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 timeCalendar time

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

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

Forward Default Intensities

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 ….

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

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

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

Truncation Error Adjustment Constructing a Business Time Path

Truncation Error Adjustment 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 = / Expected Mean = Constructing a Business Time Path

…Testing Business Time Path Continued Variance = Expected Variance = Constructing a Business Time Path

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

Default Times from Business Time Survival Probability: Default Time:

Tranche pricer Calculate cashflows resulting from defaults Validation: reprice CDS (N=1) EDU>> roundtriptest(100,100000); closed form vfix = , vflt = Gaussian vfix = , vflt = IG vfix = , vflt = input spread = 100, gaussian spread = , IG spread = Validation: recover survival curve

Survival Curve

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

Fast IG Approximation Comparison TrancheFast IGFull IG 0-3% % % % %00

Fast Approx – Both Constant λ i TrancheFast IGFull IG 0-3% % % % %00

Fast Approx – Const λ i, Uniform Default Times TrancheFast IGFull IG 0-3% % % % %00

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: = = = = 0.003

Correlation Skew

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