1. Pricing CDOs using Intensity Gamma Approach Christelle Ho Hio Hen Aaron Ipsa Aloke Mukherjee Dharmanshu Shah

2. 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

3. Business time vs. Calendar time Business timeCalendar time

4. Block diagram 6mo 1y 2y.. 5y name1. name2. name125 CDS spreads Survival Curve Construction IG Default Intensities Calibration Parameter guess Business time path generator Default time calculator Tranche pricer Objective function 0-3% … 3-6% … 6-9% …. Market tranche quotes Err<tol? NO YES

5. Advantages of Intensity Gamma Market does not believe in the Gaussian Copula Pricing non-standard CDO tranches Pricing exotic credit derivatives Time homogeneity

6. 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 – λ T1, λ T2, λ T3 ….. for (0,T 1 ), (0,T 2 ), (0,T 3 )

7. Forward Default Intensities

8. Bootstrapping the Survival Curve Assume a value for λ T1 X(0,T 1 ) = exp(- λ T1. T 1 ) Price CDS of maturity T 1 Use a root solving method to find λ T1 Assume a value for λ T2 Now X(0,T 2 ) = X(0,T 1 ) * exp(- λ T2 (T 2 -T 1 )) Price CDS of maturity T2 Use root solving method to find λ T2 Keep going on with T 3, T 4 ….

9. Constructing a Business Time Path Business time modeled as two Gamma Processes and a drift.

10. Constructing a Business Time Path Characteristics of the Gamma Process  Positive, increasing, pure jump  Independent increments are Gamma distributed:

11. 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

12. Truncation Error Adjustment Constructing a Business Time Path

13. Truncation Error Adjustment Constructing a Business Time Path

14. 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

15. 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

16. …Testing Business Time Path Continued Variance = 522.3 Expected Variance = 527.8 Constructing a Business Time Path

18. IG Forward Intensities c i (t) In IG model survival probability decays with business time Inner calibration: parallel bisection Note that one parameter redundant

19. Default Times from Business Time Survival Probability: Default Time:

20. 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

21. Survival Curve

22. A Fast Approximate IG Pricer Constant default intensities λ i Probability of k defaults given business time I T Price floating and fixed legs by integrating over distribution of I T

23. Fast IG Approximation Comparison TrancheFast IGFull IG 0-3%14291778 3-7%135187 7-10%1429 10-15%15 15-30%00

24. Fast Approx – Both Constant λ i TrancheFast IGFull IG 0-3%14291573 3-7%135133 7-10%1413 10-15%11 15-30%00

25. Fast Approx – Const λ i, Uniform Default Times TrancheFast IGFull IG 0-3%15841573 3-7%144133 7-10%1413 10-15%11 15-30%00

26. Calibration Unstable results => need for noisy optimization algorithm. Unknown scale of calibration parameters => large search space. Long computation time => forbids Genetic Algorithm Simulated Annealing

27. Calibration Redundant drift value => set a = 1 Two Gamma processes: = 0.2951 = 0.2838 = 0.0287 = 0.003

28. Correlation Skew

29. Future Work Performance improvements  Use “Fast IG” as Control Variate  Quasi-random numbers Not recommended for pricing different maturities than calibrating instruments  Stochastic delay to default Business time factor models