Put-Call Parity for Actuaries: Sample Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 1

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.

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%