An Elementary Introduction to Mathematical Finance

An option gives one the right, but not the obligation, to buy or sell a security under specified terms. A call option is one that gives the right to buy, and a put option is one that gives the right to sell the security. Both types of options will have an exercise price and an exercise time. In addition, there are two standard conditions under which options operate: European options can be utilized only at the exercise time, whereas American options can be utilized at any time up to exercise time. Thus, for instance, a European call option with exercise price K and exercise time t gives its holder the right to purchase at time t one share of the underlying security for the price K, whereas an American call option gives its holder the right to make the purchase at any time before or at time t.

A prerequisite for a strong market in options is a computationally efficient way of evaluating, at least approximately, their worth; this was accomplished for call options (of either American or European type) by the famous Black–Scholes formula. The formula assumes that prices of the underlying security follow a geometric Brownian motion. This means that if S(y) is the price of the security at time y then, for any price history up to time y, the ratio of the price at a specified future time t + y to the price at time y has a lognormal distribution with mean and variance parameters tμ and tσ2, respectively.