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Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 considered relatively expensive because a higher standard deviation is required to justify its price. The analyst might consider buying the option with the lower implied volatility and writ- ing the option with the higher implied volatility. The Black-Scholes call-option valuation formula, as well as implied volatilities, are eas- ily calculated using an Excel spreadsheet, as in Figure 15.4. The model inputs are listed in 15 Option Valuation 545 TABLE 15.2 (concluded) dN(d ) dN(d ) dN(d ) 0.06 0.5239 0.86 0.8051 1.66 0.9515 0.08 0.5319 0.88 0.8106 1.68 0.9535 0.10 0.5398 0.90 0.8159 1.70 0.9554 0.12 0.5478 0.92 0.8212 1.72 0.9573 0.14 0.5557 0.94 0.8264 1.74 0.9591 0.16 0.5636 0.96 0.8315 1.76 0.9608 0.18 0.5714 0.98 0.8365 1.78 0.9625 0.20 0.5793 1.00 0.8414 1.80 0.9641 0.22 0.5871 1.02 0.8461 1.82 0.9656 0.24 0.5948 1.04 0.8508 1.84 0.9671 0.26 0.6026 1.06 0.8554 1.86 0.9686 0.28 0.6103 1.08 0.8599 1.88 0.9699 0.30 0.6179 1.10 0.8643 1.90 0.9713 0.32 0.6255 1.12 0.8686 1.92 0.9726 0.34 0.6331 1.14 0.8729 1.94 0.9738 0.36 0.6406 1.16 0.8770 1.96 0.9750 0.38 0.6480 1.18 0.8810 1.98 0.9761 0.40 0.6554 1.20 0.8849 2.00 0.9772 0.42 0.6628 1.22 0.8888 2.05 0.9798 0.44 0.6700 1.24 0.8925 2.10 0.9821 0.46 0.6773 1.26 0.8962 2.15 0.9842 0.48 0.6844 1.28 0.8997 2.20 0.9861 0.50 0.6915 1.30 0.9032 2.25 0.9878 0.52 0.6985 1.32 0.9066 2.30 0.9893 0.54 0.7054 1.34 0.9099 2.35 0.9906 0.56 0.7123 1.36 0.9131 2.40 0.9918 0.58 0.7191 1.38 0.9162 2.45 0.9929 0.60 0.7258 1.40 0.9192 2.50 0.9938 0.62 0.7324 1.42 0.9222 2.55 0.9946 0.64 0.7389 1.44 0.9251 2.60 0.9953 0.66 0.7454 1.46 0.9279 2.65 0.9960 0.68 0.7518 1.48 0.9306 2.70 0.9965 0.70 0.7580 1.50 0.9332 2.75 0.9970 0.72 0.7642 1.52 0.9357 2.80 0.9974 0.74 0.7704 1.54 0.9382 2.85 0.9978 0.76 0.7764 1.56 0.9406 2.90 0.9981 0.78 0.7823 1.58 0.9429 2.95 0.9984 0.80 0.7882 1.60 0.9452 3.00 0.9986 0.82 0.7939 1.62 0.9474 3.05 0.9989 0.84 0.7996 1.64 0.9495 Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 546 column B, and the outputs are given in column E. The formulas for d 1 and d 2 are provided in the spreadsheet, and the Excel formula NORMSDIST(d 1 ) is used to calculate N(d 1 ). Cell E6 contains the Black-Scholes call option formula. To compute an implied volatility, we can use the Solver command from the Tools menu in Excel. Solver asks us to change the value of one cell to make the value of another cell (called the target cell) equal to a specific value. For ex- ample, if we observe a call option selling for $7 with other inputs as given in the spreadsheet, we can use Solver to find the value for cell B2 (the standard deviation of the stock) that will make the option value in cell E6 equal to $7. In this case, the target cell, E6, is the call price, and the spreadsheet manipulates cell B2. When you ask the spreadsheet to “Solve,” it finds that a standard deviation equal to .2783 is consistent with a call price of $7; therefore, 27.83% would be the call’s implied volatility if it were selling at $7. 7. Consider the call option in Example 15.2 If it sells for $15 rather than the value of $13.70 found in the example, is its implied volatility more or less than 0.5? The Put-Call Parity Relationship So far, we have focused on the pricing of call options. In many important cases, put prices can be derived simply from the prices of calls. This is because prices of European put and call EXCEL Applications www.mhhe.com/bkm Black-Scholes Option Pricing Figure 15.4 captures a portion of the Excel model “B-S Option.” The model is built to value puts and calls and extends the discussion to include analysis of intrinsic value and time value of op- tions. The spreadsheet contains sensitivity analyses on several key variables in the Black-Scholes pricing model. You can learn more about this spreadsheet model by using the interactive version available on our website at www .mhhe.com/bkm. > Concept CHECK > FIGURE 15.4 Spreadsheet to calculate Black-Scholes call-option values Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 options are linked together in an equation known as the put-call parity relationship. Therefore, once you know the value of a call, put pricing is easy. To derive the parity relationship, suppose you buy a call option and write a put option, each with the same exercise price, X, and the same expiration date, T. At expiration, the payoff on your investment will equal the payoff to the call, minus the payoff that must be made on the put. The payoff for each option will depend on whether the ultimate stock price, S T , exceeds the exercise price at contract expiration. S T Յ XS T Ͼ X Payoff of call held 0 S T Ϫ X ϪPayoff of put written Ϫ(X Ϫ S T )0 Total S T Ϫ XS T Ϫ X Figure 15.5 illustrates this payoff pattern. Compare the payoff to that of a portfolio made up of the stock plus a borrowing position, where the money to be paid back will grow, with interest, to X dollars at the maturity of the loan. Such a position is a levered equity position in 15 Option Valuation 547 FIGURE 15.5 The payoff pattern of a long call–short put position S T Payoff Payoff Payoff X S T S T X Long call + Short put = Leveraged equity Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 which X/(1 ϩ r f ) T dollars is borrowed today (so that X will be repaid at maturity), and S 0 dol- lars is invested in the stock. The total payoff of the levered equity position is S T Ϫ X, the same as that of the option strategy. Thus, the long call–short put position replicates the levered equity position. Again, we see that option trading provides leverage. Because the option portfolio has a payoff identical to that of the levered equity position, the costs of establishing them must be equal. The net cash outlay necessary to establish the option position is C Ϫ P: The call is purchased for C, while the written put generates income of P. Likewise, the levered equity position requires a net cash outlay of S 0 Ϫ X/(1 ϩ r f ) T , the cost of the stock less the proceeds from borrowing. Equating these costs, we conclude C Ϫ P ϭ S 0 Ϫ X/(1 ϩ r f ) T (15.2) Equation 15.2 is called the put-call parity relationship because it represents the proper re- lationship between put and call prices. If the parity relationship is ever violated, an arbitrage opportunity arises. Equation 15.2 actually applies only to options on stocks that pay no dividends before the maturity date of the option. It also applies only to European options, as the cash flow streams from the two portfolios represented by the two sides of Equation 15.2 will match only if each position is held until maturity. If a call and a put may be optimally exercised at different times 548 Part FIVE Derivative Markets put-call parity relationship An equation representing the proper relationship between put and call prices. 15.3 EXAMPLE Put-Call Parity Suppose you observe the following data for a certain stock. Stock price $110 Call price (six-month maturity, X ϭ $105) 17 Put price (six-month maturity, X ϭ $105) 5 Risk-free interest rate 10.25% effective annual yield (5% per 6 months) We use these data in the put-call parity relationship to see if parity is violated. C Ϫ P ՘ S 0 Ϫ X/(1 ϩ r f ) T 17 Ϫ 5 ՘ 110 Ϫ 105/1.05 12 ՘ 10 This result, a violation of parity (12 does not equal 10) indicates mispricing and leads to an arbitrage opportunity. You can buy the relatively cheap portfolio (the stock plus borrowing position represented on the right-hand side of the equation) and sell the relatively expensive portfolio (the long call–short put position corresponding to the left-hand side, that is, write a call and buy a put). Let’s examine the payoff to this strategy. In six months, the stock will be worth S T . The $100 borrowed will be paid back with interest, resulting in a cash outflow of $105. The writ- ten call will result in a cash outflow of S T Ϫ $105 if S T exceeds $105. The purchased put pays off $105 Ϫ S T if the stock price is below $105. Table 15.3 summarizes the outcome. The immediate cash inflow is $2. In six months, the various positions provide exactly offsetting cash flows: The $2 inflow is realized risklessly with- out any offsetting outflows. This is an arbitrage opportunity that investors will pursue on a large scale until buying and selling pressure restores the parity condition expressed in Equa- tion 15.2. Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 before their common expiration date, then the equality of payoffs cannot be assured, or even expected, and the portfolios will have different values. The extension of the parity condition for European call options on dividend-paying stocks is, however, straightforward. Problem 22 at the end of the chapter leads you through the extension of the parity relationship. The more general formulation of the put-call parity con- dition is P ϭ C Ϫ S 0 ϩ PV(X) ϩ PV(dividends) (15.3) where PV(dividends) is the present value of the dividends that will be paid by the stock dur- ing the life of the option. If the stock does not pay dividends, Equation 15.3 becomes identi- cal to Equation 15.2. Notice that this generalization would apply as well to European options on assets other than stocks. Instead of using dividend income in Equation 15.3, we would let any income paid out by the underlying asset play the role of the stock dividends. For example, European put and call options on bonds would satisfy the same parity relationship, except that the bond’s coupon income would replace the stock’s dividend payments in the parity formula. Let’s see how well parity works using real data on the Microsoft options in Figure 14.1 from the previous chapter. The April maturity call with exercise price $70 and time to expira- tion of 105 days cost $4.60 while the corresponding put option cost $5.40. Microsoft was sell- ing for $68.90, and the annualized 105-day interest rate on this date was 1.6%. Microsoft was paying no dividends at this time. According to parity, we should find that P ϭ C ϩ PV(X) Ϫ S 0 ϩ PV(dividends) 5.40 ϭ 4.60 ϩϪ68.90 ϩ 0 5.40 ϭ 4.60 ϩ 69.68 Ϫ 68.90 5.40 ϭ 5.38 So, parity is violated by about $0.02 per share. Is this a big enough difference to exploit? Prob- ably not. You have to weigh the potential profit against the trading costs of the call, put, and stock. More important, given the fact that options trade relatively infrequently, this deviation from parity might not be “real” but may instead be attributable to “stale” (i.e., out-of-date) price quotes at which you cannot actually trade. Put Option Valuation As we saw in Equation 15.3, we can use the put-call parity relationship to value put options once we know the call option value. Sometimes, however, it is easier to work with a put option 70 (1.016) 105/365 15 Option Valuation 549 TABLE 15.3 Arbitrage strategy Cash Flow in Six Months Position Immediate Cash Flow S T Ͻ 105 S T Ն 105 Buy stock Ϫ110 S T S T Borrow X/(1 ϩ r f ) T ϭ $100 ϩ100 Ϫ105 Ϫ105 Sell call ϩ17 0 Ϫ(S T Ϫ 105) Buy put Ϫ5 105 Ϫ S T 0 Total 2 0 0 Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 valuation formula directly. The Black-Scholes formula for the value of a European put op- tion is 3 P ϭ Xe ϪrT [1 Ϫ N(d 2 )] Ϫ S 0 e Ϫ␦T [1 Ϫ N(d 1 )] (15.4) Equation 15.4 is valid for European puts. Listed put options are American options that offer the opportunity of early exercise, however. Because an American option allows its owner to exercise at any time before the expiration date, it must be worth at least as much as the corre- sponding European option. However, while Equation 15.4 describes only the lower bound on the true value of the American put, in many applications the approximation is very accurate. 15.4 USING THE BLACK-SCHOLES FORMULA Hedge Ratios and the Black-Scholes Formula In the last chapter, we considered two investments in Microsoft: 100 shares of Microsoft stock or 700 call options on Microsoft. We saw that the call option position was more sensitive to swings in Microsoft’s stock price than the all-stock position. To analyze the overall exposure to a stock price more precisely, however, it is necessary to quantify these relative sensitivities. A tool that enables us to summarize the overall exposure of portfolios of options with various exercise prices and times to maturity is the hedge ratio. An option’s hedge ratio is the change in the price of an option for a $1 increase in the stock price. A call option, therefore, has a pos- itive hedge ratio, and a put option has a negative hedge ratio. The hedge ratio is commonly called the option’s delta. If you were to graph the option value as a function of the stock value as we have done for a call option in Figure 15.6, the hedge ratio is simply the slope of the value function evaluated at the current stock price. For example, suppose the slope of the curve at S 0 ϭ $120 equals 0.60. As the stock increases in value by $1, the option increases by approximately $0.60, as the figure shows. For every call option written, 0.60 shares of stock would be needed to hedge the investor’s portfolio. For example, if one writes 10 options and holds six shares of stock, according to the hedge ratio of 0.6, a $1 increase in stock price will result in a gain of $6 on the stock holdings, 550 Part FIVE Derivative Markets 3 This formula is consistent with the put-call parity relationship, and in fact can be derived from it. If you want to try to do so, remember to take present values using continuous compounding, and note that when a stock pays a continuous flow of income in the form of a constant dividend yield, ␦, the present value of that dividend flow is S 0 (1 Ϫ e Ϫ␦T ). (Notice that e Ϫ␦T approximately equals 1 Ϫ␦T, so the value of the dividend flow is approximately ␦TS 0 .) 15.4 EXAMPLE Black-Scholes Put Option Valuation Using data from the Black-Scholes call option in Example 15.2 we find that a European put option on that stock with identical exercise price and time to maturity is worth $95e Ϫ.10 ϫ .25 (1 Ϫ .5714) Ϫ $100(1 Ϫ .6664) ϭ $6.35 Notice that this value is consistent with put-call parity: P ϭ C ϩ PV(X) Ϫ S 0 ϭ 13.70 ϩ 95e Ϫ.10 ϫ .25 Ϫ 100 ϭ 6.35 As we noted traders can do, we might then compare this formula value to the actual put price as one step in formulating a trading strategy. hedge ratio or delta The number of shares of stock required to hedge the price risk of holding one option. Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 while the loss on the 10 options written will be 10 ϫ $0.60, an equivalent $6. The stock price movement leaves total wealth unaltered, which is what a hedged position is intended to do. The investor holding both the stock and options in proportions dictated by their relative price movements hedges the portfolio. Black-Scholes hedge ratios are particularly easy to compute. The hedge ratio for a call is N(d 1 ), while the hedge ratio for a put is N(d 1 ) Ϫ 1. We defined N(d 1 ) as part of the Black- Scholes formula in Equation 15.1. Recall that N(d ) stands for the area under the standard nor- mal curve up to d. Therefore, the call option hedge ratio must be positive and less than 1.0, while the put option hedge ratio is negative and of smaller absolute value than 1.0. Figure 15.6 verifies the insight that the slope of the call option valuation function is less than 1.0, approaching 1.0 only as the stock price becomes extremely large. This tells us that option values change less than one-for-one with changes in stock prices. Why should this be? Suppose an option is so far in the money that you are absolutely certain it will be exercised. In that case, every $1 increase in the stock price would increase the option value by $1. But if there is a reasonable chance the call option will expire out of the money, even after a moder- ate stock price gain, a $1 increase in the stock price will not necessarily increase the ultimate payoff to the call; therefore, the call price will not respond by a full $1. The fact that hedge ratios are less than 1.0 does not contradict our earlier observation that options offer leverage and are sensitive to stock price movements. Although dollar move- ments in option prices are slighter than dollar movements in the stock price, the rate of return volatility of options remains greater than stock return volatility because options sell at lower prices. In our example, with the stock selling at $120, and a hedge ratio of 0.6, an option with exercise price $120 may sell for $5. If the stock price increases to $121, the call price would be expected to increase by only $0.60, to $5.60. The percentage increase in the option value is $0.60/$5.00 ϭ 12%, however, while the stock price increase is only $1/$120 ϭ 0.83%. The ratio of the percent changes is 12%/0.83% ϭ 14.4. For every 1% increase in the stock price, the option price increases by 14.4%. This ratio, the percent change in option price per percent change in stock price, is called the option elasticity. The hedge ratio is an essential tool in portfolio management and control. An example will show why. 15 Option Valuation 551 FIGURE 15.6 Call option value and hedge ratio Value of a call (C ) S 0 40 20 0 120 Slope = .6 option elasticity The percentage increase in an option’s value given a 1% increase in the value of the underlying security. Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 8. What is the elasticity of a put option currently selling for $4 with exercise price $120, and hedge ratio ؊0.4 if the stock price is currently $122? Portfolio Insurance In Chapter 14, we showed that protective put strategies offer a sort of insurance policy on an asset. The protective put has proven to be extremely popular with investors. Even if the asset price falls, the put conveys the right to sell the asset for the exercise price, which is a way to lock in a minimum portfolio value. With an at-the-money put (X ϭ S 0 ), the maximum loss that can be realized is the cost of the put. The asset can be sold for X, which equals its original price, so even if the asset price falls, the investor’s net loss over the period is just the cost of the put. If the asset value increases, however, upside potential is unlimited. Figure 15.7 graphs the profit or loss on a protective put position as a function of the change in the value of the underlying asset. While the protective put is a simple and convenient way to achieve portfolio insurance, that is, to limit the worst-case portfolio rate of return, there are practical difficulties in trying to insure a portfolio of stocks. First, unless the investor’s portfolio corresponds to a standard market index for which puts are traded, a put option on the portfolio will not be available for purchase. And if index puts are used to protect a nonindexed portfolio, tracking error can re- sult. For example, if the portfolio falls in value while the market index rises, the put will fail to provide the intended protection. Tracking error limits the investor’s freedom to pursue ac- tive stock selection because such error will be greater as the managed portfolio departs more substantially from the market index. Moreover, the desired horizon of the insurance program must match the maturity of a traded put option in order to establish the appropriate protective put position. Today, long-term index options called LEAPS (for Long-Term Equity AnticiPation Securities) trade on the Chicago Board Options Exchange with maturities of several years. However, in the mid- 1980s, while most investors pursuing insurance programs had horizons of several years, ac- tively traded puts were limited to maturities of less than a year. Rolling over a sequence of short-term puts, which might be viewed as a response to this problem, introduces new risks because the prices at which successive puts will be available in the future are not known today. Providers of portfolio insurance with horizons of several years, therefore, cannot rely on the simple expedient of purchasing protective puts for their clients’ portfolios. Instead, they follow trading strategies that replicate the payoffs to the protective put position. 552 Part FIVE Derivative Markets 15.5 EXAMPLE Portfolio Hedge Ratios Consider two portfolios, one holding 750 IBM calls and 200 shares of IBM and the other holding 800 shares of IBM. Which portfolio has greater dollar exposure to IBM price move- ments? You can answer this question easily using the hedge ratio. Each option changes in value by H dollars for each dollar change in stock price, where H stands for the hedge ratio. Thus, if H equals 0.6, the 750 options are equivalent to 450 (ϭ 0.6 ϫ 750) shares in terms of the response of their market value to IBM stock price movements. The first portfolio has less dollar sensitivity to stock price change because the 450 share-equivalents of the options plus the 200 shares actually held are less than the 800 shares held in the second portfolio. This is not to say, however, that the first portfolio is less sensitive to the stock’s rate of re- turn. As we noted in discussing option elasticities, the first portfolio may be of lower total value than the second, so despite its lower sensitivity in terms of total market value, it might have greater rate of return sensitivity. Because a call option has a lower market value than the stock, its price changes more than proportionally with stock price changes, even though its hedge ratio is less than 1.0. Concept CHECK > portfolio insurance Portfolio strategies that limit investment losses while maintaining upside potential. Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 Here is the general idea. Even if a put option on the desired portfolio with the desired ex- piration date does not exist, a theoretical option-pricing model (such as the Black-Scholes model) can be used to determine how that option’s price would respond to the portfolio’s value if the option did trade. For example, if stock prices were to fall, the put option would increase in value. The option model could quantify this relationship. The net exposure of the (hypo- thetical) protective put portfolio to swings in stock prices is the sum of the exposures of the two components of the portfolio: the stock and the put. The net exposure of the portfolio equals the equity exposure less the (offsetting) put option exposure. We can create “synthetic” protective put positions by holding a quantity of stocks with the same net exposure to market swings as the hypothetical protective put position. The key to this strategy is the option delta, or hedge ratio, that is, the change in the price of the protective put option per change in the value of the underlying stock portfolio. 15 Option Valuation 553 EXAMPLE 15.6 Synthetic Protective Puts Suppose a portfolio is currently valued at $100 million. An at-the-money put option on the portfolio might have a hedge ratio or delta of Ϫ0.6, meaning the option’s value swings $0.60 for every dollar change in portfolio value, but in an opposite direction. Suppose the stock port- folio falls in value by 2%. The profit on a hypothetical protective put position (if the put existed) would be as follows (in millions of dollars): Loss on stocks 2% of $100 ϭ $2.00 ϩGain on put: 0.6 ϫ $2.00 ϭ 1.20 Net loss $0.80 We create the synthetic option position by selling a proportion of shares equal to the put option’s delta (i.e., selling 60% of the shares) and placing the proceeds in risk-free T-bills. The rationale is that the hypothetical put option would have offset 60% of any change in the stock portfolio’s value, so one must reduce portfolio risk directly by selling 60% of the equity and FIGURE 15.7 Profit on a protective put strategy 0 0 ϪP Cost of put Change in value of protected position Change in value of underlying asset Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V. Derivative Markets 15. Option Valuation © The McGraw−Hill Companies, 2003 The difficulty with synthetic positions is that deltas constantly change. Figure 15.8 shows that as the stock price falls, the absolute value of the appropriate hedge ratio increases. There- fore, market declines require extra hedging, that is, additional conversion of equity into cash. This constant updating of the hedge ratio is called dynamic hedging, as discussed in Section 15.2. Another term for such hedging is delta hedging, because the option delta is used to de- termine the number of shares that need to be bought or sold. Dynamic hedging is one reason portfolio insurance has been said to contribute to market volatility. Market declines trigger additional sales of stock as portfolio insurers strive to in- crease their hedging. These additional sales are seen as reinforcing or exaggerating market downturns. In practice, portfolio insurers do not actually buy or sell stocks directly when they update their hedge positions. Instead, they minimize trading costs by buying or selling stock index fu- tures as a substitute for sale of the stocks themselves. As you will see in the next chapter, stock prices and index future prices usually are very tightly linked by cross-market arbitrageurs so that futures transactions can be used as reliable proxies for stock transactions. Instead of 554 Part FIVE Derivative Markets putting the proceeds into a risk-free asset. Total return on a synthetic protective put position with $60 million in risk-free investments such as T-bills and $40 million in equity is Loss on stocks: 2% of $40 ϭ $0.80 ϩLoss on bills: 0 Net loss ϭ $0.80 The synthetic and actual protective put positions have equal returns. We conclude that if you sell a proportion of shares equal to the put option’s delta and place the proceeds in cash equivalents, your exposure to the stock market will equal that of the desired protective put position. FIGURE 15.8 Hedge ratios change as the stock price fluctuates 0 Value of a put (P) S 0 Low slope = Low hedge ratio Higher slope = High hedge ratio dynamic hedging Constant updating of hedge positions as market conditions change. [...]... is the number of contracts outstanding (Long and short positions are not counted separately, meaning that open interest can be defined as either the number of long or short contracts outstanding.) The clearinghouse s position nets out to zero, and so it is not counted in the computation of open interest When contracts begin trading, open interest is zero As time passes, open interest increases as progressively... basis is currently $5 Tomorrow, the spot price might increase to $294, while the futures price increases to $2 98. 50, so the basis narrows to $4.50 The investor s gains and losses are as follows: Gain on holdings of gold (per ounce): basis The difference between the futures price and the spot price basis risk Risk attributable to uncertain movements in the spread between a futures price and a spot price... zero sum game, with losses and gains to all positions netting out to zero Every long position is offset by a short position The aggregate profits to futures trading, summing over all investors, also must be zero, as is the net exposure to changes in the commodity price Figure 16.3, panel A, is a plot of the profits realized by an investor who enters the long side of a futures contract as a function of. .. Companies, 2003 16 Futures Markets 571 16 Futures Markets Profit Profit Payoff Profit X PT PT F0 A Long futures profit = PT – F0 F0 B Short futures profit = F0 – PT C Buy a call option F I G U R E 16.3 Profits to buyers and sellers of futures and options contracts A: Long futures position (buyer) B: Short futures position (seller) C: Buy call option currencies, and financial futures (fixed-income securities... market strategies for hedging or speculative purposes Compute the futures price appropriate to a given price on the underlying asset Design arbitrage strategies to exploit futures market mispricing Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition V Derivative Markets 16 Futures Markets Related Websites http://www.options.about.com/money/options http://www.numa.com The above sites are good places... Companies, 2003 577 16 Futures Markets daily settlement Therefore, taxes are paid at year-end on cumulated profits or losses regardless of whether the position has been closed out 16.3 FUTURES MARKET STRATEGIES Hedging and Speculation Hedging and speculating are two polar uses of futures markets A speculator uses a futures contract to profit from movements in futures prices, a hedger to protect against... markets taught some practitioners—including banks and securities firms that were hedging options sales to hedge funds and other investors—the same painful lessons of earlier portfolio insurers: Deltahedging can break down in volatile markets, just when it is needed most How you delta-hedge depends on the bets you’re trying to hedge For instance, delta-hedging would prompt options sellers to sell into... the mechanics of trading in these markets We show how futures contracts are useful investment vehicles for both hedgers and speculators and how the futures price relates to the spot price of an asset Finally, we take a look at some specific financial futures contracts— those written on stock indexes, foreign exchange, and fixed-income securities F Bodie−Kane−Marcus: Essentials of Investments, Fifth Edition... Dividend payouts The Value of a Put Option Decreases Increases Increases Increases/Uncertain* Decreases Increases *For American puts, increase in time to expiration must increase value One can always choose to exercise early if this is optimal; the longer expiration date simply expands the range of alternatives open to the option holder, thereby making the option more valuable For a European put, where... the price of the asset on the maturity date Notice that profit is zero when the ultimate spot price, PT , equals the initial futures price, F0 Profit per unit of the underlying asset rises or falls one-for-one with changes in the final spot price Unlike the payoff of a call option, the payoff of the long futures position can be negative: This will be the case if the spot price falls below the original . that protective put strategies offer a sort of insurance policy on an asset. The protective put has proven to be extremely popular with investors. Even if the asset price falls, the put conveys. The asset can be sold for X, which equals its original price, so even if the asset price falls, the investor s net loss over the period is just the cost of the put. If the asset value increases,. protective put position with $60 million in risk-free investments such as T-bills and $40 million in equity is Loss on stocks: 2% of $40 ϭ $0 .80 ϩLoss on bills: 0 Net loss ϭ $0 .80 The synthetic

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