The Black-Scholes Theory of Option Prices

Abstract

We now turn to the determination of the prices of derivative securities such as forwards, futures, or options in the presence of fluctuations in the price of the underlying. Such investments for speculative purposes are risky. Bachelier’s work on futures already shows that for relative prices, even the deterministic movements of the derivative are much stronger than those of the bond, and it seems clear that an investment into a derivative is then associated with a much higher risk (see also Bachelier’s evaluation of success rates) than in the underlying security, although the opportunities for profit would also be higher.

Derivative prices depend on certain properties of the stochastic process followed by the price of the underlying security. Remember from Chap. 2 that options are some kind of insurance: the price of an insurance certainly depends on the frequency of occurrence of the event to be insured. We therefore introduce the standard model of stock prices, as used in textbooks of quantitative finance [10], [12]–[16] and place this model in a more general context of stochastic processes.