Advanced Futures Strategies

Transcription

1 Advanced Futures Strategies 1. Suppose IBC stock is selling at $80 per share. Also, the T-bill rate with 180 days to maturity is 5%. Construct a synthetic futures contract with maturity in 180 days and futures price of $82. In equilibrium, what should be the price of this futures contract? What does that tell us about the expected return from holding the stock? What difference would it make if the stock were expected to pay dividends? 2. Again, suppose IBC stock is selling at $80 per share, and the T-bill rate with 180 days to maturity is 5%. In the previous problem, you constructed a synthetic futures contract with maturity in 180 days and futures price of $82. Now, create a portfolio of put and call options that is equivalent to the futures contract. Suppose the standard deviation for the stock s return is 25% over the period involved here. Use the options calculator to compute the value of the put and the call in your equivalent portfolio. 3. Again, suppose IBC stock is selling at $80 per share, the standard deviation for the stock is 25%, and the T-bill rate with 180 days to maturity is 5%. Construct a synthetic put option with 180 days to expiration and exercise price of $40. Suppose you wanted to use this in a manner equivalent to having an insurance policy to protect against the stock price falling below $40. Explain how you would maintain the protection with a $40 floor as the stock price moves. Repeat with a floor of $ Refer again to problem 3. Could you establish insurance with a floating protection level, so that the present value of the exercise price is always exactly half the current stock price? Would this be harder or easier to maintain than the insurance strategy described in problem 3? 5. Attached is an article from the Wall Street Journal (3/27/01: C5) about a proposal to offer futures contracts on individual stocks. These so-called Universal Stock Futures (USFs) are described as providing a useful new tool for investors, particularly for individual investors who might find it difficult to sell stocks short. Theoretically, one could replicate these futures contracts synthetically (refer to problem 1) or with option spreads (see problem 2). Discuss the practical difficulties of each of these alternatives. 6. Explain how the repurchase agreement plays a role in the pricing of futures contracts. What is the implied repo rate? 7. On January 28, the T-bill futures contract expiring on March 17 (not a leap year) was priced at 93 (IMM Index). The T-bill maturing at that time was priced at a discount of 7.10 and the T-bill maturing on June 16 at a discount of Determine the implied repo rate. Is there an arbitrage opportunity? 8. Suppose you are managing a diversified stock portfolio with beta of Describe how you could use index futures contracts to change the beta of your position to Prof. Kensinger page 1

2 9. Suppose you are managing a stock portfolio worth $12,500,000. It has a beta of During the next three months, you fear there might be a correction in the market that could take the market down about 5%. To reduce the risk of loss if this happens, you would like to reduce the beta of your portfolio to 1. A stock index futures contract with the appropriate expiration is priced at 1250 with a multiplier of 250. a) Should you buy or sell futures? How many contracts should you use? b) Suppose that after a while a correction similar to the one you feared actually occurs. Your portfolio has fallen in value to $11,750,000. The price in the market index future you sold has fallen to , and you unwind the hedge. Determine the profit on the futures contract and the overall portfolio return over the life of the hedge. How close did you come to the desired result? 10. Suppose you are a multi-national company with supplies of crude oil stored in several nations. You observe the following prices: Euro per Dollar exchange rate is 0.60 spot and 0.62 for 180-day forward. German interest rate is 5% compounded daily. U.S. interest rate is 3% compounded daily. Explain how you could take advantage of this situation using just your oil inventories, without changing the total amount of oil you own worldwide. You won t need to borrow money or invest in bonds in either country. Just assume that oil futures markets are in equilibrium in both countries. Prof. Kensinger page 2

3 Nasdaq, Liffe Plan Stock-Futures Venture By SILVIA ASCARELLI Staff Reporter of THE WALL STREET JOURNAL March 27, 2001 Page C5 LONDON-The Nasdaq Stock Market and the London International Financial Futures and Options Exchange, or Liffe, said they will jointly introduce single-stock futures in the U.S. later this year. The rare trans-atlantic partnership depends on an anticipated change in U.S. regulations, which is expected by year end. U.S. investors have been barred from investing in single-stock futures, known at Liffe as Universal Stock Futures, or USFs, because of fears of market manipulation. With such contracts, investors commit themselves to buying or selling a security at a set price on a certain date. Among other things, investors can use singlestock futures for short sales. In a short sale, an investor borrows a security, sells it and hopes to buy it back later at a lower price, pocketing the difference. Attracting Nasdaq is a coup for Liffe, which only three years ago was in disarray after the bulk of trading in futures on German government bonds, Liffe's biggest product, shifted to the rival Swiss-German Eurex exchange. The accord with Nasdaq gives Liffe access to a huge new group of potential investors in single-stock futures, which were introduced by Liffe in London two months ago. Trading currently averages about 200 contracts a day in each of 30 USFs currently listed on Liffe, or about half the average daily volume in each of the 98 individual equity options contracts now traded on the exchange. That is small stuff for Liffe, which trades nearly 750,000 contracts every day, mostly tied to bonds and money markets, but Which is trying to boost its stock-reiated business. It also represents a new foothold in Europe for Nasdaq, which is expected to announce its intent to take a majority stake in Easdaq, a floundering pan-european market patterned after Nasdaq, later this week. But Nasdaq has otherwise struggled to establish itself in Europe, and securities firms refused to back an earlier plan to launch a new pan-european exchange that would have been known as Nasdaq Europe. Executives from the two exchanges said they could eventually take their partnership to Asia, where Nasdaq is already operating a stock market with the Osaka Stock Exchange. The UPS currently traded on Liffe are based on some of the biggest stocks in Britain, continental Europe and the U.S., including Nasdaq-listed stocks like Microsoft Corp. and Cisco Systems Inc. The exchange plans to add an additional 10 next month. Liffe Chairman Brian Williamson said the exchange is aiming for about 100 USFs by year end. UPS are aimed at individual investors, who generally can't short stocks, as well as at institutional investors. The partnership will use Liffe's Connect electronic trading system and will trade during the U.S. business day. Executives declined to disclose other terms of the deal. While Mr. Williamson confidently predicted that USFs mark a revolution in equity trading, other exchanges have struggled with similar concepts. Eurex, the world's biggest Prof. Kensinger page 3

4 derivatives market, lists about 90 "Lepos," or low exercise price options, which it says function in the same way as USFs. Originally launched in Switzerland as a way to avoid stamp duty, the options have attracted no trading for a few years, a spokesman said. Prof. Kensinger page 4

5 Financial Derivatives Solutions: Problem Set 2 Spring We can use the put-call parity relationship to reason our way through this question. First, recognize that a long futures contract is equivalent to a long call and a short put with exercise prices the same as the futures price. We start with the put-call parity relationship: C(S,X,t) + B(X,t) = S + P(S,X,t) This rearranges to the following: C(S,X,t) P(S,X,t) = S B(X,t) We can calculate the present value of $82 to be received in 180 days, finding it to be $80. Then, substituting the known values we find: C(S,X,t) P(S,X,t) = = 0 If the futures price is an accurate predictor of what the spot price will be at expiration, the conclusion is that the expected return from holding the stock is the risk-free rate. If the stock were expected to pay dividends, then the storage cost would in fact be a benefit to the holder. Expected dividends would be included in the normal cost-of-carry basis. Any risk-adjustment done in the discounting would be applied to the dividend portion only. 2. We have already done the equivalent portfolio (a long futures contract equals a long call with a short put). The values of the put and call are almost identical (approximately $5.59 for each). The difference between the two is less than 3/10ths of a penny, which equals the difference between the current stock price and the exact present value of the exercise price. You can try different volatilities, but will find exactly the same result. Conclusion: In a normal cost-of-carry situation, the equilibrium futures price is the present value of the futures price (discounted at the risk-free rate) minus the spot price adjusted for storage costs. 3. Once again we can use the put-call parity relationship to reason our way through this question. We start with the put-call parity relationship: C(S,X,t) + B(X,t) = S + P(S,X,t) This rearranges to the following: C(S,X,t) + B(X,t) S = P(S,X,t) Now, we also know that: C(S,X,t) = S * N(d 1 ) B(X,t) * N(d 2 ) Substituting this into the put-call parity relationship, we find: P(S,X,t) = S * N(d 1 ) B(X,t) * N(d 2 ) + B(X,t) S This simplifies as follows: P(S,X,t) = S * (N(d 1 ) 1) + B(X,t) * (1 N(d 2 )) Therefore, the synthetic put can be constructed with the right combination of selling stock and purchasing bonds. The amounts are defined by N(d 1 ) and N(d2). With stock price of $80 and exercise price of $40, N(d 1 ) and N(d 2 ) both approach 1. The put is nearly without value, and need not be synthesized. If the exercise price were $60, N(d 1 ) would be.9690 and N(d 2 ) would be Then for each put to be synthesized, you would sell.031 shares of stock and buy.0454 bonds. 4. Such a policy would be easier to maintain because the prime input into calculating N(d 1 ) and N(d 2 ) would be held constant. This prime ingredient is the ratio of the stock price to the present value of the exercise price. 5. For class discussion. 6. The implied repo rate for a given term is the futures price minus the spot price Prof. Kensinger

6 Financial Derivatives Solutions: Problem Set 2 Spring 2015 divided by the spot price. In equilibrium, this is the internal rate of return from a repo agreement. If equilibrium were not maintained, one could arbitrage repo agreements against futures contracts. 7. With IMM Index of 93, the futures price is the following: f = 100 (100 93) (90/360) = For a million-dollar contract, then, the futures price would be $982,500 (this contract is for delivery of a specified contract on the expiration date). If the delivery in the futures contract is for a T-bill with 90 days to maturity, an arbitrage might be available. There are 48 days from Jan 28 to Mar 17, and 91 days between Mar 17 and June 16 (assuming it isn t a leap year). From Jan 28 to June 16 there are 139 days. Let us consider selling bills maturing March 17, selling the futures contract, and buying the bills maturing June 16 (which would provide the deliverable instrument in the futures contract). Then for each $1,000,000 of face value, the price paid for the June bills would be $973, on Jan 28. We can borrow against the $982,500 to be received March 17 by selling March bills short, providing $973,370 on Jan 28. This leaves an immediate profit of $303. The implied repo rate is the rate earned from buying at the spot price and selling at the futures price. On January 28 the futures price is 982,500 and the spot price is 973,067 with 48 days to expiration of the futures contract. We have FV, PV, and N, so we can use the financial calculator to measure the implicit interest rate. It comes out as an APR of 7.34% with daily compounding. This is out of equilibrium with the other quoted discounts. 8. The desired beta is half the beta for the portfolio. If the beta for the futures contract is one, the amount involved in the futures position would be half the value of the portfolio. The beta of the futures contract may not be exactly one, so to be precise in that event, multiply 0.5 times the beta of the futures contract The beta of the futures contract on the stock market index is e (r-d)t 9. Part a: This is a specific application of the principal you developed in the previous problem. Here the desired beta divided by the portfolio beta is 1/1.25. Let s keep it simple and say the beta for the futures contract is one, so the amount of the futures position would be ((1/1.25) 1)12,500,000 = 2,500,000 Divide this by the index and then by the multiplier in order to translate the amount involved into the number of contracts needed. The answer is 8 contracts. Part b: The index has dropped 5.5%, and the portfolio has dropped 5.5% from their original values (6/5.5 is only 1.09, so the drop in the portfolio was not as great as its beta would predict, and so there may have been some alpha capture). Profit from the futures position would be the number of contracts times the multiplier times So, the profit would be $137,500. Added to the new value of the portfolio, you would have $11,887,500, reducing the loss to 4.9% of the original value of the portfolio. This is a bit less than the 5.5% drop on the index, which is consistent with a beta of 1 combined with some alpha capture. Prof. Kensinger

7 Financial Derivatives Solutions: Problem Set 2 Spring Here you could convert the strategy from Set 1, problem 13. An arbitrage to take advantage of this involves the following steps: a. Sell oil worth 600,000 in Germany and buy futures contracts to replace the oil. b. Convert the Euros to Dollars and buy $1,000,000 worth of oil in the U.S. Sell futures contracts to reduce the price risk of selling the oil again in 180 days. c. Contract to exchange the future value of your Dollars for Euros at $1 = 0.62 in 180 days. Assuming that the basis reflects the interest rates and storage costs in the country where delivery will occur, you can expect the same profit from this strategy as you would have from the bond-based arbitrage in problem 13 of the first problem set. Prof. Kensinger