More Exam-Style Questions on Put-Call Parity and Arbitrage

This is Section 14 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.

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 MESQPCPA1.

Similar to Question 16 from the Casualty Actuarial Society's Fall 2007 Exam 3:

The stock of Precarious Co. does not pay any dividends. It currently trades at $565 per share, and the annual continuously compounded interest rate is 20%. European call and put options on Precarious Co. stock are available with strike price of $596, expiring in 3 years. There are no arbitrage opportunities in the pricing of these options. Digtammar decides to purchase 782 call options and sell 782 put options on Precarious Co. stock. What is the net cost of this transaction?

Solution MESQPCPA1. 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.

We note that Digtammar will pay 782[C(K, T) - P(K, T)] for this transaction, which is the same as 782[S₀ - e^-rTK] = 782[565 - e^-0.2*3596] = 186044.2631. So the net cost of the transaction to Digtammar is $186,044.2631 (i.e., this is the amount he pays).

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

Problem MESQPCPA2.

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

Spurious, Inc., stock currently trades for $63 per share. Call and put options on Spurious, Inc., stock with strike price of $76 and time to expiration of 17 months are currently available. The call price is $10, and the put price is $19. The annual continuous risk-free rate is 12%, while the annual continuous dividend yield is 7%. Tadart notices that arbitrage profit is possible under these conditions. Calculate the amount of arbitrage profit per share.

Solution MESQPCPA2. We use the put-call parity formula

C(K, T) - P(K, T) = S₀e^-dT - e^-rTK and notice that here the equality need not hold. We calculate each side of the equation separately and then calculate the positive difference between them, which will be the arbitrage profit per share.

C(K, T) - P(K, T) = 10 - 19 = -9

S₀e^-dT - e^-rTK = 63e^-0.07*17/12 -76e^-0.12*17/12 = -7.066244962

The positive difference between these two values is -7.066244962 - (-9) = $1.933755038, which is the arbitrage profit per share.

Problem MESQPCPA3.

Similar to Question 4 from the Casualty Actuarial Society's Spring 2007 Exam 3:

Tedious LLC will pay three dividends of $4 on its stock - one in 5 years, another in 19 years, and a third in 32 years. The current price of Tedious LLC stock is $101 per share. All continuously compounded risk free interest rates are 1%. The price of a European call option on Tedious LLC stock with a strike price of $157 and expiring in 41 years is $20. Find the price of a European put option on Tedious LLC stock with the same strike price and time to expiration.

Solution MESQPCPA3.

We use the put-call parity formula

C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e^-rTK, which we rearrange thus:
P(K, T) = C(K, T) - S₀ + PV_0,T(Div) + e^-rTK

Here, C(K, T) = 20 and S₀ = 100, while e^-rTK = e^-0.01*41157 = 104.1930893

Now we find PV_0,T(Div) = 4(e^-0.01*5 + e^-0.01*19 + e^-0.01*32) = 10.01735038

So P(K, T) = 20 - 100 + 104.1930893 + 10.01735038 = P(K, T) = 34.21043968

Problem MESQPCPA4.

Similar to Question 12 from the Casualty Actuarial Society's Spring 2007 Exam 3:

Evaluate the truth or falsehood of each of these propositions:
(a) The premiums for an American put decrease when the stock price increases.
(b) The premiums for a European call decrease when the strike price increases.

(c) Premiums on American options are smaller than premiums for otherwise equivalent European options.

Solution MESQPCPA4.

(a) is true. The payoff on a put option is max[0, K - S]. The higher S is, the smaller K - S becomes, and so we should expect the premium to decline.

(b) is true. The payoff on a call option is max[0, S - K]. As K increases, S - K decreases, so we should expect the premium to decline.

(c) is false. An American option is always at least as expensive as an otherwise equivalent European option, because the American option also gives the holder the choice to exercise at any time before expiration.

Problem MESQPCPA5.

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

Spacious Co. pays no dividends on its stock, which is currently worth $450 per share. For a strike price of $440 and time to expiration of 1 month, the price of a European call on Spacious Co. stock is $23, while the price of a European put is $11. A synthetic T-Bill can be created using these options. Find the annual risk-free continuously compounded rate on said T-Bill.

Solution MESQPCPA5.

This is a put-call parity problem phrased in slightly different terms. We seek to find r, given everything else. 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. We can rearrange the formula thus:

e^-rTK = S₀ - C(K, T) + P(K, T). We know that T = 1/12, S₀ = 450, K = 440, C(K, T) = 23, and P(K, T) = 11. Thus,

e^-r/12440 = 450 - 23 + 11.

Thus, e^-r/12440 = 438 and e^-r/12 = 438/440

Hence, r = -12ln(438/440) = r = 0.0546697984

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