simulation
and Outputs
−3
x 10
0.00375
0.00375
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0126
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0126
0.00295
0.00295
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0126
0.00455
0.00455 0.0126
0.00535
0.00535
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0126
16
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0126
0.00696
0.00696
0.0102
0.0102
0.011
0.011
0.0118
0.0118
0.0126
0.0126
0.00214
0.00214
0.00616 0.00616
0.00937 0.00937
0.00616
0.006960.00616
0.00696
0.00616
0.00616
0.00696
0.00696
0.00616
0.00616
0.00776
0.00776
0.00696
0.00696
0.00616
0.00616
0.00696
0.00696
0.00616
0.00616
0.00696
0.00696
0.00616
0.00616
0.00696
0.00696
0.00616
0.00616
0.00857
0.00857
0.00535
0.00535
0.00616
0.00616
0.00776
0.00776
0.00776
0.00776 0.00535
0.00535
0.00616
0.00616
0.00455
0.00455
0.00535
0.00535
0.00455
0.00455
0.00535
0.00535
0.00455
0.00455
0.00535
0.00535
0.00776
0.00776
0.00535
0.00535
0.00696
0.00696
0.00535
0.00535
0.00857
0.00857 0.00535
0.00535
0.00616
0.00616
0.00535
0.00535
0.00616
0.00616
0.00535
0.00535
0.00616
0.00616
0.00535
0.00535
0.00616
0.00616
0.00857
0.00857
0.00616
0.00616
0.00696
0.00696
0.00937
0.0102
0.0102
0.00937
0.00696
0.00696
0.00776
0.00776
0.011
0.011
0.00696
0.00696
0.00776
0.00776
0.0118
0.0118
0.00696
0.00696
0.00776
0.00776
0.00616
0.00616
0.0118
0.0118
0.00937
14
0.011
0.011
12
10
8
6
4
2
0.00776
0.00776
5
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15
20
25
30
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
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Equity Model
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
The equity indices follow geometric Brownian motion with stochastic
interest rates and deterministic volatility,
dSi (t)
= r (t) dt + σi (t) dWi (t),
Si (t)
Simulation
and Outputs
Quality Tests
Disclaimer
i = 1, . . . , N,
(2)
where N is the number of indices modeled, and the (instantaneous)
correlation between equity processes is E[ dWi (t) dWj (t)] = ρi,j dt.
Contact
Bibliography
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Equity Indices
ESG
Calibration
Report
...
Model IT
ID
Index Name
Market/Region
Economic
Conditions
SPX
S&P 500
US
Calibration
FTSE
FTSE 100
UK
N225
NIKKEI 225
Japan
OMXS30
OMX 30
Sweden
OMXH25
HEX 25
Finland
MIASJ0000PUS
iShares MSCI All Country Asia ex Japan
Asia ex Japan
Disclaimer
BVSPUSD
Ibovespa Index Sao Paolo SE
Brazil
Contact
IRTS
RTSI Index
Russia
Bibliography
NIFTY
S&P CNX NIFTY
India
HSCEI
Hang Seng China Enterprise
China
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
.
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Equity Implied Volatilities
Spot Vol
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Calibration
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Model IT
...
The implied volatilities for different maturities are derived from ATM
equity options for the equity indices:
Economic
Conditions
ATM Implied Volatilities for Equity Indices as of 31.3.2014
50
SPX
FTSE
N225
OMXS30
OMXH25
MIASJ0000PUS
BVSPUSD
IRTS
NIFTY
HSCEI
Calibration
45
Interest Rates
Equity
Property
Inflation
Quality Tests
35
Volatility (%)
Simulation
and Outputs
40
Disclaimer
30
25
20
15
Contact
10
Bibliography
5
0
0
1
2
3
4
5
Time (years)
6
7
8
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Equity Implied Volatilities
Forward Vol
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
The implied volatilities (of the previous slide) for different maturities at
t = 0 are spot volatilities; these need to be converted to corresponding
forward volatilities for simulation time steps:
√
σT2 2 T2 − σT2 1 T1
σT1 →T2 =
T2 − T1
Disclaimer
Contact
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Equity Implied Volatilities
Currency Volatility
ESG
Calibration
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Model IT
Economic
Conditions
...
For indices that are not denominated in Euros, a simple volatility
adjustment is applied to approximate the effect of currency volatility, i.e.,
the additional return volatility arising from changes in exchange rates:
√
2
2 + 2ρ
σtot = σeq
eq,fx σeq σfx + σfx ,
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
where
σeq is the equity index volatility,
σfx is the currency volatility (volatility associated with EUR/CCY
exchange rate for currency CCY), and
ρeq,fx is the correlation between equity index and currency returns
Disclaimer
Here it is assumed that ρeq,fx = 0 for each index, i.e., the
movements in equity and its currency are uncorrelated
Contact
Bibliography
The currency volatility adjustment is applied to each point in the forward
vol curve as a simple “add-on”, for each equity index denominated in a
currency other than EUR.
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Equity Implied Volatilities
FX-adjusted Fwd Vols
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Model IT
FX−adjusted Forward Volatilities for Equity Indices
50
Economic
Conditions
Calibration
Simulation
and Outputs
40
35
Volatility (%)
Interest Rates
Equity
Property
Inflation
SPX
FTSE
N225
OMXS30
OMXH25
MIASJ0000PUS
BVSPUSD
IRTS
NIFTY
HSCEI
45
30
25
20
Quality Tests
15
Disclaimer
10
5
Contact
0
Bibliography
0
10
20
30
Time (years)
40
50
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
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Property Model
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Property (real-estate) index follows geometric Brownian motion with
stochastic interest rates and constant volatility,
dS(t)
= r (t) dt + σ dW (t),
S(t)
Simulation
and Outputs
Quality Tests
(3)
where the volatility is assumed to be 12 % p.a. (σ = 0.12).
Disclaimer
Contact
Bibliography
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
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Inflation Model
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
Bibliography
...
Inflation rate follows Cox-Ingersoll-Ross (CIR) mean-reverting square-root
diffusion,
√
di(t) = α(θ − i(t)) dt + σ i(t) dW (t),
(4)
where
θ = mean-reversion level (long-term mean)
α = mean-reversion speed
σ = process volatility
The long-term mean is taken to correspond to the inflation target of the
European Central Bank (ECB), while the other parameters are estimated
from historical data using maximum likelihood estimation, conditional on θ
being fixed.
Data on Finnish inflation as provided by Statistics Finland
ˆσ
Parameter estimates: θˆparams = (α,
ˆ θ,
ˆ ) = (0.143, 0.020, 0.106)
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
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Simulation Outputs
Non-MATLAB users
ESG
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Model IT
...
The following entries and simulated values are written to the
Economic Scenario output file (e.g., .csv, .xlsx):
Economic
Conditions
Field Name
Description
simulation
Scenario number
period
Simulation time (in months), with 0 corresponding to valuation date
short rate
Short rate (nominal)
nom rate Tyr
Nominal T-year interest rate
Quality Tests
ba
Cumulative value of bank account process (1/discount factor)
Disclaimer
cpi
Inflation rate
real rate Tyr
Real T-year interest rate
X eq index
Cumulative value of equity index X ∈ {eur, fin, wexeur, em}
X eq return
1-period return of equity index X ∈ {eur, fin, wexeur, em}
prop index
Cumulative value of property index
prop return
1-period return of property index
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Contact
Bibliography
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Simulation Outputs
MATLAB users
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Alternatively, the Economic Scenarios can be provided as a
MATLAB structure for easy and convenient access to data
directly from MATLAB.
This is the preferred method for cFrame users.
Quality Tests
Disclaimer
Contact
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LMM Simulation
Technicalities 1/2
ESG
Calibration
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
...
Recall from (1) that in the Libor market model for discretely compounded
forward rates, each of the modeled forward rates Lj evolves according to
the SDE
dLj (t)
= µj (L, t) dt + σj (t) dWj (t),
Lj (t)
where the drift terms depend on the measure used, and are determined
from no-arbitrage arguments.
For generating economic interest rate scenarios, the spot (-Libor) measure
is used; that is, the discretely compounded bank acount is taken as the
num´eraire asset:
Disclaimer
Contact
Bibliography
Measure, Q
Spot Measure
Num´
eraire, N(t)
∏η(t)−1
B(t) = Pη(t) (t) i=0 [1 + δi Li (Ti )]
Drift, µj (t)
δj Lj (t)
i=η(t) 1+δj Lj (t) ρi,j (t)σi (t)σj (t)
∑j
where η(t) is the index of the next forward rate to reset (η(t) = j if
Tj−1 ≤ t < Tj ), and Pj (t) is the value at time t of a zero-coupon bond
.
.. ..
maturing at Tj .
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LMM Simulation
Technicalities 2/2
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Calibration
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
...
Greater accuracy is achieved when the LMM implementation is done using
log-forward rate rather than directly the forward itself (see, e.g., [GLA04]);
simulating the log-process has also the benefit that the diffusion coefficient
becomes state-independent.
Using It´
o’s lemma, d ln L = dL
− 12 σ 2 dt, and we get, for a time step from t to u
L
∫ u
∫
∫ u
1 u 2
ln Lj (u) = ln Lj (t) +
σj (s) ds +
σj (s) dWj (s),
(5)
µj (s) ds −
2 t
t
t
Note that the drifts µj (t) of the forward rates Lj (t) are indirectly stochastic, as
they depend explicitly on the stochastic L themselves, also continuously evolving
from t to u. This state-dependency makes discretization of the drift for
simulation purposes hard.
The most straightforward (and standard) way is to use Euler stepping,
Contact
µ
¯j =
Bibliography
j
∑
i=η(t)
δi Li (t)
ρi,j (t)σi (t)σj (t)
1 + δi Li (t)
(6)
to approximate the integral (i.e., the rates L are freezed to be constant at their
(known) starting values over the interval [t, u]).
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LMM Simulation
More technicalities: Predictor-Corrector
ESG
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
...
The accuracy of the simple Euler-stepping for drift can be increased by using the
so-called predictor-corrector (PC) scheme (see [BDJ08]). The idea is to simulate
the forward rates one step ahead keeping all the state variables constant,
recalculate the drift term using these evolved rates, and take the average of the
two drifts. The actual time step is then performed using the same random
numbers and this PC drift.
More precisely, to implement the PC scheme, we use single Euler step (with µ)
¯ to
ˆ(u), which are used to
get initial estimates of the 1-step ahead fwd rates, L
calculate new drift terms, µ.
ˆ The adjusted drifts are then obtained as
[t,u]
µ
˜j
∫
:=
t
u
j
1 ∑
µj (s) ds ≈
2
i=η(t)
[
]
ˆi (u)
δi Li (t)
δi L
+
ρ (t)σi (t)σj (t),
ˆi (u) i,j
1 + δi Li (t)
1 + δi L
Contact
Bibliography
and these µ
˜ are used to generate the “actual” vector of fwd rates L(u) using the
ˆ(u) in the first step.
same set of random numbers used to generate the estimates L
The accuracy of the PC method has been shown to
. be
. . good
. . . . in
. .the
. . literature.
. . . .
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Nominal & Real Interest Rates
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Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
For the economic scenario output file, spot rates are derived from the
forward rates simulated using LMM. These are nominal interest rates.
The real rates are obtained using the relationship:
Simulation
and Outputs
1 + rRe =
1 + rNom
,
1+i
Quality Tests
Disclaimer
Contact
where subscripts Re and Nom refer to real and nominal interest
rates, respectively, and i is the inflation rate.
Bibliography
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Equity Simulation
ESG
Calibration
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Model IT
...
The equity price paths are generated in a standard way by simulating
GBMs based on drawing correlated standard Normal (N(0, 1))
random variables.1
Economic
Conditions
Volatility is deterministic and depends on the simulation time
Calibration
The drift is the stochastic short interest rate
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Therefore, the interest rate (LMM) simulation has to be run
simultaneously or (for efficiency) before the equity simulation in
practice
Equity indices (see p.27) are simulated jointly, and four baskets are
formed from the simulated indices as follows:
Disclaimer
Contact
Bibliography
Equity Baskets
Name
Weights
SPX
FTSE
N225
OMXS30
OMXH25
MIASJ0000PUS
BVSPUSD
IRTS
NIFTY
HSCEI
Euro
0
0.85
0
0.15
0
0
0
0
0
0
Finland
0
0
0
0
1
0
0
0
0
0
0.75
0.13
0.12
0
0
0
0
0
0
0
0
0
0
0
World ex Euro
Emerging Markets
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. . . . . . 0.2
. . . . .0.2. . . 0.15
.
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Simulation and Outputs
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.
Property Simulation
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
The property index price paths are generated by simulating GBM,
similarly to equity processes.
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
Bibliography
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Inflation Simulation
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Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
The inflation rate paths are generated by simulating the
Cox-Ingersoll-Ross process.
The values are obtained exactly for each simulation point by
sampling from the exact transition law of the CIR process (essentially,
sampling from non-central χ2 df; see [GLA04]).
Quality Tests
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Contact
Bibliography
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
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Quality Tests
Checking Market Consistency
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Model IT
...
Market consistency tests:
Yield curve replication
Economic
Conditions
Calibration
Does the average simulated yield curve match the initial curve?
Martingale (1 = 1) tests
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Do all the asset classes earn, on average, the risk-free return?
Volatility replication
Is the average scenario volatility of each asset equal to the market
Quality Tests
(input) volatility?
Disclaimer
Contact
Pricing test
Are traded instruments priced correctly using the simulated scenarios?
Bibliography
In practice, suitable variance reduction and adjustment techniques can be
used to improve the fit to (the consistency with)
. . market
. . . . . inputs/prices.
. . . . . . . .
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Checking the Market Consistency
Note
ESG
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...
The most important criteria for market-consistent scenarios is
that they price traded instruments correctly (by definition).
Model IT
Economic
Conditions
Interest rate scenarios ⇒ reproduce observed bond prices and
swaption prices (volatilities)
Index asset scenarios ⇒ reproduce index asset option prices2
Calibration
Interest Rates
Equity
Property
Inflation
Most other tests, such as martingale (1st moment) test and
volatility (2nd moment) test, serve more as sanity checks
Simulation
and Outputs
Assuming the simulation procedures employed are correct, these
tests concern the (possible) discretization error in simulating
SDEs, and the (inevitable) sampling error in using Monte-Carlo
with finite sample size
In other words, these tests check that model output = model
input
Quality Tests
Disclaimer
Contact
Bibliography
2
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.
Variance Reduction 1/2
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
When using a small number of scenarios (such as 1′ 000 or even 10′ 000),
the observed sampling errors can be large.
Obvious solutions: use 1) more scenarios, or 2) quasi-random
numbers/low-discrepancy sequences (such as Sobol)
But: 1) raising scenario number not always feasible, and 2) use of
quasi-random numbers becomes problematic in very high dimensional
spaces (such as here)
Disclaimer
Contact
Bibliography
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Variance Reduction 2/2
ESG
Calibration
Report
Model IT
...
Variance reduction techniques used in practice with economic scenario
data:
Economic
Conditions
Sampling–selection: run several simulations and select the “most
market-consistent” (as measured by the tests)
Calibration
Interest Rates
Equity
Property
Inflation
In effect: try to find the set of pseudo-random numbers (or the seed
of your rng) that produces best convergence
Simulation
and Outputs
Moment-matching: use simple linear transformations to ensure the
market-consistency of (index asset) scenarios
Quality Tests
Disclaimer
When generating economic scenarios, “true” model values are
Contact
known/given by market: to minimize the size of sampling error,
Bibliography
moment-matching3 is used.
3
The term “moment matching” is also used by Towers Watson, while Barrie&Hibbert calls this “path
.
.. ..
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Initial Yield Curve Replication
ESG
Calibration
Report
...
Model IT
Initial Spot Curve vs. Average Simulated Spot Curve
3.5
Economic
Conditions
3
Calibration
Interest Rates
Equity
Property
Inflation
Spot Rate (%)
2.5
Simulation
and Outputs
2
1.5
Initial
Sim Mean
Quality Tests
1
Disclaimer
0.5
Contact
Bibliography
0
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Zero-Coupon Bond Price Replication
Initial Discount Factor Curve Replication
ESG
Calibration
Report
...
Model IT
Zero−Coupon Bond Prices
1
Economic
Conditions
0.9
Calibration
0.8
Interest Rates
Equity
Property
Inflation
Initial
Sim Mean
P(0,T)
0.7
Simulation
and Outputs
0.6
0.5
Quality Tests
0.4
Disclaimer
0.3
Contact
0.2
Bibliography
0.1
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Equity Indices
Martingale (1=1) Test & Volatility Replication
ESG
Calibration
Report
...
Model IT
Martingale Test: Equity Indices
EQ Index Volatilities; Input vs. Scenario
1.05
Economic
Conditions
1.04
Calibration
1.03
Interest Rates
Equity
Property
Inflation
0.5
0.4
1.02
1.01
0.3
1
Simulation
and Outputs
0.99
Quality Tests
0.98
Disclaimer
0.97
Contact
0.96
Bibliography
0.95
0.2
Line 1=1
SPX
FTSE
N225
OMXS30
OMXH25
MIASJ0000PUS
BVSPUSD
IRTS
NIFTY
HSCEI
0
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Time (years)
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Equity Baskets
Martingale (1=1) Test
ESG
Calibration
Report
...
Model IT
Martingale Test: Equity Baskets
1.05
Economic
Conditions
1.04
Calibration
1.03
Interest Rates
Equity
Property
Inflation
Line 1=1
Euro_EQ
FI_EQ
World_ex_Euro_EQ
EM_EQ
1.02
1.01
1
Simulation
and Outputs
0.99
Quality Tests
0.98
Disclaimer
0.97
Contact
0.96
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Property
Martingale (1=1) Test & Volatility Replication
ESG
Calibration
Report
...
Model IT
Martingale Test: Property Index
Property Volatility; Input vs. Scenario
1.05
Economic
Conditions
1.04
Calibration
1.03
Interest Rates
Equity
Property
Inflation
0.125
Line 1=1
Property
Input
Scenario
1.02
1.01
1
Simulation
and Outputs
0.99
Quality Tests
0.98
Disclaimer
0.97
Contact
0.96
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0.95
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Calibration & Simulation Errors
LMM Simulation - Swaption Prices
ESG
Calibration
Report
...
Mean absolute relative errors:
Calibration error (Market→Model): 1.44 %
Model IT
Simulation error (Model→Scenario): 1.39 %
Economic
Conditions
Total error4 (Market→Scenario): 2.02 %
Calibration
Interest Rates
Equity
Property
Inflation
Total error
Simulation
and Outputs
Simulations from the
Quality Tests
Market. prices
Disclaimer
Calibration
Model of the
Market
Simulation
Model of the
Market
Contact
Bibliography
Calibration error
Simulation error
4
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Note that the simulation and calibration errors are generally
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Swaption Prices
Relative Differences – Payer Swaptions
ESG
Calibration
Report
...
Relative difference between swaption scenario price and market price;
Payer swaptions (tenors (x−axis) × expiries (y−axis))
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
50
−0.06−0.03−0.03−0.02−0.01 0.01 0.01 0.01 0.01 0.01 NaN NaN NaN NaN NaN NaN NaN NaN NaN
45
−0.04−0.03−0.03−0.03−0.04−0.03−0.04−0.05−0.05−0.06−0.05−0.02NaN NaN NaN NaN NaN NaN NaN
40
−0.01−0.01−0.01−0.01−0.01 0.00 −0.01−0.01−0.01−0.01−0.01 0.02 0.01 NaN NaN NaN NaN NaN NaN
35
−0.01−0.01−0.01−0.01−0.02−0.02−0.03−0.04−0.05−0.05−0.04−0.01−0.01−0.04NaN NaN NaN NaN NaN
30
0.01 0.03 0.02 0.01 0.01 0.01 −0.01−0.01−0.02−0.03−0.03−0.01 0.02 0.00 −0.02NaN NaN NaN NaN
25
0.02 0.02 0.01 −0.00−0.02−0.02−0.03−0.04−0.05−0.06−0.05−0.02 0.01 0.00 −0.00−0.02NaN NaN NaN
20
0.01 0.03 0.04 0.03 0.02 0.02 0.01 0.00 −0.01−0.01 0.00 0.02 0.04 0.03 0.03 0.03 0.01 NaN NaN
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−0.04−0.02−0.01−0.02−0.02−0.02−0.03−0.04−0.04−0.04−0.07−0.05−0.06−0.06−0.05−0.04−0.04−0.03−0.03
45
−0.06 0.01 0.01 0.01 −0.01−0.01−0.01−0.01−0.02−0.02−0.04−0.02−0.04−0.04−0.04−0.04−0.05−0.04−0.05
1
40
−0.03 0.01 0.02 0.00 −0.01−0.02−0.02−0.02−0.02−0.02−0.04−0.03−0.05−0.05−0.06−0.06−0.07−0.06−0.07
2
35
3
Bibliography
25
−0.03 0.00 0.02 0.01 0.01 0.00 0.00 0.00 0.01 0.01 0.01 0.01 −0.01−0.01−0.02−0.03−0.04−0.03−0.05
30
−0.00 0.02 0.04 0.04 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.04 0.03 0.03 0.02 0.01 0.01 −0.01
4
20
−0.02 0.01 0.02 0.02 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 −0.01
5
Contact
15
−0.02−0.01−0.01−0.01−0.01−0.01−0.01−0.02−0.02−0.02−0.02−0.01−0.01−0.00 0.00 −0.00−0.00 0.00 −0.01
6
12
7
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9
−0.00 0.00 0.01 0.00 0.01 0.01 0.01 0.00 −0.00−0.01−0.01−0.00 0.00 −0.00−0.00−0.01−0.01−0.01−0.03
10
8
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0.02 0.01 0.02 0.02 0.03 0.02 0.02 0.02 0.01 0.00 0.01 0.01 0.01 0.01 0.01 0.00 0.00 −0.00−0.02
Quality Tests
7
−0.00−0.00 0.02 0.02 0.01 0.01 0.01 −0.00−0.01−0.02−0.02−0.01−0.00 0.00 0.00 −0.01−0.00−0.01−0.03
9
6
10
5
Simulation
and Outputs
4
−0.01 0.01 0.03 0.02 0.02 0.02 0.02 0.01 0.00 −0.00 0.01 0.02 0.03 0.03 0.02 0.02 0.01 0.00 NaN
3
0.01 0.02 0.01 0.00 −0.00−0.01−0.02−0.03−0.04−0.05−0.04−0.02 0.00 0.01 0.01 0.02 0.01 0.00 NaN
12
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Swaption Prices
Relative Differences – Receiver Swaptions
ESG
Calibration
Report
...
Relative difference between swaption scenario price and market price;
Receiver swaptions (tenors (x−axis) × expiries (y−axis))
Model IT
0.06
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
50
−0.08−0.05−0.05−0.04−0.03−0.01−0.01−0.01−0.02−0.03NaN NaN NaN NaN NaN NaN NaN NaN NaN
45
−0.04−0.03−0.06−0.07−0.08−0.08−0.09−0.09−0.09−0.09−0.08−0.06NaN NaN NaN NaN NaN NaN NaN
40
−0.04−0.04−0.06−0.06−0.07−0.06−0.07−0.08−0.09−0.10−0.10−0.08−0.09NaN NaN NaN NaN NaN NaN
35
0.01 0.01 −0.00 0.00 −0.01−0.01−0.03−0.04−0.05−0.06−0.06−0.04−0.06−0.10NaN NaN NaN NaN NaN
30
0.02 0.02 0.00 −0.00−0.01−0.02−0.03−0.04−0.05−0.06−0.06−0.05−0.04−0.06−0.08NaN NaN NaN NaN
25
0.02 −0.00−0.03−0.04−0.06−0.06−0.07−0.08−0.09−0.09−0.09−0.06−0.04−0.05−0.07−0.08NaN NaN NaN
20
−0.00−0.02−0.03−0.04−0.05−0.06−0.07−0.08−0.09−0.09−0.08−0.06−0.04−0.06−0.06−0.07−0.09NaN NaN
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0.02
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−0.04−0.01 0.01 0.01 0.02 0.02 0.02 0.01 0.02 0.02 0.00 0.03 0.03 0.03 0.04 0.05 0.05 0.06 0.06
45
−0.06 0.02 0.02 0.02 0.01 0.00 −0.00−0.01−0.01−0.02−0.04−0.03−0.05−0.05−0.05−0.05−0.06−0.05−0.06
1
40
−0.02 0.03 0.05 0.05 0.04 0.03 0.03 0.03 0.03 0.02 0.00 0.00 −0.02−0.02−0.03−0.03−0.04−0.03−0.04
2
35
3
25
−0.03 0.02 0.04 0.04 0.03 0.03 0.03 0.03 0.04 0.04 0.03 0.03 0.01 0.00 0.00 −0.01−0.02−0.01−0.03
30
−0.00 0.04 0.05 0.05 0.04 0.04 0.04 0.05 0.05 0.05 0.05 0.05 0.05 0.04 0.04 0.03 0.02 0.02 0.01
4
Bibliography
20
5
Contact
15
−0.01 0.03 0.04 0.04 0.04 0.03 0.03 0.03 0.04 0.03 0.03 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.00
12
−0.02 0.02 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.02 0.01 0.01 0.01 0.00 0.00 −0.00−0.02
6
9
7
Disclaimer
10
Quality Tests
−0.01 0.03 0.04 0.03 0.04 0.04 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.00 −0.01
8
0.02 0.03 0.04 0.04 0.04 0.04 0.04 0.04 0.03 0.02 0.02 0.02 0.01 0.01 0.01 −0.00−0.00−0.01−0.03
8
7
0.00 0.02 0.05 0.05 0.05 0.05 0.05 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 −0.00−0.02
9
6
10
5
Simulation
and Outputs
4
−0.01 0.02 0.03 0.03 0.03 0.02 0.02 0.01 0.00 −0.01−0.00 0.01 0.02 0.02 0.01 0.00 −0.00−0.02NaN
3
0.01 0.02 0.01 −0.00−0.01−0.03−0.04−0.05−0.07−0.07−0.06−0.04−0.03−0.02−0.02−0.02−0.03−0.04NaN
12
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Swaption Prices
Option Price Surfaces
ESG
Calibration
Report
...
Payer: Sim
Payer: Mkt
Relative diff. Payer
Model IT
Economic
Conditions
0.2
0.2
0.1
0.1
0.1
0
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
0
50
0
50
−0.1
50
50
Swap
Maturity
50
Swap
Maturity
0 0
Option
Expiry
Receiver: Sim
50
Swap
Maturity
0 0
Option
Expiry
Receiver: Mkt
0 0
Option
Expiry
Relative diff. Receiver
0.2
0.2
0.1
0.1
0.1
0
Disclaimer
Contact
Bibliography
0
50
0
50
−0.1
50
50
Swap
Maturity
0 0
Option
Expiry
50
Swap
Maturity
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Expiry
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Maturity
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Expiry
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Stressed Interest Rate Scenarios
Spot Curves
ESG
Calibration
Report
...
Model IT
Spot Curves in Stress Scenarios
4.5
Economic
Conditions
4
Calibration
3.5
Interest Rates
Equity
Property
Inflation
Interest Rate (%)
3
Simulation
and Outputs
2.5
2
1.5
Quality Tests
Up Initial
Up Sim Mean
1
Disclaimer
Base Initial
Base Sim Mean
Contact
Down Initial
0.5
Bibliography
0
Down Sim Mean
0
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Stressed Interest Rate Scenarios
Forward Curves
ESG
Calibration
Report
...
Forward Curves in Stress Scenarios
Model IT
6
Economic
Conditions
5
Calibration
Interest Rates
Equity
Property
Inflation
Interest Rate (%)
4
Simulation
and Outputs
3
2
Up Initial
Quality Tests
Up Sim Mean
Base Initial
Disclaimer
Base Sim Mean
1
Down Initial
Down Sim Mean
Contact
0
Bibliography
0
10
20
30
Time (years)
Note: Stressed forwards calculated from yearly spot rates.
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
.
.. ..
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Disclaimer
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
This document is an example of an actual ESG calibration
report, with one set of models used for illustrative purposes.
The report and its contents are to be used for marketing
purposes only.
Simulation
and Outputs
Quality Tests
Disclaimer
The actual Economic Scenarios provided to each client are
tailored to meet the specific needs of the client.
Contact
Bibliography
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
.
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Model IT Ltd
Contact Information
ESG
Calibration
Report
Model IT
Economic
Conditions
Calibration
...
Model IT Ltd
Unioninkatu 13, 7th Floor
00130 Helsinki, FINLAND
www.modelit.fi
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
For more information, contact
Quality Tests
Disclaimer
Contact
Matias Leppisaari
[email protected]
Timo Salminen
[email protected]
Bibliography
Lasse Koskinen
[email protected]
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Table of Contents
ESG
Calibration
Report
...
1.
Economic Conditions
2.
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
3.
Simulation and Outputs
Quality Tests
4.
Quality Tests
5.
Disclaimer
6.
Contact
7.
Bibliography
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Disclaimer
Contact
Bibliography
.
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Bibliography I
ESG
Calibration
Report
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
Quality Tests
Disclaimer
Contact
Bibliography
...
[BME06] D. Brigo and F. Mercurio (2006):
Interest Rate Models – Theory and Practice, 2nd Ed.
Springer Finance, Heidelberg.
[RJA03] R. Rebonato and P. J¨
ackel (2003):
Linking Caplet and Swaption Volatilities in a BGM Framework:
Approximate Solutions
Journal of Computational Finance, Vol.6, No.4, pp.41–60.
[GLA04] P. Glasserman (2004):
Monte Carlo Methods in Financial Engineering
Springer.
[BDJ08] C. Beveridge, N. Denson and M. Joshi (2008):
Comparing Discretization of the LIBOR Market Model in the Spot
Measure
Working paper, University of Melbourne.
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Bibliography
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Bibliography II
ESG
Calibration
Report
...
Model IT
Economic
Conditions
Calibration
Interest Rates
Equity
Property
Inflation
Simulation
and Outputs
[SAL07] T. Salminen (2007):
Calibration of a Forward Rate Market Model
Master’s Thesis, Degree Programme in Engineering Physics and
Mathematics, Helsinki University of Technology.
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