Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant Institute) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.1/35
Outline Probabilistic approach to the heat equation Fokker-Planck equation and the boundary value problem Matching exit probability - free boundary formulation Application: modeling credit risk in finance Poisson jump-diffusion Partial integro-differential equation formulation Analysis issues Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.2/35
Probabilistic Approach to the Heat Equation Random walk: step sizes from normal distribution N(0,T) X: new position of the particle starting from 0 P[X (x,x + dx)] = 1 2πT e x2 2T dx One step of random walk replaced by N steps X n+1 = X n + ǫ n, ǫ n N(0,T/N), n = 0,1,...,N 1, X 0 = 0, ǫ n independent P[X N (x,x + dx)] same as above Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.3/35
Probabilistic Approach to the Heat Equation (Continued) Let N, X t follows the process dx t = dw t, X 0 = 0 W t : standard Brownian motion u(x,t) = 1 2πt e x2 2t = P[Xt (x,x + dx)]/dx satisfies Heat kernel u t = 1 2 u xx, u(x,0) = δ(x) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.4/35
More general cases Itô process: dx t = a(x t,t)dt + σ(x t,t)dw t, X 0 = 0. Distribution of particles satisfies the Fokker-Planck equation u t = 1 2 (σ2 u) xx (au) x u(x,0) = δ(x) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.5/35
Killing of Particles x path t particle deleted! x = b 0 Boundary condition for u at x = b 0 : u(b 0,t) = 0, t > 0 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.6/35
Solution to the Model Problem Consider the special case a = 0, σ = const: u t = 1 2 σ2 u xx, x > b 0 u(x,0) = δ(x), u(b 0,t) = 0. Solution u(x,t) = 1 2πσ 2 t ] [e x2 2σ 2 t e (x 2b 0 )2 2σ 2 t, x b 0 Explicit solutions also available for linear b Standard existence theory for general barrier b(t) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.7/35
Survival Probability and Free Boundary Probability of survival up to t: Q(t) = b(t) u(x,t)dx, Q (t) < 0 Probability that X t has exited the barrier by t: Exit probability density: P(t) = 1 Q(t) = 1 b(t) udx P (t) = Q (t) = b(t) u t dx = 1 2 x (σ2 u) x=b(t) Can we find b(t) to generate a given exit probability density? Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.8/35
Application in Finance: Distance-to-default Model (Ref. Avellaneda and Zhu, Risk, 2001) Distance-to-default: X t = V t b(t) 0, V t,b(t): generalized value of the firm and liability dx t = b (t)dt + σ(x t,t)dw t Barrier interpretation: b(t), t 0 to be determined First exit time: τ = inf {t 0 : X t 0} u(x, t): survival probability density at t: u(x,t)dx = P [X t (x,x + dx), t < τ], x 0 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.9/35
Schematic illustration: credit index path t X t : distance-to-default barrier credit event Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.10/35
P (t) with Linear Barrier 0.1 0.09 Default Probability Density with Linear Barriers a=2.5, b=0 a=1.5, b=1 0.08 0.07 Prob Density 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (years) b(t) = α βt Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.11/35
Control Problem for u(x, t) Determine b so that the solution to u t = 1 2 (σ2 u) xx + b u x, x > 0, t > 0 with initial and boundary conditions: u t=0 = δ(x + b(0)) u x=0 = 0, t > 0 satisfies the additional BC data fitting 1 2 [ x ( σ 2 u )] x=0 = P (t), t > 0 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.12/35
Calibration: Free Boundary Problem Difficulties: Starting from a δ-function initial data Need to determine the correct b(t) Approaches: Initial layer: b(t),0 t t 0 for small t 0 approximated by a linear barrier Second-order finite difference method to solve for t > t 0, using u(x,t 0 ) from the initial layer as initial condition Matching two solutions at t = t 0 : Choose α, β so that P(t 0 ) = P(t 0 ), P (t 0 ) = P (t 0 ) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.13/35
Examples Default probabilities for the bank industry with ratings in AAA and BAA1. year AAA BAA1 1 0.0073 0.0222 2 0.0136 0.0285 3 0.0166 0.0315 4 0.0190 0.0339 5 0.0210 0.0360 6 0.0229 0.0380 7 0.0246 0.0396 8 0.0264 0.0415 9 0.0284 0.0437 10 0.0307 0.0466 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.14/35
Default Barriers for AAA and BAA1 Companies -1-1.5 AAA BAA1-2 b(t) -2.5-3 -3.5-4 -4.5 0 2 4 6 8 10 t Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.15/35
Questions Existence of solutions? P (t) > 0, P(t) 1, for t < T u 0 for all x 0? Stability of the barrier? Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.16/35
Blowup of Default Barrier b(t) 8 7 6 5 4 3 2 1 0-1 -2 0 2 4 6 8 10 t P = 0.1 P = 0.2 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.17/35
Linear Stability Analysis Small perturbations to P lead to small changes in b(t)? Perturbation analysis: for small ǫ P(t) = P 0 (t) + ǫp 1 (t) + O(ǫ 2 ) u(x,t) = u 0 (x,t) + ǫu 1 (x,t) + O(ǫ 2 ) b(t) = b 0 (t) + ǫb 1 (t) + O(ǫ 2 ) u 0 satisfies the equations with extra condition P 0 Goal: bound b 1 (t) in terms of P 1 (t) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.18/35
Perturbation Equation Assume σ = 1 u 1, if exists, should satisfy v t = 1 2 v xx + b 0v x + b 1u 0,x v x=0 = 0 v t=0 = 0 v x x=0 = 2P 1(t) b 1 chosen based on P 1(t) Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.19/35
Heat Kernel Consider problem for w(x,t,s), for arbitrary b 1 : w t = 1 2 w xx + b 0w x, t > s w x=0 = 0 w t=s = b 1(s)u 0,x (x,s) Express the solution w(x,t,s) = K b0 ( b 1(s)u 0,x (x,s) ) K b0 is the general time-dependent heat kernel Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.20/35
Duhamel s Principle Represent solution Compute u 1 x(0,t): u 1 (x,t) = t 0 w(x,t,s)ds t t 2P 1(t) = 0 w x (0,t,s)ds = 0 K b0,u 0 (t,s)b 1(s)ds b 1 and P 1 (t) related through an integral equation Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.21/35
Discussions Challenges: Extremely low default probability for short time horizon Propositions: Introduce time dependent volatility σ(t) Allow the barrier to be stochastic (Pan, 2001) Start from a probability distribution (Imperfect information) Add Poisson jumps Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.22/35
Compound-Poisson Process Features: Shocks (jumps) coming at uncertain times Markov process Z(t) : total number of occurrences before t, Intensity λ: P n (t) = P{Z(t) = n} = (λt)n n! e λt for small h, P 1 (h) = λh + o(h), P 0 (h) = 1 λh + o(h) Applications in finance: stock option pricing (Merton, 1976), reduced-form models Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.23/35
Combined Wiener-Poisson Process Discontinuous process: dx t = b dt + σdw t + dq t q t experiences jumps with intensity λ, Once a jump occurs, probability measure of the jump amplitude G(x,dy) = P[x (y,y + dy)] is given Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.24/35
Sample Paths 3 2.5 Typical Path for Poisson Process lambda=10 lambda=2 2 1.5 position 1 0.5 0-0.5-1 -1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.25/35
Infinitesimal Generator of the Process For the Poisson process, with small t > 0 E x [f(x t )] λt f(y)g(x,dy) + (1 λt)f(x) For the combined process E x [f(x t )] f(x) Af(x) = lim t 0+ t = 1 2 σ2 f xx b f x + λ (f(y) f(x)) G(x,dy). Kolmogorov backward equation: f t = Af Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.26/35
Forward Equation with Boundary Condition Adjoint operator (assuming G(x, dy) = g(x, y)dy): A u = 1 ( σ 2 u ) ] 2 xx + b (t)u x + λ[ u(y,t)g(y,x)dy u Boundary condition Forward equation (Fokker-Planck) u t = A u, x > 0, u(x,t) x=0 = 0, t 0. 0 Killing of particles: Q (t) = σ2 2 u x(0,t)+λ 0 [ u(y, t) 0 ] g(y,x)dx 1 dy < 0 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.27/35
Partial Integro-Differential Equation Assume g(y,x) = g(x y), u t = 1 2 (σ2 u) xx + b (t)u x λu + λ 0 u(y)g(x y)dy, x > 0 Boundary condition: u(0, t) = 0 Probability density function g(x) for jump size g(x) = 1 2πβ 2 e (x µ) 2β 2 2 Example: λ = λ 0 e t, µ = e 2t, β = 1 2 e 0.2t Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.28/35
Default Probability Density with Poisson Jumps dp/dt 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Default Probability Density 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t λ 0 = 0 λ 0 = 1 λ 0 = 2 λ 0 = 3 λ 0 = 4 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.29/35
Matching Condition at the Boundary Matching condition is nonlocal 1 σ 2( 2 u ) λ x x=0 0 0 u(y)g(x y)dydx = λ(p(t) 1)+P (t), t > 0 Finite difference-nyström approximation of the partial integro-differential equation Integral term treated as a source term Initial layer: a linear barrier for 0 < t < t 0 is sought to match data (P(t 0 ) and P (t 0 )), numerical solutions used Similar shooting technique to determine the free boundary for t > t 0 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.30/35
Example: Jump-diffusion Bank industry with AAA ratings year May 2004 Dec 2001 1 0.0058 0.0073 2 0.0094 0.0136 3 0.0115 0.0166 4 0.0135 0.0190 5 0.0155 0.0210 6 0.0171 0.0229 7 0.0190 0.0246 8 0.0210 0.0264 9 0.0232 0.0284 10 0.0256 0.0307 Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.31/35
Barriers with Jump-diffusion (1) b(t) 0-1 -2-3 AAA Banks, May 2004 λ 0 = 0 λ 0 = 1 λ 0 = 2-4 -5 0 2 4 6 8 10 t Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.32/35
Barriers with Jump-diffusion (2) AAA Banks, Dec 2001 0-1 λ 0 = 0 λ 0 = 1 λ 0 = 2 b(t) -2-3 -4-5 0 2 4 6 8 10 t Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.33/35
Survival density 20 15 solution profiles barrier 10 5 0-5 -10 0 2 4 6 8 10 12 t Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.34/35
Summary Allow jumps in distance-to-default Additional parameters to fit the data Partial integro-differential equation formulation Efficient and stable numerical solutions Stability analysis needed Study the equation with random killing term Build in correlation structures Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation p.35/35