Maximum and Minimum Option Prices: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 8

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 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.