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# P1.T4.816. Black-Scholes-Merton (BSM) for dividend-paying stocks and the early exercise decision for American-style options (Hull Ch.15)

#### Nicole Seaman

##### Director of FRM Operations
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Learning objectives: Explain how dividends affect the decision to exercise early for American call and put options. Compute the value of a European option using the Black-Scholes-Merton model on a dividend-paying stock.

Questions:

816.1. Brian is a Risk Analyst who is using the Black-Scholes-Merton (BSM) option pricing model to value an European-style at-the-money (ATM) call option with one year maturity on a stock while its current price is $53.00. The stock's volatility is 41.0% per annum and he assumes the risk-free rate is 4.0%; i.e., S(0) = K = 53.00, , σ = 0.410, Rf = 0.040, and T = 1.0 year. Because he estimates N(d1) = 0.61890 and N(d2) = 0.45720, his estimate for the call option's value is$9.520.

However, he neglected to consider that the company recently announced a new dividend policy. Going forward, the expected dividend yield is 2.0% per annum with continuous compounding (the dividends will be paid discretely, of course, but he translated them into their continuous equivalent). The re-calculated cumulative normal distribution function values are given by N(d1) = 0.60020 and N(d2) = 0.43790. Consequently, which is nearest to the revised value of the call option on the dividend-paying stock?

a. $7.43 b.$8.88
c. $9.51 d.$10.16

816.2. Each of the following is true about American-style options EXCEPT which is false?

a. If the stock pays a dividend, it might be optimal to early exercise both a call and a put
b. If the stock does NOT pay a dividend, it might be optimal to early exercise a put, however it is never optimal to early exercise the call
c. Black's approximation determines the price of an American call option as the maximum of either its European equivalent with identical maturity, T, or its European equivalent with maturity at its final ex-dividend date, t(n)
d. Ceteris paribus a higher dividend yield (compared to the riskless interest rate) favors early exercise of a put option, but a higher riskless interest rate (compared to the dividend yield) favors early exercise of a call option

816.3. Consider an American call option on a stock. The stock price is $115.00, the time to maturity is one year (12 months), the risk-free rate of interest is 3.0% per annum, the exercise price is$100.00, and the volatility is 28.0% per annum. Dividends of \$1.10 are expected in three months (+0.25 years) and again in nine months (+0.75 years). In theory, when is it (or will it be) optimal to exercise the option? Note: this is variation on Hull's EOC Question 15.21.

a. The option should be exercised immediately
b. The option probably should be exercised immediately prior to the ex-dividend date for the dividend payable in three months (+0.25 years)
c. The option probably should be exercised immediately prior to the ex-dividend date for the dividend payable in nine months (+0.75 years)
d. It will never be optimal to early exercise this option