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Department of Accounting and Finance
M.Sc. Finance
M.Sc. Investment and Finance
M.Sc. International Banking and Finance
And
M.Sc. International Accounting and Finance
40917: Options and Futures
Thursday 20th^ May 2010 10.30am – 12.00pm (1½ hours)
Instructions for Candidates
Answer THREE Questions
ONE from Section A, ONE from Section B and
ONE further Question from either Section A or B
[Failure to comply will result in papers not being marked]
Calculators must not be used to store text and/or formulae nor be capable of communication. Invigilators may require calculators to be reset. Please write clearly as illegible writing cannot be marked. Failure to follow these requirements will lead to a deduction of marks.
To Be Issued: Standard Normal Distribution Tables and Formulae sheet
To Be Completed (please write clearly)
Registration Number 2008
Family Name:
Other Name:
Course (please indicate by ticking appropriate box)
Finance Investment & Fin. Int. Banking & Fin
Int. Accounting & Fin
Please Note: This question paper is to be returned along
with your examination answer booklet, you should
complete the above and slip this paper into your answer
booklet. Under no circumstances is a copy of this paper
to leave the exam room.
SOLUTIONS
the high demand for short futures will push the price down to its equilibrium level. Arbitrageurs will then close their short positions by going long, and taking the gain.
c) Discuss whether it makes sense for a coffee producer to hedge using coffee futures. (6 marks) Hedging coffee is hedging an asset you don’t yet have, but expect to have in the future. Eg if the crop fails you have lower than expected quantity of the asset, and the losses on the hedge F 0 F 02 D FT may be large. In addition you have a much smaller quantity of coffee to sell – albeit at a higher price due to shortage. Consider, eg weather futures.
d) Explain the purpose of a short hedge with futures, and the outcome of a perfect hedge. (6 1/3 marks) The short hedge protects the value of assets you hold (and perhaps plan to sell). The hedge works because you have the assets and the quantity will not diminish over time. The perfect hedge, where the asset in the contract is identical to the asset being hedged, and where the hedge period is the same as the length of the futures contract, will earn the risk free rate. (this can be demonstrated) (TOTAL 33 1/3 MARKS)
Q2. a) TABLE 2 Short term interest rates and futures prices on 2nd February 2010
Spot rates 3 months 6 months 1 year LIBOR 0.62% 0.84% 1.26% Futures prices 17 th^ March contract 43 days
16 th^ June contract 134 days
15 th^ Sept contract 225 days 3-month sterling
One contract is for £500,000 nominal
A firm expects to make a 3 month deposit of £150m in mid-September, and decides to hedge against possibility of a fall in rates. The deposit is made in mid-September when 3-month is LIBOR is 1.5%. September futures contracts are close to expiration and the futures price is 1 tick above the spot. Demonstrate the outcome for the firm, and comment on the result. (10 marks) 300 LONG futures contracts @ 98.92 (implicit rate 1.08%) In mid-September futures is 1 tick above spot rate, ie 98. Loss on futures = [(F (^) T – F 0 ) F 0B 4 100] ticks F 0B 4 £12.50 F 0B 4 300 contracts = [(98.51 – 98.92) F 0B 4 100] ticks F 0B 4 £12.50 F 0B 4 300 contracts = F 02 D£153,750. Therefore only has £149.84625m to deposit @ 1.5% (= 0.375% over 3 months) 149.84625(1.00375) = £150. This is equivalent to a 3-mth return of 0.00272% on £150m, = 1.088% pa This is slightly above the rate in the futures contract.
b) TABLE 3 Long gilt futures on 2nd February 2010
Futures price for the June 2010 contract =114. 1 contract is for £100,000 nominal Deliverable bonds Price conversion factors Treas 8 % 2021 1. Treas 4% 2022 0.
A firm plans to borrow £250m in June by an issue of 9-year corporate bonds. At present a coupon rate of 2.25% pa is available on corporate bonds. The treasurer decides to hedge against the possibility of a higher coupon rate in June.
The treasurer closes the futures position in early June by delivering the cheapest to deliver bond. The June futures price has fallen to 112.274, and the clean prices of the deliverable bonds are Treas 8% 130.19 and Treas 4% 93.508. The coupon rate for a 9-year loan is now 2.5%. Show the outcome for the firm and compare the full cost of the loan taken in June what was available on 2 nd^ February. (15 marks) In £100,000 = 2500 contracts @ 114.25. June delivery: Treas 8% (112.234 F 0B 4 1.15955) F 02 D 130.19 = F 02 D0. Treas 4% (112.234 F 0B 4 0.83285) F 02 D 93.508 = F 02 D0. Treas 4% are cheapest to deliver. Loss on delivery is F 02 D0.0006 F 0B 4 2500 contracts F 0B 4 100 bonds per contract = F 02 D £ This is a trivial amount in the context of a £250m loan. The margin account shows a gain as follows (equivalent to a reversing trade) F 0 F 02 D F (^) T Gain = (114.25 F 02 D 112.274) F 0B 4 100(ticks) F 0B 4 £10 F 0B 4 2500 = £4.94m
Firm now only needs to borrow £245.06m. Annual interest is 2.5% = £6.1265m. This is more than what was available in February, ie 2.25% of £250m = £5.625m pa. The cost of a 9-year loan in February was 2.25%. The present value of the loan discounted at 2.25%, by definition, was £250m.
Hence, find the PV of the June loan, discounted at 2.25% PV = (6.1265 F 0B 4 AF, 2.25%, 9) + (245.06 F 0B 4 DF, 2.25%, 9) = £250.0015m Hence, the full long term cost is the same (marginally more), so the hedge is effective, ie it is equivalent to borrowing £250m @ 2.25%.
c) Explain the logic of the spot-futures parity theorem. Outline the adjustments that are necessary if the underlying is a consumption asset. (8 1/3 marks)
Section B Answer at least ONE question from this section
TABLE 4 BT equity options on 8 th^ February 2010
CALLS PUTS
19 Mar 39 days
18Jun 130 days
17 Sep 221 days
19 Mar 39 days
18 Jun 130 days
17 Sep 221 days
BT 100 31.25 32.5 33.5 0.25 1.5 3.
(*131) 130 6.5 10.25 12 5 8.75 13
160 0.25 2.5 4 29 30.25 33.
Refer to table 4 as required when answering this question.
a) Identify the options with the highest and lowest time values, and explain the reasons. (6 1/3 marks) Sept 130 put: highest time value = 13 (intrinsic value 0), close to ‘at the money’ + long time to expiration. The equivalent call is similar with time value of 11 (intrinsic value 1) Mar 160 put: lowest time value = 0 (intrinsic value 29), deep in the money
- short time to expiration. The 100 March call is similar with a time value of only 0.25.
b) A speculator writes September 160 puts on 100,000 shares. Explain the risks involved and illustrate with two contrasting possible outcomes. (8 marks) Speculation, belief that share price will rise during the 221 days to expiration. If the price rises to 160 or more, she gains by the amount of the full premium = £3375. She will only lose from the speculation if S (^) T is below 126.25. Eg if S (^) T is 100, she will lose 60p per share = £6000, reduced by the premiums to an overall loss of £2625. The biggest theoretical loss is £12,
c) Construct a synthetic short stock with September 130 options and compare the outcome with short selling the share. Assume an annual risk free rate of 0.8%. (10 marks) Synthetic = short call + long put. Demonstrate the payoffs at expiration in comparison with short selling the stock. Sept 160 sell call 4 buy put F 02 D33. Cost of synthetic = 29.75 Short stock premium 131 Payoffs and profit at expiration
SYNTHETIC SHORT STOCK
If ST > K the call is exercised If K > S (^) T the put exercised SHORT STOCK
S T
short call payoff
long put payoff Synthetic profit = Payoff F 02 Dcost ie payoff F 02 D 29. Profit = gain(loss) on short sale
F 0 2 D^40 0 F 0 2 D69. F 0 2 D^59
The short stock earns 10.75p more than the synthetic. However, this does not take into account the different investment outlays any dividends dividend paid to the stock.
The synthetic costs 160.75 more than short stock. So it should earn the risk free rate on the difference. 160.75(e0.008 × 221/365^ − 1) = 0. This implies that the short stock position paid a dividend to the share owner of 10.75 + 0.78 = 11.53. Result, in terms of profit or loss, should be: Synthetic gain = short stock gain F 02 D D + opp cost of investment difference = Eg 20.25 = 31 F 02 D 11.53 + 0.
d) Use June options to construct either a bull spread with puts, or a butterfly spread with calls. Show the possible outcomes from the strategy and explain its objective. (9 marks) Illustrate either strategy – SEE NOTES ON HOW THEY WORK. (TOTAL 33 1/3 MARKS)
Q5. Refer to table 4 (See Q4) as required when answering this question.
Dividend FV(D) Payoff from long put K F 02 D ST Repay loans F 02 DFV(D) F 02 D K Net position = 0
If ST = K Neither option is exercised Outcome: Long stock K (as S (^) T = K) Dividend FV(D) Repay loans F 02 DFV(D) F 02 D K Net position = 0
c) Use September 130 to compare a protective put with a short hedge using futures. The September futures price is 120.58. The annual risk free rate is 0.8%. The dividends due on the shares are worth 11.05 at the expiration date. Illustrate each strategy using S (^) T = 100 and S (^) T = 160. Comment on the results. (11 marks)
Protective put cost = 131 + 13 = 144 Hedge with short futures = 131 + zero cost contract @ 120.58.
S (^) T = 100 Short futures: Convergence F (^) T = ST = 100 Gain on futures = F 0 F 02 D F (^) T = 120.58 F 02 D 100 = 20. loss on share = F 02 D 31 Dividend = 11. NET OUTCOME = 0. Protective put: (K = 130) exercise option to sell @ 130 Loss on PP = F 02 D 14 Dividend = 11. NET OUTCOME = F 02 D2.
S (^) T = 160 Short futures: loss on futures = 120.58 F 02 D 160 = F 02 D39. Gain on share = 29 Dividend = 11. NET OUTCOME = 0. Protective put: Do not exercise Gain 160 F 02 D 144 = 26 Dividend = 11. NET OUTCOME = 37.
Discuss the different outcomes (short hedge always earns risk free rate) (TOTAL 33 1/3 MARKS)
TABLE 5 These figures relate to some of the BT options in Table 4
S = 131
Option N(d 1 ) Gamma Vega
March 100 0.994 0.024 1.
June 130 0.473 0.0134 29.
September 160 0.226 0.008 28.
Q6. Refer to tables 4 and 5 as required when answering this question.
a) Explain the reasons for the significant differences between the Greek letters for March 100 and June 130 options (Table 5). (8 marks) Discuss the moneyness and time to expiration of the options. March 100 has a delta close to 1 which means the value of a call will change by almost the same amount as the underlying, as it is already deep in the money with a short time to expiration. The June 130 has a much lower delta: it is close to at the money with a relatively long time to expiration. The gamma is small because delta can only change by small amounts, eg is share price rises further delta will move towards 1 (max value) and gamma will move towards zero. The June 130 is close to at the money, and delta is around 0. indicating that it will move towards 1 as it goes in the money, or towards zero as it moves out of the money. Gamma is big because the changes in delta can be large resulting from relatively small movements in S. Vega (volatility) has a much lesser effect on options that are deep in the money.
b) (^) A portfolio containing BT shares and options has a delta of F 02 D80,000 and a gamma of 350. What is the predicted impact on the value of the portfolio if the price of the underlying asset falls by 2p? (5 marks) value of portfolio changes by F 02 D80,000 F 0B 4^ F 02 D2p = £1,600. But the delta changes by F 02 D 2 F 0B 4 350 = F 02 D700. So incremental impact is F 02 D 700 F 0B 4^ F 02 D2p = £14.
c) Outline the manner and frequency with which traders use the Greek letters to hedge their short term inventory positions in stocks and options. (8 marks) Discuss the ‘realities of hedging’, eg as discussed in textbooks such as Hull.
d) Explain the biases in the Black-Scholes option pricing model as revealed by empirical evidence. (12 1/3 marks) Volatility smiles define the relationships between BS implied volatility and the exercise price. Based on empirical evidence on equity options, BS appears to systematically underprice deep in the money calls and out of the money puts. The smiles are different depending on the underlying asset. Students may illustrate diagrammatically. Whaley’s (1982) research indicated that formulas that allow for early exercise do better at pricing than BS. However, these equations are highly complex. BS does worse at predicting prices of high dividend payout stocks, whereas the true American call option formula does equally well for high or low dividend stocks. Rubenstein (1994) documented a decline in the efficiency of the BS model. In particular, options on the same stock with the same exercise price should have the same implied volatilities but they are progressively different Also found low exercise price options appeared to have higher prices (and
