In this section, I review the assumptions used by Black-Scholes to obtain their famous option pricing formulae and explain the implications of violating them. These assumptions can be more succinctly summarized as follow:

Stock Price Process

In modeling stock price movements, Black-Scholes assumed that

(7.1)

where

S is the price of the stock.

μ is the annualized, continuously compounded growth rate. σ is the annualized volatility of the stock price return. dz is the random variable drawn from a standard normal probability density function.

dS is the small change in the stock price over a small time interval. μ and σ are both assumed to be constants.

TABLE 7.8 Settlement Associated with a Delta-Hedging Program for Vanilla European-Style Put Option on a Dividend-Paying Stock (In-The-Money Expiry)

As mentioned previously, the implication of this assumption is that In ST is normally distributed with a mean of , and a variance of where

St is the price of the stock at time t. t is the current time.

T is the future time.

The lognormal distributional assumption used to describe future stock price movements is only an assumption because:

■ Stock prices do not exactly follow this distribution in practice.

■ μt,T and σt,T are never constants.

Given this backdrop, it is not unreasonable for a practitioner to use:

■ Implied market information so as to ensure that the simulated stock price is consistent with the market view (i.e., future stock prices are not lognormally distributed).

■ Historical information to estimate parameters of a guessed probability density function (pdf) for price returns or trying to fit distributions on price-return data so as to derive an empirical pdf that would serve as a great proxy for future price returns.

■ A mixture of pdfs (e.g., a mixture of lognormals or mean-reverting double lognormals, and so on) or the standard diffusion equation for movements in stock price but with expressions for μ and σ that are functions of the underlying stock price and time.

Whatever the methodology, the reader should note that the consequences of using a different return distribution to simulate future stock price movements are completely different simulated stock price paths and hence different option values.

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