Parity of Options on Stocks: Practice Problems and Solutions

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^-rTK

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:

Problem POS1. The stock price of Carl Menger, Inc., is 98 per share. The company will pay two dividends of 3 per share, one six months from now and the other one year from now. A call option on Carl Menger, Inc., stock expiring one year from now with a certain strike price has a price of 23. A put option with the same strike price and expiration date has a price of 12. The annual continuously compounded rate of interest is 0.056. Find the strike price of the options.

Solution POS1. C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e^-rTK

Here, C(K, 1) = 23; P(K, 1) = 12; S₀ = 98; r = 0.056. We want to find K.

We note that PV_0,T(Div) = 3*e^-0.056/2 + 3*e^-0.056 = 5.753782508

Thus, 23 - 12 = [98 - 5.753782508] - e^-0.056K

11 = 92.24621749- e^-0.056K

81.24621749 = e^-0.056K

81.24621749 = 0.9455391359K

K = 85.92581143

Problem POS2. Hayekian LLC pays dividends on its stock on a continuously compounded basis with annual dividend yield of 0.02. One share of Hayekian LLC stock is currently worth 43. The monthly effective interest rate is 0.003. A put option on Hayekian LLC stock with strike price 45 currently has price 5. Find the price of the call option on Hayekian LLC stock with strike price 45. Both the call and the put option expire in 15 months.

Solution POS2. Since we want to find C(K, T), it is in this case more useful to employ McDonald's form of the parity equation:

C(K, T) = P(K, T) + S₀e^-dT - PV_0,T(K)

Here, T = 15/12 = 1.25

We also note that the present value discount factor for a time of 15 months is 1.003^-15 = 0.9560618851.

From our given information, P(45, 1.25) = 5; S₀ = 43; K = 45; d = 0.02

Thus, C(45, 1.25) = 5 + 43e^-0.02*1.25 - 0.9560618851*45

C(45, 1.25) = 3.915541388

Problem POS3. A one-year forward contract on the stock of Mises, Rothbard, & Associates has price 132. The annual effective interest rate is 0.054. Mises, Rothbard, & Associates pays monthly dividends of 2 per share, starting one month from now. Find the current price of a share of stock of Mises, Rothbard, & Associates.

Solution POS3. We use the equation e^-rTF_0,T = [S₀ - PV_0,T(Div)], which we can transform into

S₀ = e^-rTF_0,T + PV_0,T(Div) and solve for S₀.

We were not given a continuously compounded interest rate, but we note that e^-rT is identical to (1+i)^-T, where i is the annual effective interest rate. Here, i = 0.054.

To find the present value of the dividend stream, we first need to find the monthly effective interest rate m, which is 1.054^1/12 - 1 = 0.0043923223.

Recalling all that fun financial mathematics from Exam 2 / FM, we can treat this dividend stream as an annuity-immediate for 12 time periods, paying 2 per time period, with effective interest rate per time period equal to m = 0.0043923223.

Thus, PV_0,1(Div) = 2(1-1.0043923223^-12)/0.0043923223 = 23.32861453.

From the given information, F_0,1 = 132.

Hence, 1.054^-1*132 + 23.32861453 = S₀ = 148.5658062

Problem POS4. The stocks of two corporations - Reliable, Inc., and Equally Reliable, Inc. - have identical prices of 934 per share today, but different dividend structures. Reliable, Inc. pays dividends of 4 every 2 months starting 2 months from now at an annual effective interest rate of 0.032. Equally Reliable, Inc., pays dividends on a continuously compounded basis. Find the annual continuously compounded dividend yield for the stock of Equally Reliable, Inc.

Solution POS4. Since the two stock prices are identical, we can use the equation

S₀ - PV_0,T(Div) = S₀e^-dT, with S₀ = 934.

We can let T equal anything we want, so - to avoid annuity calculations - we can let T = 1/6 and only involve one of Reliable's dividend payments in our present value calculations.

Thus, PV_0,1/6(Div) = 1.032^-1/6*4 = 3.979055913

Hence, 934 - 3.979055913 = 934e^-d/6

Thus, 930.0209441/934 = e^-d/6 = 0.9957397688

-6*ln(0.9957397688) = d = 0.0256159909

Problem POS5. Hazlitt Enterprises has stock price of 275 per share. A call option on Hazlitt Enterprises expiring in one year and with strike price of 325 has price 59. The price of a call option on Bastiat Corp. stock having strike price of 23 and expiring in one year is 6. One put option on Bastiat Corp. stock with the same strike price and expiration date has exactly one-tenth the price of one put option on Hazlitt Enterprises. Hazlitt Enterprises pays no dividends, while Bastiat Corp. pays a single dividend of 38 at the end of the year (this company believes in returning the vast majority of its profits to its shareholders as soon as they accrue). The continuously compounded rate of interest is 0.17. Find the current price of one share of Bastiat Corp. stock.

Solution POS5. This is a multi-step problem; first we want to find the price of a Hazlitt Enterprises put option; then we will use this to determine the price of Bastiat Corp. stock.

For Hazlitt, we use the equation C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e^-rTK and rearrange it, taking into consideration the fact that Hazlitt pays no dividends.

P(K, T) = C(K, T) - S₀ + e^-rTK

For Hazlitt, we have T = 1; K = 325; C(325, 1) = 59; S₀ = 275; r = 0.17.

Thus, P(325, 1) = 59 - 275 + e^-0.17325 = 58.19106539

A put option on Bastiat Corp. stock costs one-tenth of the price of a put option on Hazlitt Enterprises. Thus, for Bastiat, we have P(23, 1) = 5.819106539; C(23, 1) = 6; K = 23; Div = 38; r = 0.17.

We will rearrange the relevant equation to make it more convenient to find S₀:

S₀ = C(K, T) - P(K, T) + PV_0,T(Div) + e^-rTK

Since we are dealing with a continuously compounded rate of interest, PV_0,T(Div) = e^-0.1738 = 32.05926303. Moreover, e^-rTK = e^-0.1723 = 19.40429078

Thus, S₀ = 6 - 5.819106539 + 32.05926303 + 19.40429078 = S₀ = 51.64444727.

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