During the past 10 years, options that possess path-dependent features have become mainstays of the over-the-counter derivatives market. Among such features are:
· barriers which must or must not be crossed for a standard option to pay off,
· terminal values calculated on the basis of averages of the underlying asset price over a specified period, and
· payoffs which depend explicitly on extreme values (maxima or minima) of the underlying price process.
If the underlying asset price follows a geometric Brownian motion with constant parameters and sampling for extrema or averages is assumed continuous, closed form solutions for many of these options may be obtained. Unfortunately, most real-world options contracts specify discrete, often infrequent, fixing dates for these path dependencies. Further, parameters such as rates and volatilities often cannot be assumed constant.
In this paper, the pricing of a variety of discretely sampled path-dependent options is examined in an extended Black-Scholes-Merton framework that explicitly permits time-dependent parameters. First, formal closed-form results for a number of extremum-dependent options are obtained; these depend explicitly on multivariate cumulative normal integrals and are evaluated essentially exactly for cases of few (n < 6) sample points. It is found that discrete sampling leads to large price effects. Second, it is shown that the independent increments property of the price process allows a factorization of the payoff formulae for many of these options that permits its evaluation using repeated convolutions. Third, this approach is implemented using the fast Fourier transform to obtain rapidly convergent values. It is shown that discreteness effects can be large even for frequent (daily) sampling of extrema.
An interesting by-product of this approach is an explicit numerical representation of the probability density of discretely sampled extrema.
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