Continuous Time Option Pricing
The binomial option pricing model (BOPM), produces reasonably accurate option values if the user has accurate beliefs about the values of u and d. Also, the BOPM is extremely flexible. For example, the BOPM can be used to value both European and American puts and calls, either with or without dividend payments. The possibility of early exercise can be accounted for in each period, and the values of u, d, and r can change over time. The primary drawback of the BOPM is that a computer must be used to estimate option values when the time to maturity is carved into many small periods.
Under certain assumptions and when time to maturity is carved into an infinite number of subintervals, the BOPM will converge to the option pricing model attributed to Fischer Black and Myron Scholes. The Black-Scholes call option pricing model (henceforth, the BSOPM) provides an explicit solution to the problem of option pricing.1 By “explicit,” we mean that an equation is obtained.
Although the BSOPM is relatively simple to use, it is important to be aware that the model can accurately value only European options, or American calls on non-dividend-paying stocks. As the possibility of early exercise becomes more likely, the BSOPM produces increasingly inaccurate values. Moreover, an important assumption of the BSOPM is that the wanderings of the stock price through time follow one particular type of pricing process, which cannot change over time.
In this chapter, we first present a set of sufficient assumptions that will lead to the derivation of the BSOPM. Then, the model itself is stated, and a numerical example is shown in detail. Following that, we examine a few details concerning the assumed stochastic process guiding the value of the underlying asset. It is important to realize that a model generally will produce useful results only if the assumptions behind it are realistic. If the assumptions are violated in the real world, the model will frequently provide poor predictions.
Although other sets of assumptions have been used to derive the BSOPM, the following assumption list is sufficient.