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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 nondividendpaying 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 exdividend 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 SoeqT 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 noarbitrage arguments (see Problem 14.36). PutCall Parity
Replacing So by SoeqT in equation (9.3), we obtain puteall parity for a stock providing
a dividend yield equal to q:
(14.3)
This result can also be proved using noarbitrage arguments (see Problem 14.36). 14.2 OPTION PRICING FORMULAS
By replacing So by Soe qT in the BlackScholes 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 = SoeqTN(d 1)  KerT N(d2 )
p = Ke rT N(d2 )  SoeqTN(d 1)
c Since (14.4)
(14.5) qT Soe )
So
In ( ~ =In KqT 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 exdividend 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): 14183. 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 RiskNeutral 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 riskneutral 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 riskneutral valuation procedure, described in Section 13.7, can be used.
In a riskneutral 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 riskneutral 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 SOirq)T. A similar analysis to that in the appendix of Chapter 13 gives
the expected payoff in a riskneutral 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 riskneutral probability p of an up
movement is chosen so that the expected return is r  q. This means that
pSu + (1 p)Sd = e(rq)/!,r or ad p= ud where
a = e(rq)/!,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 ineet 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 76301510 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
feblllOp 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 LEAPSLONG 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 500CB 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 shortdated options, the exchanges
trade longermaturity contracts known as LEAPS. The acronym LEAPS stands for
"longterm 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 puteall 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 exdividend date occurs during the life of the option.
In the United States exdividend 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 exdividend
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 riskfree 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 0.706geo.03x2/]2  900 x 0.6782eo.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 BlackScholes
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 Americanstyle
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 welldiversified
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 longtenn investor you should buy stocks rather than bonds. Consider a US fund manager who is trying to persuade investors to buy as a
longtenn 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 riskfree bonds over the next 10 years.
Historically stocks have outperformed bonds in the United States over almost any
lOyear 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 10year riskfree 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 riskfree 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, longdated 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 riskfree interest rate equals beta times the excess return of a market
index over the riskfree 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 riskfree 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:
Riskfree interest rate:
Excess return from index
over riskfree interest rate:
Excess return from portfolio
over riskfree 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 overthecounter 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 exchangetraded market is much smaller than the overthecounter 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 exchangerate 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 exchangerate
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 riskfree 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 ~ SoerfT  Ke rT p ~ Ke rT _ SoerfT Equation (14.3), with q replaced by Ii, provides the puteall 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 = SaerfTN(d))  Ke rT N(d2 )
(14.7)
p = Ke rT N(d2 )  SaerfTN(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 crJT = 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 4month European call option on the British pound. Suppose that the
current exchange rate is 1.6000, the exercise price is 1.6000, the riskfree interest rate
in the United States is 8% per annum, the riskfree 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 = erT[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 highinterest currencies and put options on Options' on Stock Indices, Currencies, and Futures 323 lowinterest currencies are the most likely to be exercised prior to maturity. The reason
is that a highinterest currency is expected to depreciate and a lowinterest currency is
expected to appreciate. Binomial trees can be used to value Ainericanstyle 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 Americanstyle 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
maturesnot 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 midcurve 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, sugarworld, crude oil, natural gas,
gold, Treasury bonds, Treasury notes, 5year Treasury notes, 30day federal funds,
Eurodollars, Iyear and 2year midcurve 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
sixtyfourth of 1%. Table 14.4 gives the price of the March call futures option on a
Treasury bond on February 4, 2004, as 206, 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
CAUSSETTLE
PUTSSETTlE
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 CALLSSETTLE PUTSSETTLE 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 SugarWorld 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 CAllSSETTLE PUTSSETTLE GasolineUnlead (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
CattleFeeder (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~ 261puts
Op int Tues 3,298 calls 5,427 puts  CattleLive (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 ...
 Hogslean (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 J4.4Colltillued
STRIKE CAUSSmLE PUTSSmLE 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
TBonds 049 Apr
HI
230
3.j)4 May
229 348 ~~~ t~~ Tu vol 14,191 calls 17,000 puts
Op Int Tues 412,644 calls 444,891 puts TNotes $100,000; points and 64lhs of 100%
PrIce
Mar Apr May Mar Apr
112
2.j)Q 130 I52 020 125
113
117 100
.• 037 I58
114
044 041 059 100
115
020 024 040 140
116
aos 014 016 228
026
117
0.j)3 008
_
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
146 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. MidCurve 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 PUTSSmLE 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
121 CAUSSmlE 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. MidCurve 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 116 049 062 015 H8
11200 056 036
.• 022 127
11250 036 025
.• 034
11300 022 017
_ 052
11350 Oll 011
.• 110
11400 006
_
_ 136
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 PUTSSmLE $ million; pis. of 100%
PrIce
Feb Mar Apr
9825
.• 5.90 EuroBUND (CBT) minus
Mar
.117
.062
.007
.002 CAUSSmLE Euribor (CBT) $100,000; points and 64lhs of 100%
PrI",
Mar Apr May Mar
110
2.j)6 2lI3 2·35 036
111
128 136
058
112
058 111 142 124
113
034 054
200 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 shortterm 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
longterm 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 3month 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 shortterm 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 3month 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 9609 (or 96 = 96.28125). The yield on longterm /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 104 (or I ~ = 1.0625% of the
principal). If longterm rates fall to 6% per annum and the Treasury bond
futures price rises to 10000, 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 livehogs 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'utCall Parity
In Chapter 9, we derived a puteall parity relationship for European stock options. We
now present a similar argument to derive a puteall 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 riskfree 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 riskfree 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 markingto';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 nondividendpaying stock in
equation (9.3) except that the stock price is replaced by the futures price times erT
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 riskfree
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  8.00eO. 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 upfront 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 Imonth 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 samethat 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 riskfree interest rate of 6%. The value of the portfolio}oday must be or ~ = 1.6eO.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 riskfree 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 = erT[p!', + (1 P)fd] (14.12) where ld (14.13) P= ud 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 = eO.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 Americanstyle 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 RISKNEUTRAL 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 riskneutral world a futures price behaves in the same
way as a stock paying a dividend yield at the domestic riskfree 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 puteall 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 riskneutral
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 veryshortterm (btperiod) interest rate at CHAPTER 14 332 time 0, riskneutral valuation gives the value of the contract at time 0 as
er/:,.tE[F/:,.t  FoJ where E denotes expectations in a riskneutral 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 riskneutral 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 nondividendpaying 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 riskneutral world where the numeraire is the money market account" A zerodrift stochastic
process is known as a martingale. A forward price is a martingale in a different riskneutral world. This is one
where the numeraire is a zerocoupon 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 = erT[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
riskfree 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 = eO.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 riskfree
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, lowinterest 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 highinterest 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 BlackScholes formula for valuing European options on a nondividendpaying
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 riskfree interest rate. . 3. A futures price is analogous to a stock providing a dividend yield where the dividend yield is equal to the domestic riskfree interest rate.
The extension to BlackScholes 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): 14183.
Bodie, Z. "On the Risk of Stocks in the Long Run," Financial Analysts Journal, 51, 3 (1995):
1822.
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): 31029.
Biger, N., and J. C. Hull. "The Valuation of Currency Options," Financial Management, 12
(Spring 1983): 2428. 336 CHAPTER 14
Garman, M. B., and S. W. Kohlhagen. "Foreign Currency Option Values," Journal of
International Money and Finance, 2 (December 1983): 23137.
Giddy, 1. H. and G. Dufey. "Uses and Abuses of Currency Options," Journal of Applied
Corporate Finance, 8, 3 (1995): 4957.
Grabbe, J. O. "The Pricing of Call and Put Options on Foreign Exchange," Journal of
International Money and Finance, 2 (December 1983): 23953.
Jorion, P. "Predicting Volatility in the Foreign Exchange Market," Journal of Finance 50,
2 (1995): 50728.
On Options on Futures
Black, F.,"The Pricing of Commodity Contracts," Journal of Financial Economics, 3 (March
1976): 16779.
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): 6186.
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), 3359. 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
riskfree interest rate is 8% per annum. What is a lower bound for the price of a 6month
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 riskfree interest rates are
6% and 8%, respectively. What is the value of a 2month 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 3month atthemoney European call option on a stock index
when the index is at 250, the riskfree 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 8month 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 riskfree interest rate is 4% per annum, and the foreign riskfree 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 6month European call option
with a strike price of 50?
14.13. Calculate the value of a 5month European put futures option when the futures price
is $19, the strike price is $20, the riskfree 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
riskfree rate of interest is 7% per annum and the index provides a dividend yield of
4% per annum. Calculate the value of a 3month European put with strike price 700.
14.16. What is the puteall parity relationship for European currency options?
14.17. A foreign currency is currently worth $1.50. The domestic and foreign riskfree interest
rates are 5% and 9%, respectively. Calculate a lower bound for the value of a 6month
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 riskfree rate is 6% per annum. A 3month European call option
on the index with a strike price of 245 is currently worth $10. What is the value of a
3month 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 riskfree 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 livecattle 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 2month call futures option with a strike price of 40 when the riskfree
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 4month put futures option with a strike price of 50 when the riskfree
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 3month
periods it will either rise by 10% or fall by 10%. The riskfree interest rate is 8% per
annum. What is the value of a 6month 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 6month 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 puteall parity relationships.
14.29. A futures price is currently 25, its volatility is 30% per annum, and the riskfree interest
rate is 10% per annum. What is the value of a 9month 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 riskfree interest
rate is 6% per annum. What is the value of a 5month 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
riskfree interest rate is 10% per annum. Identify an arbitrage opportunity. Both options
have 1 year to maturity.
14.32. "The price of an atthemoney European call futures option always equals the price of a
.similar atthemoney European put futures option." Explain why this statement is true.
14.33. Suppose that a futures price is currently 30. The riskfree interest rate is 5% per annum.
A 3month American call futures option with a strike price of 28 is worth 4. Calculate
bounds for the price of a 3month 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 dollaryen 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 exchangetraded 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 riskfree 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 riskfree rate
Portfolio B: an American put option plus eqT 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 riskfree 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 riskfree 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 riskfree 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 riskfree 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 puteall 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 riskfree interest rate is 8% and the
dividend yield on the index is 3%. What is the value of a 6month 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 riskfree 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 puteall 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 3month European call option on the futures with
a strike price of 42 if the riskfree 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
Riskfree 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 riskfree
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.
 Spring '11
 DonBlasius
 Math

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