Pricing Options on Dividend-Paying Stocks: Practice Problems and Solutions

This is Section 55 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here. See Section 49 here. See Section 50 here. See Section 51 here. See Section 52 here. See Section 53 here. See Section 54 here.

We consider American options where the underlying stock pays a dividend D at time t₁ (the expiration time of the compound option). We can either exercise the option at the stock price right before the dividend is paid (S_{t_1} + D) or we can hold the option until expiration. The underlying option will be priced on the basis of the stock price after the dividend is paid (S_{t_1}).

The regular call option payoff at time t₁ is max[C(S_{t_1}, T - t₁), S_{t_1} + D - K].

The value of the unexercised call is

C(S_{t_1}, T - t₁) = P(S_{t_1}, T - t₁) + S_{t_1} - Ke^-r(T-t_1).

The call option payoff at time t₁ can thus be written as

S_{t_1} + D - K + max[P(S_{t_1}, T - t₁) - D + K(1 - e^-r(T-t_1)) , 0]

The current value of the call option is the present value of this amount.

We note that it is possible to express P(S_{t_1}, T - t₁) - D + K(1 - e^-r(T-t_1)) above as the value of a compound CallOnPut option with strike price D - K(1 - e^-r(T-t_1)).

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 14, p. 455.

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

Problem PODPS1. 4 months from now, the stock of Ludicrous Co. will pay a dividend of $2 per share. A certain Call Option F on Ludicrous Co. stock has a strike price of $33 and time to expiration of 6 months. You know with certainty that the stock will have a price of $35 4 months from now after the dividend is paid and that Call Option F will have a price of $3 4 months from now after the dividend is paid. What will be the value of Call Option F 4 months from now?

Solution PODPS1. The value of the call option will be

max[C(S_{t_1}, T - t₁), S_{t_1} + D - K]. We know that C(S_{t_1}, T - t₁) = 3, S_{t_ 1}= 35, D = 2, K = 33. Thus, the value of the call option will be max(3, 35 + 2 - 33) = max(3, 4) = $4.

Problem PODPS2. 4 months from now, the stock of Ludicrous Co. will pay a dividend of $2 per share. A certain Call Option F on Ludicrous Co. stock has a strike price of $33 and time to expiration of 6 months. You know with certainty that the stock will have a price of $35 4 months from now after the dividend is paid and that Call Option F will have a price of $3 4 months from now after the dividend is paid. The annual continuously compounded risk-free interest rate r is 0.03. If the call option is unexercised, what will be the value 4 months from now of a put option on Ludicrous Co. stock with a strike price of $33 and time to expiration of 6 months?

Solution PODPS2. The value of the unexercised call is

C(S_{t_1}, T - t₁) = P(S_{t_1}, T - t₁) + S_{t_1} - Ke^-r(T-t_1).

We know from the given information that C(S_{t_1}, T - t₁) = 3, S_{t_1} = 35, K = 33, r = 0.03, T = ½, t₁ = 1/3. We rearrange the formula:

P = C - S_{t_1} + Ke^-r(T-t_1) = P = 3 - 35 + 33e^{-0.03(1/2-1/3)} = P = $0.8354118134

Problem PODPS3. You have perfect knowledge that 1 year from now, the stock of Imperious LLC will pay a dividend of $10 per share. Right after it pays the dividend, the stock will be worth $100 per share. A certain put option on this stock has a strike price of $102, time to expiration of 2 years, and will have a price of $12 in 1 year if unexercised. The annual continuously compounded risk-free interest rate r is 0.08. What will be the price in 1 year (after the dividend is paid) of a call option on this stock with a strike price of $102 and time to expiration of 2 years?

Solution PODPS3. We use the call option payoff expression

S_{t_1} + D - K + max[P(S_{t_1}, T - t₁) - D + K(1 - e^-r(T-t_1)) , 0], where S_{t_1} = 100, D = 10, K = 102, T = 2, t₁ = 1, r = 0.08, and P = 12. P(S_{t_1}, T - t₁) - D + K(1 - e^-r(T-t_1)) =

12 - 10 + 102(1 - e^-0.08) = 9.842132669 > 0. Thus, the call option price will be

100 + 10 - 102 + 9.842132669 = $17.84213267

Problem PODPS4. You have perfect knowledge that 1 year from now, the stock of Imperious LLC will pay a dividend of $10 per share. Right after it pays the dividend, the stock will be worth $100 per share. A certain put option on this stock has a strike price of $102, time to expiration of 2 years, and will have a price of $12 in 1 year if unexercised. The annual continuously compounded risk-free interest rate r is 0.08. You can use a compound CallOnPut option on the Imperious LLC put to determine the price in 1 year (after the dividend is paid) of a call option on this stock with a strike price of $102 and time to expiration of 2 years. What is the strike price of such a CallOnPut option?

Solution PODPS4. The strike price of such a CallOnPut option is D - K(1 - e^-r(T-t_1)) = 10 - 102(1 - e^-0.08) = $2.157867331.

Problem PODPS5. You have perfect knowledge that 1 year from now, the stock of Imperious LLC will pay a dividend of $10 per share. Right after it pays the dividend, the stock will be worth $100 per share. A certain put option on this stock has a strike price of $102, time to expiration of 2 years, and will have a price of $12 in 1 year if unexercised. The annual continuously compounded risk-free interest rate r is 0.08. You can use a compound CallOnPut option on the Imperious LLC put to determine the price in 1 year (after the dividend is paid) of a call option on this stock with a strike price of $102 and time to expiration of 2 years. What will be the payoff in 1 year on such a compound CallOnPut option on the Imperious LLC put?

Solution PODPS5. The payoff on this CallOnPut option is

P(S_{t_1}, T - t₁) - D + K(1 - e^-r(T-t_1)) = P(S_{t_1}, T - t₁) - Strike. It is given that P = 12 and Strike = 2.157867331. Thus, P(S_{t_1}, T - t₁) - Strike = 12 - 2.157867331 = $9.842132669

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