## Multiperiod Options

We now extend the solution method to multiperiod options by working backward one step at a time,

A two-stage lattice representing a two-period call option is shown in Figure 12.6. It is assumed as before that the initial price of the stock is S, and this price is modified by the up and down factors u and d while moving through the lattice, The values shown in the lattice are those of the corresponding call option with strike price K and expiration time corresponding to the final point in the lattice The value of the option is known at the final nodes of the lattice. In particular,

We again define the risk-neutral probability as

R — d where R is the one-period return on the risk-free asset, Then, assuming that we do not exercise the option early (which we already know is optimal, but will demonstrate 12 .6 MULTIPERIOD OPTIONS 3.31

again shortly), we can find the values of C„ and Cd from the single-period calculation given earlier Specifically,

Then we find C by another application of the same risk-neutral discounting formula Hence,

For a lattice with more periods, a similar procedure is used. The single-period, risk-free discounting is just repeated at every node of the lattice, starting from the final period and working backward toward the initial time.

Example 12.3 (A 5-month call) Consider a stock with a volatility of its logarithm of a — ,20 The current price of the stock is \$62 The stock pays no dividends. A certain call option on this stock has an expiration date 5 months from now and a strike price of \$60 The current rate of interest is 10%, compounded monthly We wish to determine the theoretical price of this call using the binomial option approach

First we must determine the parameters for the binomial model of the stock price fluctuations. We shall take the period length to be 1 month, which means A/ — 1/12, • The parameters are found from Eqs (11 1) to be it = eaJTt = 1,05943

R = 1 + .1/12 = 1.00833. Then the risk-neutral probability is q = (R — c!)/(u -d) - .55770,,

We now form the binomial lattice corresponding to the stock price at the beginning of each of six successive months (including the current month) This lattice ts shown in Figure 12.7, with the number above a node being the stock price at that node. Note that an up followed by a down always yields a net multiple of 1.

Next we calculate the call option price. We start at the final time and enter the expiration values of the call below the final nodes.. This is the maximum of 0 and S — K For example, the entry for the top node is 82 75 — 60 = 22,75,

The values for the previous time are found by the single-step pricing relation. The value of any node at this time is the discounted expected value of two successive values at the next time. The expected value is calculated using the risk-neutral

 78.11 73.72 ^ ^18.60 69.59^ ^14 71^ \ 69 59 1119" ^ 65.68^ 0.08 62^ ^ 6 96 ^ v62 4 ^ 58 52 ^ 14 55.24 ^ -^55.24 96^ FIGURE 12.7 5-month call using a binomial lattice., The upper numbers are the stock prices, the lower numbers the option values The option values are found by working backward through the lattice probabilities q and 1 — q. For example, the value at the top node is [,,5577 x 22.75 + (1 - .5577) x 13.7.2]/I 008.33 = 18,60, We work toward the left, one period at a time, until finally the initial value is reached, In this case we conclude that the price of the option computed this way is \$5,85 Note that the entire process is independent of the expected growth rate of the stock This value only enters the binomial model of the stock through the probability p\ but this probability is not used in the option calculation. Instead it is the risk-neutral probability q that is used. Note, however, that this independence results from using the small At approximation for parameter matching. And indeed, in practice this approximation is almost invariably used (even for At equal to 1 year) If the more general matching formula were used, the growth rate would (slightly) influence the result 