Solution Guide to Exercises for Chapter 20 Option markets III: applications

Transcription

1 THE ECONOMIC OF FINANCIAL MARKET R. E. BAILEY olution Guide to Exercises for Chapter 20 Option markets III: applications 1. uppose that the futures price of a contract to deliver 1 kilogram of tin twelve months from the present equals $4.40. uppose also that European style put and call options each with an exercise price of $4.40 can be traded in the futures contract, each with a premium of $1.00 per contract. Assume that the interest factor (for twelve months from the present) equals 1.10 and that the differences between forward and futures contracts can be ignored. (a) Describe how, in the absence of market frictions, a portfolio of put and call option can be constructed to have the same payoff as the futures contracts. i. Long futures position. Consider the purchase of one futures contract at price f = The payoff at maturity, T, equals f T Now purchase one call option and write one put option. The payoff on the call option at expiry is c T max[0, f T 4.40]. The payoff on the put option at expiry is p T max[0, 4.40 f T ]. (Notice the minus sign, which appears because the put option has been written.) In words: if f T 4.40, the put option is discarded and its payoff is zero; if f T < 4.40, the payoff is (4.40 f T ), a loss. Hence, if f T < 4.40, the call option is discarded and the put option has a payoff of f T 4.40, equal to that on the futures contract. Alternatively, if f T > 4.40, the put option is discarded and the payoff on the call option is f T 4.40, equal to that on the futures contract. If f T = 4.40, the payoff on both options equals zero, the same as the futures contract. ii. hort futures position. Consider the sale of one futures contract at price f = The payoff at maturity, T, equals 4.40 f T. Now write one call option and purchase one put option. The payoff on the call option at expiry is c T max[0, f T 4.40]. (Notice the minus sign, which appears because the call option has been written.) In words: if f T 4.40, the call option is discarded and its payoff is zero; if f T > 4.40, the payoff is (f T 4.40), a loss. The payoff on the put option at expiry is p T max[0, 4.40 f T ]. Hence, if f T < 4.40, the call option is discarded and the put option has a payoff of 4.40 f T, equal to that on the futures contract. Alternatively, if f T > 4.40, the put option is discarded and the payoff on the call option is (f T 4.40) = 4.40 f T, equal to that on the futures contract. If f T = 4.40, the payoff on both options equals zero, the same as the futures contract. (b) uppose that the put option premium happens to equal $1.50, the other information remaining the same. how that this set of prices allows an arbitrage opportunity. 1

2 Consider writing (selling) one put option, buying one call option, selling one futures contract and lending the difference between the put and the call premia. The payoffs from this strategy are as follows: Initial At expiry, T outlay f T > 4.40 f T 4.40 Write one put option f T 4.40 Buy one call option 1.00 f T ell one futures contract f T 4.40 f T Lend: Notice that the strategy can be scaled-up by any proportion (e.g put options written, 1000 call options purchased, 1000 futures options sold, and 500 loaned out). Hence, an arbtirarily large, risk-free profit can be made with zero initial outlay. This is an arbitrage opportunity. 2. Graph the payoffs (at the expiry date) for each of the following strategies as functions of the underlying asset price: (a) A long straddle. (b) A long strip. (c) A long strap. (d) A long strangle. (e) A bull spread. (f) A bear spread. (g) A butterfly spread. ee the diagrams on the following pages. 2

3 Figure 1: Long traddle X p Put payoff Call payoff p c X Figure 2: Long trip Put payoff Call payoff X c 2p 3

4 Figure 3: Long trap X p Put payoff Call payoff p X 2c Figure 4: Long trangle Put payoff Call payoff p c X 1 X 2 4

5 Figure 5: Bull pread c 2 hort call payoff X 1 Long call payoff c 1 X 2 Figure 6: Bear pread hort call payoff c 2 c 1 X 1 X 2 Long call payoff 5

6 Figure 7: Butterfly pread 2c 2 hort calls payoff Long call payoff X 3 c c 3 1 X 1 X 2 6

7 3. Your company is going to make a 3-month deposit of 1m on 1st July. There is a currently premium of 26 ticks for a call option on the short sterling futures contract with exercise price 95, expiring at the end of June. (a) Explain how the purchase of two call options could be used to place a floor of 4.74% under the interest rate to be received on the deposit. (Note that the face value of the notional deposit in the futures contract is 500,000.) (i) The purchase of two call options costs: = 650. Note that: 650 = 0.26% of 1m for 3 months. Consider the following range of interest rates ruling at the end of June (of course, it cannot be known which row will occur when the options are purchased): Interest Futures Gain (+) % gain (+) Lending Rate Price or loss ( ) or loss ( ) Rate 8.00% % = 7.74% 7.00% % = 6.74% 6.00% % = 5.74% 5.00% % = 4.74% 4.00% % = 4.74% 3.00% % = 4.74% 2.00% % = 4.74% (ii) If the interest rate at the end of June is less than 5%, the option is exercised to place a floor of 4.74% under the interest rate. If the interest rate is 5% or greater, the option is allowed to die and the firm benefits from higher interest rates (though with a cost of 0.26%, the call option premium. (b) uppose that the deposit is not to be made until 1st eptember. How would your strategy change? Consider buying 2 call options with eptember expiry and then sell (or exercise) early. This will not generate a perfect floor because, in general, the futures price (on 1st eptember for end of eptember maturity) will not exactly reflect interest rates on 1st eptember. But, with only one month to maturity, the relationship should be close. (c) uppose that the deposit is to be made on 1st July for 6-months, not 3-months. How would your strategy change? (i) As before, buy two call options expiring at the end of June. Once again the strategy is not perfect in achieving a floor. (ii) If the 6-month deposit is at a floating (not fixed) interest rate, consider buying two eptember call options as well as the two June call options. This would place a floor under the interest rate for the second three-month period. (d) uppose that the deposit is to be made on 1st July for 3-months, but is for 800,000 not 1m. How would your strategy change? In this case it would be worth considering the purchase of one call rather than two. Once again, the strategy is not perfect in generating a floor. 7

8 (e) Discuss the complications which arise if it is believed that the interest rate to be received on the deposit is not perfectly correlated with the interest rate implicit in the futures contract. In this circumstance it would be useful to obtain evidence (i.e. past data) on how the interest rate (say, i t ) which your company will receive on its deposit varies with the rate which is implicit in the futures contract (say, r t ). The statistical relationship between the two interest rates could then be used to estimate an optimal (risk minimising) hedge ratio. This hedge ratio will be the regression coefficient with i t as the dependent variable and r t as the independent variable (regressor). ***** 8