Title:

Numerical solutions to a class of stochastic partial differential equations arising in finance

We propose two alternative approaches to evaluate numerically credit basket derivatives in a Nname structural model where the number of entities, N, is large, and where the names are independent and identically distributed random variables conditional on common random factors. In the ﬁrst framework, we treat a Nname model as a set of N Bernoulli random variables indicating a default or a survival. We show that certain expected functionals of the proportion L_{N} of variables in a given state converge at rate 1/N as N [right arrow  infinity]. Based on these results, we propose a multilevel simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a meansquare error of epsilon ^{2} and computational complexity of order epsilon^{−2}, independent of N. In particular, this optimal complexity order also holds for the inﬁnitedimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives. In the second framework, we extend the approximation of Bush et al. [13] to a structural jumpdiﬀusion model with discretely monitored defaults. Under this approach, a Nname model is represented as a system of particles with an absorbing boundary that is active in a discrete time set, and the loss of a portfolio is given as the function of empirical measure of the system. We show that, for the inﬁnite system, the empirical measure has a density with respect to the Lebesgue measure that satisﬁes a stochastic partial differential equation. Then, we develop an algorithm to efficiently estimate CDO index and tranche spreads consistent with underlying credit default swaps, using a ﬁnite diﬀerence simulation for the resulting SPDE. We verify the validity of this approximation numerically by comparison with results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the ﬂexibility of the model and its extensions to match CDO spreads from precrisis and crisis periods.
