73323289-14-Options-on-Indices-Currencies

73323289-14-Options-on-Indices-Currencies - O ptions on...

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Unformatted text preview: O ptions on Stock Indices, Currencies, and Futures In this chapter we tackle the problem of valuing options on stock 'indices, currencies, and futures contracts. As a first step, we produce results for options on a stock paying a known dividend yield. We then argue that stock indices, currencies, and futures prices are analogous to stocks paying known dividend yield. This enables the results for options on a stock paying a dividend yield to be applied to value options on these other assets. 14.1 RESULTS FOR A STOCK PAYING A KNOWN DIVIDEND YIELD This section provides 'a simple rule .that enables results produced for European options on a non-dividend-paying stock to be extended so that they apply to European options on a stock paying a known dividend yield. Dividends cause stock prices to reduce on the ex-dividend date by the amount of the dividend payment. The payment of a dividend yield at rate q therefore causes the growth rate in the stock price to be less than it would otherwise be by an amount q. If, with a dividend yield of q, the stock price grows from So today to ST at time T, then in the absence of dividends it would grow from So today to STe qT at time T. Alternatively, in the absence of dividends, it would grow from Soe-qT today to ST at time T. This argument shows that we get the same probability distribution for the stock price at time T in each of the following two cases: 1. The stock starts at price So and provides a dividend yield at rate q. 2. The stock starts at price Soe- qT and pays no dividends. This leads to a simple rule. When valuing a European option lasting for time T on a stock paying a known dividend yield at rate q, we reduce the current stock price from So to Soe -qT and then value the option as though the stock pays no dividends. 313 314 CHAPTER 14 Lower Bounds for Option Prices As a first application of this rule, consider the problem of determining bounds for the price of a European option on a stock providing a dividend yield equal to q. Substituting Soe- qT for So in equation (9.1), we see that the lower bound for the European call option price c is (14.1) To obtain a lower bound for a European put option, we can similarly replace So by Soe- qT in,equation (9.2), to get (14.2) These results can also be proved using no-arbitrage arguments (see Problem 14.36). -Put-Call Parity Replacing So by Soe-qT in equation (9.3), we obtain put-eall parity for a stock providing a dividend yield equal to q: (14.3) This result can also be proved using no-arbitrage arguments (see Problem 14.36). 14.2 OPTION PRICING FORMULAS By replacing So by Soe- qT in the Black-Scholes formulas, equations (13.20)and (13.21), we obtain the price c of a European call and the price p of a European put on a stock providing a dividend yield at rate q as = Soe-qTN(d 1) - Ke-rT N(d2 ) p = Ke- rT N(-d2 ) - Soe-qTN(-d 1) c Since (14.4) (14.5) qT Soe- ) So In ( ~ =In K-qT the parameters d 1 and d2 are given by 2 d _ In(So/ K) + (1" - q + a /2)T 1- d2 _ In(So/ K) - avT + (1" - avT 2 q - a /2)T _ d _ - 1 [;:r;T av 1 These results were first derived by Merton. 1 As discussed in Section 13.12, the word "dividend" should be defined as the reduction of the stock price on the ex-dividend date arising from any dividends declared. If the dividend yield is not constant during the life of the option, equations (14.4) and (14.5) are still true, with q equal to the average I Se~ R. Merton, "Theory of Rational Option Pricing," Bell Journal oj Economics and lvlanagemenl Science, 4 (Spring 1973): 141-83. Options on Stock Indices, Currencies, and Futures 315 annualized dividend yield during the life of the option. The dividend yield should be expressed with continuous compounding (see Section 5.6). Differential Equation and Risk-Neutral Valuation To prove the results in equations (14.4) and (14.5) more formally, we can either solve the differential equation that the option price must satisfy or use risk-neutral valuation. When we include a dividend yield of q in the analysis in Section 13.6, the differential equation (13.16) becomes 2 af . at + (I - af + 2:0'2 S 2 asf 1 a2 q)S as 2 = rf (14.6) Like equation (13.16), this does not involve any variable affected by risk preferences. Therefore, the risk-neutral valuation procedure, described in Section 13.7, can be used. In a risk-neutral world, the total return from the stock must be r. The dividends provide a return of q. The expected growth rate in the stock price must therefore be r - q. So the risk-neutral process for the stock price is given by dS = (r - q)Sdt + O'Sdz (14.7) To value a derivative dependent on a stock that provides a dividend yield equal to q, we set the expected growth rate of the stock equal to r - q and discount the expected payoff at rate r. When the expected growth rate in the stock price is r - q, the expected stock price at time T is SOir-q)T. A similar analysis to that in the appendix of Chapter 13 gives the expected payoff in a risk-neutral world as where d] and d2 are defined as above. Discounting at rate r for time T leads to equation (14.4). Binomial Trees Binomial trees can be used to value an option on a stock paying a known dividend yield in the way described in Chapter 11. To match the stock price volatility, we set u and d = e -rr-/M = err-/M where l:i.t is the length of the time step. The risk-neutral probability p of an up movement is chosen so that the expected return is r - q. This means that pSu + (1- p)Sd = e(r-q)/!,r or a-d p=-- u-d where a = e(r-q)/!,r This was the result we used in Section 11.9. 1 See Technical Note 6 on the author's website for a proof of this. 3 16 14.3 CHAPTER 14 OPTIONS ON STOCK INDICES As discussed in Chapter 8, several exchanges trade options on stock indices. Some of the indices track the movement of the market as a whole. Others are based on the performance of a particular sector (e.g., computer technology, oil and gas, transportation, or telecommunications). Quotes Table 14.1 shows quotes for options on the Dow Jones Industrial Average (DJX) and S&P 500 (SPX) as they appeared in the Money and Investing section of the Wall Street Table 14.1 Quotes for stock index options from the Wall in-eet Journal, February 5, 2004. Wednesday, Feb. 4, 2004 Mar 108 ( lB2 Mar l08p 41 Apr 108 p 40 Volume figures are unofficial. Feb l12p 23 Open interest reflects previous Call Vol._8,25l trading day. p·Put c.call. The tDtals for call and put volume are Put Vol._ 14,484 Volume. last. net change and open interest fOf a\l COntracts. midday figures CHICAGO , STRJKE DJ Mar Mar Apr Mar Feb Mar Feb Feb Apr Feb Mar Apr Mar Mar Feb Apr Feb Feb Mar Mar Apr Feb Feb Mar Mar Apr Apr Feb Feb Mar Apr Feb Mar Apr Apr Feb Feb NET OPEN VOL LAST CNG lIlT INDUS AVG(DJX) 90p 5 015 -o.OS 5,844 92p 105 0.20 0.05 14,161 92p 1 055 0.05 40 96p 310 0.40 _11.814 98p 40 010 -o.OS 7,602 98p 775 0.60 0.05 4,211 99 ( 10 5.90 010 328 99p 200 015 _ 2,190 99p 3 135 0.05 606 lOOp 179 0.20. _ 6,935 lOOp 3 0.90 0.0521,574 lOOp 3 135 -o.OS 2,248 101 ( 10 4.80 -0.70 3,075 101 p 3 1 -0.10 4,772 102p 151 0.40 -010 2,925 102p 2,133 2 015 2,206 104 ( 40 175 -0.05 5,265 104 p 422 lOS 015 7,282 104 ( 378 250 -030 11.255 104 p 458 210 0.20 12,458 104 p 5 285 010 1,799 105 (2,068 105 -015 13,467 105 p 2,335 150 010 15,555 lOS ( 646 190 -015 39,444 105 P 122 235 0.05 21,489 105 ( 200 275 0.05 1.914 105 p 102 3 _ 895 106 (LOn 0.65 -010 4,647 106 p 65 210 015 3,485 106p 30 3 015 5,426 106 p 5 3.80 0.20 1.547 107 ( 118 035 -010 4,414 107p 2 350 _ 125 107 ( 10 150 -0.75 6V 107p 5 430 010 742 108 ( 6 010 -0.10 3,305 l08p 2 3.90 DAO 2,585 0.85 .• 11.4n 4 .• 614 5 0.20 88 730 010 435 Open Inl313,904 Open Inl.370,073 S & P 500(SPX) Feb 850 p 10 0.05 _ 1,434 Mar 850 p 430 0.40 010 29,388 Apr 850p 10 105 •. 311 Feb 875 p 5 0.05 _ 613 Apr 875 p 5 165 010 16 Mar 900 p 5 0.80 _ 37,089 Apr 900 p 85 190 -015 2 Feb 925 ( 14019950 250 690 Feb 925 p 4 010 -0.05 3,579 Mar 925 p 96 105 O.OS 14,592 Feb 950 p 200 DAO 030 17,129 Feb 975 p 2,090 015 0.10 lB,301 Mar 975 ( 10155 650 9,7lB Mar 975 p 360 205 0.05 40,001 Apr 975 p 26 5.20 110 2,027 Feb 995 p 23 030 _ 13,445 Mar 995 p 2,004 2.70 010 27,317 Apr 995 p 4 5.70 _. 2,658 Feb 1005 p 256 035 O.OS 36,093 Mar 1005 ( 11125 -1250 2,370 Mar 100S p LID 290 -0.10 25,947 Feb 1025 ( 1010050 -950 5,757 Feb lO25p 6;117 0.60 O.OS 45,995 Mar 1025p 515 4.60 05055,930 Apr 1025 p ill 9 100 5,171 Feb 1035 p 306 0.70 _ 2,864 Feb 1040p 10 1 015 4,270 Feb 1050 ( 1.789 7630-1510 9,986 Feb 1050 p 1.929 110 _ 42,107 Mar 1050 ( 10 84 -5.80 19,676 Mar 1050p 36 6.90 0.90 48,190 Feb 1055 p 130 140 010 3,134 Mar 1060 ( 1 73 -650 3,391 Mar 1060 p 2,305 810 130 7,m Mar 1070p 600 9 0.90 7,919 Feb 1075 ( 27 57.80 -3.70 11.711 Feb 1075 p11.023 2.70 0.60 28,638 Mar 1075 ( 16 6450 -350 33,m Mar 1075 p 519 10.40 180 38,840 Apr 1075 p lB5 16.40 130 1.138 Feb 1085 ( 4 48.80 -0.20 204 Feb 1085 p 583 3.70 100 6,492 Mar 1085 p 305 12 250 5,608 Feb 1090 ( 30 Feb 1090 p 85 feb 1100 ( 447 Feb 1100 p 2,617 Mar 1100 ( 33 Mar 1100p 4,203 Apr 1100 ( 32 Apr 1100p 8,895 feb 1105 p 124 Feb 1110 ( 6 feb-lllOp 3,828 Mar 1110 ( 11 Mar 1110 p 688 Feb lll5 ( 4 feb lll5 p 115 feb l120 ( 93 Feb l120 p 255 feb 1125 ( LB03 Feb 1125 p l,570 Mar 1125 (4,980 Mar 1125 p 4,764 Apr 1125 ( 36 Apr 1125 p 327 Feb 1130 (1.156 feb 1130p 2,693 Mar 1130 ( 2,829 Mar 1130 p 2,864 feb 1135 ( 322 feb 1135 p 396 Mar 1135 ( 413 Mar 1135 p 851 Feb 1140 (1.779 Feb 1140 p 948 Mar 1140 c 1.401 Mar 1140p 1 Feb 1145 ( 52 Feb 1145 p 47 Feb 1150 (3,479 Feb 1150 p 943 Mar 1150 ( 520 Mar 1150p 52 Apr 1150 ( 23 Feb 1155 ( 179 Feb 1160 (l,351 Feb 1160 p 126 Mar 1160 ( 402 Mat 1160 p 1 Feb WO (3,054 Feb WO p 13 Feb 1175 (1.617 feb 1175 p 55 43.60 -5.40 319 4 0.70 5,371 3130 -810 21,191 6.40 ill 32,392 4050 -650 40,878 15.80 180 44,776 50 -450 462 24 4.00 9,380 7.20 170 1.734 2650 -550 13 850 2.20 6,048 34040 -9.80 20,786 lB10 210 lB,829 20.60 -5.20 973 1030 310 9,530 lB -6.00 ill 12.10 350 6,774 14 -7.00 19,486 1450 450 32,lB5 24.70 -4.90 80,288 25 4.00 78,162 32.90 -4.80 2,641 32 310 2,931 1130 ·5.20 4,741 16.60 4.20 11.001 2150 -5.00 12,667 27 3.80 13,475 9 -450 l,262 19.90 5.60 2,600 19.90 -410 9,978 30 5.00 9,651 7 -5.00 6,401 22 550 8,040 lB -3.00 3,698 30 200 2,151 6.60 -2.90 944 26 5.00 1.584 4.20 -330 26,943 28.70 6.20 6,483 13 -3.70 35,491 38 5.00 23,226 20.60 -3.40 2,122 3.80 -2.00 1.557 255 -185 6,062 37 7.80 1.159 10 -2.90 2,098 45 100 14 150 -0.90 6,733 42.80 3.80 261 ll5 -0.85 28,065 49 6.00 2,196 Mar 1175 ( 614 6 -2.00 26,761 Mar 1175 p 3 5710 8.10 2,304 Apr W5 ( 558 12.20 -150 2,095 Feb 1180 ( 420 0.80 -055 1.543 Feb ill5 (,. 7 0.85 -015 731 Feb 1190 ( 86 050 -035 2,597 Mar 1190 ( 104 4.70 _ Feb 1200 (1.259 035 -0.20 22,677 Feb 1200 p 16 70.70 5.70 315 Mar 1200 (1,965 3 -050 23,307 Mar 1200 p 1 73.40 4.90 463 Apr 1200 ( 25' 7 -0.60 3,481 Feb 1210 ( 10 015 -010 1,424 Feb 1215 ( 13 015 -015 963 Feb 1225 ( n 010 -010 5,124 Mar 1225 ( 1 110 -030 3,OlB Mar 1225p 2 96.90 -4.60 II Apr 1225 ( 20 310 -050 2,845 Feb 1250 ( 55 0.05 -0.10 8,403 Mar 1250 ( 14 055 -0.0511.441 Mar 1250 p 30120 050 515 Apr 1250 ( 3 150 -0040 410 Call Vol. __ 37,739 Open Inl!,2OO,OO3 Put Vol_ 85,508 Open 1nll,976,864 LEAPS-LONG TERM OJ INDUS AVG • CB Dee05 76p 10 2 .. _ Dee 05 104 ( 1 9.20 0.40 11.701 Dee 05 108 (500 6.90 0.40 82 Dee05 lOBp 500 10 100 20 Call Vol._ 501 Open Inl.13,617 PutVol._SiO Op<nlnl_32,357 S & P 500-CB Dec 04 80 ( 60 3310 .. 7,B95 Dec 05 80 p 10 175 0.05 12,238 Dec 04 90 P 132 160 010 38,870 Dec 05 90 (6126 .• 24,696 Dec 05 90 P 3 350 050 18,4lB Dee 04 95 P 10 2.20 '\0.05 5,595 Dec 04 100 p 87 3.10 015 25,728 Dec 05 lOOp 10 5 030 28,m Dec 04 105 p 8 430 010 2,324 Dee 04 110 p 12 5.60 0.60 25,390 Dee 06 llO ( 11 14.70 -100 4,328 Dec 06 110p 10 9.10 0.70 33,8ll Dec 04 120 ( 4 3.40 -010 9,911 Call Vol._136 Open Inl480,710 Put Vol. _ 282 OpenJnl435,054 Source: Reprinted by permission of Dow Jones, Inc., via Copyright Clearance Center, Inc. © 2004 Dow Jones & Company, Inc. All Rights Reserved Worldwide. Options on Stock Indices, Currencies, and Futures 317 Journal on Thursday February 5, 2004. The Wall Street Journal also shows quotes for options on a number of other indices including the Nasdaq lOQ (NDX), Russell 2000 (RUT), and S&P 100 (OEX). All the options trade on the Chicago Board Options Exchange and all are European, except the contract on the S&P 100, which is American. The quotes refer to the price at which the last trade was made on Wednesday, February 4, 2004. The closing prices of the DJX and SPX on February 4, 2004, were 104.71 and 1,126.52, respectively. One index option contract is on 100 times the index. (Note that the Dow Jones index used for index options is 0.01 times the usually quoted Dow Jones index.) Index options are settled in cash. This means that, on exercise of the option, the holder of a call option contract receives (S - K) x 100 in cash and the writer of the contract pays this amount in cash, where S is the value of the index at the close of trading on the day of the exercise and K is the strike price. Similarly, the holder of a put option contract receives (K - S) x 100 in cash and the writer of the contract pays this amount in cash. Table 14.1 shows that, in addition to relatively short-dated options, the exchanges trade longer-maturity contracts known as LEAPS. The acronym LEAPS stands for "long-term equity anticipation securities" and was originated by the CBOE. LEAPS are exchang~-traded options that last up to 3 years. (Note when interpreting Table 14.1 that the S&P 500 index is divided by 10 for the purpose of defining LEAPS contracts.) The usual expiration month for LEAPS on indices is December. As mentioned in Chapter 8, the CBOE and several other exchanges also trade LEAPS on many individual stocks. These have expirations in January. The CBOE also tradesfiex options on indices. As mentioned in Chapter 8, these are options where the trader can choose the expiration date, the strike price, and whether the option is American or European. Valuation In valuing index futures in Chapter 5, we assumed that the index could be treated as a security paying a known dividend yield. In valuing index options, we make similar assumptions. This means that equations (14.1) and (14.2) provide a lower bound for European index options; equation (14.3) is the put-eall parity result for European index options; and equations (14.4) and (14.5) can be used to value European options on an index. In all cases, So is equal to the value of the index, (J is equal to the volatility of the index, and q is equal to the average annualized dividend yield (continuously compounded) on the index during the life of the option. The calculation of q should include only dividends whose ex-dividend date occurs during the life of the option. In the United States ex-dividend dates tend to occur during the first week of February, May, August, and November. At any given time, the correct value of q is therefore likely to depend on the life of the option. This is even more true for some foreign indices. In Japan, for example, all companies tend to use the same ex-dividend dates. Example 14.1 Consider a European call option on the S&P 500 that is 2 months from maturity. The current value of the index is 930, the exercise price is 900, the risk-free interest rate is 8% per annum, and the volatility of the index is 20% per annum. Dividend yields of 0.2% and 0.3% are expected in the first month and the second month, 318 CHAPTER 14 respectively. In this case, So = 930, K = 900, r = 0.08, ( j = 0.2, and T = 2/12. The total dividend yield during the option's life is 0.2 + 0.3 = 0.5%. This is 3% per annum. Hence, q = 0.03, and d] = In(930/900) + (0.08 - 0.03 + 0.22/2) x 2/12 = 0.5444 0.2)2/12 2 d? = In(930/900) + (0.08 - 0.03 - 0.2 /2) x 2/12 = 0.4628 - 0.2)2/12 N(d]) = 0.7069, so that the call price c is given by equation (14.4) as c = 930 x 0.706ge-o.03x2/]2 - 900 x 0.6782e-o.08x2/12 = 51.83 One contract would cost $5,183. If the absolute amount of the dividend that will be paid on the stockS underlying the index (rather than the dividend yield) is assumed to be known, the basic Black-Scholes formula can be used with the initial stock price being reduced· by the present value of the dividends. This is the approach recommended in Chapter 13 for a stock paying known dividends. However, it may be difficult to implement for a broadly based stock index because it requires a knowledge of the dividends expected on every stock underlying the index. Binomial Trees In some circumstances it is optimal to' exercise American put and call options on an index prior to the expiration date. Binomial trees can be used to value American-style index options as discussed in Section 11.9. An example of the use of binomial trees for index options is in Example 11.1 and Figure 11.11. Portfolio Insurance Portfolio managers can use index options to limit their downside risk. Suppose that the value of an index today is So. Consider a manager in charge of a well-diversified portfolio whose beta is 1.0. A beta of 1.0 implies that the returns from the portfolio mirror those from the index. Assuming the dividend yield from the portfolio is the same as the dividend yield from the index, the percentage changes in the'value of the portfolio can be expected to be approximately the same as the percentage changes in the value of the index. Each contract on the S&P 500 is on 100 times the index. It follows that the value of the portfolio is protected against the possibility of the index falling below K if, for each 100So dollars in the portfolio, the manager buys one put option contract with strike price K. Suppose that the manager's portfolio is worth $500,000 and the value of the index is 1,000. The portfolio is worth 500 times the index. The Jllanager can obtain insurance against the value of the portfolio dropping below $450,000 in the next 3 months by buying five put option contracts with a strike price 319 Options on Stock Indices, Currencies, and Futures Business Snapshot 14.1 Can We Guarantee that Stocks Will Beat Bonds in the Long Run? ~ It is often said that if you are a long-tenn investor you should buy stocks rather than bonds. Consider a US fund manager who is trying to persuade investors to buy as a long-tenn investment an equity fund that is expected to mi!ror the S&P 500. The manager might be t.empted to offer purchasers of the fund a guarantee that their return will be at least as good as the return on risk-free bonds over the next 10 years. Historically stocks have outperformed bonds in the United States over almost any lO-year period. It appears that the fund manager would not be giving much away. In fact, this type of guarantee is surprisingly expensive. Suppose that an equity index is 1,000 today, the dividend yield on the index is 1% per annum, the volatility of the index is 15% per annum, and the 10-year risk-free rate is 5% per annum. To outperfonn bonds, the stocks underlying the index must earn more than 5% per annum. The dividend yield will provide 1% per annum. The capital gains on the stocks must therefore provide 4% per annum. This means that we require the index level to be at least 1,000eO.04xlO = 1,492 in 10 years. A guarantee that the return on $1,000 invested in the index will be greater than the return on $1,000 invested in bonds over the next 10 years is therefore equivalent to the right to sell the index for 1,492 in 10 years. This is a European put option on the index and can be valued from equation (14.5) with So = 1,000, K = 1,492, r = 5%, (J = 15%, T = 10, and q = 1%. The value oUhe put option is 169.7. This shows that the guarantee contemplated by the fund manager is worth about 17% of the fundhardly something that should be given away! of 900. Suppose that the risk-free rate is 12%, the dividend yield on the index is 4%, and the volatility of the index is 22%. The parameters of the option are: So = 1000, . K = 900, r = 0.12, (J = 0.22, T = 0.25, q = 0.04 From equation (14.5), the value of the option is $6.48. The cost of the insurance is therefore 5 x 100 x 6.48 = $3,240. To illustrate how the insurance works, consider the situation where the index drops to 880 in 3 months. The portfolio will be worth about $440,000. The payoff from the options will be 5 x (900 - 880) x 100 = $10, 000, bringing the total value of the portfolio up to the insured value of $450,000 (or $446,760 when the cost of the options are taken into account). It is sometimes argued that the return from stocks is certain to beat the return from bonds in the long run. If this were true, long-dated portfolio insurance where the strike price equaled the future value of a bond portfolio would not cost very much. In fact, as indicated in Business Snapshot 14.1, it is quite expensive. When the Portfolio's Beta Is Not 1.0 If the portfolio's returns are not expected to equal those of an index, the capital asset pricing model can be used. This model asserts that the expected excess return of a portfolio over the risk-free interest rate equals beta times the excess return of a market index over the risk-free interest rate. Suppose that the $500,000 portfolio just considered 320 CHAPTER 14 Relationship between value of index and value of portfolio for beta = 2.0. Table 14.2 Value of index in 3 months Value of portfolio in 3 months ($) 1,080 1,040 1,000 960, 920 880 570,000 530,000 490,000 450,000 410,000 370,000. has a beta of 2.0 instead of 1.0. As before, we assume that the S&P 500 index is currently 1,000, the risk-free rate is 12% and the dividend yield on the index is 4%. Table 14.2 shows the expected relationship between the level of the index and the value of the portfolio in 3 months. To illustrate the sequence of calculations necessary to derive Table 14.2, Table 14.3 shows the calculations for the case when die value of the index in 3 months proves to be 1,040. Suppose that Sa is the value of the index. It can be shown that, for each 100Sa dollars in the portfolio, a total of beta put contracts should be purchased. The strike price should be the value that the index is expected to have when the value bf the portfolio reaches the insured value. Assume that the required insured value is $450,000, as in the beta = 1.0 case. Table 14.2 shows that the appropriate strike price for the put options purchased is 960. The option parameters are: S = 1000, K = 960, r = 0.12, (J = 0.22, T = 0.25, q = 0.04 and equation (14.5) gives the value of the option as $19.21. In this case, 100Sa = $100,000 and beta = 2.0, so that two put contracts are required for each $100,000 in the portfolio. Table 14.3 Calculations for Table 14.2 when the value of the index is 1,040 in 3 months. Value of index in 3 months: Return from change in index: Dividends from index: Total return from index: Risk-free interest rate: Excess return from index over risk-free interest rate: Excess return from portfolio over risk-free interest rate: Return from portfolio: Dividends from portfolio: Increase in value of portfolio: Value of portfolio: 1,040 40/1,000, or 4% per 3 months 0.25 x 4 = 1% per 3 months 4+ 1 = 5% per 3 months 0.25 x 12 = 3% per 3 months 5 - 3 = 2% per 3 months 2 x 2 = 4% per 3 months 3 + 4 = 7% per 3 months 0.25 x 4 = 1% per 3 months 7 - 1 = 6% per 3 months $500, 000 x 1.06 = $530, 000 Options on Stock Indices, Currencies, and Futures 321 Since the portfolio is worth $500,000, a total of 10 contracts should be purchased. The total cost of the insurance is therefore 10 x 100 x 19.21 = $19,~1O. To illustrate that the required result is obtained, consider what happens if the value of the index falls to 880. As shown in Table 14.2, the value of the portfolio is then about $370,000. The put options payoff (960 - 880) x 10 x 100 = $80,000, and this is exactly what is necessary to move the total value of the portfolio manager's position up from $370,000 to the required level of $450,000. (After the cost of the options are taken into account the value of the portfolio is $430,790.) There are two reasons why the cost of hedging increases as the beta of a portfolio increases: more put options are required, and they have a higher strike price. 14.4 CURRENCY OPTIONS Currency options are primarily traded in the over-the-counter market. The advantage of this market is that large trades are possible with strike prices, expiration dates, and other features tailored to meet the needs of corporate treasurers. European and American options do trade on the Philadelphia Stock Exchange in the United States, but the exchange-traded market is much smaller than the over-the-counter market. For a corporation wishing to hedge a foreign exchange exposure, foreign currency options are an interesting alternative to forward contracts. A company due to receive sterling at a known time in the future can hedge its risk by buying put options on sterling that mature at that time. The strategy guarantees that the value of the sterling will not be less than the strike price, while allowing the company to benefit from any favorable exchange-rate movements. Similarly, a company due to pay sterling at a known time in the future can hedge by buying calls on sterling that mature at that time. The approach guarantees that the cost of the sterling will not be greater than a certain amount while allowing the company to benefit from favorable exchange-rate movements. Whereas a forward contract locks in the exchange rate for a future transaction, an option provides a type of insurance. This insurance is not free. It costs nothing to enter into a forward transaction, whereas options require a premium to be paid up front. Valuation To value currency options, we define So as the spot exchange rate. To be precise, So is the value of one unit of the foreign currency in US dollars. As explained in Section 5.10, a foreign currency is analogous to a stock paying a known dividend yield. The owner of foreign currency receives a yield equal to the risk-free interest rate, Ii, in the foreign currency. Equations (14.1) and (14.2), with q replaced by Ii, provide bounds for the European call price, C, and the European put price, p: C ~ Soe-rfT - Ke- rT p ~ Ke- rT _ Soe-rfT Equation (14.3), with q replaced by Ii, provides the put-eall parity result for currency options: 322 CHAPTER 14 Finally, equations (14.4) and (14.5) provide the pricing formulas for currency options when q is replaced by Ii: c = Sae-rfTN(d)) - Ke- rT N(d2 ) (14.7) p = Ke- rT N(-d2 ) - Sae-rfTN(-d}) (14.8) where d} = d2 = 1n(SalK) + (r - Ii r;:r; + cr2/2)T cryT 1n(Sal K) + (r - 2 Ii - cr /2)T cr-JT = d {;;; } - cry T Both the domestic interest rate, r, and the foreign interest rate, rt, are the,rates for a maturity T. Put and call options on a currency are symmetrical in that a put option to _sell currency A for currency B at an exercise price K is the same as a call option to buy B with A at 1/ K. Example 14.2 Consider a 4-month European call option on the British pound. Suppose that the current exchange rate is 1.6000, the exercise price is 1.6000, the risk-free interest rate in the United States is 8% per annum, the risk-free interest rate in Britain is 11 % per annum, and the option price is 4.3 cents. In this case, So' == 1.6, K = 1.6, r = 0.08, If = 0.11, T = 0.3333, and c = 0.043. The implied volatility can be calculated by trial and error. A volatility of 20% gives an option price of 0.0639, a volatility of 10% gives an option price of 0.0285, and so on. The implied volatility is 14.1 %. From equation (5.9), the forward rate Fa for a maturity T is given by Thus, equations (14.7) and (14.8) can be simplified to = e-rT[FaN(d}) - KN(d2 )] P = e- rT [KN(-d2 ) - FaN(-d))] c (14.9) (14.10) where d} = In(Fal K) + cr2T 12 r;:r; cryT 2 d? - - In(Fal K) - cr TI2 _ d _ - r;:r; cryT -) {;;;T cry 1 Note that, for equations (14.9) and (14.10), to be the correct equations for valuing a European option on the spot foreign exchange rate, the maturities of the forward contract and the option must be the same. Binomial Trees In some circumstances it is optimal to exercise American currency options prior to maturity. Thus, American currency options are worth more than their European counterparts. In general, call options on high-interest currencies and put options on Options' on Stock Indices, Currencies, and Futures 323 low-interest currencies are the most likely to be exercised prior to maturity. The reason is that a high-interest currency is expected to depreciate and a low-interest currency is expected to appreciate. Binomial trees can be used to value Ainerican-style currency options as described in Section 11.9. An example of the valuation of a currency option is given in Example 11.2 and Figure 11.12. 14.5 FUTURES OPTIONS Options on futures contracts, or futures options, are now traded on many different exchanges. They are American-style options and require the delivery of an underlying futures contract when exercised. If a call futures option is exercised, the holder acquires a long position in the underlying futures contract plus a cash amount equal to the most recent settlement futures price minus the strike price. If a put futures option is exercised, the holder acquires a short position in the underlying futures contract plus a cash amount equal to the strike price minus the most recent settlement futures price. As the following examples show, the effective payoff from a call futures option is the futures price at the time of exercise less the strike price; the effective payoff from a put futures option is the strike price less the futures price at the time of exerc~se. Example 14.3 Suppose it is August 15 and an investor has one September futures call option contract on copper with a strike price of 70 cents per pound. One futures contract is on 25,000 pounds of copper. Suppose that the futures price of copper for delivery in September is currently 81 cents, and at the close of trading on August 14 (the last settlement) it was 80 cents. If the option is exercised, the investor receives a cash amount of 25,000 x (80 - 70) cents = $2,500 plus a long position in a futures contract to buy 25,000 pounds of copper in September. If desired, the position in the futures contract can be closed out immediately. This would leave the investor with the $2,500 cash payoff plus an amount 25,000 x (81 - 80) cents = $250 reflecting the change in the futures price since the last settlement. The total payoff from exercising the option on August 15 is $2,750, which equals 25,000(F - K), where F is the futures price at the time of exercise and K is the strike price. Example 14.4 An investor has one December futures put option on corn with a strike price of 200 cents per bushel. One futures contract is on 5,000 bushels of corn. Suppose that the current futures price of corn for delivery in December is 180, and the most recent settlement price is 179 cents; If the option is exercised, the investor receives a cash amount of 5,000 x (200 - 179) cents = $1,050 plus a short position in a futures contract to sell 5,000 bushels of corn in December. 324 CHAPTER 14 If desired, the position in the futures contract can be closed out. This would leave the investor with the $1,050 cash payoff minus an amount 5,000 x (180 - 179) cents = $50 reflecting the change in the futures price since the last settlement. The net payoff from exercise is $1,000, which equals 5,000(K - F), where F is the futures price at the time of exercise and K is the strike price. Quotes' Futures options are referred to by the month in which the underlying futures contract matures-not by the expiration month of the option. As mentioned eaflier, futures options are American. The expiration date of a futures option contract is usually on, or .a few days before, the earliest delivery date of the underlying futures contract. (For example, the CBOT Treasury bond futures option expires on the Friday preceding by at least two business days the end of the month before the futures contract expiration month.) An exception is the CME mid-curve Eurodollar contract, where the futures contract expires either one or two years after the options contract. Table 14.4 shows quotes for futures options as they appeared in the Wall Street Journal on February 5, 2004. The most popular contracts (as measured by open interest) are those on com, soybeans, cotton, sugar-world, crude oil, natural gas, gold, Treasury bonds, Treasury notes, 5-year Treasury notes, 30-day federal funds, Eurodollars, I-year and 2-year mid-curve Eurodollars, Euribor, Eurobunds, and the S&P 500. Options on Interest Rate Futures The most actively traded interest rate options offered by exchanges in the United States are those on Treasury bond futures, Treasury note futures, and Eurodollar futures. Table 14.4 shows the closing prices for these instruments on February 4, 2004. A Treasury bond futures option is an option to enter a Treasury bond futures contract. As mentioned in Chapter 6, one Treasury bond futures contract is for the delivery of $100,000 of Treasury bonds. The price of a Treasury bond futures option is quoted as a percentage of the face value of the underlying Treasury bonds to the nearest sixty-fourth of 1%. Table 14.4 gives the price of the March call futures option on a Treasury bond on February 4, 2004, as 2-06, or 2 -A % of the bond principal, when the strike price is 110. This means that one contract costs $2,093.75. The quotes for options on Treasury notes are similar. An option on Eurodollar futures is an option to enter into a Eurodollar futures contract. As explained in Chapter 6, when the Eurodollar futures quote changes by 1 basis point, or 0.01 %, there is a gain or loss on a Eurodollar futures contract of $25. Similarly, in the pricing of options on Eurodollar futures, 1 basis point represents $25. The Wall Street Journal quote for the CME Eurodollar futures contract in Table 14.4 should be multiplied by 10 to get the CME quote in basis points. For example, the 5.90 quote for the CME March call futures option when the strike price is 98.25 in Ta~le 14.4 indicates that the CME quote is 59.0 basis points and one contract costs 59.0 x $25 = $1,475.00. 325 O ptions'on Stock Indices, Currencies, and Futures Table 14.4 Closing prices of futures options on February 4, 2004. Wednesday, February 4, 2004 final or "ttlement prices of "Iected contJilCts. Volume and open interest are totals in all contract months. STRIKE Grain and Oilseed 50,000 Ibs; cents per lb. Price Jly Mar Mar May 2.44 5.85 713 19 67 68 L64 5.21 6.47 39 .90 4.60 5.86 69 .65 70 .46 4.04 518 ill 71 18 354 4.75 2.03 72 .15 3.07 415 2.90 Est vol 9,021 Tu 8,443 calls 5,904 puts Op int Tues 217,446 calls 113,615 puts Corn (CBn 5,000 bu; cents per bu. STRIKE CAUS-SETTLE PUTS-SETTlE Jly Mar May Jly Price Mar May 260 11875 20.250 26.750 L625 5150 8500 5500 14.750 21150 5.250 9500 13500 270 2.250 10500 17150 12.000 15500 19.000 280 .750 7375 14.000 20500 22.000 25.625 290 300 150 5125 ill75 30.000 29.625 32.625 310 .125 3500 9150 Est vol 14,610 Tu 8,885 calls 6,364 puts Op int Tues 323,990 calls 227,010 puts Soybeans (esn 5,000 bu; cents per bu. Jly Mar May Price Mar May 760 47500 58500 60.000 L875 13.000 780 3L250 46500 50150 5500 20.750 800 18.875 36150 42.000 13125 30.750 820 10150 28500 35.000 24.500 42.250 840 5125'22.000 29500 39375 56.000 860 2500 17.000 24.750 56.625 70.750 Est vol 17,482 Tu 16,204 calls 6,863 puts Op int Tues 153,237 calls 125,007 puts Soybean Meal (CST) 100 tons; $ per ton Price Mar May Jly JIy 28500 38500 50.000 62.750 77.000 92.000 Mar May JIy 235 240 9.00 1350 1450 2.00 715 1175 245 250 3.75 930 10.90 6.75 12.60 18.25 255 260 135 650 850 14.40 19.75 25.70 Est vol 2,445 Tu 2,767 calls 2,418 puts Op int Tues 39,831 calls 36,748 puts Soybean Oil (CBT) 60,000 Ibs; cents per lb. Price Mar May Jly Mar 290 1080 1.770 2.070 150 295 .750 1545 L870 .400 300 550 1325 L700 .700· 305 310 .250 LOOO L410 315 .Est vol 6,036 Tu 2,484 calls 2,045 puts Op int Tues 55,851 calls 44,819 puts May Jly 1000 1620 L280 l.57ll 2.240 -- Wheat (CST) 5,000 bU; cents per bu. Price Jly Mar May Mar May 360 19.250 32.375 34500 3.250 10.000 370 12.750 26500 29.750 6.750 14.000 380 8.000 21500 25150 12.000 19.000 390 4500 17.500 21500 18.500 25.000 400 2.500 14125 18150 26375 3L500 410 1375 ll250 15500 35150 38.625 Est vol 4,768 Tu 2,369 calls 1,615 puts Op int Tues 76,609 calls 56,869 puts Jly 17150 22.500 28.000 34.250 4LOOO 48.000 Wheat (K() 5,000 bU; cents per bu. Price Jly Mar Mar May 360 22.500 30.625 36375 2.000 370 15.000 24.875 31250 4500 380 9125 20.000 26.625 8.625 390 5150 16125 22.750 14.750 400 2.875 14.000 19.375 22.375 410 2.000 10.500 16500 3LOOO Est vol 2,045 Tu 437 calls 315 puts Op int Tues 2l,347 calls 19,365 puts Jly 16.000 20.750 26125 32.l25 38.750 May 10.250 14500 19500 25.625 32.500 CALLS-SETTLE PUTS-SETTLE Food and Fiber Cotton (NYCE) May L60 L95 2.34 2.78 3.27 3.80 Jly L87 2.20 2.58 3.00 3.46 3.95 Orange Juice (NyeE) 15,000 Ibs; cents per lb. Jly Mar May Price Mar May 1165 14045 17.10 50 .05 .15 55 6.75 9.75 12.60 10 .40 60 2.40 5.75 815 .75 135 65 .45 3.05 5.05 3.50 3.50 70 .15 155 2.95 835 7.05 75 .10 .80 170 1335 11.40 Est vol 412 Tu 1,547 calls 843 puts Op int Tues 42,351 cal~ 14,369 puts Jly .25 .75 L40 3.00 5.90 955 Coffee «(5CE) 37,500 /bS; cents per lb. Price Mar Apr May Mar 675 5.40 817 9.06 030 70 335 638 753 0.75 72.5 1.85 4.94 614 175 75 LOO 3.82 518 330 775 0.49 2.98 430 539 80 013 2.34 358 7.63 Est vol 9.420 Tu 2,864 calls 2,718 puts Op int Tues 78,119 calls 38,500 puts Sugar-World May 194 2.90 410 552 7.14 8.91 Apr 0.01 0.02 0.09 030 0.67 ill May 0.02 0.06 017 039 0.74 li7 CAllS-SETTLE PUTS-SETTLE Gasoline-Unlead (NYM) 42,000 gal; $ per gal. Price Mar Apr May Mar 97 .0462 .0932 .0305 98 .0409 .0842 .0877 .0352 .0361 99 .0823 .0404 .0318 .0725 .0773 .0461 100 101 .0279 .0671 .0724 .0522 102 .0243 .0619 .0680 .0586 Est vol 2,854 Tu 1,831 cal~ 1,008 puts Op int Tues 2l,736 calls 17,368 puts .0661 .0727 .0795 .0866 Apr .0301 .0338 .0378 .0421 .0466 .0514 Natural Gas (NYM) 10,000 MMSru; $ per MMSru. Price Mar Apr May Mar Apr 555 178 .486 382 176 560 358 160 138 304 519 565 330 553 334 144 570 313 130 .210 359 589 193 .216 197 389 .625 575 580 175 103 185 0421 .662 Est vol 37,627 Tu 17,1ll calls 19,795 puts Op int Tues 316,788 calls 386,608 puts Brent Crude (IPE) 1,000 net bbls; $ per bbl. Apr May Mar Price Mar Data not available from source. May .0484 .0528 .0574 .0623 .0674 May Apr May Apr May 310 «(5CE) 112,000 Ibs; cents per Ib. Price Mar Apr May Mar 450 li9 139 1.40 0.01 500 0.69 0.89 0.93 0.01 550 015 0.47 055 0.07 600 0.02 0.18 017 034 650 0.01 0.05 012 0.83 700 0.01 0.01 0.06 1.33 Est vol 2,533 Tu 1,814 calls 1,889 puts Op int Tues 154,632 calls ll2,414 puts Cocoa Apr 0.98 1.85 2.79 417 5.82 7.68 STRIKE 89 .0330 .0348 .0275 .0333 .0290 .0314 .0248 .0393 90 91 .0245 .0282 .0224 .0448 92 .0210 .0254 .0513 Est vol ,ill Tu 800 cal~ 300 puts Op int Tues 27,374 ralls 19,492 puts «(5CE) 10 metric tons; $ per ton Mar Apr May Mar Pri" 3 1500 84 108 133 42 78 105 1550 11 1600 81 33 14 54 1650 4 36 61 73 24 45 120 1700 1 1750 1 15 34 170 Est vol 1,663 Tu 439 calls 341 puts Op int Tues 18,472 calls 15,125 puts Apr 39 59 85 ill 155 196 May 64 86 ill 142 176 214 Livestock Cattle-Feeder (CAlE) 50,000 Ibs.; cents per lb. Price Mar Apr May Mar 4.00 550 6.28 3.00 8000 8100 350 8200 2.50 4.00 2.10 410 8300 1.60 3.10 4.00 4.60 8400 120 510 8500 Est vol 534 Tu 183 ;;;I~ 261-puts Op int Tues 3,298 calls 5,427 puts - Cattle-Live (CME) 40,000 lbs.; cents per lb. Feb Mar Apr Price Feb 73 150 2.00 050 0.80 74 L70 0.80 035 75 150 135 76 0.18 L25 2.18 77 0.08 LOS 3.08 78 0.03 0.85 4.03 Est vol 1,903 Tu 690"calls 855 puts Op int Tues 40,381 calls 42,076 puts .- Petroleum Crude Oil (NYM) 1,000 bbls.; $ per bbl. Apr Price Mar Apr May Mar 3200 ill 136 139 0.43 137 3250 120 ill li8 0.60 163 3300 0.91 0.93 LOO 0.81 L94 3350 0.66 0.75 0.84 106 2.26 3400 0.49 0.61 0.70 139 2.62 033 050 0.00 1.73 3.00 3450 Est vol 43,517 Tu 13,264 calls 17,244 puts Op int Tues 341,383 calls 486,295 puts Est vol Tu calls puts Op int Tues calls puts May 2.07 2.36 2.67 3.01 3.37 Heating Oil No.2 (NYM) 42,000 gal; $ per gal. Mar Apr May Mar Apr May Price 87 .0437 .0426 .0335 .0240 .0540 88 .0381 .0386 .0303 .0284 .0600 .0850 ... - Hogs-lean (eME) 40,000 IbS; cents per lb. Price Feb Apr May Feb 2.63 3.80 010 57 58 178 318 4.80 035 59 108 2.65 0.65 60 053 2.l5 3.63 liO 61 0.28 173 62 015 135 2.68 2.73 Est vol 243 Tu 207 cal~ 358 puts Op int Tues 5,619 calls 6,176 puts Colltilllled 011 2.80 330 3.90 4.40 4.80 Mar Apr 4.05 4.75 5.55 618 7.08 7.88 .- ... _. .... May Apr L93 2.30 2.78 318 2.l8 lIext page 2.98 326 CHAPTER 14 Table J4.4-Colltillued STRIKE CAUS-SmLE PUTS-SmLE Metals Copper (00) 25,000 Ibs.; cents per lb. PrIce Mar Apr May Mar 2.Apr May 114 5.00 6.00 7.15 1.55 ~90 4.65 116 3.70 4.90 6.15 2.25 3.80 5.60 1J.B 2.55 3.95 510 310 4.85 6.65 UO 175 310 4.40 430 6.00 7.85 122 115 180 3.70 5.70 9.65 9.10 U4 0.70 100 3.05 715 13.90 1050 Est vol 1,650 T~,~47 calls 23 puts Op int Tues 12,,,,", calls 3,638 puts Gold (CMX) 100 troy ounces; $ pm' troy ounce Price Mar Apr Jan Mar Apr 390 1350 16.80 2180 190 5.10 395 10.00 13.70 19.00 330 7.00 400 7.00 1100 1750 530 930 405 4.80 8.80 1430 810 mo 410 310 6.60 12.50 11.50 14.90 415 UO 550 10.80 15.40 18.80 Est vol 18,000 Tu 4,487 calls 5,463 puts Op Int Tues 306,159 cal~ 227,854 puts Silver Jan 910 1140 14.90 16.60 '19.70 23.00 (00) 5,000 troy ounces; cts per troy ounte PrIce Mar Apr May Mar 610 2030 3050 38.40 1550 620 15.90 2630 3430 2110 625 14.00 24.40 32.40 2410 630 12.40 22.70 30.70 27.60 640 9.70 1950 2750 34.90 650 7.60 16.80 24.70 42.70 Est vol 1,800 Tu 3,474 calls 3,954 puts Op int Tues 66,669 calls 26,556 puts Apr 24.40 30.10 3310 36.40 4310 5050 May 32.20 38.l0 4UO 44.40 5UO 58.40 Interest Rate T-Bonds 0-49 Apr HI 2-30 3.j)4 May 2-29 3-48 ~~~ t~~ Tu vol 14,191 calls 17,000 puts Op Int Tues 412,644 calls 444,891 puts T-Notes $100,000; points and 64lhs of 100% PrIce Mar Apr May Mar Apr 112 2.j)Q 1-30 I-52 0-20 1-25 113 1-17 1-00 .• 0-37 I-58 114 0-44 0-41 0-59 1-00 115 0-20 0-24 0-40 1-40 116 a-os 0-14 0-16 2-28 0-26 117 0.j)3 0-08 _ Est vol 150,806 Tu 63,052 calls 65,301 puts Op int Tues 3,045,055 calls 1,083,950 puts 5 Yr Treas Notes May 1-46 Eurodollar (CME) Feb Mar zii ~:~~ ~:~~ ~~~~ O§s i~~ 0.45 0.05 0.12 .• 165 9900 .• 0.05 0.02 _ 4.10 9925 .• 0.00 9950 _ 0.00 •. 6.60 Est vol 288,753; Tu vol 83,303 calls 142,595 puts Op int Tues 4,268,863 calls 4,408,535 puts 1. Yr. Mid-Curve Eurodlr m 30 Day Federal Funds Mar .002 .007 .017 tj~ 457 6.07 l~~~ A~t~~ _ _ Swiss Franc Apr (CME) 036 0.91 ~:it tl~ 166 157 1860 0.08 0.96 142 331 Est vol 755 Tu 242 calls 625 puts Op int Tues 6;157 calls 5,097 puts Im Apr 2.95 138 (CME) 2.53 2.04 1820 1830 Sop 5.40 113 0.68 176 2.27 2.83 125,000 francs; tents per franc PrIce Feb Mar Apr Feb 7900 110 170 _ 0.08 7950 0.69 139 _ 0.17 8000 037 III _ 035 8050 0.18 0.89 .• 0.66 8100 0.10 0.71 .• lOS 8150 0.05 055 .• 1.53 Est vol 189 lu 44 tails 384 puts Op Int Tues l,690 calls 2,356 puts Mar 0.68 0.87 109 137 169 2.03 Apr Mar 116 117 Apr 191 2.14 2.96 Euro Fx (CME) 125.000 eurns; cents per eurn PrI", Feb Mar Apr Feb U400 135 2.36 2.82 0.15 U450 0.98 2.07 2.55 0.28 lli~~' ~:~~ tl~ Po~ ~:n H~ t~ Apr 0.02 0.06 0.13 014 035 0.48 (EUREX) Apr 100 128 161 198 2.39 2.82 317 May U4 1.50 183 ill 2.56 Japanese Yen 12,500.000 yen; PrIce Feb 9400 103 9450 0.60 9500 030 DJ Industrial Avg (CBOT) $100 timos premium mr zl~:'36 4tfb I~ J~ 20~O 103 2100 30.00 3550 7.00 104 1450 24.00 29.75 1050 105 9.00 18.50 2415 15.00 106 550 14.00 1950 2150 107 3.00 10.00 _ 29.00 Est vol U4 Tu ill calls 72 puts Op int Tues 5,861 calls 5,480 puts 1615 20.00 2450 30.00 S&P 500 Stock Index $250 times premium Price Feb Mar 1115 19.70 29.90 1120 16.60 26.90 1125 13.80 24.00 1130 1130 2140 APT 37.80 34.90 32.00 2930 Feb 10.80 12.70 14.90 17.40 (CME) Mar 2100 23.00 25.10 2750 ApT 29.90 32.00 34.10 36.40 41.30 Op int Tues 88,723 calls 228,763 puts «ME) cents per 100 Mar Apr 172 2.30 144 2.03 119 178 m i1: Canadian Dollar 100,000 Can.$, cents per PrIce Feb Mar 7400 _ 13S 7450 051 104 Index tpJvol 14,455 Tu~:~~ calls 10.464 puts~~~~ ~}~ ~~1~ ~~~ Est 4,759 yen Feb 0.06 0.13 033 Mar 0.75 0.97 122 ~~~ I.84 _ 0.67 _ Est vol 1,352 Tu 3,271 cails 531 puts Op int Tues 23,459 calls 20,676 puts Apr .005 .007 .017 BritiSh Pound 0.92 122 157 196 (UFFE) 100,MO; pis. In 100% PrIce Mar Apr May Mar 11350 101 0.78 102 015 11400 0.68 056 0.78 0.42 11450 0.42 039 0.61 0.66 11500 012 016 0.46 0.96 11550 0.11 0.17 034 135 11600 0.06 0.10 _ 180 Vol Wd 35,857 calls 42,186 puts Op int Tues 366,384 calls 479,IB8 puts ~~ 0.04 PUTS-SmLE 62,500 pounds; cents per pound m ~~ (CME) Eurn 1.000,000 PrIce Feb Mar Apr Feb Mar 9n50 0.18 019 0.17 0.00 97875 0.06 0.07 0.08 0.00 0.01 98000 0.01 0.03 0.03 0.07 0.09 981Z5 0.01 0.01 0.19 010 98250 0.00 0.00 031 032 98375 .• _ 0.00 0.44 0.44 Vol Wd 327,805 calls 29,IB1 puts Op int Tues 5,655,304 calls 1,807,541 puts ~~~ 9650 (CBT) $5,000,000; 100 PrIce Feb 988750 .127 989375 .065 990000 .007 990625 .• May 1-21 CAUS-SmlE 12600 016 134' 187 106 U4 12650 0.15 114 168 145 2.44 Est vol 3,767 Tu 3,252 calls 2,088 puts Op Int Tues 39,137 calls 43,286 puts 2 Yr. Mid-Curve Eurodlr $L000,000 conlrad units; pis. of 100% Price Mat Jan Sop Mar Jan ~:~ 4.05 ~~ 9625 2.45 2.82 2.90 195 9650 127 185 _ 317 9675 0.60 1.12 .• 5.10 9700 0.17 050 _ 7.17 Est vol 800 Tu 8,400 calls_O puts Op int Tues 158,035 calls 33,178 puts Apr 0.12 (CME) $Looo,ooo conlrad units; Pis. of 100% Price Feb Mar Apr Feb Mar 9725 4.02 4.65 2.75 0.17 0.80 9150 2.05 2.87 160 0.70 152 9715 0.70 155 0.82 185 2.70 9800 0.15 0.65 035 3.80 430 9825 0.02 010 0.15 .• 635 9850 0.00 0.05 .• _ .• Est vol 210,600 Tu 61,545 calis ll9,840 puts Op int lu", 934,544 cails 932,093 puts ~~~ STRIKE 7500 013 0.77 103 038 1550 0.10 057 _ 0.75 7600 0.05 0.42 .• 119 7650 0.02 031_ 167 Est vol 419 Tu 219 calls 163 puts Op int Tues 12,761 calls 9,409 puts Currency (CBT) $100,000; points and 64lhs of 100% PrIce Mar Apr May Mar Apr 11150 1-16 0-49 0-62 0-15 H8 11200 0-56 0-36 .• 0-22 1-27 11250 0-36 0-25 .• 0-34 11300 0-22 0-17 _ 0-52 11350 O-ll 0-11 .• 1-10 11400 0-06 _ _ 1-36 Est vol 17,994 Tu 4,736 calls 25,086 puts Op int Tues ill,023 calls 426,615 puts dally average Apr Feb .120 .002 .060 .002 .007 .007 .002 PUTS-SmLE $ million; pis. of 100% PrIce Feb Mar Apr 9825 .• 5.90 Euro-BUND (CBT) minus Mar .117 .062 .007 .002 CAUS-SmLE Euribor (CBT) $100,000; points and 64lhs of 100% PrI", Mar Apr May Mar 110 2.j)6 2-lI3 2·35 0-36 111 1-28 1-36 0-58 112 0-58 1-11 1-42 1-24 113 0-34 0-54 2-00 ill vol 23,701; ~~ ~t~ Est STRIKE 991Z50 .002 .002 991875 _ _ .• _ Est vol 330 Tu 1399 calls 3,303 puts Op int Tues ll8,420 calls 162,873 puts Other Options Nasdaq 1.00 (CME) $100 times NASDAQ 100 Index PrI,e Feb Mar Apr 1460 __ .• Est vol 41 Tu 3 calls 2 puts Op int Tues 2,185 calls 958 puts NYSE Composite (CME) Can.$ Apr Feb _ 0.07 '. 016 Apr 104 127 152 Mar 050 0.69 $50 timos premium PrIce Feb' Mar Apr 6500 7450 moo 16400 Est vola Tu 3 calls 20 puts Op int Tues 1 calls 9,514 puts Mar Apr (NYfE) Feb Mar Apr 6500 11150 16450 Source: Reprinted by permission of Dow Jones, Inc., via Copyright Clearance Center, Inc. © 4004 Dow Jones & Company, Inc. All Rights Reserved Worldwide. Options on Stock Indices, Currencies, and Futures 327 Interest rate futures option contracts work in the same way as the other futures options contracts discussed in this chapter. For example, the payoff from a call is max(F - K, 0), where F is the futures price at the time of exerCise and K is the strike price. In addition to. the cash payoff, the option holder obtains a long position in the futures contract when the option is exercised and the option writer obtains a corresponding short position. Interest rate futures prices increase when bond prices increas·e (i.e., when interest rates fall). They decrease when bond prices decrease (i.e., when interest rates rise). An investor who thinks that short-term interest rates will rise can speculate by buying put options on Eurodollar futures, whereas an investor who thinks the rates will fall can speculate by buying call options on Eurodollar futures. An investor who thinks that long-term interest rates will rise can speculate by buying put options on Treasury note futures or Treasury bond futures, whereas an investor who thinks the rates will fall can speculate by buying call options on these instruments. Example 14.5 It is February and the futures price for the June Eurodollar contract is 93.82 (corresponding to a 3-month Eurodollar interest rate of 6.18% per annum). The price of a call option on the contract with a strike price of 94.00 is quoted at the CME as 0.1, or 10 basis points (corresponding to a' Wall Street Journal quote of 1.00). This option could be attractive to an investor who feels that interest rates are likely to come down. Suppose that short-term interest rates do drop by about 100 basis points and the investor exercises the call when the Eurodollar futures price is 94.78 (corresponding to a 3-month Eurodollar interest rate of 5.22% per annum). The payoff is 25 x (94.78 - 94.00) = $1,950. The cost of the contract is 10 x 25 = $250. The investor's profit is therefore $1,700. Example 14.6 It is August and the futures price for the December Treasury bond contract traded on the. CBOT is 96-09 (or 96 = 96.28125). The yield on long-term /2 government bonds is about 6.4% per annum. An investor who feels that this yield will fall by December might choose to buy December calls with a strike price of 98. Assume that the price of these calls is 1-04 (or I ~ = 1.0625% of the principal). If long-term rates fall to 6% per annum and the Treasury bond futures price rises to 100-00, the investor will make a net profit per $100 of bond futures of 100.00 - 98.00 - 1.0625 = 0.9375 Since one option contract is for the purchase or sale of instruments with a face value of $100,000, the investor would make a profit of $937.50 per option contract bought. Reasons for the Popularity of Futures Options It is natural to ask why people choose to trade options on futures rather than options on the underlying asset. The main reason appears to be that a futures contract is, in many circumstances, more liquid and easier to trade than the underlying asset. Furthermore, a futures price is known immediately from trading on the futures exchange, whereas the spot price of the underlying asset may not be so readily available. CHAPTER 14 328 Consider Treasury bonds. The market for Treasury bond futures is much more active than the market for any particular Treasury bond. Moreover, a Treasury bond futures price is known immediately from trading on the CBOT. By contrast, the current market price of a bond can be obtained only by contacting one or more dealers. It is not surprising that investors would rather take delivery of a Treasury bond futures contract than Treasury bonds. Futures on commodities are also often easier to trade than the· commodities themselves. For example, it is much easier and more convenient to make or take delivery of a live-hogs futures contract than it is to make or take delivery of the hogs themselves. An important point about a futures option is that exercising it does not usually lead to delivery of the underlying asset. This is -because, in most circumstances, the underlying futures contract is closed out prior to delivery. Futures options are therefore normally eventually settled in cash. This is appealing to many investors, particularly - those with limited capital who may find it difficult to come up with the funds to buy the underlying asset when an option is exercised. Another advantage sometimes cited for futures options is that futures and futures options are traded in pits side by side in the same exchange. This faCilitates hedging, arbitrage, and speculation. It also tends to make the markets more efficient. A final point is that futures options tend to entail lower transactions costs than spot options in many situations. p'ut-Call Parity In Chapter 9, we derived a put-eall parity relationship for European stock options. We now present a similar argument to derive a put-eall parity relationship for European futures options on the assumption that there is no difference between the payoffs from futures and forward contracts. Consider European call and put futures options, both with strike price K and time to expiration T. We can form two portfolios: Portfolio A: a European call futures option plus an amount of cash equal to Ke -rT Portfolio B: a European put futures option plus a long futures contract plus an amount of cash equal to Foe -rT In portfolio A, the cash can be invested at the risk-free rate r and will grow to K at time T. Let FT be the futures price at maturity of the option. If FT > K, the call option in portfolio A is exercised and portfolio A is worth FT. If FT ~ K, the call is not exercised and portfolio A is worth K. The value of portfolio A at time T is therefore given by max(FT, K) In portfolio B, the cash can be invested at the risk-free rate to grow to Fo at time T. The put option provides a payoff ofmax(K - FT , 0). The futures contract provides a payoff of F T - Fo. The value of portfolio B at time T is therefore given by Fo + (FT - Fo) + max(K - FT, 0) = max(FT, K) Since the two portfolios have the same value at time T and there are no early exercise 329 Opti01is on Stock Indices, Currencies, and Futures opportunities, it follows that they are worth the same today. The value of portfolio A today is c + Ke- rT where c is the price of the call futures option. The marking-to';market process ensures that the futures contract in portfolio B is worth zero today. Therefore, portfolio B is worth where p is the price of the put futures option. Hence, c + Ke- rT = p + Foe- rT (14.11) This is the same as put-<:all parity for options on a non-dividend-paying stock in equation (9.3) except that the stock price is replaced by the futures price times e-rT For American options, the put-<:all parity relationship is (see Problem 14.38) Example 14.7 Suppose that the price of a European call option on silver futures for delivery in 6 months is $0.56 per ounce when the exercise price is $8.50. Assume that the silver futures price for delivery in 6 months is currently $8.00 and the risk-free interest rate for an investment that matures in 6 months is 10% per annum. From a rearrangement of equation (14.11), the price of a European put option on silver futures with the same maturity and exercise price as the call option is 0.56 + 8.50e- O. 1x0.5 - 8.00e-O. 1x0.5 = 1.04 14.6 VALUATION OF FUTURES OPTIONS USING BINOMIAL TREES This section examines, more formally than in Chapter 11, how binomial trees can be used to price futures options. The key difference between futures options and stock options is that there are no up-front costs when a futures contract is entered into. Suppose that the current futures price is 30 and it is expected to move either up to 33 or down to 28 over the next month. We consider a I-month call option on the futures with a strike price of 29 and ignore daily settlement. The situation is shown in Figure 14.1. If the futures price proves to be 33, then the payoff from the option is 4 and the value of the futures contract is 3. If the futures price proves to be 28, then the payoff from the option is zero and the value of the futures contract is _2. 3 To set up a riskless hedge, we consider a portfolio consisting of a short position in one option contract and a long position in !:l futures contracts. If the futures price moves up to 33, the value of the portfolio is 3!:l - 4; if it moves down to 28, the value of the portfolio is -2!:l. The portfolio is riskless when these are the same-that is, There is an approximation here in that the gain or loss on the futures contract is not realized at time T. It is realized day by day between time 0 and time T. However, as the length of the time step in a binomial tree becomes shorter, the approximation becomes better, and in the limit, as the time step tends to zero, an accurate answer is obtained. 3 330 CHAPTER 14 Figure 14.1 Futures price movements in numerical example. 33 30 28 when 3~ -4= -2~ 0.8. For this value of ~, we know the portfolio will be worth 3 x 0.8 - 4 = -1.6 in 1 month. Assume a risk-free interest rate of 6%. The value of the portfolio}oday must be or ~ = -1.6e-O.06xO.08333 = -1.592 The portfolio consists of one short option and ~ futures contracts. Since the value of the futures contract today is zero, the value of the option today must be 1.592. A Generalization We can generalize this analysis by considering a futures price that starts at Fo and is anticipated to rise to FOll or move down to Fod over the time period T. We consider a derivative maturing at the end of the time period, and we suppose that its payoff is III if the futures price moves up and Id if it moves down. The situation is summarized in Figure 14.2. The riskless portfolio in this case consists of a short position in one option combined with a long position in ~ futures contracts, where Figure 14.2 Fa f Futures price and option price in general situation. Options 011 Stock Indices, Currencies, and Futures 331 The value of the portfolio at the end of the time period, then, is always Denoting the risk-free interest rate by r, we obtain the value of the portfolio today as Another expression for the present value of the portfolio is - f, where f is the value of the option today. It follows that -f = [(Fou - Fo)b - !,,]e- rT Substituting for b and simplifying reduces this equation to f = e-rT[p!', + (1- P)fd] (14.12) where l-d (14.13) P=-- u-d In the n,umerical example in Figure 14.1, u = 1.1, d 1" = 4, and fd = O. From equation (14.13), we have = 0.9333, r = 0.06, T = 0.08333, 1 - 0.9333 P = 1.1 - 0.9333 = 0.4 and, from equation (14.12), f = e-O.06xO.08333(OA x 4 + 0.6 x 0) = 1.592 This result agrees with the answer obtained for this example earlier. Multistep Trees In practice, trees are. used to value American-style futures options in the same way as they are used to value options on stocks. Tllis is explained in Section 11.9. An example is in Example 11.3 and Figure 11.13. 14.7 THE DRIFT OF FUTURES PRICES IN A RISK-NEUTRAL WORLD There is a general result that allows us to use the analysis in Section 14.1 for futures options. This result is that in a risk-neutral world a futures price behaves in the same way as a stock paying a dividend yield at the domestic risk-free interest rate r. One clue that this might be so is given by noting that the equation for p in a binomial tree for a futures price is the same as that for a stock paying a dividend yield equal to q when q = r. Another clue is that the put-eall parity relationship for futures options prices is the same as that for options on a stock paying a dividend yield at rate q when the stock price is replaced by the futures price and q = r. To prove the result formally, we calculate the drift of a futures price in a risk-neutral world. We define F c as the futures price at time t. If we enter into a long futures contract today, its value is zero. At time bt (the first time it is marked to market) it provides a payoff of F D.C - Fo. If r is the very-short-term (bt-period) interest rate at CHAPTER 14 332 time 0, risk-neutral valuation gives the value of the contract at time 0 as e-r/:,.tE[F/:,.t - FoJ where E denotes expectations in a risk-neutral world. We must therefore have e- r/:,.( E(F/:,.( - Fo) = 0 showing that E(Ft:. t) = Fo Similarly; E(F2/:"t) = Ft:. t , E(F3t:. t) = F2t:. t, and so on. Putting many results like this together, we see that for any time T The drift of the futures price in a risk-neutral world is therefore zero. From equalion (14.7), then, the futures price behaves like a stock providing a dividend yield q equal to r. This result is a very general one. It is true for all futures prices and does not depend on any assumptions about interest rates, volatilities, etc. 4 The usual assumption made for the process followed by a futures price F in the riskneutral world is dF =aFdz (14.14) where a is a constant. Differential Equation For another way of seeing that a futures price behaves like a stock paying a dividend yield at rate q, we can derive the differential equation satisfied by a derivative dependent on a futures price in the same way as we derived the differential equation for a derivative dependent on a non-dividend-paying stock in Section 13.6. This iss 2 af I a f -a + '-) - - 7 a 2 F 2 =1'f t aF- (14.15) It has the same form as equation (14.6) with q set equal to r. This confirms that, for the purpose of valuing derivatives, a futures price can be treated in the same way as a stock providing a dividend yield at rate r. 14.8 BLACK'S MODEL FOR VALUING FUTURES OPTIONS European futures options can be valued by extending the results we have "'produced. Fischer Black was the first to show this in a paper published in 1976. 6 The underlying As we will discover in Chapter 25, a more precise statement of the result is: "A futures price has zero drift in the traditional risk-neutral world where the numeraire is the money market account" A zero-drift stochastic process is known as a martingale. A forward price is a martingale in a different risk-neutral world. This is one where the numeraire is a zero-coupon bond maturing at time T. 4 5 See Technical Note 7 on the author's website for a proof of this. See F. Black, "The Pricing of Commodity Contracts," Jot/mal of Financial Economics, 3 (March 1976): 167-:79. 6 Options on Stock Indices, Currencies, and Futures 333 assumption is that futures prices have the same lognormal property that we assumed for stock prices in Chapter 13. The European call price c and the European put price p for a futures option are given by equations (14.4) and (14.5(with So replaced by Fo and q = r: c = e-rT[FoN(d l ) - rT p = e- [KN(-d2 ) KN(d2 )] - (14.16) FoN(-d l )] (14.17) where dl = d2 In(Fo/ K) - + a 2 T /2 r;r; avT 2 _In(Fo/K) - a T/2 _ d a.JT - r;;:; 1 -ayT and a is the volatility of the futures price. When the cost of carry and the convenience yield are functions only of time, it can be shown that the volatility of the futures price is the same as the volatility of the underlying asset. Note that Black's model does not require the option contract and the futures contract to mature at the same time. Example 14.8 Consider a European put futures option on crude oil. The time to the option's maturity is 4 months, the current futures price is $20, the exercise price is $20, the risk-free interest rate is 9% per annum, and the volatility of the futures price is 25% per annum. In this case, Fo = 20, K = 20, r = 0.09, T = 4/12, a = 0.25, and In(Fo/ K) = 0, so that d1 = d2 N(-d 1) = a; = 0.07216 a.JT = -0.07216 = 0.4712, N( -d2 ) = 0.5288 and the put price p is given by p = e-O.09x4/12(20 x 0.5288 - 20 x 0.4712) = 1.12 or $1.12. 14.9 FUTURES OPTIONS vs. SPOT OPTIONS In this section we compare options on futures and options on spot when they have the same strike price and time to maturity. An option 011 spot or spot optioll is a regular option to buy or sell the underlying asset in the spot market. The payoff from a European spot call option with strike price K is max(ST - K, 0) where ST is the spot price at the option's maturity. The payoff from a European futures call option with the same strike price is max(FT - K, 0) CHAPTER 14 334 where F T is the futures price at the option's maturity. If the European futures option matures at the same time as the futures contract, F T = ST and the two options are in theory equivalent. If the European call futures option matures before the futures contract, it is worth more than the corresponding spot option in a normal market (where futures prices are higher than spot prices) and less than the corresponding spot· option in an inverted market (where futures prices are lower than spot prices). Similarly, a European futures put option is worth the same as its spot option counterpart when the futures option matures at the same time as the futures contract. If the European put futures option matures before the futures contract, it is worth less than the corresponding spot option in a normal market and more· than the corresponding spot option in an inverted market. Results for American Options _ Traded futures options are, in practice, usually American. Assuming that the risk-free rate of interest, r, is positive, there is always some chance that it will be optimal to exercise an American futures option early. American futures optioIls are, therefore, . worth more than their European counterparts. It is not generally true that an American futures option is worth the same as the corresponding American spot option when the futures and options contracts have the same maturity. Suppose, for example, that there is a normal market with futures prices consistently higher than spot prices prior to maturity. This is the ·case with most stock indices, gold, silver, low-interest currencies, and some commodities. An American call futures option must be worth more than the corresponding American spot call option. The reason is that in some situations the futures option will be exercised early, in which case it will provide a greater profit to the holder. Similarly, an American put futures option must be worth less than the corresponding American spot put option. If there is an inverted market with futures prices consistently lower than spot prices, as is the case with high-interest currencies and some commodities, the reverse must be true. American call futures options are worth less than the corresponding American spot call option, whereas American put futures options are worth more than the corresponding American spot put option. The differences just described between American futures options and American spot options hold true when the futures contract expires later than the options contract as well as when the two expire at the same time. In fact, the differences tend to be greater the later the futures contract expires. SUMMARY The Black-Scholes formula for valuing European options on a non-dividend-paying stock can be extended to cover European options on a stock providing a known dividend yield. This is a useful result because a number of other assets on which options are written can be considered to be analogous to a stock providing a dividend yield. In particular: 1. An index is analogous to a stock providing a dividend yield. The dividend yield is . the average dividend yield on the stocks composing the index. 335 Opti01is on Stock Indices, Currencies, and Futures 2. A foreign currency is analogous to a stock providing a dividend yield where the dividend yield is the foreign risk-free interest rate. . 3. A futures price is analogous to a stock providing a dividend yield where the dividend yield is equal to the domestic risk-free interest rate. The extension to Black-Scholes can, therefore, be used to value European options on indices, foreign currencies, and futures contracts. Index options are .settled in cash. Upon exercise of an index call option, the holder receives the amount by which the index exceeds the strike price at close of trading. Similarly, upon exercise of an index put option, the holder receives the amount by which the strike price exceeds the index at close of trading. Index options can be used for portfolio insurance. If the portfolio has a f3 of 1.0, it is appropriate to buy one put option for each 100So dollars in the portfolio, where So is the value of the index; otherwise, f3 put options should be purchased for each 100So dollars in the portfolio, where f3 is the beta of the portfolio calculated using the capital asset pricing model. The strike price of the put options purchased should reflect the level of insurance required. Currency options are traded both on organized exchanges and over the counter. They can be used by corporate treasurers to hedge foreign exchange exposure. For example, a US corporate treasurer who knows that sterling will be received at a certain time in the future can hedge by buying put options that mature at that time. Similarly, a US corporate treasurer who knows that the company will be paying sterling at a certain time in the future can hedge by buying call options that mature at that time. Futures options require the delivery of the underlying futures contract upon exercise. When a call is exercised, the holder acquires a long futures position plus a cash amount equal to the excess of the futures price over the strike price. Similarly, when a put is exercised, the holder acquires a short position plus a cash amount equal to the excess of the strike price over the futures price. The futures contract that is delivered typically expires slightly later than the option. If we assume that the two expiration dates are the same, a European futures option is worth exactly the same as the corresponding European spot option. However, this is not true of American options. If the futures market is normal, an American call futures option is worth more than the corresponding American spot call option, while an American put futures is worth less than the corresponding American spot put option. If the futures market is inverted, the reverse is true. FURTHER READING General Merton, R. C. "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science, 4 (Spring 1973): 141-83. Bodie, Z. "On the Risk of Stocks in the Long Run," Financial Analysts Journal, 51, 3 (1995): 18-22. On Options on Currencies Amin, K., and R. A. larrow. "Pricing Foreign Currency Options under Stochastic Interest Rates," Journal of International A;foney and Finance, IO (1991): 310-29. Biger, N., and J. C. Hull. "The Valuation of Currency Options," Financial Management, 12 (Spring 1983): 24--28. 336 CHAPTER 14 Garman, M. B., and S. W. Kohlhagen. "Foreign Currency Option Values," Journal of International Money and Finance, 2 (December 1983): 231-37. Giddy, 1. H. and G. Dufey. "Uses and Abuses of Currency Options," Journal of Applied Corporate Finance, 8, 3 (1995): 49-57. Grabbe, J. O. "The Pricing of Call and Put Options on Foreign Exchange," Journal of International Money and Finance, 2 (December 1983): 239-53. Jorion, P. "Predicting Volatility in the Foreign Exchange Market," Journal of Finance 50, 2 (1995): 507-28. On Options on Futures Black, F.,"The Pricing of Commodity Contracts," Journal of Financial Economics, 3 (March 1976): 167-79. Hilliard, J. E., and J, Reis. "Valuation of Commodity Futures and Options',under Stochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot," Journal of Financial and Quantitative Analysis, 33, I (March 1998): 61-86. Miltersen, K. R., and E. S. Schwartz. "Pricing of Options on Commodity Futures with Stochastic - Term Structures of Convenience Yields and Interest Rates," Journal of Financial and Quantitative Analysis, 33, I (March 1998), 33-59. Questions and Problems (Answers in Solutions Manual) 14.1. A portfolio is currently worth $10 million and has a beta of 1.0. The S&P 100 is currently standing at 500. Explain how a puJ option on the S&P 100 with a strike of 480 can be used to provide portfolio insurance. 14.2. "Once we know how to value options on a stock paying a dividend yield, we know how to value options on stock indices, currencies, and futures." Explain this statement. 14.3. A stock index is currently 300, the dividend yield on the index is 3% per annum, and the risk-free interest rate is 8% per annum. What is a lower bound for the price of a 6-month European call option on the index when the strike price is 290? 14.4. A currency is currently worth $0.80. Over each of the next 2 months it is expected to increase or decrease in value by 2%. The domestic and foreign risk-free interest rates are 6% and 8%, respectively. What is the value of a 2-month European call option with a strike price of $0.80? 14.5. Explain the difference between a call option on yen and a call option on yen futures. 14.6. Explain how currency options can be used for hedging. 14.7. Calculate the value of a 3-month at-the-money European call option on a stock index when the index is at 250, the risk-free interest rate is 10% per annum, the volatility of the index is 18% per annum, and the dividend yield on the index is 3% per annum. 14.8. Consider an American call futures option where the futures contract and the option contract expire at the same time. Under what circumstances is the futures option worth more than the corresponding American option on the underlying asset? 14.9. Calculate the value of an 8-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free interest rate is 4% per annum, and the foreign risk-free interest rate is 8% per annum. 14.10. Why are options on bond futures more actively traded than options on bonds? Optioizs on Stock Indices, Currencies, Gild Futures 337 14.11. "A futures price is like a stock paying a dividend yield." What is the dividend yield? 14.12. A futures price is currently 50. At the end of 6 months it will be ~ither 56 or 46. The riskfree interest rate is 6% per annum. What is the value of a 6-month European call option with a strike price of 50? 14.13. Calculate the value of a 5-month European put futures option when the futures price is $19, the strike price is $20, the risk-free interest rate is 12% per annum, and the volatility of the futures price is 20% per annum. 14.14. A total return index tracks the return, including dividends, on a certain portfolio. Explain how you would value (a) forward contracts and (b) European options on the index. 14.15. The S&P 100 index currently stands at 696 and has a volatility of 30% per annum. The risk-free rate of interest is 7% per annum and the index provides a dividend yield of 4% per annum. Calculate the value of a 3-month European put with strike price 700. 14.16. What is the put--eall parity relationship for European currency options? 14.17. A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9%, respectively. Calculate a lower bound for the value of a 6-month call option on the currency with a strike price of $1.40 if it is (a) European and (b) American. 14.18. Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum and the risk-free rate is 6% per annum. A 3-month European call option on the index with a strike price of 245 is currently worth $10. What is the value of a 3-month European put option on the index with a strike price of 245? 14.19. Would you expect the volatility of a stock index to be greater or less than the volatility of a typical stock? Explain your answer. 14.20. Does the cost of portfolio insurance increase or decrease as the beta of the portfolio increases? Explain your answer. 14.21. Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide protection against the value of the portfolio falling below $54 million in 1 year's time? 14.22. Consider again the situation in Problem 14.21. Suppose that the portfolio has a beta of 2.0, that the risk-free interest rate is 5% per annum, and that the dividend yield on both the portfolio and the index is 3% per annum. What options should be purchased to provide protection against the value of the portfolio falling below $54 million in 1 year's time? 14.23. Suppose you buy a put option contract on October gold futures with a strike price of $400 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $377 and the most recent settlement price is $380? 14.24. Suppose you sell a call option contract on April live-cattle futures with a strike price of 70 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is exercised when the futures price is 76 cents and the most recent settlement price is 75 cents? 14.25. Consider a 2-month call futures option with a strike price of 40 when the risk-free interest rate is 10% per annum. The current futures price is 47. What is a lower bound for the value of the futures option if it is (a) European and (b) American? 338 CHAPTER 14 14.26. Consider a 4-month put futures option with a strike price of 50 when the risk-free interest rate is 10% per annum. The current futures price is 47. What is a lower bound for the value of the futures option if it is (a) European and (b) American? 14.27. A futures price is currently 60. It is known that over each of the next two 3-month periods it will either rise by 10% or fall by 10%. The risk-free interest rate is 8% per annum. What is the value of a 6-month European call option on the futures with a strike price of 60? If the call were American, would it ever be worth exercising it early? 14.28. In Problem 14.27, what is the value of a 6-month European put option on futures with a strike price of 60? If the put were American, would it ever be worth exercising it early? Verify that the call prices calculated in Problem 14.27 and the put prices calculated here satisfy put-eall parity relationships. 14.29. A futures price is currently 25, its volatility is 30% per annum, and the risk-free interest rate is 10% per annum. What is the value of a 9-month European call on the futures with a strike price of 26? . 14.30. A futures price is currently 70, its volatility is 20% per annum, and the risk-free interest rate is 6% per annum. What is the value of a 5-month European put on thl: futures with a strike price of 65? 14.31. Suppose that a futures price is currently 35. A European call option and a European put option on the futures with a strike price of 34 are both priced at 2 in the market. The risk-free interest rate is 10% per annum. Identify an arbitrage opportunity. Both options have 1 year to maturity. 14.32. "The price of an at-the-money European call futures option always equals the price of a .similar at-the-money European put futures option." Explain why this statement is true. 14.33. Suppose that a futures price is currently 30. The risk-free interest rate is 5% per annum. A 3-month American call futures option with a strike price of 28 is worth 4. Calculate bounds for the price of a 3-month American put futures option with a strike price of 28. 14.34. Can an option on the yenjeuro exchange rate be created from two options, one on the dollarjeuro exchange rate, and the other on the dollar-yen exchange rate? Explain your answer. ·14.35. A corporation knows that in 3 months it will have $5 million to invest for 90 days at LIBOR minus 50 basis points and wishes to ensure that the rate obtained will be at least 6.5%. What position in exchange-traded interest rate options should it take? 14.36. Prove the results in equations (14.1), (14.2), and (14.3) using the following portfolios: Portfolio A: one European call option plus an amount of cash equal to Ke- rT Portfolio B: e- qT shares, with dividends being reinvested in additional shares Portfolio C: one European put option plus e- qT shares, with dividends on'\the shares being reinvested in additional shares Portfolio D: an amount of cash equal to Ke- rT 14.37. Show that, if C is the price of an American call with strike price K and maturity T on a stock providing a dividend yield of q, and P is the price of an American put on the same stock with the same strike price and exercise date, then Soe- qT - K::::; C - P::::; So - Ke- rT where So is the stock price, r is the risk-free interest rate, and r > O. (Hint: To obtain the O ptionsoll Stock Indices, Currencies, and Futures 339 first half of the inequality, consider possible values 0(: Portfolio A: a European call option plus an amount K invested at the risk-free rate Portfolio B: an American put option plus e-qT of stock with dividends being reinvested ~~~~ / To obtain the second half of the inequality, consider possible values of: Portfolio C: an American call option plus an amount Ke- rT invested at the risk-free rate Portfolio D: a European put option plus one stock, with dividep.ds being reinvested in the stock.) 14.38. Show that, if C is the price of an American call option on a futures contract when the strike price is K and the maturity is T, and P is the price of an American put on the same futures contract with the same strike price and exercise date, then Foe- rT K:::;: C - P:::;: Fo - Ke- rT where Fo is the futures price and r is the risk-free rate. Assume that r > 0 and that there is no difference between forward and futures contracts. (Hint: Use an analogous approach to that indicated for Problem 14.37.) 14.39. If the price of currency A expressed in terms of the price of currency B follows the process dS = (rB - rA)S dt + as dz - where rA is the risk-free interest rate in currency A and rB is the risk~free interest rate in currency B. What is the process followed by the price of currency B expressed in terms of currency A? Assignment Questions 14.40. Use the DerivaGem software to calculate implied volatilities for the March 104 call and the March 104 put on the Dow Jones Industrial Average (DJX) in Table 14.1. The value of the DJX on February 4, 2004, was 104.71. Assume that the risk-free rate was 1.2% and that the dividend yield was 3.5%. The options expire on March 20, 2004. Are the quotes for the two options consistent with put-eall parity? 14.41. A stock index currently stands at 300. It is expected to increase or decrease by 10% over each of the next two time periods of 3 months. The risk-free interest rate is 8% and the dividend yield on the index is 3%. What is the value of a 6-month put option on the index with a strike price of 300 if it is (a) European and (b) American? 14.42. Suppose that the spot price of the Canadian dollar is US $0.75 and that the Canadian dollar/US dollar exchange rate has a volatility of 4% per annum. The risk-free rates of interest in Canada and the United States are 9% and 7% per annum, respectively. Calculate the value of a European call option to buy one Canadian dollar for US $0.75 in 9 months. Use put-eall parity to calculate the price of a European put option to sell one Canadian dollar for US $0.75 in 9 months. What is the price of a call option to buy US $0.75 with one Canadian dollar in 9 months? 14.43. A mutual fund announces that the salaries of its fund managers will depend on the performance of the fund. If the fund loses money, the salaries will be zero. If the fund makes a profit, the salaries will be proportional to the profit. Describe the salary of a fund manager as an option. How is a fund manager motivated to behave with this type of remuneration package? 340 CHAPTER 14 14.44. A futures price is currently 40. It is known that at the end of 3 months the price will be either 35 or 45. What is the value of a 3-month European call option on the futures with a strike price of 42 if the risk-free interest rate is 7% per annum? 14.45. Calculate the implied volatility of soybean futures prices from the following information concerning a European put on soybean futures: Current futures price Exercise price Risk-free rate Time to maturity Put price 525 525 6% per annum 5 months 20 14.46. Use the DerivaGem software to calculate implied volatilities for the July options on corn futures in Table 14.4. Assume the futures prices in Table 2.2 apply and that the risk-free rate is 1.1 % per annum. Treat the options as American and use 100 time steps. The options mature on June 19, 2004. Can you draw any conclusions from the pattern of implied volatilities you obtain? ...
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This note was uploaded on 01/30/2012 for the course MATH 174 taught by Professor Donblasius during the Spring '11 term at UCLA.

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