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, ...

We test six term structure models in the Heath, Jarrow, and Morton (1992) class using Eurodollar futures and options data from 1987-1992. We study the time series of implied interest rate volatilities from these models. Using one-day lagged implied volatilities, our one-and two-parameter models simultaneously price zn average of 18.5 options each day with an average absolute error of one-and-a-half to two basis points. Although the models fit well, we document systematic strikeprice and time-to-maturity biases for all models. We also implement simple trading strategies to test whether the models identify genuine biases.

Abstract. Option market activity increases by more than 10 percent in the four days before quarterly earnings announcements. We show that the direction of this preannouncement trading foreshadows subsequent earnings news. Specifically, we find option traders initiate a greater proportion of long (short) positions immediately before “good” (“bad”) earnings news. Midquote returns to active‐side option trades are positive during nonannouncement periods and are significantly higher immediately prior to earnings announcements. Bid‐ask spreads for options widen during the announcement period, but traders do not gravitate toward high delta contracts. Collectively, the evidence shows option traders participate generally in price discovery (the incorporation of private information in price), and more specifically in the dissemination of earnings news.

We use an extension of the equilibrium framework of Rubinstein (1976) and Brennan (1979) to derive an option valuation formula when the stock return volatility is both stochastic and systematic. Our formula incorporates a stochastic volatility process as well as a stochastic interest rate process in the valuation of options. If the "mean," volatility, and "covariance" processes for the stock return and the consumption growth are predictable, our option valuation formula can be written in "preference-free" form. Further, many popular option valuation formulae in the literature can be written as special cases of our general formula. IN THIS PAPER, WE investigate the valuation of individual stock options when the volatility of the underlying stock return is stochastic and has a systematic component which is related to the stochastic volatility of the consumption growth or the market return. Such an investigation is timely and of significant interest because there is now a considerable amount of evidence that the volatility of stock prices is not only stochastic, but that it is also highly correlated with the volatility of the market as a whole. (For example, see Black (1975), Conrad, Kaul, and Gultekin (1991), Jarrow and Rosenfeld (1984), Jorion (1988), and Ng, Engle, and Rothschild (1992). Furthermore, empirical work by Black and Scholes (1972), Gultekin, Rogalski, and Tinic (1982), and Whaley (1982) has shown that the empirical biases inherent in the Black-Scholes option prices are different for options on high and low risk stocks. Because low risk stocks are mainly large firm stocks with stochastic return volatilities that are highly correlated with the return volatility of the market, and high risk stocks are mainly small firm stocks with a lessimportant systematic volatility component, incorporating systematic volatility effects in option valuation models has the potential to reduce the empirical biases exhibited by prices computed from the Black-Scholes formula.We use equilibrium arguments that extend the consumption-based representative agent framework of Rubinstein (1976), Brennan (1979), and Stapleton and Subrahmanyam (1984) to allow individual asset returns and the consumption growth to exhibit stochastic volatility. In particular, we permit the volatility of individual asset returns to consist of a systematic component that is related to the consumption (or market) volatility in addi-*School of Business Administration, University of Michigan. We are grateful to Robert Engle, Robert Jarrow, Stanley Kon, and Dilip Madan for valuable discussions. The usual disclaimer applies. 882The Journal of Finance tion to an idiosyncratic component. We also permit our framework to include a broad class of discrete-time processes that do not necessarily converge to diffusion processes.' Within this framework, we derive pricing formulae for calls and puts based on equilibrium arguments. Equilibrium arguments are necessary for option valuation since a perfect hedge for an option cannot be constructed in our general discre...

In this paper, we build a general framework to price contingent claims on foreign currencies using the Heath et al. (1987) model of the term structure. Closed form solutions are obtained for European options on currencies and currency futures assuming that the volatility functions determining the term structure are deterministic. As such, this paper provides an example of a bond price process (for both the domestic and foreign economies) consistent with Grabbe's (1983) formulation of the same problem.