Parity of Options on Bonds: Practice Problems and Solutions

This is Section 5 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here.

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

We calculate PV_0,T(Coupons), which we consider as the present value of an annuity paying $10 per period for ten periods. Thus, PV_0,T(Coupons) = 10(1-1.03^-10)/0.03 = 85.30202837

Thus, C(K, T) - P(K, T) + PV_0,T(Coupons)] + PV_0,T(K) = 20 - 3 + 85.30202837 + 148.818783 =

B₀ = 251.1208114

Problem POB3. Irregular LLC issues coupon bonds which pay an annual coupon of X for 76 years. The annual continuously compounded interest rate is 0.05. The bond currently sells for 87. A call option on the bond with strike price 200 expiring in 76 years has price 5. A put option with the same strike price and time to expiration has price 2. Find X.

Solution POB3. We use the formula C(K, T) - P(K, T) = [B₀ - PV_0,T(Coupons)] - PV_0,T(K)

and rearrange it as follows:

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

K = 200; T = 76; PV_0,T(K) = e^-76*0.05*200 = 4.474154371

C(K, T) = 5; P(K, T) = 2; B₀ = 87.

Thus, PV_0,T(Coupons) = 87 - 4.474154371 - 2 + 5 = 85.52584563

85.52584563 = X(1 - e^-0.05*76)/0.05

Thus, 85.52584563 = 19.55258456X

So X = 4.374145288

Problem POB4. Amorphous Industries issues a bond with price 100 and annual coupons of 2, paid for 23 years. The annual effective interest rate is 0.04. A put option with a certain strike price and expiring in 23 years has price 5, whereas a call option with the same strike price and time to expiration has price 3. Find the strike price of both options.

Solution POB4. We use the formula C(K, T) - P(K, T) = [B₀ - PV_0,T(Coupons)] - PV_0,T(K)

and rearrange it as follows:

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

Here, PV_0,T(Coupons) = 2(1-1.04^-23)/0.04 = 29.71368333.

C(K, T) = 3; P(K, T) = 5; T = 23; B₀ = 100.

PV_0,T(K) = 100 - 29.71368333 - 3 + 5 = 72.28631667

Thus, 72.28631667 = (1.04^-23)K

72.28631667/(1.04^-23) = K = 178.1652082

Problem POB5. Multinational Corp. issues a bond that pays annual coupons of 5 and has a price of 90. A call option on the bond expiring in one year and with a strike price of 100 has price 6, while a put option with the same strike price and time to expiration has price 7. Find the annual effective interest rate.

Solution POB5. This problem would be notoriously messy if we had to consider more than one coupon payment period. But, fortunately, the coupons are annual and the time to expiration is one year. Thus, T = 1 and PV_0,T(Coupons) = 5(1+i)^-1, where i is the desired interest rate.

Furthermore, PV_0,T(K) = 100(1+i)^-1. Thus, PV_0,T(Coupons) + PV_0,T(K) = 105(1+i)^-1, which is a single term we can isolate on one side of our equation, which becomes

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

Here, B₀ = 90; C(K, T) = 6; P(K, T) =7.

Thus, 105(1+i)^-1 = 90 - 6 + 7

Thus, 105(1+i)^-1 = 91 and 91(1+i) = 105

Hence, i = 105/91 - 1 = i = 0.1538461538

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