Early Exercise on American Options: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 9

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 9 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here.

Early exercise is never optimal for American call options on non-dividend-paying stock. But it can be optimal for a dividend-paying stock just before dividends are paid.

Early exercise for American calls is not optimal at any time where K - PV_t,T(K) > PV_t,T(Div).

But early exercise can be optimal if PV_t,T(Div) ≥ K - PV_t,T(K). The best time to exercise in this case is just before the ex-dividend date so as to have the benefit of receiving all the interest accumulated prior to that time.

Early exercise for American puts can be optimal even on non-dividend-paying stock. If the interest rate is positive, exercising early will enable one to receive the strike price K now, whereas the value of holding on to the put is only PV_t,T(K) < K.

But an American put will never be exercised if P > K - S. If P ≤ K - S, early exercise is possible. The possibility of early exercise does not necessarily imply that doing so will be optimal whenever P ≤ K - S is the case.

Explanation of Varuables:
P = American put price.
K = strike price.

t = the present time, from our viewpoint.

T = time at expiration.

PV_t,T(Div) = present value of dividends.

PV_t,T(K) = present value of strike price.

S = stock price.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 294-297.

Problem EEAO1. Oblivious Co. pays monthly dividends of $4 per share, starting one month from now. American call options on Oblivious Co. are issued for a strike price of $23. The annual effective interest rate is 0.03. Is it ever optimal to exercise such American call options early? Demonstrate why or why not using the appropriate inequality.