Path-Dependent Option Valuation When the Underlying Path Is Discontinuous

The payo s of path-dependent options depend not only on the nal values, but also on the sample paths of the prices of the underlying assets. A rigorous modeling of the underlying asset price processes which can appropriately describe the sample paths is therefore critical for pricing path-dependent options. This paper allows for discontinuities in the sample paths of the underlying asset prices by assuming that these prices follow jump di usion processes. A general yet tractable approach is presented to value a variety of path-dependent options with discontinuous processes. The numerical examples show that ignoring the jump risk may lead to serious biases in pathdependent option pricing. Jumps are an important feature of the price processes of nancial assets, and are especially pronounced for certain types of assets, such as small-cap stocks and exchange rates. 1 For this reason, Merton's (1976) groundbreaking work, which explicitly admits jumps in the underlying asset prices for pricing standard European options, has generated a profound impact on the nance profession. Merton's work was done a couple of decades ago. In recent years, the variety of new option contracts has increased enormously. Many types of so-called exotic options are now popular items in the over-the-counter market. Most exotic options, such as barrier options, lookbacks, Asian options, American capped options, and many others, are path-dependent, that is, their payo s depend not only on the nal values, but also on the sample paths of the underlying asset prices. For these options, a rigorous consideration of jumps in the sample paths of the underlying asset prices seems more important than for the standard non-pathdependent options in Merton's world. But these new path-dependent options are often so complicated that they cannot be priced in Merton's framework. For this reason, this paper presents a new general yet tractable approach to value a variety of path-dependent options whose underlying prices follow jump-di usion processes. 2 If a jump-di usion process of the underlying asset price is misspeci ed as a pure diffusion process, what will happen to the pricing performance of the option pricing model? For a given option, say a down-and-out call, will it be overpriced, or underpriced by the model due to the misspeci cation of the price process? To answer these questions, several numerical examples are provided on barrier options and lookback options. The examples show that for the path-dependent options, ignoring the jump risk often leads to serious biases in the pricing of both short maturity and long maturity options. The examples also show some interesting and sometimes surprising maturity patterns of pricing errors See, Kon (1984), Ball and Torous (1985), Jorion (1988), and Bates (1991, 1996), Das, Foresi and Sundaram(1996). In a recent paper, Amin (1993) provided a discrete time approach to value one kind of path dependent option| an American option under jump di usion processes. The approach presented here is much di erent from Amin's approach.