Early Exercise on American Options: Practice Problems and Solutions

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.

Solution EEAO1. We compare K - PV_t,T(K) with PV_t,T(Div). Here, the monthly effective interest rate is 1.03^1/12 - 1 = 0.00246627, so PV_t,T(Div) = 4(1 - 1.00246627^-12)/0.00246627 = 47.239298667.

On the other hand, PV_t,T(K) = 23/1.03 = 22.33009709.

So K - PV_t,T(K) = 23 - 22.33009709 = 0.6699029126.

0.6699029126 < 47.239298667, so K - PV_t,T(K) < PV_t,T(Div). Thus, it may be optimal to exercise such options early.

Problem EEAO2. Delirious, Inc., stock currently trades for $96 per share. Put options are written for a strike price of $100. What is the maximum price of the put options at which early exercise might be optimal?

Solution EEAO2. If P ≤ K - S, early exercise is possible. Here, K = 100 and S = 96. So if P ≤ 100 - 96, i.e., if P ≤ 4, early exercise is possible. Thus, the maximum put price at which early exercise might be optimal is $4.

Problem EEAO3. Insidious LLC pays annual dividends of $5 per share of stock, starting one year from now. For which of these strike prices of American call options on Insidious LLC stocks might early exercise be optimal? All of the call options expire in 5 years. More than one answer may be correct.

(a) $756

(b) $554
(c) $396
(d) $256
(e) $43
(f) $10

Solution EEAO3. We will compare K - PV_t,T(K) with PV_t,T(Div). We calculate that

PV_t,T(Div) = 5(1-1.03^-5)/0.03 = 22.89853594. Whenever K - PV_t,T(K) < 22.89853594, early exercise may be optimal.

For (a), K - PV_t,T(K) = 756 -756/1.03⁵ = 103.867759 > 22.89853594, so early exercise is never optimal.

For (b), K - PV_t,T(K) = 554 -554/1.03⁵ = 76.11473345 > 22.89853594, so early exercise is never optimal.

For (c), K - PV_t,T(K) = 396 -396/1.03⁵ = 54.40692138 > 22.89853594, so early exercise is never optimal.

For (d), K - PV_t,T(K) = 256 -256/1.03⁵ = 35.1721512 > 22.89853594, so early exercise is never optimal.

For (e), K - PV_t,T(K) = 43 -43/1.03⁵ = 5.907822271< 22.89853594, so early exercise is possible.

For (f), K = 10, so K - PV_t,T(K) < 10 < 22.89853594, so early exercise is possible.

So (e) and (f) are correct answers.

Problem EEAO4. Insidious LLC pays annual dividends of $5 per share of stock, starting one year from now and ending five years from now. Thereafter, Insidious LLC will never again pay dividends. For which of these strike prices and times to expiration of American call options on Insidious LLC stocks might early exercise be optimal? More than one answer may be correct.

(a) $756, 2 months

(b) $554, 4 months
(c) $396, 1 year
(d) $256, 2 years
(e) $43, 19 years
(f) $10, 123 years

Solution EEAO4. We will compare K - PV_t,T(K) with PV_t,T(Div). Just as in problem EEAO3, re calculate that PV_t,T(Div) = 5(1-1.03^-5)/0.03 = 22.89853594. Whenever K - PV_t,T(K) < 22.89853594, early exercise may be optimal.

For (a), K - PV_t,T(K) = 756 -756/1.03^1/6 = 3.71525004 < 22.89853594, so early exercise is possible.

For (b), K - PV_t,T(K) = 554 -554/1.03^1/3 = 5.431722337 < 22.89853594, so early exercise is possible.

For (c), K - PV_t,T(K) = 396 -396/1.03 = 11.53398058 < 22.89853594, so early exercise is possible.

For (d), K - PV_t,T(K) = 256 -256/1.03² = 14.69544726 < 22.89853594, so early exercise is possible.

For (e), K - PV_t,T(K) = 43 -43/1.03¹⁹ = 18.47770085 < 22.89853594, so early exercise is possible.

For (f), K - PV_t,T(K) = 10 -10/1.03¹²³ = 9.736353894 < 22.89853594, so early exercise is possible.

So for all of these cases, early exercise is at least conceivable. Therefore, all answers are correct.

Problem EEAO5. American put options on Ingenious Co. currently cost $56 per option. We know that it is never optimal to exercise these options early. Which of these are possible values of the strike price K on these options and the stock price S of Ingenious Co. stock? More than one answer may be correct.

(a) S = 123, K = 124

(b) S = 430, K = 234
(c) S = 234, K = 430
(d) S = 1234, K = 1275
(e) S = 500, K = 600
(f) S = 850, K = 800

Solution EEAO5. Since early exercise is never optimal, we know that P > K - S, i.e.,

56 > K - S.

For (a), K - S = 1 < 56, so (a) is a possible answer.

For (b), K - S = 234 - 430 < 0 < 56, so (b) is a possible answer.

For (c), K - S = 430 - 234 = 196 > 56, so (c) is not a possible answer.

For (d), K - S = 1275 - 1234 = 41 < 56, so (d) is a possible answer.

For (e), K - S = 600 - 500 = 100 > 56, so (e) is not a possible answer.

For (f), K - S = 800 - 850 = - 50 < 56, so (f) is a possible answer.

Thus, (a), (b), (d) and (f) are correct answers.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.