Arrow-Debreu Prices for Affine Models

Abstract

We put forward a general methodology to price arbitrary payoffs linked to the realization of interest rates, asset prices, or other variables driven by the multivariate Affine Jump-Diffusion process of Duffie and Kan (1996). We attack and solve the basic problem of computing the Arrow-Debreu state prices or, equivalently, Green's functions associated with the process. Given the Arrow-Debreu state prices, one can price derivative instruments with payoffs of arbitrary complexity. Within this framework, we also develop a scheme to price derivatives with early exercise at intermediate dates. To derive Arrow-Debreu state prices we exploit the basic observation that the integral of the overnight interest rate is itself affine. We augment the state space to add the integral of the overnight rate and we use transform methods to compute the density of the augmented affine process to calculate Arrow-Debreu prices. The main goal of the paper is to provide a viable numerical implementation of the proposed methodology, and we illustrate with applications the concepts introduced below. Our primary interest lies in exploring the viability of the numerical implementation, and we will measure advantages and disadvantages of our approach in the associated metric. The method is well suited to price payoffs for which transform methods as, e.g., in Chacko and Das (1999) and Duffie, Pan, and Singleton (1998), cannot be applied. This is typically the case when payoffs are non-linear or non-loglinear in the underlying factors. While the techniques we exploit rely in essence on transform methods, this paper should be of interest also to researchers who prefer simulation or tree-based implementations. A scheme for improving the accuracy of tree-based methods is presented. In a similar vein, we suggest a simulation procedure for the general Affine Jump-Diffusion model, which recovers arbitrage-free prices regardless of the time step. In this context, the proposed methodology can serve as a tool to detect problems in alternative implementations. Consider the case of a jump for instance; our method suggests that the resulting distribution can be multimodal. It is difficult to envision that a tree-based implementation would easily recover the correct state prices without some form of tinkering with the implementation.