Compound Options: Practice Problems and Solutions

This is Section 54 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.

An option on an option is called a compound option. A compound option has two strikes and two expirations associated with it - one each for the underlying option and for the compound option itself.

At a compound call option's expiration, the payoff of the compound call option on a call option (a CallOnCall option) is

max[C(S_{t_1}, K, T - t₁) - x , 0]

Meaning of Variables:
C(S_{t_1}, K, T - t₁) = underlying call option price.

x = compound option's strike price.

T = time to expiration of underlying call option.

t₁ = time to expiration of compound call option. Note: t₁ < T

K = strike price of underlying call option.

S_{t_1} = underlying stock price at time t₁.

Now we let S* be the critical stock price above which the compound option gets exercised. Then it follows that C(S*, K, T - t₁) = x, and the option gets exercised for all prices S_{t_1} > S*.

Two conditions must exist for a compound CallOnCall option to be valuable ultimately:

1. S_{t_1} > S* at time t₁.

2. S_T > K at time T.

There four types of compound options: CallOnCall, CallOnPut, PutOnCall, and PutOnPut. The naming convention for these options is XOnY, where Y is the underlying option type and X is the type of the option on the underlying option.

Compound Option Parity

The following relationship holds among prices of CallOnCall, PutOnCall, and ordinary call options (as priced by the Black-Scholes formula):

CallOnCall(S, K, x, σ, r, t₁, t₂, δ) - PutOnCall(S, K, x, σ, r, t₁, t₂, δ) + xe^-rt_1 =

BSCall(S, K, σ, r, t₂, δ)

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 14, pp. 453-454.

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

Problem CO1. The stock of Vicious Co. currently trades for $53 per share. You (but only you) have perfect knowledge that the stock will trade for $66 per share in 6 months. (The rest of the market does not have this knowledge.) A certain call option on the stock of Vicious Co. has a strike price of $58 at time to expiration of 6 months. You have perfect knowledge that this option will trade in 3 months at one-fifth of what you know its ultimate payoff will be. A CallOnCall option on this call option has a strike price of $1.33 and time to expiration of 3 months. What will the payoff on this CallOnCall option be in 3 months?

Solution CO1. The payoff of the CallOnCall option is max[C(S_{t_1}, K, T - t₁) - x , 0], where x = 1.33, T

= ½, and t₁ = ¼. We know that C(S_{t_1}, K, T - t₁) = (1/5)(66-58) = 1.6. Thus, the payoff on the CallOnCall option will be 1.6 - 1.33 = $0.27

Problem CO2. The Black-Scholes price of Call Option Q - which expires is 2 years - is $44. The annual continuously-compounded risk-free interest rate is 0.03. The price of a PutOnCall option on Option Q with a strike price of $50 and expiring in 1 year is $10. What is the price of a CallOnCall option on Option Q with a strike price of $50 and expiring in 1 year?

Solution CO2. We use the formula

CallOnCall(S, K, x, σ, r, t₁, t₂, δ) - PutOnCall(S, K, x, σ, r, t₁, t₂, δ) + xe^-rt_1 =

BSCall(S, K, σ, r, t₂, δ), where PutOnCall = 10, BSCall = 44, x = 50, r = 0.03, and t₁ = 1.

We rearrange the formula thus:

CallOnCall = BSCall + PutOnCall - xe^-rt_1 = 44 + 10 - 50e^-0.03 =

CallOnCall = $5.477723323

Problem CO3. The annual continuously-compounded risk-free interest rate is 0.12. A CallOnCall option on Call Option Ϋ has a price of $53. A PutOnCall option on Option Ϋ has a price of $44. Both compound options have a strike price of $100 and time to expiration of 2 years. Find the Black-Scholes price of the underlying Option Ϋ.

Solution CO3. We use the formula

CallOnCall(S, K, x, σ, r, t₁, t₂, δ) - PutOnCall(S, K, x, σ, r, t₁, t₂, δ) + xe^-rt_1 =

BSCall(S, K, σ, r, t₂, δ), where CallOnCall = 53, PutOnCall = 44, x = 100, r = 0.12, and t₁ = 2.

Thus, BSCall = 53 - 44 + 100e^-0.12*2 = BSCall = $87.66278611

Problem CO4. The Black-Scholes price of Call Option Ї is $12. A CallOnCall option on Call Option Ї has a price of $5. A PutOnCall option on Option Ї has a price of $3.33. Both compound options have a strike price of $14 and expire in 3 years. Find the annual continuously-compounded interest rate.

Solution CO4. We use the formula

CallOnCall(S, K, x, σ, r, t₁, t₂, δ) - PutOnCall(S, K, x, σ, r, t₁, t₂, δ) + xe^-rt_1 =

BSCall(S, K, σ, r, t₂, δ), where BSCall = 12, CallOnCall = 5, PutOnCall = 3.33, x = 14, and t₁ = 3. We rearrange the formula thus: xe^-rt_1 = BSCall - CallOnCall + PutOnCall, so

100e^-3r = 12 - 5 + 3.33 and e^-3r = 0.1033 and r = -ln(0.1033)/3 = r = 0.7567059676

Problem CO5. Call Option Q on Stock F expires in 1 year and has a strike price of $66. A CallOnCall option on Call Option Q expires in 6 months. The critical stock price for this option is $45. For which of these combinations of stock prices S_1/2 6 months from now and S₁ 1 year from now will the compound CallOnCall option be ultimately valuable? More than one answer is possible.

(a) S_1/2 = 43, S₁ = 79
(b) S_1/2 = 46, S₁ = 65
(c) S_1/2 = 46, S₁ = 79
(d) S_1/2 = 67, S₁ = 47
(e) S_1/2 = 99, S₁ = 79

Solution CO5. Two conditions must exist for a compound CallOnCall option to be valuable:

1. S_{t_1} > S* at time t₁.

2. S_T > K at time T.

Here, S* = 45, and K = 66. Thus, in order for the option to be ultimately valuable, it must be that S_1/2 > 45 and S₁ > 66. The requirement that S_1/2 > 45 rules out choice (a).

The requirement that S₁ > 66 rules out choices (b) and (d). The remaining choices, (c) and (e), fulfill the requirements that S_1/2 > 45 and S₁ > 66. Thus, (c) and (e) are correct answers.

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