In this chapter, we examine the pricing of derivatives with nonlinear payoffs, a special case of which are the payoffs of standard call and put options on stocks. Since the valuation methods are the same for all types of options, we will focus on stock options to understand better the pricing properties of general derivatives with nonlinear payoffs.


An option is a contract that gives the buyer the right, but not the obligation, to buy or to sell a particular asset (the underlying asset) on or before the option's expiration date, at an agreed price. The agreed price is called the strike price or exercise price. A call option is the right to buy the underlying asset and a put option is the right to sell it. The act of buying or selling the underlying asset by using the option is referred to as exercising the option. An American option can be exercised any time prior to or on the expiration date. A European option can be exercised only on the expiration date. Hence, there are four main types of options: an American call, an American put, a European call, and a European put.1

The option buyer must pay the seller of the option (also called the option writer) a fee to enter into the agreement. This fee is called the option price or option premium. Notice two important differences between a futures or forward contract compared to an option. First, in an option one of the parties must always compensate ...

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