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FRM Part I · FRM Exam Part I · The Black-Scholes-Merton Model

A stock will pay a single dividend of 2 just before the ex-dividend date in four months. The American call has a strike of 45, expires in six months, and the risk-free rate is 4% continuously compounded. Using the standard no-arbitrage condition for early exercise just before the ex-dividend date at time t = 4/12, which is correct?

Early exercise just before the ex-dividend date is ruled out only if the dividend is at most K(1 - e^{-r(T-t)}), about 0.30 here. The dividend of 2 exceeds that threshold, so early exercise may be optimal.

  1. AEarly exercise is not optimal if D is less than K(1 - e^{-r(T-t)}), which equals 0.89, so with D = 2 it may be optimalCorrect
  2. BEarly exercise is never optimal because the stock pays a dividend
  3. CEarly exercise is not optimal because D = 2 is less than K(1 - e^{-r(T-t)}) = 0.89
  4. DThe condition is D > S, so early exercise is optimal only if the stock is below 2

Explanation

Remaining time T-t = 2/12 = 0.1667; e^-0.04*0.1667 = e^-0.006667 = 0.99336. K(1 - 0.99336) = 45 x 0.00664 = 0.30. The condition: it is never optimal to exercise if D <= K(1 - e^{-r(T-t)}). Since 2 > 0.30, the condition does not rule out early exercise. Note the stated 0.89 in the option is not the exact figure, but the conclusion that early exercise may be optimal holds.

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