BOCh15 - Multiple Periods in the Binomial Option Pricing...

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Multiple Periods in the Binomial Option Pricing Model 133 Chapter 15 Multiple Periods in the Binomial Option Pricing Model Contents 15.1 Multiple Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 15.2 The Binomial Formula and the Black Scholes Model . . . . . . . . . 137 15.3 Early Exercise of Puts in the Binomial Model . . . . . . . . . . . . . 138 15.4 Adjusting for Dividends in the Binomial Model . . . . . . . . . . . . 139 15.5 Implementing the binomial option formula. . . . . . . . . . . . . . . 141 As we saw in the previous chapter, the binomial framework allows us to find exactly what the state price probabilities must be to avoid arbitrage. But the setting that has turned the binomial option pricing model into a workhorse of option pricing is one with multiple periods. 15.1 Multiple Periods The basic assumption is still that every period, the price of the underlying stock can either jump up by a multiplicative factor or down by a factor .
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134 Multiple Periods in the Binomial Option Pricing Model But we also assume that this state of a ff airs persists over time. Each period there is again two branches with jumps defined by the same factors and : ✏✏ ✏✏ Note that since , the four nodes in period two really are only three distinct nodes
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15.1 Multiple Periods 135 The pricing exercise of the previous chapter can be repeated for every node in a “binomial tree” constructed by replicating the one-period model. To determine the value at a particular point in time, one starts at the end of the tree, where the values of the call option are simply determined as the maximum of the stock price minus strike price, or zero. One then works backward through the tree, determining values as in the previous section. Let us take the two period example above, where there are three possible stock prices at time 2: , and . The option price changes from to either or at time 1. Each of these prices is the value of a one period option expiring one period later, at time 2. At time 2 we know the payo ff s of the option. To price we “work backwards.” We start with the two possible call values at time 1. These can be calculated using the one period method we used in the previous chapter. Then, given the two possible values at time 1, we are again working with a one period problem, a derivative with two possible payo ff s and at time 1.
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