Credit Risk: Pricing, Measurement, and Management

By David Lando | Go to book overview

Appendix E
A Term-Structure Workhorse

This appendix gives completely explicit closed-form solutions to certain functionals of a univariate affine jump-diffusion with exponentially distributed jumps. It includes as special cases the classical models of Vasicek (1977) and Cox et al. (1985), along with the solutions found in the appendix of Duffie and Gârleanu (2001). The tools for the analysis are provided in Duffie et al. (2000), where multivariate affine models are also covered, but the explicit closed-form solutions presented here are, as far as we know, due to Christensen (2002). While it is reasonably fast to implement the ODEs which appear in affine modeling, it is still much less of a hurdle and faster computationally to simply copy down a closed-form expression given here. We also provide a completely closed-form solution for a characteristic function that is important for determining option prices and threshold-dependent payoffs.

We take as given a probability space (Ω, F, P). P can be both a physical measure governing the true transition dynamics of our state variable, or it can be a martingale measure used for computing prices. Whenever we work with affine models (or quadratic models) and are concerned both with pricing and estimation, we typically pick the density process in such a way that the dynamics are affine (or quadratic) under both the physical measure and the martingale measure. Hence the formulas derived here will be useful for calculating actual survival probabilities, implied survival probabilities, bond prices, and prices of derivative securities.

The workhorse is the following affine jump-diffusion:

where

and Nt is a Poisson process with intensity

is a sequence of i.i.d. exponentially distributed random variables with

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