Jump Diffusion Option Valuation in Discrete Time

Abstract

We develop a simple, discrete time model to value options when the underlying process follows a jump diffusion process. Multivariate jumps are superimposed on the binomial model of Cox, Ross, and Rubinstein (1979) to obtain a model with a limiting jump diffusion process. This model incorporates the early exercise feature of American options as well as arbitrary jump distributions. It yields an efficient computational procedure that can be implemented in practice. As an application of the model, we illustrate some characteristics of the early exercise boundary of American options with certain types of jump distributions. JUMP DIFFUSION PROCESSES ARE popular in option valuation models. These processes were incorporated by Merton (1976) into the theory of option valuation and constitute an important alternative to the lognormal diffusion process assumed in the Black and Scholes (1973) model. In this paper, we develop a tractable discrete time model for valuing options with these processes.Many authors have suggested that incorporating jumps in option valuation models may explain some of the large empirical biases exhibited by the Black-Scholes model.1 However, in many cases, closed form solutions for valuing options are not available. For example, this is true when there is a positive probability of early exercise, when the jump distribution is neither lognormal nor discrete, or when the option is based on the term structure of interest rates (see Ahn and Thompson (1988) and Jarrow and Madan (1991)). In these important cases, the model developed in this paper can be used to value options.Another aim of this paper is to provide a model which relies on only simple mathematics to price options with jump diffusion processes. It is hoped that the key economic relationships underlying jump diffusion models are transparent in this model. We provide enough details to make the procedure readily accessible. two anonymous referees, the editor, and the participants of the Finance Workshop at the University of Michigan and the 1992 Western Finance Association Meetings are sincerely appreciated. Taeyoung Chung and Charles Jones provided valuable research assistance. The usual disclaimer applies. For example, see Ball and Torous (1985), Jarrow and Rosenfeld (1984), and Jorion (1988). 1833 1834 The Journal of Finance To develop our model, we use the binomial model in Cox, Ross, and Rubinstein (1979), hereafter the CRR model, as the starting point. As in the CRR model, we assume that the stock price can either increase by an "uptick" or decrease by a "downtick" in each discrete period, where we define a tick to be the minimum possible change in the stock price. We term these stock price changes "local" price changes. In the continuous time case, a jump can instantaneously cause the stock price to undergo a large change of random size. Therefore, in the discrete time model, we also permit the stock price to change by multiple ticks in a single period. We term these price changes as discrete time jumps. In the discrete time context, ...

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A Jump-Diffusion Approach to Modeling Credit Risk and Valuing Defaultable Securities

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