Exam-Style Questions on Put-Call Parity and Arbitrage for Actuaries

This is Section 13 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.

The problems in this section were designed to be similar to problems from past versions of Exam 3F / Exam MFE. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Problem ESQPCP1.

Similar to Question 1 from the Society of Actuaries' Sample MFE Questions and Solutions:

Subterranean, Inc., pays no dividends on its stock, which currently trades for $123 per share. You know that the European call option on Subterranean, Inc., stock is $3 more expensive than the European put option with the same strike price and time to expiration. The strike price of the options is $145. The options expire in 9 years. Calculate the annual continuously compounded risk-free interest rate.

Note: The test will give you five multiple-choice possibilities for answers to this kind of problem. See if you can do without them. I will only include multiple-choice possibilities when they are necessary to answering the question or would make doing so more difficult rather than less.

Solution ESQPCP1. We use the put-call parity formula

C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e^-rTK, which, with no dividends, simplifies to

C(K, T) - P(K, T) = S₀ - e^-rTK.

Here, T = 9, C(K, T) - P(K, T) = 3, S₀ = 123, K = 145, and we want to find r.

So 3 = 123 - 145e^-9r

Thus, 145e^-9r = 120 and -9r = ln(120/145).

Hence -ln(120/145)/9 = r = 0.0210268888

Now try the corresponding test question, if you have not done so already. Do this after you do each of the problems here.

Problem ESQPCP2.

Similar to Question 13 from the Casualty Actuarial Society's Fall 2007 Exam 3:

You are given the price of an underlying asset (currently $300), the strike prices (K) of various American and European call options on this asset, their time to expiration (T), and their type. Given that none of them has the same price as any of the others, which of these options can be expected to have the highest price today?

(a) American, K = 326, T = 19.6 years
(b) European, K = 324, T = 17.3 years
(c) American, K = 324, T = 19.6 years
(d) European, K = 324, T = 19.6 years
(e) American, K = 324, T = 17.3 years

Solution ESQPCP2. We recall that American options are always at least as valuable as European options with the same strike price and time to expiration. So we know that (c) > (d) and (e) > (b). Furthermore, we recall that American options with longer time to expiration are always worth more than otherwise equivalent American options with shorter time to expiration. Thus, (c) > (e) > (b). Furthermore, call options with a smaller strike price are always worth more than otherwise equivalent call options with a larger strike price. Thus, (c) > (a) and the price of (c) is the largest one.

Problem ESQPCP3.

Similar to Question 14 from the Casualty Actuarial Society's Fall 2007 Exam 3:

Pernicious Co. will pay a $99.23 dividend on each share of its stock in three years. Currently, Pernicious Co. stock is worth $23000 per share. European call options on Pernicious Co. stock have the following properties: call price = $234, strike price = $95600, time to expiration = 9 years. The annual continuously compounded risk-free interest rate is 14%. Find the price of a European put option on Pernicious Co. stock with the same strike price and time to expiration.

Solution ESQPCP3.

We use the put-call parity equation C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e^-rTK, rearranging it as follows:
P(K, T) = C(K, T) - S₀ + PV_0,T(Div) + e^-rTK

Here, r = 0.14 and PV_0,T(Div) = 99.23e^-0.14*3 = 65.19875593

T = 9, so e^-rTK = 95600e^-0.14*9 = 27117.32493

Since C(K, T) = 234 and S₀ = 23000, it follows that

P(K, T) = 234 - 23000 + 65.19875593 + 27117.32493 =

P(K, T) = $4416.523689

Problem ESQPCP4.

Similar to Question 15 from the Casualty Actuarial Society's Fall 2007 Exam 3:

Terlefian nivands (TNV) trade for Roblanian dufords (RD) at an exchange rate of 6 TNV/RD. A call option denominated in TNV expiring 4 years from now and having a strike price of 9 TNV currently trades for 1 TNV. A TNV-denominated put option with the same strike price and time to expiration currently trades for 3 TNV. The annual continuously-compounded risk-free rate on Terlefian nivands is 13%. Find the annual continuously-compounded risk-free rate on Roblanian dufords.

Solution ESQPCP4.

The formula for parity of options on currencies is

C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT

We rearrange this equation: x₀e^-uT = C(K, T) - P(K, T) + Ke^-rT.

Here, T = 4, C(K, T) = 1, P(K, T) = 3, x₀ = 6, r = 0.13. We want to find u.

Thus, 6e^-4u = 1 - 3 + 6e^-0.13*4

6e^-4u = 1.567123288

Hence, e^-4u = 0.2611872146

Thus, u = -ln(0.2611872146)/4 = u = 0.3356294578

Problem ESQPCP5.

Similar to Question 2 from the Society of Actuaries' Sample MFE Questions and Solutions:

For six European options on Invincible Co. stock, all expiring T years from now, you have the following information:

For strike price $100, call price = $21 and put price = $5.

For strike price $120, call price = $14 and put price = $7.

For strike price $130, call price = $12 and put price = $10.

Carus thinks that an arbitrage opportunity is possible through purchasing one $100-strike call option, selling 5 $120-strike call options, and purchasing a certain number of $130-strike call options - along with lending $1.

Timdon thinks that an arbitrage opportunity is possible through purchasing one $100-strike call and selling one $100-strike put option, purchasing 6 $120-strike puts and selling 6 $120-strike calls, and purchasing some number of $130-strike calls while selling the same number of $130-strike puts.

Hingab believes that Carus and Timdon are sadly mistaken in thinking that their proposals would result in arbitrage profit.

Who is correct?

(a) Nobody is correct.

(b) Only Hingab is correct.

(c) Only Timdon is correct.

(d) Only Carus is correct.

(e) Both Timdon and Carus are correct.

Solution ESQPCP5.

One of the requirements for an arbitrage position is that it costs nothing on net to enter into. The second requirement is that it will make the owner a profit, irrespective of future price movements.

How might it be possible for Carus's series of transactions to cost nothing to enter into?

Purchasing one $100-strike call costs $21, while selling 5 $120-strike call options gives him $14*5 = $70. He also buys X $130-strike calls at a price of $12 each and lends out $1. So his net cost is -21 + 70 - 1 - 12X or 48 - 12X. In order for 48 - 12X to be 0, X must equal 4.

At expiration time T, given interest rate r, Carus will have the following gains.

If the stock price S < 100, all the calls are worthless and Carus only gets e^rT > 0from lending the $1.

If 100 ≤ S < 120, Carus gets S - 100 from the $100-strike call, along with the e^rT from lending the $1. His profit is S - 100 + e^rT > 0.

If 120 ≤ S < 130, Carus gets S - 100 from the $100-strike call, along with the e^rT from lending the $1. But he must also pay 5(S - 120) for his 5 sold $120-strike calls. His net profit is

S - 100 + e^rT - 5S + 600 = 500 - 4S + e^rT. Note that if r is small so that e^rT is approximately 1, then it is possible (if, for instance, S = 129) for 4S to be greater than 500 + e^rT and for

500 - 4S + e^rT< 0. Hence, for some of the higher stock prices in the range 120 ≤ S < 130, Carus could actually lose money at expiration. Thus, Carus's idea of an arbitrage opportunity is not valid.

Now we consider Timdon's proposal:

How might it be possible for Timdon's series of transactions to cost nothing to enter into?

Purchasing one $100-strike call and selling one $100-strike put option costs 21 - 5 = $16.

Purchasing 6 $120-strike puts and selling 6 $120-strike calls brings in 6(14 - 7) = $42.

Lending $2 costs $2 at time = 0. This leaves a cost 42 - 16 - 2 = 24 to be accounted for in

purchasing X $130-strike calls and selling X $130-strike puts. Doing so costs X(12 - 10) = 2X. Thus, 2X = 24 and X =12.

At expiration time T, depending on the stock price S, Timdon will have a payoff of

2e^rT from lending $2.

(S - 100) from purchasing one $100-strike call and selling one $100-strike put option.

6(120 - S) for purchasing 6 $120-strike puts and selling 6 $120-strike calls.

12(S - 130) for purchasing 12 $130-strike calls and selling 12 $130-strike puts.

Thus, his total payoff is 2e^rT + S - 100 + 720 - 6S + 12S - 1560 =

2e^rT + 7S - 740. We note that in order for 7S to be greater than 740, S must be greater than 105.7142857. If S goes substantially below 105.7142857, it is possible that even the accumulated value of the lent-out $2 will not be enough to preserve a profit for Timdon. Thus, this position is only profitable for S above a certain value (which is slightly less than 105.7142857, depending on the interest rate) and Timdon's proposal is not a genuine arbitrage opportunity.

So neither Carus nor Timdon is correct, meaning that Hingab is right to think that they have not stumbled upon arbitrage opportunities. Thus, (b) is the correct answer.

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