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Thursday, February 28th M375T=M396C Introduction to Actuarial Financial Mathematics Spring 2013, The University of Texas at Austin In-Term Exam I Instructor: Milica ˇCudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes TRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE MULTIPLE CHOICE 1 (5) a b c d e 2 (5) a b c d e 3 (5) a b c d e 4 (5) a b c d e 5 (5) a b c d e

FOR GRADER’S USE ONLY:

Standing assumptions: • No-arbitrage

• All options are European in style DEFINITIONS

1. (5 points) Write the definition of an arbitrage opportunity. Solution:

TRUE/FALSE QUESTIONS

Please, circle the correct answer on the front page of this exam. 1. (2 pts) Source: Sample FM(DM) Problem #27.

The position consisting of one long homeowners insurance contract benefits from falling prices in the underlying asset.

Solution: TRUE

Recall our comparison of the homeowner’s insurance policy to the put option. The payoff of the put option is decreasing in the price of the underlying asset.

2. (2 pts) Consider a portfolio consisting of the following four European options with the same expiration date T on the underlying asset S:

• long one call with strike 40, • long two calls with strike 50,

• short one call with strike 65. Let S(T ) = 69. Then, the payoff from the above position at time T is less than 60.

Solution: FALSE The payoff is

FREE-RESPONSE PROBLEMS

1. (20 points) Consider a European call option and a European put option on a non-dividend-paying stock. Assume:

(1) The current price of the stock is $55.

(2) The call option currently sells for $0.15 more than the put option. (3) Both options expire in 4 years.

(4) Both options have a strike price of $70.

Calculate the continuously compounded risk-free interest rate r. Solution: In our usual notation,

S(0) = 55, VC(0) − VP(0) = 0.15, T = 4, K = 70. We employed a no-arbitrage argument to get the put-call parity:

2. (20 points) Let the initial price of a non-dividend-paying stock be $20 and the risk-free continuously compounded interest rate be r = 0.05.

Assume that the current premium for an at-the-money European put on this asset with expiration date in one year equals $0.50. The premium for the European call with the same strike and expiration date and on the same asset is $1.50.

Is there an arbitrage opportunity? If your answer is affirmative, provide an arbitrage portfolio and show that it is an arbitrage portfolio. If your answer is negative, justify it!

Solution: One equality which is always true in arbitrage-free market-models is the put-call parity. Let us examine if it holds for the above data. In our usual notation:

VC(0) − VP(0) = 1.5 − 0.5 = 1,

S(0) − Ke−rT = 20(1 − e−0.05) = 0.9754.

So, the put-call parity is violated. This observation helps us construct the arbitrage portfolio. Noticing that the portfolio consisting of the outright purchase of the asset and borrowing K−rT can be considered to be relatively “cheap” as compared to the portfolio consisting of the long call and the short put, we decide to do the following at time−0:

(1) buy one share of stock,

(2) borrow K−rT at the risk free rate to be repaid at time T ,

(3) write a European call option on this asset with strike K and exercise date T , and (4) buy a European put option on this asset with strike K and exercise date T .

The initial cost of this portfolio is:

S(0) − Ke−rT − (VC(0) − VP(0)) < 0.

The negative initial cost means that we initially receive money. This money can be invested at the risk-free rate (thus creating a fully-leveraged portfolio) or just kept (at the zero interest rate).

At time−T , the payoff/worth of our portfolio is always: S(T ) − K − (S(T ) − K)++ (KS(T ))+ = 0.

3. (20 points) Which of the positions listed will benefit from the underlying asset’s price decline? Draw the payoff curves for each position and justify your answer.

(i) Short put (ii) Long put (iii) Short call (iv) Short stock

(v) Short forward contract

4. (8 points) A stock currently sells for $32.00. A 6 month call option with strike $35.00 has a premium of $2.27. Assuming a 4% continuous dividend yield, what is the price of the associated put option as dictated by put-call parity?

Solution: We have:

VP(0, 35, 0.5) = VC(0, 35, 0.5) − e−δTS0+ e−rT35

MULTIPLE CHOICE QUESTIONS

Please, circle the correct answer on the front page of this exam.

1. The initial price of the market index is $900. After 3 months the market index is priced at $920. The nominal rate of interest convertible monthly is 4.8%.

The premium on the long call, with a strike price of $930, is $2.00. What is the profit or loss at expiration for this long call?

(a) $2.00 loss (b) $2.02 loss (c) $2.02 gain (d) $2.00 gain

(e) None of the above. Solution: (b)

In our usual notation, the profit is

(S(T ) − K)+− C · (1 + j)3

with C denoting the price of the call and j the effective monthly interest rate. We get (920 − 930)+− 2 · 1.043 ≈ −2.02.

2. The premium on a 2-month call option on the market index with an exercise price of 1050 is $9.30 when originally purchased. After 2 months the position is closed and the index spot price is 1072. If interest rates are 0.5% effective per month, what is the call’s profit?

(a) $9.30 (b) $9.39 (c) $12.61 (d) $22.00

(e) None of the above. Solution: (c)

The value at expiration of the cost of the call is 9.30 · 1.0052 _{≈ 9.39.}
The payoff of the call is 1072 − 1050 = 22.

So the profit is 22 − 9.39 = 12.61.

3. Jafee Corp. common stock is priced at $36.50 per share. The company just paid its $0.50 quarterly dividend. Interest rates are 6.0%. A $35.00 strike European call, maturing in 6 months, sells for $3.20. What is the price P of a 6-month, $35.00 strike put option?

(a) 0 ≤ P < $1.25 (b) $1.25 ≤ P < $1.45

(c) $1.45 ≤ P < $1.55 (d) $1.55 ≤ P < $1.66

Solution: (d)

VP(0) = VC(0) + Ke−rT − F0,TP (S) = VC(0) + Ke−rT − S(0) + De−rt1 + De−rt2

4. (5 points) Consider an investment in S&P 500 Index futures contracts at a price of $1000. The initial margin requirement is 15.0% of the notional value. The maintenance margin is $100. If the continuously compounded interest rate is 5.0% what will the futures price need to be for a margin call to occur 10 days from now? Assume no settlement within the 10 days (i.e., the futures price does not change within the 10 days).

(a) $939.79 (b) $940.79 (c) $949.79 (d) $950

(e) None of the above. Solution: (c)

Per futures contract, the initial deposit into the margin account is 1000 · 0.15 = 150.

Over the course of the next 10 days, interest is accrued and the balance in the account at the end of the 10 days is

150e0.05·(10/365) ≈ 150.21.

So, the price of an index futures contract should drop by 150.21 − 100 = 50.21 to cause a margin call. In other words, the index futures price needs to be 1000 − 50.21 = 949.79. 5. A certain common stock is priced at $74.20 per share. The company just paid its

$1.10 quarterly dividend. Assume that the interest rate is r = 6.0%. Consider a $70 strike European call, maturing in 6 months which currently sells for $6.50. How much (arbitrage) profit/loss is made by shorting the corresponding European put whose premium is $2.50?

(a) $0.15 loss (b) $0.15 gain

(c) $0.36 loss (d) $0.36 gain

(e) None of the above. Solution: (b) or (e)

We can obtain the no-arbitrage premium of the corresponding put as dictated by put-call parity, as follows:

VP(0, K = 70, T = 0.5) = VC(K = 70, T = 0.5) + e−rTK − S(0) + P V0,T(Dividends) = 6.50 + e−0.03· 70 − 74.20 + e−0.06·0.25· 1.10 + e−0.06·0.5· 1.10 = −67.70 + 0.97 · 70 + 0.98 · 1.10 + 0.97 · 1.10

= −67.70 + 68.97 + 1.08 = 2.38.