Parity of Options on Stocks: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 2

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 2 of the Study Guide. See Section 1 here.

The equation expressing put-call parity for European options on stocks is

C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e-rT

K 

With respect to the forward contract on the stock, the following relationship holds between forward price and stock price:

e^-rTF_0,T = [S₀ - PV_0,T(Div)]

When dividends are paid on the basis of a continuously compounded rate d, then

S₀ - PV_0,T(Div) = S₀e^-dT

and

C(K, T) - P(K, T) = S₀e^-dT - 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.

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

Div = the stream of dividends paid on the stock.

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.

S₀ = the current stock price.

F_0,T = forward price for the underlying asset (in this case, the stock).

d = the continuously compounded dividend yield.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 283.

Note: McDonald uses the equation form C(K, T) = P(K, T) + [S₀ - PV_0,T(Div)] - e^-rTK. I prefer subtracting the put price from the call price so that the equation can be treated analogously to the formula for put-call parity. McDonald also uses "delta" in place for d, but symbolic constraints do not permit me to do so here.

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