Pricing CDOs using Intensity Gamma Approach
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Transcript of Pricing CDOs using Intensity Gamma Approach
Pricing CDOs using Intensity Gamma Approach
Christelle Ho Hio HenAaron IpsaAloke MukherjeeDharmanshu 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
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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)
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Forward Default Intensities Survival Curve in terms of λ
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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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Series Representation of a Gamma Process (Cont and Tankov)
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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 +/- 0.072Expected Mean = 63.333
Constructing a Business Time Path
1.,3.,1,5. 2121
…Testing Business Time Path Continued
Variance = 522.3Expected 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 = 0.0421812Gaussian vfix = 0.0422499, vflt = 0.0428865IG vfix = 0.0429348, vflt = 0.0422907input 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
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Implied survival curveGaussian copulaIntensity 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 17783-7% 135 1877-10% 14 2910-15% 1 515-30% 0 0
Fast Approx – Both Constant λi
Tranche Fast IG Full IG 0-3% 1429 15733-7% 135 1337-10% 14 1310-15% 1 115-30% 0 0
Fast Approx – Const λi, Uniform Default TimesTranche Fast IG Full IG 0-3% 1584 15733-7% 144 1337-10% 14 1310-15% 1 115-30% 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 Correlation0-3% 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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Correlation Skew
Comparison of Base Correlations
0,00%
20,00%
40,00%
60,00%
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100,00%
120,00%
0-3% 3-7% 7-10% 10-15% 15-30%Tranches
Bas
e C
orre
latio
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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