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  1. 1. Chapter Twenty Four ANSWERS TO QUESTIONS 1. Writing index calls does not influence the value of the stock portfolio itself in any way. However, writing calls can result in a financial liability if they close in-the- money. Calls become in-the-money when the stock market rises, so presumably the stock portfolio rose as well if the index calls became valuable. It is likely that the “loss” on the calls is largely offset by a gain on the stock, so in practice the price appreciation of the total portfolio is limited. 2. The lower the striking price the higher the premium income, but the greater the likelihood of exercise. Logically, high striking prices yield little income, but are seldom exercised. High striking prices also have low deltas, which means they provide only a modest adjustment to position delta. In-the-money options, which have high deltas, are powerful tools in altering the portfolio risk. 3. You could also write index or equity puts. 4. Individual equity options are useful in hedging company-specific risk. They are also useful for generating income in small portfolios that are not sufficiently valuable to provide the necessary collateral for option writing. 5. There is disagreement over which is riskier, writing individual calls or index calls. A good argument can be made that writing individual equity calls is riskier, as a single security sometimes has a spectacular one-day gain, rising by perhaps 50% because of takeover news. The stock market has never experienced a gain of this size. Option writers are hurt by rapidly rising prices, and because equity options can rise faster than index options, a good argument can be made that the margin requirements should be stiffer on individual options. (If the equity option is covered, of course, the risk of a rapid price rise is eliminated.) 6. First, figure out the market value and beta of the portfolio to be hedged. Then select a striking price for the put depending on the level of protection desired. Next, calculate the delta of the desired option. (This may require calculating implied volatility first.) Finally, calculate the hedge ratio as shown on page 547. 7. At-the-money futures puts and calls sell for the same price. 8. Writing calls is a bearish strategy when done in isolation. From the writer’s perspective, gains accrue when the value of the underlying asset declines. Combining this strategy with a long position in the underlying asset reduces the market exposure in the underlying asset, because the short option will offset some gains that might be experienced without the options.
  2. 2. Chapter Twenty Four 9. Option prices are not a linear function of the underlying stock price. The 0.5% decline results in the option falling in value by $0.93, while a 0.5% rise results in an option price rise of only $0.50. This difference is primarily because the option was not initially at-the-money. ANSWERS TO PROBLEMS 1. Student response. 2. The income shortfall in the example beginning on page 536 is $16,624. The AUG 315 calls have a premium of $1.75. The number of contracts is therefore $16,624 = 95 $1.75 x 100 3. As shown in the example, the market value of the stock portfolio is equal to about 36 OEX contracts. If fewer than 36 contracts were written, the maximum value of the stock portfolio would be theoretically unlimited. Writing 36 or more contracts puts a limit on the portfolio price appreciation, with the limit occurring at the option striking price. The index can rise from 298.96 to 315.00 (a rise of 5.36%) before the options become in-the-money. Therefore, the maximum value of the equity portfolio is approximately ($996,975 x 1.0536) + $3,025 = $1,053,438 (stock) (cash) This assumes the option premium income was spent and that the stock portfolio tracks the OEX index perfectly. 4. a. 10% rise stock: $996,975 x 0.10 x 1.08 = 107,673 gain calls: 100 x 56 x ($3.40 - 20.68) = 96,768 loss net $10,905 gain $996,975 + 10,905 + 3,025 = $1,010,905 b. 10% decline stock: $996,975 x -0.10 x 1.08 = 107,673 loss calls: 100 x 56 x ($3.40 - 0) = 19,040 gain
  3. 3. Chapter Twenty Four net $88,633 loss $996,975 - 88,633 + 3,025 = $911,367 5 - 7. Student response 8. These delta values assume the futures contract sells for 100. Calls Puts NOV DEC MAR NOV DEC MAR 96 .941 .865 .754 -.055 -.127 -.226 98 .784 .711 .635 -.212 -.281 -.345 100 .503 .503 .501 -.493 -.489 -.479 102 .224 .298 .369 -.772 -.694 -.611 9. At-the-money futures puts and calls should sell for the same price. Therefore, the put should also sell for $2. 10. With a striking price of 99, an underlying asset price of 100, a riskfree interest rate of 5%, a call premium of $2, and one-twelfth of a year until expiration, use the futures put-call parity relationship: P = C - e-Rt(F-K) = $2 - e-(.05)(.0833)($100 - 99) = $2.00 - $1.00 = $1.00 Portfolio size 0.035 x $223 million 11. HR = x beta = x 1.0 = 24.02 Futures size 1300 x 250 Assume a long position in 24 SPX futures contracts to remove the cash drag. 12. CFA Guideline Answer (reprinted with permission from the CFA Study Guide, Association for Investment Management and Research, Charlottesville, VA. All Rights Reserved. A. The problem here is to sell equities and reinvest the proceeds with the skilled fixed-income manager, without changing the split between the existing allocations to the two asset classes. The solution is to turn to derivative financial instruments as the means to the end: selling enough of the existing fixed-income exposure and bringing in enough of the equity exposure to get the desired mix result.
  4. 4. Chapter Twenty Four The following are distinct derivatives strategies that the board could use to increase the Fund’s allocation to the fixed-income manager without changing the present fixed-income index/equity proportions. Strategy 1. Use futures. One strategy would be to sell futures on a fixed-income index and buy futures on an equity index. Selling the futures eliminates the fixed-income index return and risk, while keeping the skilled fixed-income manager’s extra return. By being long the equity index, the portfolio obtains the index return and risk, keeping its exposure to the equity market. Strategy 2. Use swaps. A second strategy would be to use over-the-counter swaps. BI would swap a fixed-income index return for an equity index return in a notional amount large enough to keep the skilled manager’s extra return while eliminating the fixed-income market return and replacing it with the equity market return. Strategy 3. Use option combinations. A third strategy would use put and call options to create futures-like securities. Buying put options and selling call options on a fixed-income index, while selling put options and buying call options on a stock index, would achieve the same result as the appropriate futures position. B. The following are advantages and disadvantages of each strategy identified and explained in Part A: Strategy 1. Use futures Advantages: 1. Futures contracts are liquid instruments. 2. Transaction costs are low. 3. Credit risk is negligible because the securities are marked to market daily. Disadvantages: 1. If the holding period is long, rollover (transaction) costs are incurred. 2. Standard contract forms are limited, so contracts may not exist on the index or instrument needed.
  5. 5. Chapter Twenty Four Strategy 2. Use swaps Advantages: 1. Swaps can be tailored to fit the desired investment horizon, eliminating (or reducing) rollover costs. 2. Swaps can be contracted for a specific index (like the performance benchmark) even if there is no futures contract on it. 3. The desired adjustment goal can be accomplished through a single transaction. Disadvantages: 1. A counterparty credit risk is created that can be much larger than with other types of instruments. 2. Swap agreements are illiquid instruments, and disposals can be both difficult and expensive. 3. Transaction costs are large because of typical “tailoring” of a given swap. Strategy 3. Use option combinations Advantages: 1. Transaction costs are low. 2. Credit risks are small. Disadvantages: 1. Rollover(s) may be necessary. 2. The “right” option may not be available when needed or at all. 3. Holders may exercise the put option and end the hedge. A generic disadvantage of any strategy is that returns are automatically eroded by the costs of establishing and maintaining the strategies, of meeting margin requirements, if any, and of unwinding them, if necessary.
  6. 6. Chapter Twenty Four