The Black-Scholes Option Model

The Black-Scholes model is neat and intuitive. It describes a process for calculating the fair value of a European call option, but one of its many attractions is that it can easily be modified to handle other types, such as foreign-exchange or interest rate options. 

Incorporated in the model are certain assumptions. For instance, apart from the price of the underlying asset, S, and the time, t, all the variables in the model are assumed to be constant, including, most crucially, the volatility variable. In addition, the following assumptions are made  

The behavior of underlying asset prices follows a geometric Brownian motion, or Weiner process, with a variance rate proportional to the square root of the price. This is stated formally in (8.11). 

S = the underlying asset price a = the expected return on the underlying asset b = the standard deviation of the asset s price returns t = time 

The following section presents an intuitive explanation of the B-S model, in terms of the normal distribution of asset price returns. 

Most option pricing models use one of two methodologies, both of which are based on essentially identical assumptions. The first method, used in the Black-Scholes model, resolves the asset-price model s partial differential equation corresponding to the expected payoff of the option. The second is the martingale method, first introduced in Harrison and Kreps (1979) and Harrison and Pliska (1981). This derives the price of an asset at time 0 from its discounted expected future payoffs assuming risk-neutral probability. A third methodology assumes lognormal distribution of asset returns but follows the two-step binomial process described in chapter 11. 

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