modelling credit spread behaviour - free
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Modelling Credit Spread Behaviour
FIRST
FIRST Credit, Insurance and RiskAngelo Arvanitis, Jon Gregory, Jean-Paul Laurent
ICBI Credit, Counterparty & Default Risk Forum29 September 1999, Paris
FIRSTCredit,
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OverviewOverview
Part I−Need for Credit Models
Part II−Simple Binomial Model
Part III−Jump-Diffusion Modelpage 2
Part IV−Credit Migration Model
Part V−Estimating Credit Spread Volatilities
Credit Modelling
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Part INeed for credit spread modelspage 3
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Need For Credit Models (I)Need For Credit Models (I)
- Credit derivatives market- Active management of loan portfolios
Why? Growth of emerging markets
page 4
Active management of counterparty risk in standard
derivatives portfolios
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Need For Credit Models (II)Need For Credit Models (II)
Valuing credit derivatives, options on risky bonds, vulnerable derivatives
Assessing the credit risk of portfolios -
spread and event riskWhat for?
page 5
- Optimising portfolio risk / return profile- Relative value analysis
Credit Modelling
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Need For Credit Models (III)Need For Credit Models (III)
Model the evolution of the risk free rate and
the credit spread
Estimate the current risk freeand risky term structures
Calibrate to observedbond and option prices
How?
page 6
Maturity
Yiel
d
Risk-freeRisky
Credit spread
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Credit DataCredit Data
Limited / crude data available on creditMoody’s historical data (annual)
−Default probability
−Pairwise default correlation
−Credit migration
−Loss given defaultDefault correlation and recovery rate difficult to
estimateCredit crashes - high default correlation
0 20%≤ ≤q kl
0 25%≤ ≤pi
0 5%≤ ≤ρ ij
0 100%≤ ≤li
page 7
Credit Modelling
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Credit spread for an AA bondCredit spread for an AA bond
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90
Jump
page 8
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Properties of Credit SpreadsProperties of Credit Spreads
Jump Component- Discrete change in default
probability- Credit migration
Continuous Component- Mean reverting
- Change in market price of risk - risk premia
mean reversiondowngrade
Time
more volatile
Cre
dit s
prea
d
page 9
Credit Modelling
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Modelling Credit SpreadModelling Credit Spread
r rrisky risk free= + ~λ
Credit Spread
ConstantSimple binomial model
(Part II)
Continuous and jump components
Jump-diffusion model (Part III)
Model underlyingcredit migration process
(Part IV)
page 10
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Part IISimple Binomial Model
page 11
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Simple Binomial Model (I)Simple Binomial Model (I)
- Constant risk free term structure
- Constant recovery rate
- Constant credit spread if no default
- Jump in credit spread if default occurs
page 12
- Derive risk neutral default probabilities from risky and
risk- free bond prices
Risk neutral default probabilities - Actual default probabilities
- Risk premia- Liquidity
- Uncertainty over recovery rate
Credit Modelling
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Simple Binomial Model (II)Simple Binomial Model (II)
( )[ ]v T p T q T q T( ) ( ) ~( , ) ~( , )= − +1 0 0 δ
probability of default
default
v T( )risky bond price
1
δ
1 0− ~ ( , )q T
~ ( , )q T0
promised amount
recovery rate
no default
risk free bond price
~( , )( ) / ( )( )
q Tv T p T
01
1=
−− δ
- price any product with payoff contingent on default event
page 13
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Part IIIJump-Diffusion Model
page 14
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Jump-Diffusion ModelJump-Diffusion Model
Continuous component - Positive and mean reverting- Correlated with interest rates
Jump Component- Jumps of random size occur at random times
- Jumps in only one direction
Risk-free interest rate-Continuous and mean
reverting
- Standard implementation and calibration
- Standard numerical pricing algorithms can be used
page 15
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Risk Free Term Structure (I)Risk Free Term Structure (I)
Assumptions on the future evolution of the instantaneous risk free rate
−Volatility (normal, lognormal, square root, … )−Drift / mean reversion
Long term meanRate of mean reversion
σ r r t( )
r t( )k tr ( )
page 16
( )r t r k t r t r t dt r t dW tr
t
r r
t
( ) ( ) ( ) ( ) ( ) ( ) ( )= + − +∫ ∫00 0
σ
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Risk Free Term Structure (II)Risk Free Term Structure (II)
σ r rk= =0 1 2. , σ r rk= =0 1 1 0. ,
σ r rk= =0 1 2. , σ r rk= =0 2 2. ,
page 17
Credit Modelling
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Credit Spread Term Structure (I)Credit Spread Term Structure (I)
~( ) ( ) ( )λ ρt r t x t= + ( )corr r t x t( ), ( ) = 0
determines correlationbetween and
~( )λ t r t( )- Uncorrelated with interest rates- Continuous and jump component
page 18
θ θe zz− >, 0
e n nn− = =λτ λτ τ( ) / ! , , , . . . tim e interval0 1 2
Random jump size z, exponentially distributed
Random number of jumps - follows Poisson process
Credit Modelling
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Credit Spread Term Structure (II)Credit Spread Term Structure (II)
( )x t x k s x s x s dsx
t
( ) ( ) ( ) ( ) ( )= + −∫00
+ ∫σ x x
t
s x s dW s( ) ( ) ( )0
+≤
∑Z ii i t
( ); ( )τ
Credit spread component uncorrelated with the risk free interest rate
page 19
Credit Modelling
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Credit Spread Term Structure (III)Credit Spread Term Structure (III)
more frequentand larger jumps
page 20
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Part IVCredit Migration Model
page 21
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Credit Migration ModelCredit Migration Model
- Model jointly assets in various credit classes
- Portfolio management and risk analysis
- Jumps modelled as changes in credit ratings and defaults
- Continuous part modelled as continually changing risk premia
page 22
- Flexible in terms of data requirements and
number of states
- Calibration - incorporate economic and historical
information
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Markov Chains - Generator Matrix (I)Markov Chains - Generator Matrix (I)
absorbing state (default)
constantover time
~
~ ~ . ~ ~~ ~ . ~ ~
. . . . .~ ~ . ~ ~
.
,
,
, , ,
Λ =
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
−
−
− − − −
λ λ λ λλ λ λ λ
λ λ λ λ
1 12 1 1 1
21 2 2 1 2
1 1 1 2 1 1
0 0 0 0
K K
K K
K K K K K
1 2 K-1 K
12
K-1K
Continuous time Markov chainDiscrete state space
page 23
Credit Modelling
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Markov Chains - Generator Matrix (II)Markov Chains - Generator Matrix (II)
, transition matrix over short period dtI dt+ ~Λ
, non-negative transition probabilities
, sum of all probabilities equals 1
A state i+1 is always more risky than state i
λij ≥ 0
~ ~λ λi ijij i
K
= −=≠
∑1
~ ~ , ,,λ λijj i
K
i jj k
i k k i≥
+≥
∑ ∑≤ ∀ ≠ +1 1
page 24
Credit Modelling
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Markov Chains - Transition MatrixMarkov Chains - Transition Matrix
Transition matrix for the period t to TExplicit computation
[ ]~( , ) exp ( )Q t T D T t= −−Σ Σ1
~Λ Σ Σ= −1D
~( , )
~ ( , ) . ~ ( , ) ~ ( , )~ ( , ) . ~ ( , ) ~ ( , )
. . . .~ ( , ) . ~ ( , ) ~ ( , )
.
,
,
, ,
Q t T
q t T q t T q t Tq t T q t T q t T
q t T q t T q t T
K K
K K
K K K K
=
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
−
−
− − −
1 1 1 1
21 2 1 2
1 1 1 1
0 0 1
page 25
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Model Structure (I)Model Structure (I)
States : uniquely determine default probabilityCredit ratings - can incorporate past credit
rating transitions - non-Markovian model
- Risk neutral generator matrix
- constant
~Λ
~Λ
page 26
jump in credit spread due to downgrade
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Model Structure (II)Model Structure (II)
Incorporate stochastic risk premia
continuous process for risk premia
~Λstochastic
jump in credit spread due to downgrade
~ ~ ( )Λ Λstochastic U t= ×
page 27
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Stochastic Generator Matrix Stochastic Generator Matrix
Stochastic generator matrix arises from randomly changing risk premia
~ ~ ( )Λ Λstochastic U t= ×
( )U t U a kU t dt U t dWt
t
t
( ) ( ) ( ) ( )= + − +∫ ∫00 0
σ
Closed form formulae for bond prices
Stochastic component
Mean reverting process
page 28
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Stochastic Risk PremiaStochastic Risk Premia
If eigenvectors are constant, can poseΛ Σ Σ( , ) ( )t T t= −1D
Possible evolution of eigenvaluesdX a b X dt dw t X tj j j j j j j
K= − + = =( ) , ( ) diag( ( ))σ D 1
Pricing equation is now modified to
q t T X s ds tiK ij jt
T
jKj
K
( , ) ( ) exp ( ) ( )= ⎛⎝⎜
⎞⎠⎟
⎡⎣⎢
⎤⎦⎥
−
=∫∑ Σ Σ1
1
E Dpage 29
Expectation has closed (algebraic) form− depends on parameters a b σ and on D(t)
Credit Modelling
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Calibration (I)Calibration (I)
Prices of risky bonds for various credit
classes and maturities- Least squares estimation- Adjust historical generator matrix to fit market prices- Achieve fit closest to historical data
B Ti ( , )0
- Historical generator matrix (estimated from one year transition
matrix)- Credit spread historical time series
Λ
~
( , , , )ΛΛ
stochastica k σ
Price exoticstructures
Simulate Credit Spread
Simulate Credit Spread
page 30
Credit Modelling
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Calibration (II)Calibration (II)
priorgenerator matrix
( )min ( ) ( ; ~ )
~
~,Λ
ΛP F h v hji
ji
h
Ti
j
J
i
Kij ij
iji j
Ki
−⎛⎝⎜
⎞⎠⎟ +
−⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪=== =∑∑∑ ∑
1
2
11
2
1
λ λβ
market price of coupon bond for
class i
coupon at date h
Least squares fit to match directly observed coupon bond prices (any number)
confidence level
page 31
Obtain solution closest to the historical generator matrix - stable calibrationΛ
Credit Modelling
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Calibration - Emerging Markets (II)Calibration - Emerging Markets (II)
0
50
100
150
200
250
300
350
400
450
500
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
3 states
5 states
Actual
page 32
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Calibration - Corporate Market (II)Calibration - Corporate Market (II)
0
200
400
600
800
1000
1200
1400
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
AAA
AA
A
BBB
BB
B
CCC
Cre
dit s
prea
d (b
p)
Maturity (years)
page 33
Credit Modelling
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Calibration - US Industrials (I)Calibration - US Industrials (I)
0
20
40
60
80
100
120
140
160
180
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
AAA
AA
A
Cre
dit S
prea
d (b
p’s)
Maturity (years)
page 34
Credit Modelling
FIRSTCredit,
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Calibration - US Industrials (II)Calibration - US Industrials (II)
0
200
400
600
800
1000
1200
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
BBB
BB
B
CCC
Cre
dit S
prea
d (b
p’s)
Maturity (years)
page 35
Credit Modelling
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Part VEstimating Credit Spread Volatility
page 36
Credit Modelling
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Credit spread volatilities estimatesCredit spread volatilities estimates
74 Bonds67 Investment grade (Baa and above) US Industrial bonds7 Speculative grade (Ba and below) Emerging Market bonds
74 Bonds67 Investment grade (Baa and above) US Industrial bonds7 Speculative grade (Ba and below) Emerging Market bonds
8 Model states1 Moody’s Aaa 5 Moody’s Ba1 - Ba32 Moody’s Aa1 - Aa3 6 Moody’s B1 - B33 Moody’s A1 - A3 7 Moody’s CCC4 Moody’s Baa1 - Baa3 8 Default
8 Model states1 Moody’s Aaa 5 Moody’s Ba1 - Ba32 Moody’s Aa1 - Aa3 6 Moody’s B1 - B33 Moody’s A1 - A3 7 Moody’s CCC4 Moody’s Baa1 - Baa3 8 Default
page 37
Historical generator matrix from Moody’s average 1y transition matrix 1920 - 1996Historical generator matrix from Moody’s average 1y transition matrix 1920 - 1996
Credit Modelling
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Estimated short term spreadsEstimated short term spreads
0 50 100 150 200 2500
0.05
0.1
0.15
0.2
Credit ratings
AA ABBB BB
B CCCpage 38
Lower ratings have− higher spread− higher volatility
Credit Modelling
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Eigenvalues of generator matrixEigenvalues of generator matrix
0 50 100 150 200 250-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0 50 100 150 200 250-0 05
0
0.05
0 50 100 150 200 250-0.05
0
0.05
0 50 100 150 200 250-0.05
0
0.05
0 50 100 150 200 250-0.05
0
0.05
PrincipalcomponentsEigenvalues
- first 3 principal components account
for 99% of the variance
page 39
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Advanced Modelling IssuesAdvanced Modelling Issues
Stochastic Recovery Rates- Recovery rates are random with
high variance- Exogenous- Endogenous - depend
on the severity of default
Credit Events Correlatedwith Interest Rates- Credit migration and defaults
depend on interest rates- Joint state variables for interest
rates and credit spreads- Incorporate business cycles
Second Generation Products- Basket options - credit spread,
default correlations- Multiple Currencies- Quantos
Non - Markovian Bankruptcy Process- Autocorrelated migration
process- Markovian in state space
augmented with laggedvalues
page 40
Credit Modelling