In the previous sections, I discussed the use of the Black-Scholes model and how it can be adapted to value options on indices, currencies, swaps, and bonds. In all these discussions, one common underlying thread was that the options could only be exercised on the option maturity date. In this section, I will discuss the valuation of early-exercise options restricting my discussion to nondividend-paying stocks.

Valuing options that allow for early exercise can be a tricky proposition since one has to decide at the time of exercise if it is economically better to exercise the option and collect the proceeds or delay the exercise. Given this backdrop, one can expect the valuation of early-exercise features to be more numerically intensive, simply because this amounts to solving a multidimensional integration problem. To see this, note that in valuing European-style options, one calculates the expectation of the option payoff with respect to the stock price on option maturity – which is essentially a single-variate integration problem. When early-exercise features are permitted, one would need to compute the value of the payoffs at the times of exercise (which would imply the need to integrate with respect to the stock prices at such times) resulting in a multivariate integration problem.^{[1]}

Cox, Ross, and Rubinstein (1979) solved the issue of early exercise using an intuitive approach. By breaking down stock price movements over time

FIGURE 3.1 Illustration of a Stock-Price Tree

into a two-state (up and down) process, and assuming that stock prices recombine (i.e., going up and then coming down is the same as going down and then up), they construct a stock-price tree to value the option. Mathematically they assumed that a stock price at time τ (i.e., ST where t < τ <T) can either move up to a level uST with probability p or to a level dST with probability (1 – p) at time τ + At. By matching the moments of the distribution, they go on to show that

(3.14a)

(3.14b)

(3.14c)

Figure 3.1 shows the generation and the recombination of the stock- price tree for the first three time steps At, 2At, and 3At.

Table 3.11 shows the implementation of the stock-price tree illustrated in Figure 3.1.

The stock price tree can now be used to value the early exercise feature as shown in Table 3.12.

As can be seen in Table 3.12, the payoffs in the option tree at time to maturity (i.e., when time is two years) is simply the payoff associated with holding the option to expiry. Using that as an initial point, the authors use backward induction to work back in time by rolling down the tree. Thus, at the times prior to the option maturity (e.g., when time is 0.66 years), the option holder gets to either exercise (in which case the option pays off) or

TABLE 3.11 Stock Price Tree

TABLE 3.12 Option Price Tree

continue (in which case it would be simply holding the value of the option). At the initial time node (when time is 0 years), no exercise is carried out and hence this is simply the present value of the probability-weighted continuation along the tree (holding value of the option). It should be noted that this binomial tree could also be used to value European-style options by simply forcing the exercise value to be 0 at times 0.66 years and 1.32 years. The final option premium is obtained by making the tree more dense (i.e., Δt → 0). The reader is referred to Hull (2012) for details.

To extend the methodology to value the early-exercise features for options on other underlyings, one has to make the appropriate changes in the way the variables u, d, and p are formulated.

[1] To see this, consider the simple example when there is only one exercise time (on option maturity, T). The payoff to the call option holder at this time is. Thus to compute the option value, one has to compute, whererepresents the conditional density of the stock price at time T given the value of the stock price at time t (which is the standard probability density function for ST) – a single variate integral. Now suppose that the option holder is also allowed to exercise at time u, where. Then at time u, the option holder's payoff is Thus, the value of the option is given by the expression – a bivariate integral. It can be easily inferred that as the number of exercise points increase so does that dimensionality of the integral – making this a computationally intensive proposition.

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