Option Prices and Time to Expiration: Practice Problems and Solutions

This is Section 10 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. See Section 9 here.

Principles:

"An American call with more time to expiration is at least as valuable as an otherwise identical call with less time to expiration." Let t and T be times to expiration and t < T.

Then C_Amer(K, T) ≥ C_Amer(K, t).

"A longer-lived American put is always worth at least as much as an otherwise equivalent American put." Let t and T be times to expiration and t < T. Then P_Amer(K, T) ≥ P_Amer(K, t).

"A European call on a non-dividend-paying stock will be at least as valuable as an otherwise identical call with a shorter time to expiration." But with a dividend-paying stock, it may be possible for a shorter-lived European option to be worth more than a long-lived European option.

So if the stock does not pay dividends and t and T are times to expiration such that t < T, then C_Eur(K, T) ≥ C_Eur(K, t) and P_Eur(K, T) ≥ P_Eur(K, t). But if the stock pays dividends, this need not be the case.

Problem OPTE1. Which of these statements about call and put options on Representative Co. is always true? More than one answer may be correct.

(a) An American call with a strike price of $43 expiring in 2 years is worth at least as much as an American call with a strike price of $52 expiring in 1 year.
(b) An American call with a strike price of $43 expiring in 2 years is worth at least as much as an American call with a strike price of $43 expiring in 1 year.
(c) An American call with a strike price of $43 expiring in 2 years is worth at least as much as a European call with a strike price of $43 expiring in 2 years.
(d) An American put option with a strike price of $56 expiring in 3 years is worth at least as much as an American put option with a strike price of $56 expiring in 4 years.
(e) A European put option with a strike price of $56 expiring in 3 years is worth at least as much as a European put option with a strike price of $56 expiring in 2 years.

Solution OPTE1.

(a) is not always true; the equation C_Amer(K, T) ≥ C_Amer(K, t) only always holds if the strike prices of the options are the same.

(b) is true, because C_Amer(43, 2) ≥ C_Amer(43, 1).

(c) is true because C_Amer(43, 2) ≥ C_Eur(43, 2), from Section 8.

(d) is false; indeed, P_Amer(56, 4) ≥ P_Amer(56, 3).

(e) is false, because it may be the case that Representative Co. is bankrupt, in which case the put options will be worth the present value of the strike price; with the same strike price and longer time to expiration, the longer-term European put option will have a smaller present value factor by which the strike price is multiplied.

Thus, (b) and (c) are correct answers.

Problem OPTE2. European put options are traded on the stock of Chronic Bankruptcy, Inc. Chronic Bankruptcy, Inc., is bankrupt and currently has a stock price of $0 per share. What is the price of a put option that has strike price $92 and expires 2 years from now? The annual continuously compounded interest rate is 0.21.

Solution OPTE2.

Since Chronic Bankruptcy, Inc., is bankrupt, the put option has price equal to the present value of the strike price. Thus, PV_0,T(K) = 92e^-0.21*2 = P = 60.44830743.

Problem OPTE3. European put options are traded on the stock of Chronic Bankruptcy, Inc. Chronic Bankruptcy, Inc., is bankrupt and currently has a stock price of $0 per share. The annual continuously compounded interest rate is 0.21. Which of these put options has the highest price? The strike price K is given for each option, as is the time to expiration T.
(a) K = 23; T = 0.3
(b) K = 95; T = 23

(c) K = 64; T = 2
(d) K = 42; T = 0.65
(e) K = 1000; T = 100
(f) K = 95; T = 21

Solution OPTE3.

(b) and (f) have the same strike price, but (f) expires sooner, so (f) has a higher price and we know that (b) is not the correct answer.

Since Chronic Bankruptcy, Inc., is bankrupt, the put option has price equal to the present value of the strike price.

For (a), P = PV_0,T(K) = 23e^-0.21*0.3 = 21.59569989

For (c), P = PV_0,T(K) = 64e^-0.21*2 = 42.05099647

For (d), P = PV_0,T(K) = 42e^-0.21*0.65 = 36.64106545

For (e), P = PV_0,T(K) = 1000e^-0.21*100 = 7.582560428*10^-7

For (f), P = PV_0,T(K) = 95e^-0.21*21 = 1.154741941.

Of these values, 42.05099647 is the largest. So (c) is the correct answer.

Problem OPTE4. Imprudent Industries plans to pay a liquidating dividend of $20 in one year. Currently, European calls and puts on Imprudent Industries are traded with the following strike prices (K) and times to expiration (T). Which of the following European options will have the highest value? (Assume that, for T = 1, the options can be exercised just before the dividend is paid.) The annual effective interest rate is 0.03.

(a) Call; K = 3; T = 1.1
(b) Call; K = 19; T = 1
(c) Call; K = 19; T = 1.1
(d) Put; K = 10; T = 2
(e) Put; K = 10; T = 0.4

Solution OPTE4. Both (a) and (c) are worthless, because these calls can only be exercised once the company has liquidated itself and its stock price is 0. (b) can only be exercised immediately prior to liquidation, when the stock price will be equal to 20 (right before the $20 dividend is paid out). So profit on the option in one year will be $1 and the current price of (b) is PV_0,1(1).

The price of (e) will exceed the price of (d), because both are present values of 10, except (e) involves a shorter time period and thus a larger present value factor.

The price of (e) is 10*1.03^-0.4 = 9.882461023 > 1 > PV_0,1(1). So we know the (e) exceeds both (d) and (b). Thus, (e) is the correct answer.

Problem OPTE5. Gloom and Doom, Inc. will be bankrupt in 2 years, at which time it will pay a liquidating dividend of $30 per share. You own a European call option on Gloom and Doom, Inc., stock expiring in 2 years with a strike price of $20. The annual continuously compounded interest rate is 0.02. How much is the option currently worth? (Assume that, for T = 2, the option can be exercised just before the dividend is paid.)

Solution OPTE5. In 2 years, you will be able to exercise the option right before the dividend is paid. Then, the stock will be worth $30 and you will buy it for $20 and sell it for $30 right away, getting a $10 profit. This profit is a certainty. So the call price is the present value of $10, which is 10*e^-0.02*2 = $9.607894392.

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