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EC7076

Examination 2021: SOLUTIONS

Section A

Answer all questions

1. Which of the following is not true?

(a) When traders follow the risk limits that have been specified, big mistakes cannot happen.

(b) A trader is “hedging” when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure.

(c) In a “speculation” the trader has no exposure to offset. She is betting on the future movements in the price of the asset.

(d) Arbitrage involves taking a position in two or more markets to lock in a profit.

2. A stock index currently stands at 350. The risk-free interest rate is 8% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum. What should the futures price for a 4-month contract be?

(a) $350

(b) $359.46

(c) $353.52

(d) $354.7

3. Which of the following is not true?

(a) A swap rate for a particular maturity is the average of the bid and offer fixed rates that a market maker is prepared to exchange for LIBOR in a standard plain vanilla swap with that maturity.

(b) The swap rate for a particular maturity is the LIBOR/swap par yield for that maturity.

(c) Swap rates are risk-free lending rates.

(d) Swap rates are close to risk-free lending rates.Page 2 of 7

4. Which of the following is always negatively related to the price of a European put option on a stock?

(a) The stock price

(b) The strike price

(c) The time to expiration

(d) The volatility

5. American options can be valued using a binomial tree by

(a) Checking whether early exercise is optimal at all nodes where the option is in-themoney.

(b) Checking whether early exercise is optimal at the final nodes.

(c) Checking whether early exercise is optimal at the penultimate nodes a the finalm nodes.

(d)Increasing the number of time steps on the tree.

6. When valuing a European option using the no-arbitrage approach in a one-step binomial tree,

(a) We set up a riskless portfolio consisting of a position in the call option and a position in the stock and then set the return on the portfolio equal to the risk-free interest rate.

(b) We choose probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate.

(c) We value the option by calculating its expected payoff and discounting this expected payoff at the risk free rate.

(d) None of the above.

7. A Markov process is a particular type of stochastic process which is characterized by the following feature

(a) The past history of the variable can be used to predict its future values

(b) The current value of the variable is irrelevant for predicting its future values

(c) Only the current value of the variable is relevant for predicting its future values None of the above.

8. The Black-Scholes-Merton model assumes

(a) The return from the stock in a short period of time is lognormal

(b) The stock price at a future time is lognormal

(c) The stock price at a future time is normal

(d) None of the above

9. In the Black-Sholes-Merton option pricing formula N(d1) denotes

(a) The area under the normal distribution from zero to d1

(b) The area under the normal distribution up to d1

(c) The area under the normal distribution beyond d1

(d) The area under the normal distribution between –d1 and d1

10.A European call and European put have the same strike price and time to maturity.

Which of the following is true?

(a) The gamma of a call is the same as the gamma of a put

(b) The delta of a call is the same as the delta of a put

(c) The theta of a call is the same as the theta of a put

(d) None of the above

Section B – short answers

Answer all questions

11.Explain carefully the difference between hedging, speculation and arbitrage.

Answer: A trader is hedging when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure. In a speculation the trader has no exposure to offset. She is betting on the future movements in the price of the asset.

Arbitrage involves taking a position in two or more different markets to lock in a profit.

12.Explain how you would value a swap that is the exchange of a floating rate in one currency for a fixed rate in another currency? Answer: The floating payments can be valued in currency A by (i) assuming that the forward rates are realized, and (ii) discounting the resulting cash flows at appropriate currency A discount rates. Suppose that the value is . The fixed payments can be valued in currency B by discounting them at the appropriate currency B discount rates. Suppose that the value is . If is the current exchange rate (number of units of currency A per unit of currency B), the value of the swap in currency A is.

13.Explain why margins are required when clients write options but not when they buy options.

Answer: When an investor buys an option, cash must be paid up front. There is no possibility of future liabilities and therefore no need for a margin account. When an investor sells an option, there are potential future liabilities. To protect against the risk of a default, margins are required.

14.A European call option and put option on a stock both have a strike price of $20 and an expiration date in 3 months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in 1 month. Identify the arbitrage opportunity open to a trader.

Answer: If the call is worth $3, put-call parity shows that the put should be worth This is greater than $3. The put is therefore undervalued relative to the call. The correct arbitrage strategy is to buy the put, buy the stock, and short the call. This costs $19. If the stock price in three months is greater than $20, the call is exercised. If it is less than $20, the put is exercised. In either case the arbitrageur sells the stock for $20 and collects the $1 dividend in one month. The present value of the gain to the arbitrageur is

15.Explain what is meant by perfect hedge. Does a perfect hedge always lead to a better outcome than an imperfect hedge? Explain your answer.

Answer: A perfect hedge is one that completely eliminates the hedger’s risk. A perfect hedge does not always lead to a better outcome than an imperfect hedge. It just leads to a more certain outcome. Consider a company that hedges its exposure to the price of an asset. Suppose the asset’s price movements prove to be favorable to the company. A perfect hedge totally neutralizes the company’s gain from these favorable price

movements. An imperfect hedge, which only partially neutralizes the gains, might well

give a better outcome.

Section C – longer answers

Answer any two questions

16.Companies X and Y have been offered the following rates per annum on a $5 million

10-year investment:

Company X requires a fixed-rate investment; company Y requires a floating-rate investment. Design a swap that will net a bank, acting as intermediary, 0.2% per annum and will appear equally attractive to X and Y?

Answer: The spread between the interest rates offered to X and Y is 0.8% per annum on fixed rate investments and 0.0% per annum on floating rate investments. This means that the total apparent benefit to all parties from the swap is 0.8% p.a. Of this 0.2% per annum will go to the bank. This leaves 0.3% per annum for each of X and Y. In other words, company X should be able to get a fixed-rate return of 8.3% per annum while company Y should be able to get a floating-rate return LIBOR + 0.3% per annum. The required swap is shown in figure below. The bank earns 0.2%, company X earns 8.3%, and company Y earns LIBOR + 0.3%.

17.A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. Find the value of a six-month European put option and the value of a six-month American put option if the strike price in each option is $42?

Answer: A tree describing the behavior of the stock price is shown in the figure below. The risk-neutral probability of an up move, , is given by Calculating the expected payoff and discounting, we obtain the value of the option as The value of the European option is 2.118. This can also be calculated by working back through the tree as shown in the figure. The second number at each node is the value of the European option.

At each node, upper number is the stock price, the next number is the European put price, and the final number is the American put price The value of the American option is shown as the third number at each node on the tree. It is 2.537. This is greater than the value of the European option because it is optimal to exercise early at node C.

18.Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. What is the price of the option if it is a European call and what is the price of the option if it is a European put? Answer: In this case , , , and The European call price is

or $2.52. The European put price is

or $1.05.

19.Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.

Answer: In the no-arbitrage approach, we set up a riskless portfolio consisting of a position in the option and a position in the stock. By setting the return on the portfolio equal to the risk-free interest rate, we are able to value the option. When we use risk neutral valuation, we first choose probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate. We then value the option by calculating its expected payoff and discounting this expected payoff at the risk-free interest rate.