THE IMPACT ON OPTION PRICING OF SPECIFICATION ERROR IN THE UNDERLYING STOCK PRICE RETURNS

Abstract

I. INTRODUCTION IN AN EARLIER PAPER,1 I briefly discussed the problem of errors in option pricing due to a misspecification of the stochastic process generating the underlying stock's returns. While there are many ways in which a specification error can be introduced, the particular form chosen in that paper was to compare the option prices arrived at by an investor who believes that the distribution of the unanticipated returns of the underlying stock is lognormal, and hence that he can use the classic Black-Scholes pricing formula2, with the correct option prices if the true process for the underlying stock is a mixture of a lognormal process and a jump process. This is a particularly important case because the nature of the error is not just one of magnitude, but indeed the qualitiative characteristics of the two processes are fundamentally different. In this paper, I examine the nature and magnitude of the error in a quantitative fashion using simulations. Before discussing the simulations, it is necessary to briefly summarize the option pricing results deduced in the earlier paper. At the heart of the derivation of the Black-Scholes option pricing formula is the arbitrage technique by which investors can follow a dynamic portfolio strategy using the stock and riskless borrowing to exactly reproduce the return structure of an option. By following this strategy in combination with a short position in an option, the investor can eliminate all risk from the total position, and hence to avoid arbitrage opportunities, the option must be priced such that the return to the total position must equal the rate of interest. However, for this arbitrage technique to be carried out, investors must be able to revise their portfolios frequently and the underlying stock price returns must follow a stochastic process that generates a continuous sample path. In effect, this requirement implies that over a short interval of time, the stock price cannot change by much. In my earlier paper I derived an option pricing formula when the sample path of the underlying stock returns does not satisfy the continuity property. In particular, it was assumed that the stock price dynamics can be written as a combination of two types of changes: (1) the normal vibrations in price, for examples, due to a temporary imbalance between supply and demand, changes in capitalization rates, changes in the economic outlook, or other new information that causes marginal * Professor of Finance, Massachusetts Institute of Technology. I thank J. Ingersoll for programming the simulations and general scientific assistance, and F. Black and M. Scholes for helpful discussions. Aid from the National Science Foundation is gratefully acknowledged.