Option pricing when underlying stock returns are discontinuous

The validity of the classic Black-Scholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the more-general cast when the underlying stock returns are gcncrated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black&holes formula in that it does not dcpcnd on investor prcfcrenccs or knowledge of the expcctsd return on the underlying stock. Morcovcr, the same analysis applied to the options can bc extcndcd to the pricingofcorporatc liabilities. 1. Intruduction In their classic paper on the theory of option pricing, Black and Scholcs (1973) prcscnt a mode of an:llysis that has rcvolutionizcd the theory of corporate liability pricing. In part, their approach was a breakthrough because it leads to pricing formulas using. for the most part, only obscrvablc variables. In particular, their formulas do not rcquirc knowledge of tither investors ’ tastes or their beliefs about expcctcd returns on the underlying common stock. Moreover, under specific posited conditions, their formula must hold to avoid the creation of arbitrage possibilities.’ To derive the option pricing formula, Black and Scholes ’ assume ‘ideal conditions ’ in the market for the stock and option. These conditions are: ‘An earlier version of this paper with the same title appeared as a Sloan School of Manage-ment Working Paper #787-75 (April 1975). Aid from the National Science Foundation is gratefully acknowledged. ‘For an alternative derivation of the Black-Scholes model and a discussion of option pricing models in general, see Mcrton (I973b). For applications of the Black-Scholcr tcchniquc t