Derivative securities and the fundamental theorem of asset pricing

One of the most widely studied problems in financial mathematics is the pricing of derivative securities, also known as contingent claims. These are securities whose price depends on the value of another underlying security. Financial options are the most common examples of derivative securities. For example, a European call option gives the holder the right to purchase an underlying security for a prescribed amount (called the strike price) at a prescribed time in the future, known as the expiration or exercise date. The exercise date is also known as the maturity date of the derivative security. Recall the similar definitions of European put options as well as American call and put options from Section 1.3.2.

Options are used mainly for two purposes: speculation and hedging. By speculating on the direction of the future price movements of the underlying security, investors can take (bare) positions in options on this security. Since options are often much cheaper than their underlying security, this bet results in much larger earnings in relative terms if the price movements happen in the expected direction compared to what one might earn by taking a similar position in the underlying. Of course, if one guesses the direction of the price movements incorrectly, the losses are also much more severe.