## S Z

It follows that if

D^sXd-^'"-'"-1») it is not optimal to exercise at time i„_i. Similarly, for any / < n, if

it is not optimal to exercise at time i,.

The inequality in (10.31) is approximately equivalent to

Assuming that X is fairly close to the current stock price, the dividend yield on the stock would have to be either close to or above the risk-free rate of interest for this inequality not to be satisfied. This is not usually the case.

We can conclude from this analysis that, in most circumstances, the only time that needs to be considered for the early exercise of an American call is the final ex-dividend date, t„. Furthermore, if inequality (10.31) holds for i = 1, 2,..., n — 1 and inequality (10.29) holds, we can be certain that early exercise is never optimal.

### Blacks Approximation

Black suggests an approximate procedure for taking account of early exercise.17 This involves calculating, as described earlier in this section, the prices of European options that mature at times T and f„, and then setting the American price equal to the greater of the two. This approximation seems to work well in most cases. A more exact procedure suggested by Roll, Geske, and Whaley is given in Appendix 10B.18

17See F. Black, "Fact and Fantasy in the Use of Options," Financial Analysts Journal, 31 (July-August 1975), 36-41, 61-72.

18 See R. Roll, "An Analytic Formula for Unprotected American Call Options on Stocks with Known Dividends," Journal of Financial Economics, 5 (1977), 251-58; R. Geske, "A Note on an Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends," Journal of Financial Economics, 7 (1979), 375-80; R. Whaley, "On the Valuation of American Call Options on Stocks with Known Dividends," Journal of Financial Economics, 9 (June 1981), 207-11; R. Geske, "Comments on Whaley's Note," Journal of Financial Economics, 9 (June 1981), 213-15.

### Example 10.8

Consider the situation in Example 10.7, but suppose that the option is American rather than European. In this case D\ = D2 = 0.5, S = 40, X = 40, r = 0.09, t\ occurs after 2 months and t2 occurs after 5 months.

Since this is greater than 0.5, it follows [see inequality (10.31)] that the option should never be exercised on the first ex-dividend date.

Since this is less than 0.5, it follows [see inequality (10.29)] that when it is sufficiently deeply in-the-money, the option should be exercised on its second ex-dividend date.

We now use Black's approximation to value the option. The present value of the first dividend is

so that the value of the option on the assumption that it expires just before the final ex-dividend date can be calculated using the Black-Scholes formula with S = 39.5074, X = 40, r = 0.09, a = 0.30, and T-t — 0.4167. It is $3.52. Black's approximation involves taking the greater of this and the value of the option when it can only be exercised at the end of 6 months. From Example 10.7 we know that the latter is $3.67. Black's approximation therefore gives the value of the American call as $3.67.

Whaley19 has tested empirically three models for the pricing of American calls on dividend-paying stocks: (1) the formula in Appendix 10B; (2) Black's model; and (3) the European option pricing model described at the beginning of this section. He used 15,582 Chicago Board options. The models produced pricing errors with means of 1.08 percent, 1.48 percent, and 2.15 percent, respectively. The typical bid-ask spread on a call option is greater than 2.15 percent of the price. On average, therefore, all three models work well and within the tolerance imposed on the options market by trading imperfections.

Up to now, our discussion has centered around American call options. The results for American put options are less clear cut. Dividends make it less likely that an American put option will be exercised early. It can be shown that it is never worth exercising an American put for a period immediately prior to an ex-dividend date.20 Indeed, if

19 See R. E. Whaley, "Valuation of American Call Options on Dividend Paying Stocks: Empirical Tests," Journal of Financial Economics, 10 (March 1982), 29-58.

20See H. E. Johnson, "Three Topics in Option Pricing," Ph.D. thesis, University of California, Los Angeles, 1981, p. 42.

### Example 10.8

Consider the situation in Example 10.7, but suppose that the option is American rather than European. In this case D\ = D2 = 0.5, S = 40, X = 40, r = 0.09, t\ occurs after 2 months and t2 occurs after 5 months.

Since this is greater than 0.5, it follows [see inequality (10.31)] that the option should never be exercised on the first ex-dividend date.

Since this is less than 0.5, it follows [see inequality (10.29)] that when it is sufficiently deeply in-the-money, the option should be exercised on its second ex-dividend date.

We now use Black's approximation to value the option. The present value of the first dividend is

so that the value of the option on the assumption that it expires just before the final ex-dividend date can be calculated using the Black-Scholes formula with S = 39.5074, X = 40, r = 0.09, a = 0.30, and T — t = 0.4167. It is $3.52. Black's approximation involves taking the greater of this and the value of the option when it can only be exercised at the end of 6 months. From Example 10.7 we know that the latter is $3.67. Black's approximation therefore gives the value of the American call as $3.67.

Whaley19 has tested empirically three models for the pricing of American calls on dividend-paying stocks: (1) the formula in Appendix 10B; (2) Black's model; and (3) the European option pricing model described at the beginning of this section. He used 15,582 Chicago Board options. The models produced pricing errors with means of 1.08 percent, 1.48 percent, and 2.15 percent, respectively. The typical bid-ask spread on a call option is greater than 2.15 percent of the price. On average, therefore, all three models work well and within the tolerance imposed on the options market by trading imperfections.

Up to now, our discussion has centered around American call options. The results for American put options are less clear cut. Dividends make it less likely that an American put option will be exercised early. It can be shown that it is never worth exercising an American put for a period immediately prior to an ex-dividend date.20 Indeed, if

19See R. E. Whaley, "Valuation of American Call Options on Dividend Paying Stocks: Empirical Tests," Journal of Financial Economics, 10 (March 1982), 29-58.

20See H. E. Johnson, "Three Topics in Option Pricing," Ph.D. thesis, University of California, Los Angeles, 1981, p. 42.

### Example 10.8

Consider the situation in Example 10.7, but suppose that the option is American rather than European. In this case D\ = D2 = 0.5, S = 40, X = 40, r = 0.09, t\ occurs after 2 months and t2 occurs after 5 months.

Since this is greater than 0.5, it follows [see inequality (10.31)] that the option should never be exercised on the first ex-dividend date.

Since this is less than 0.5, it follows [see inequality (10.29)] that when it is sufficiently deeply in-the-money, the option should be exercised on its second ex-dividend date.

We now use Black's approximation to value the option. The present value of the first dividend is

so that the value of the option on the assumption that it expires just before the final ex-dividend date can be calculated using the Black-Scholes formula with S = 39.5074, X = 40, r = 0.09, a = 0.30, and T — t = 0.4167. It is $3.52. Black's approximation involves taking the greater of this and the value of the option when it can only be exercised at the end of 6 months. From Example 10.7 we know that the latter is $3.67. Black's approximation therefore gives the value of the American call as $3.67.

Whaley19 has tested empirically three models for the pricing of American calls on dividend-paying stocks: (1) the formula in Appendix 10B; (2) Black's model; and (3) the European option pricing model described at the beginning of this section. He used 15,582 Chicago Board options. The models produced pricing errors with means of 1.08 percent, 1.48 percent, and 2.15 percent, respectively. The typical bid-ask spread on a call option is greater than 2.15 percent of the price. On average, therefore, all three models work well and within the tolerance imposed on the options market by trading imperfections.

Up to now, our discussion has centered around American call options. The results for American put options are less clear cut. Dividends make it less likely that an American put option will be exercised early. It can be shown that it is never worth exercising an American put for a period immediately prior to an ex-dividend date.20 Indeed, if

19See R. E. Whaley, "Valuation of American Call Options on Dividend Paying Stocks: Empirical Tests," Journal of Financial Economics, 10 (March 1982), 29-58.

20See H. E. Johnson, "Three Topics in Option Pricing," Ph.D. thesis, University of California, Los Angeles, 1981, p. 42.

an argument analogous to that just given shows that the put option should never be exercised early. In other cases, numerical procedures must be used to value a put.

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