Using Hermite Expansions for Fast and Arbitrarily Accurate Computation of the Expected Loss of a Loan Portfolio Tranche in the Gaussian Factor Model

We propose a fast algorithm for computing the expected tranche loss in the Gaussian factor model with arbitrary accuracy using Hermite expansions. No assumptions about homogeneity of the portfolio are made. The algorithm is a generalization of the algorithm proposed in [4]. The advantage of the new algorithm is that it allows us to achieve higher accuracy in almost the same computational time. It is intended as an alternative to the much slower Fourier transform based methods [2]. 1 The Gaussian Factor Model Let us consider a portfolio of N loans. Let the notional of loan i be equal to the fraction fi of the notional of the whole portfolio. This means that if loan This work was supported by the Director, Office of Science, Office of Advanced Scientific Computing Research, of the U.S. Department of Energy under Contract No. DEAC03-76SF00098. E-mail: [email protected] 1 i defaults and the entire notional of the loan is lost the portfolio loses fraction fi or 100fi% of its value. In practice when a loan i defaults a fraction ri of its notional will be recovered by the creditors. Thus the actual loss given default (LGD) of loan i is LGDi = fi(1− ri) (1) fraction or LGDi = 100fi(1− ri)% (2) of the notional of the entire portfolio. We now describe the Gaussian m-factor model of portfolio losses from default. The model requires a number of input parameters. For each loan i we are give a probability pi of its default. Also for each i and each k = 1, . . . , m we are given a number wi,k such that ∑m k=1 w 2 i,k < 1. The number wi,k is the loading factor of the loan i with respect to factor k. Let φ1, . . . , φm and φ, i = 1, . . . , N be independent standard normal random variables. Let Φ(x) be the cdf of the standard normal distribution. In our model loan i defaults if m