Pricing CDOs using Intensity Gamma Approach

Christelle Ho Hio Hen

Aaron Ipsa

Aloke Mukherjee

Dharmanshu Shah

Intensity Gamma

M.S. Joshi, A.M. Stacey “Intensity Gamma: a new approach to pricing portfolio credit derivatives”, Risk Magazine, July 2006

Partly inspired by Variance Gamma Induce correlation via business time

Business time vs. Calendar time

Business time Calendar time

Block diagram

6mo 1y 2y .. 5yname1 .name2..name125

CDS spreads

Survival Curve Construction

IG Default Intensities

Calibration

Parameter guess

Business time path generator

Default time calculator

Tranche pricer

Objective function0-3% …3-6% …6-9% …..

Market tranche quotes

Err<tol? NO

YES

Advantages of Intensity Gamma

Market does not believe in the Gaussian Copula

Pricing non-standard CDO tranches Pricing exotic credit derivatives Time homogeneity

The Survival Curve

Curve of probability of survival vs time Jump to default = Poisson process P(λ) Default = Cox process C(λ(t)) Pr (τ > T) = exp[ ] Intensity vs time –

λT1, λT2, λT3….. for (0,T1), (0,T2), (0,T3)

0

( )T

u du

Forward Default Intensities Survival Curve in terms of λ

0

0.001

0.002

0.003

0.004

0.005

0.006

0 1 2 3 4 5 6

Tim e

Inte

nsi

ty

Bootstrapping the Survival Curve

Assume a value for λT1

X(0,T1) = exp(-λT1 . T1) Price CDS of maturity T1

Use a root solving method to find λT1

Assume a value for λT2

Now X(0,T2) = X(0,T1) * exp(-λT2(T2-T1)) Price CDS of maturity T2 Use root solving method to find λT2

Keep going on with T3, T4….

Constructing a Business Time Path

Business time modeled as two Gamma Processes and a drift.

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Constructing a Business Time Path

Characteristics of the Gamma ProcessPositive, increasing, pure jump Independent increments are Gamma

distributed:

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tVariance

tMean

Series Representation of a Gamma Process (Cont and Tankov)

T,V are Exp(1), No Gamma R.V’s Req’d.

Constructing a Business Time Path

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Truncation Error Adjustment

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Constructing a Business Time Path

Truncation Error Adjustment

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Constructing a Business Time Path

Test Effect of Estimating Truncation Error in Generating 100,000 Gamma Paths

1. Set Error = .001, no adjustment Computation Time = 42 Seconds

2. Set Error = 0.05 and apply adjustment Computation Time = 34 seconds

Constructing a Business Time Path

Testing Business Time Paths Given

drift a = 1, Tenor = 5, 100,000 paths

Mean = 63.267 +/- 0.072

Expected Mean = 63.333

Constructing a Business Time Path

1.,3.,1,5. 2121

…Testing Business Time Path Continued

Variance = 522.3

Expected Variance = 527.8

Constructing a Business Time Path

Constructing a Business Time Path

IG Forward Intensities ci(t)

In IG model survival probability decays with business time

Inner calibration: parallel bisection Note that one parameter redundant

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Default Times from Business Time

Survival Probability:

Default Time:

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Tranche pricer

Calculate cashflows resulting from defaults Validation: reprice CDS (N=1)EDU>> roundtriptest(100,100000);

closed form vfix = 0.0421812, vflt = 0.0421812

Gaussian vfix = 0.0422499, vflt = 0.0428865

IG vfix = 0.0429348, vflt = 0.0422907

input spread = 100, gaussian spread = 101.507,

IG spread = 98.4998

Validation: recover survival curve

Survival Curve

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Implied survival curve

Gaussian copula

Intensity Gamma

A Fast Approximate IG Pricer

Constant default intensities λi

Probability of k defaults given business time IT

Price floating and fixed legs by integrating over distribution of IT

Fast IG Approximation Comparison

Tranche Fast IG Full IG

0-3% 1429 1778

3-7% 135 187

7-10% 14 29

10-15% 1 5

15-30% 0 0

Fast Approx – Both Constant λi

Tranche Fast IG Full IG

0-3% 1429 1573

3-7% 135 133

7-10% 14 13

10-15% 1 1

15-30% 0 0

Fast Approx – Const λi, Uniform Default Times

Tranche Fast IG Full IG

0-3% 1584 1573

3-7% 144 133

7-10% 14 13

10-15% 1 1

15-30% 0 0

Calibration

Unstable results => need for noisy optimization algorithm. Unknown scale of calibration parameters

=> large search space. Long computation time => forbids Genetic Algorithm

Simulated Annealing

Calibration

Redundant drift value => set a = 1 Two Gamma processes: = 0.2951 = 0.2838

= 0.0287 = 0.003

Tranches Market spreads Market Base Correlation Simulated Spreads Simulated Base Correlation

0-3% 12,30% 32,61% 12,57% 31,56%3-7% 0,78% 54,30% 1,96% 42,60%7-10% 0,17% 65,12% 0,31% 52,40%

10-15% 0,08% 77,64% 0,04% 66,12%15-30% 0,05% 96,32% 0% 90,67%

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ti dItctX )(exp)(

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Correlation Skew

Comparison of Base Correlations

0,00%

20,00%

40,00%

60,00%

80,00%

100,00%

120,00%

0-3% 3-7% 7-10% 10-15% 15-30%Tranches

Bas

e C

orr

elat

ion

MarketBaseCorrelation

SimulatedBaseCorrelation

Future Work

Performance improvementsUse “Fast IG” as Control VariateQuasi-random numbers

Not recommended for pricing different maturities than calibrating instrumentsStochastic delay to default

Business time factor models