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BUSINESS SCHOOL

A LEVEL 4 MODULE, SPRING SEMESTER 2022 Sample Paper

BUSI4445 DERIVATIVE INVESTMENT

Answer any Three Questions

Word limit: 1,000 words per question

Question 1

(a) Suppose that the risk-free interest rate is 10% per annum with continuous compounding and that the net convenience yield on a commodity futures contract is 4% per annum. The current commodity price is standing at 400, and the futures price for a contract deliverable in four months is 405. What arbitrage opportunities does this create? (20%)

(b) Explain the convenience yield by using an example of a futures contract on crude oil qualitatively. (20%)

(c) Explain why does the convenience yield reflect the market’s expectations concerning the future availability of the commodity? Give qualitative discussions comparing with the risk-free rate without any calculations. (30%)

(d) Suppose that corn is harvested primarily in the fall, from September through November (“production seasonality”), but consumed throughout the year (“constant demand”). In order to be consumed when it is not being produced, corn must be stored. Without any calculations, give qualitative discussions about how production seasonality and the horizon (maturity) of a corn forward contract interact with the net convenience yield. (30%)

Question 2

(a) A trader buys two July futures contracts on frozen orange juice. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. Ignore interest rates. What price change would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account? (20%)

(b) It is July 2014. A mining company has just discovered a small deposit of gold. It will take six months to construct the mine. The gold will then be extracted on a more or less continuous basis for one year. Futures contracts on gold are available with delivery months every two months from August 2014 to December 2015. Each contract is for the delivery of 100 ounces. Discuss how the mining company might use futures markets for hedging. (20%)

(c) Explain qualitatively the convenience yield by using an example of a food producer for whom corn is an essential input. (30%)

(d) Show qualitatively that, if the futures price of a commodity is greater than the spot price during the delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if the futures price is less than the spot price? Explain your answer. (30%)

Question 3.

(a) An asset S has spot price St at time t and has a constant continuous dividend yield q. The continuous risk-free rate is r. There exist European style call and put options, with prices Ct and Pt, on this asset both with exercise price X and with maturity time T. You are told that Ct = Pt. Express the exercise price in terms of the spot price St (Hint: use put call parity). (20%)

(b) The following four European options are defined on asset S:

	Type	Exercise price	Maturity time
Option 1	Call	E95	12 months
Option 2	Call	E 105	9 months
Option 3	Put	E95	12 months
Option 4	Put	E 105	9 months

Given that St = E90, q = 4%, the volatility of the asset is 40% and r = 7%, deduce the value of a portfolio that consists of Option 1 long, Option 2 short, Option 3 short and Option 4 long. Calculate the value of the portfolio at time t. (40%)

(c) Why an American option is always worth at least as much as its intrinsic value. Is the same true of a European option? Explain your answer. (20%)

(d) Explain the effects of dividends on the value of a stock option (consider both call and put options in your answer)? (20%)

Question 4.

(a) What is meant by the delta of a security and what is meant by delta neutrality of a portfolio? (Explain and define your terms.) (10%)

(b) What is meant by the gamma of a security and what is meant by gamma neutrality of a portfolio? (Explain and define your terms.) (10%)

(c) A portfolio of risky securities, all of which have a common underlying asset, S, is to be dynamically hedged. Given that you can go long or short on the asset S and that you can deposit into and borrow from a risk free cash account explain the process of dynamic hedging. (30%)

(d) You have an European put option with a maturity time of 6 months on 1000 unites of the underlying asset S. The asset S has a current spot price of £5, a continuous dividend yield of 2% p.a. and a volatility of 40% p.a. The risk free rate is 4% p.a. The exercise price of the put option is $4.90. You decide to delta- gamma hedge the portfolio using two securities A and B that has a value, delta and gamma as follows :

	Value	Delta	Gamma
Security A	0.53	-0.43	0.27
Security B	0.74	0.55	0.19

How many units of security A and security B should you take for the hedge?

You might also like to know that the delta and gamma of a put option are given by:

= ;

∆ = − −(−)(−+);

() =

− 2

(50%)

Question 5.

(a) An asset has a continuous dividend q. The current time is t and the spot price at this time is given by St. You would like a long futures contract, with maturity time T for a unit of asset.

Unfortunately, the furthest maturity time you can get is U (where: t<U<T) but you know that at time U futures contracts will be available with maturity time T. Assuming that futures contracts are available for any amount of asset, you decide to use the following simple scheme. At time t you take a futures contract for −(−) units of asset with maturity time U. Then at time U you will take a futures contract for one unit of asset maturing at time T. You will use the same margin account from time t through to time T, leaving any profit or loss, due to either futures contract, in this until time T.

Show that this strategy is equivalent to a single futures contract on a unit of asset with agreement time t and maturity time T, if such a contract had existed.

(b) A stock index is currently 990, the risk-free rate is 5%, and the dividend yield on the index is 2%. Use a three-step tree to value an 18-month European put option with a strike price of 1,000 when the volatility is 20% per annum. (40%)