Parity of Options on Bonds: Revised Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 5 (Version 2.0)

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 5 of the Study Guide. See an index of all sections by following the link in this paragraph.

The formula for parity of options on bonds is

C(K, T) - P(K, T) = [B₀ - PV_0,T(Coupons)] - PV_0,T(K)

Explanation of Variables:

K = strike price of the options.

T = time to expiration of the options.

C(K, T) = price of a European call with strike price K and time to expiration T.

P(K, T) = price of a European put with strike price K and time to expiration T.

B₀ = bond price.

PV_0,T (Coupons) = present value of the bond's coupons.

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

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 286.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem POB1. A zero-coupon bond issued by Indestructible Co. currently sells for $67. The annual effective interest rate is 0.04. A call on the bond with a strike price of $80 expiring in 9 years sells for $11.56. Find the price of a put option on the bond with the same strike price and time to maturity.

Solution POB1. We use the formula C(K, T) - P(K, T) = [B₀ - PV_0,T(Coupons)] - PV_0,T(K)

and rearrange it as follows, taking into account the absence of coupons:
C(K, T) - B₀ + PV_0,T(K) = P(K, T).

Here, C(K, T) = 11.56, B₀ = 67, T = 9, K = 80, and PV_0,T(K) = 1.04^-9*80 = 56.20693885.

Thus, 11.56 - 67 + 56.20693885 = P(K, T) = 0.7669388463

Problem POB2. A certain bond issued by Volatile Industries pays annual coupons of $10 for 10 years. The annual effective interest rate is 0.03. A call on the bond with a strike price of $200 expiring in 10 years sells for $20. A put option on the same bond with the same strike price and time to maturity sells for $3. Find the price of the bond.

Solution POB2. We use the formula C(K, T) - P(K, T) = [B₀ - PV_0,T(Coupons)] - PV_0,T(K)

and rearrange it as follows:

B₀ = C(K, T) - P(K, T) + PV_0,T(Coupons)] + PV_0,T(K)

We are given that C(K, T) = 20, P(K, T) = 3, K = 200.

We calculate PV_0,T(K) = 1.03^-10*200 = 148.818783