The Binomial Option Pricing Model
At expiration, we know that an option is worth its intrinsic value. Thus, at expiration, an option's intrinsic value serves as an exact option pricing model. However, it is also important to be able to value options before expiration. While the put-call parity relationship does generate put and call option price boundaries before expiration, an important condition of the put-call parity relationship is that a put price must be known before a call price can be generated. Thus, although put-call parity and other pricing bounds shown in Chapter 16 narrowed the range of possible option prices, we still do not have a model that provides an exact option price before expiration.
In this chapter, we derive a model that provides an exact option price before expiration. This model is called the binomial option pricing model (BOPM). One of the advantages of this particular model is its flexibility.1 The binomial option pricing model can be used to value complex options of many types, including American puts, American calls on dividend-paying stocks, options on debt instruments and interest rates, and exotic options. It can also accommodate changing conditions over time. For example, if interest rates are believed likely to change during the life of the option, or if the volatility of the underlying asset is believed likely to change, the BOPM can handle these situations quite easily. Even more, the BOPM can account for different assumed stock price movement processes, such as one in which the variance of the stock's returns is greater at lower price levels (constant elasticity of variance), or if there is a probability of a “jump” (a discontinuity, perhaps due to a possible tender offer for the stock) in the stock's price at each date.2
Studying the BOPM in detail also allows us to achieve additional understanding of several aspects of options that we have already learned. For example, the BOPM clarifies conditions under which options will be exercised early. Moreover, it validates the concept that buying a call is like buying stock and borrowing, and the concept that buying a put is like selling stock short and lending. The idea that options can be replicated with portfolios of stocks and bonds, which has revolutionized the field of finance, permits institutions to hedge their option portfolios. Option replication gave rise to the portfolio insurance industry that had an estimated size of $80 billion in September 1987 and has even been blamed for the stock market crash of October 1987.3 Finally, under certain assumptions, the BOPM can be shown to converge to several different option pricing models. These include the Black-Scholes option pricing model (BSOPM), which is the most widely used option pricing model, and the subject of the next chapter.
Before beginning with the details of the BOPM, take about 30 minutes and answer the following questions. Doing so will make the subsequent material easier to follow. The answers are at the end of the quiz, but try to work through the questions independently.