## More General Binomial Model

The Black-Scholes model (see Section 16.8) is undoubtedly the most popular option pricing model in current use. However, many find the underlying mathematics too demanding and prefer to use the simpler Binomial option pricing model, which has gained considerable popularity. The methodology for the Binomial model was developed by Cox, Ross and Rubinstein and published in 1979.

The essential difference between this model and the Black-Scholes model is in the underlying model for the evolution of stock prices over time. Recall that, it is necessary to solve equations (16.1) and (16.2), i.e. solve

According to the methodology of the Binomial model, divide the time to maturity, t, into n discrete time intervals of equal length t/h. Thus, each interval of time may be indexed from 1 up to n. Assume that during each time interval, the stock price may rise by a multiple of eu (greater than 1) with probability p, or fall by a multiple of ev (less than 1) with probability 1 — p. Then over one time interval:

St +1 = Steu with probability p

= Stev with probability 1 — p u, v and p are constants satisfying 0 < p <1 and v < tt/h < u. Over two intervals

Over two intervals there are three possible outcomes. The stock price increases in both intervals, the price increases then decreases (or decreases then increases), or the price decreases in both intervals.

Continuing the process, it can be seen that after k discrete time intervals the stock price, St + k, takes the value

St + k = Steju + (k-j)v with probability kQpj(1 -p)k-j j = 0, 1, ..., k (16.3)

where j is the number of upward jumps (thus k -j is the number of downward jumps). The parameter j has a binomial distribution (see Chapter 9 for a discussion of the binomial distribution).

Time at maturity is reached after n time intervals, therefore ST = St + n. From equation (16.3) we know that

St =Steju + (n-j)v with probability nCjp(1 -p)n-j j =0, 1, ..., n (16.4)

Equation (16.4) gives the distribution of stock prices at maturity. Taking expectations n

E [St] =Y, nCjpj(1 - p)" -jSteju +(n - j)v j = 0

and equation (16.4) together with equations (16.1) and (16.2) leads to n

Ct = e-rT nCjpj(1 - p)n -j max{0, Steju +(n -j)v - X} (16.5)

and n

Pt = e-rT Y, nCjpj(1 - p)n - j max{0, X - Steju +(n -j)v} (16.6)

Equations (16.5) and (16.6) are the call and put formulae given by the Binomial option pricing model, assuming there are no dividends payable on the underlying stock prior to maturity. To complete the model, we need to know the value of p and need to estimate the parameters u, v and n.

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