Maximum and Minimum Option Prices: Practice Problems and Solutions

This is Section 8 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.

Constraints on prices of American and European call options:

S ≥ C_Amer(S, K, T) ≥ C_Eur(S, K, T) ≥ max[0, PV_0,T(F_0,T) - PV_0,T(K)]

Constraints on prices of American and European put options:

K ≥ P_Amer(S, K, T) ≥ P_Eur(S, K, T) ≥ max[0, PV_0,T(K) - PV_0,T(F_0,T)]

Meaning of variables:
K = strike price.

T = time to expiration.

S = price of the stock.

C_Amer(S, K, T) = price of American call.

C_Eur(S, K, T) = price of European call.
P_Amer(S, K, T) = price of American put.

P_Eur(S, K, T) = price of European put.

PV_0,T(K) = present value of the strike price.

PV_0,T(F_0,T) = prepaid forward price for the stock.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 293-294.

Problem MMOP1. Sagacious Co. stock currently sells for $562 per share. Both American and European call options are written on Sagacious Co. stock for a strike price of $550. The options expire one year from now. The annual effective interest rate is 0.09. The prepaid forward price for Sagacious Co. stock is $546 (the forward contract expires one year from now). Which of these are possible prices for the American and European call options? More than one correct answer is possible.

(a) C_Amer(S, K, T) = $256, C_Eur(S, K, T) = $295

(b) C_Amer(S, K, T) = $56, C_Eur(S, K, T) = $35

(c) C_Amer(S, K, T) = $45, C_Eur(S, K, T) = $43

(d) C_Amer(S, K, T) = $990, C_Eur(S, K, T) = $270

(e) C_Amer(S, K, T) = $41.5, C_Eur(S, K, T) = $43

Solution MMOP1. We note that T = 1 and PV_0,T(K) = 550*1.09^-1 = 504.587156.

So PV_0,T(F_0,T) - PV_0,T(K) = 546 - 504.587156 = 41.41284404.

Thus, 41.41284404 is the lower bound on both call prices.

This rules out answer (b), since there the European call is priced at $35.

Furthermore, in (a) and in (e), the European call is more expensive than the American call, which is impossible.

In (d), the American call's price exceeds S = 562, which is impossible.

Answer (c) has both the American and European call price within our upper and lower bounds and has the American call priced greater than the European call. So (c) is indeed a possible case.

Problem MMOP2. Imperious LLC stock currently sells for $95 per share. Both American and European put options are written on Imperious LLC stock for a strike price of $102. The options expire 17 months from now. The annual continuously compounded interest rate is 0.06. The 17-month forward price of Imperious LLC stock is $104. Which of these are possible prices for the American and European put options? More than one correct answer is possible.

(a) P_Amer(S, K, T) = 103, P_Eur(S, K, T) = 102

(b) P_Amer(S, K, T) = 0.00005, P_Eur(S, K, T) = 0.00003

(c) P_Amer(S, K, T) = 86, P_Eur(S, K, T) = 87

(d) P_Amer(S, K, T) = 102, P_Eur(S, K, T) = 102

(e) P_Amer(S, K, T) = 30, P_Eur(S, K, T) = 34

Solution MMOP2. The strike price - 102 - is the upper bound on our put option values.

Now we determine the lower bound: PV_0,T(K) = e^(-17/12)0.06102 = 93.68825301

PV_0,T(F_0,T) = e^(-17/12)0.06104 = 95.52527758

So PV_0,T(K) - PV_0,T(F_0,T) = 93.68825301 - 95.52527758 < 0, so

max[0, PV_0,T(K) - PV_0,T(F_0,T)] = 0 and the lower bound is 0.

For (a), the American put price exceeds the strike price, which is impossible.

For (c) and (e), the European put price exceeds the American put price, which is impossible.

But with (b), because the lower bound on the put price is 0, even these very low put prices are possible - since the American put price exceeds the European put price. Furthermore, we recall that our inequality looks as follows:

K ≥ P_Amer(S, K, T) ≥ P_Eur(S, K, T) ≥ max[0, PV_0,T(K) - PV_0,T(F_0,T)]

So it is permissible for the strike price to be equal to the American put price and the European put price, and (d) is a possibility. So both (b) and (d) are correct answers.

Problem MMOP3. Devious Co. stock currently sells for $43 per share. Both American and European call and put options are written on Devious Co. stock for a strike price of $56. The options expire 1 year from now. The annual continuously compounded interest rate is 0.10. The one-year forward price of Devious Co. stock is $50. Which of these are possible prices for the American call and European put options? More than one correct answer is possible.

(a) C_Amer(S, K, T) = 47; P_Eur(S, K, T) = 45
(b) C_Amer(S, K, T) = 42; P_Eur(S, K, T) = 2

(c) C_Amer(S, K, T) = 9; P_Eur(S, K, T) = 60

(d) C_Amer(S, K, T) = 3; P_Eur(S, K, T) = 5.5

(e) C_Amer(S, K, T) = 32; P_Eur(S, K, T) = 53

Solution MMOP3. Here, we use both inequalities:

S ≥ C_Amer(S, K, T) ≥ C_Eur(S, K, T) ≥ max[0, PV_0,T(F_0,T) - PV_0,T(K)]

K ≥ P_Amer(S, K, T) ≥ P_Eur(S, K, T) ≥ max[0, PV_0,T(K) - PV_0,T(F_0,T)]

For the call options, the upper bound is the stock price 43.

For the put options, the upper bound is the strike price 56.

PV_0,T(F_0,T) = e^-0.10*50 = 45.2418709

PV_0,T(K) = e^-0.10*56 = 50.67089541

Here, PV_0,T(K) > PV_0,T(F_0,T), so the lower bound on the call price is 0, while the lower bound on the put price is 50.67089541 - 45.2418709 = 5.42902451

(a) is impossible because the call price exceeds the upper bound of 43.

(b) is impossible because the put price is less than the lower bound of 5.42902451.

(c) is impossible because the put price exceeds the upper bound of 56.

(d) has the call price between 0 and 43 and the put price between 5.42902451 and 56, so (d) is possible.

(e) has the call price between 0 and 43 and the put price between 5.42902451 and 56, so (e) is possible.

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

Note: It is possible for an American call to be less expensive than a European put, and vice versa, as this problem illustrates.

Problem MMOP4. Highly Predictable, Inc. pays no dividends on its stock. The stock currently sells for $1231 per share. You know that a certain put option on Highly Predictable, Inc., stock cannot have a price less than $32. The option expires in 43 years, and the annual effective interest rate is 0.02. What is the strike price of this put option?

Solution MMOP4. By our inequality

K ≥ P_Amer(S, K, T) ≥ P_Eur(S, K, T) ≥ max[0, PV_0,T(K) - PV_0,T(F_0,T)], we know that our lower bound max[0, PV_0,T(K) - PV_0,T(F_0,T)] = 32. Moreover, 0 is not 32, so

PV_0,T(K) - PV_0,T(F_0,T) = 32.

Since Highly Predictable, Inc. pays no dividends, the prepaid forward price is equal to the stock price. Thus, PV_0,T(F_0,T) = 1231. Thus, PV_0,T(K) = 32 + 1231 = 1263.

So 1263 = 1.02^-43*K and 1.02⁴³*1263 = K = $2959.448156

Problem MMOP5. A European put option on Transparency, Ltd., costs the same as an American call option on this firm. We know that the European put is 5 times more expensive than its lowest possible value, which is nonzero. We know that the American call is $23 more expensive than the European call, and that the European call is one-seventh of the stock price. All options expire in one year and have a strike price of $80. The annual continuously compounded interest rate is 0.04. A prepaid 1-year forward contract on one share of Transparency, Ltd., stock costs $70. Find the price of one share of Transparency, Ltd., stock.

Solution MMOP5. The lowest possible value of the European put option is

PV_0,T(K) - PV_0,T(F_0,T) = e^-0.04*80 - 70 = 6.863155132.

The price of the European put and American call is thus 5*6.863155132 = 34.31577566.

The price of the European call is 34.31577566 - 23 = 11.31577566.

So the price of one share of stock is 11.31577566*7 = $79.21042962

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