Put-Call Parity for Actuaries: Sample Problems and Solutions

Put-call parity for European options with the same strike price and time to expiration is

Call - put = present value of (forward price - strike price)

Equation for put-call parity:
C(K, T) - P(K, T) = PV_0,T(F_0,T - K) = e^-rT(F_0,T - K)

Meaning 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.

F_0,T = forward price for the underlying asset.

PV_0,T = the present value over the life of the options.

e^-rT*F_0,T = prepaid forward price for the asset.

e^-rT*K= prepaid forward price for the strike.

r = the continuously compounded interest rate.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 282.

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

Problem PCP1. The European call option on Asset Q that expires in one year has strike price 32 and option price 4. The forward price of Asset Q in one year is 36. The annual continuously compounded interest rate is 0.08. Find the price of the put option on Asset Q with strike price of 32.

Solution PCP1. We have

C(K, T) - P(K, T) = e^-rT(F_0,T - K)

We have C(32, 1) = 4, F_0,T = 36 and K = 32.

Thus, 4 - P(K, T) = e^-0.08(36 - 32)

4 - e^-0.08(4) = P(K, T) = 0.3075346145

Problem PCP2. The price for a prepaid forward contract for widgets expiring in one year is 9500. A European call option for widgets expiring in one year and with a strike price of 9432 has a price of 283, while a European put option for widgets expiring in one year and with a strike price of 9432 has a price of 125. Find the annual continuously compounded interest rate.

Solution PCP2. C(K, T) - P(K, T) = e^-rT(F_0,T - K), i.e.,

C(K, T) - P(K, T) = e^-rTF_0,T - e^-rTK

We have C(9432, 1) = 283 and P(9432, 1) = 125.

Furthermore, we have e^-rTF_0,T = 9500 and T = 1.

Thus, 283 - 125 = 9500 - e^-r*9432

158 = 9500 - e^-r*9432

9342 = e^-r*9432

e^-r = 0.9914040115

r = -ln(0.9914040115) = r = 0.0086331471 = 0.86331471%

Problem PCP3. The price for a forward contract on actuarial textbooks expiring in four months is 700. At a certain strike price, the price of a European call option expiring in four months for actuarial textbooks in 129, while the price of a European put option is 567. The annual continuously compounded interest rate is 0.023. Find the strike price of the call and put options.

Solution PCP3. We note that T here is equal to 4/12 = 1/3.

C(K, T) - P(K, T) = e^-rT(F_0,T - K). We want to find K.

C(K, 1/3) = 129; P(K, 1/3) = 567; r = 0.023

129 - 567 = e^-0.023/3(700 - K)

-438 = e^-0.023/3(700 - K)

-441.3709053 = 700 - K

700 + 441.3709053 = K = 1141.370905. (Prices like this one are the reason why you might want to use free study materials like this guide!)

Problem PCP4. A European call option on impossible-to-open medicine bottles expiring in 17 months has price 45. A European put option on impossible-to-open medicine bottles with the same strike price and expiration date has price 93. Both options have a strike price of 420. The annual continuously compounded interest rate is 0.3445. Find the 17-month forward price of impossible-to-open medicine bottles.

Solution PCP4. We note that T here is equal to 17/12.

C(K, T) - P(K, T) = e^-rT(F_0,T - K). We want to find F_0,T.

C(420, 17/12) = 45; P(420, 17/12) = 93; r = 0.3445; K = 420

Thus, 45 - 93 = e^{-(17/12)0.3445}(F_0,T - 420)

-48 = 0.6138272964(F_0,T - 420)

-78.19789098 = F_0,T - 420

F_0,T = 420 - 78.19789098 = F_0,T = 341.802109.

Problem PCP5. The prepaid forward price of pies is 3.1415926535. The contracts expire in pi half-months. A European call option on pies expiring in pi half-months has a price of √(2), while a European put option with the identical strike price has a price of e. During this time of war, hyperinflation, and radicalism, the annual effective interest rate is 0.7071067812. Find the strike price of the call and put options.

Solution PCP5. We note that T here is equal to (3.1415926535/2)/12 = 0.1308996939

C(K, T) - P(K, T) = PV_0,T(F_0,T - K). We want to find K.

We also note that PV_0,T = 1.7071067812^{-0.1308996939} = 0.9323890126

C(K, 0.1308996939) = √(2); P(K, 0.1308996939) = e.

Thus, √(2) - e = 3.1415926535 - 0.9323890126*K

-1.304068266 = 3.1415926535 - 0.9323890126*K

0.9323890126*K = 4.44566092

K = 4.768032291

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