um studied forward and futures prices on theBritish pound, Canadian dollar, German mark, Japanese yen, and Swiss franc between 1974 and1979. They found very few statistically significant differences between the two sets of prices. Theirresults were confirmed by Park and Chen, who as part of their study looked at the British pound,German mark, Japanese yen, and Swiss franc between 1977 and 1981.
French studied copper and silver during the period from 1968 to 1980. The results for silvershow that the futures price and the forward price are significantly different (at the 5% confidencelevel), with the futures price generally above the forward price. The results for copper are less52 CHAPTER 3
clear-cut. Park and Chen looked at gold, silver, silver coin, platinum, copper, and plywoodbetween 1977 and 1981. Their results are similar to those of French for silver. The forward andfutures prices are significantly different, with the futures price above the forward price. Rendlemanand Carabini studied the Treasury bill market between 1976 and 1978. They also found statisticallysignificant differences between futures and forward prices. In all these studies, it seems likely thatthe differences observed are due to the factors mentioned in the previous section (taxes, trans-actions costs, and so on).
3.10 STOCK INDEX FUTURES
 
A stock index tracks changes in the value of a hypothetical portfolio of stocks. The weight of astock in the portfolio equals the proportion of the portfolio invested in the stock. The percentageincrease in the stock index over a small interval of time is set equal to the percentage increase in thevalue of the hypothetical portfolio. Dividends are usually not included in the calculation so thatthe index tracks the capital gain/loss from investing in the portfolio.°
If the hypothetical portfolio of stocks remains fixed, the weights assigned to individual stocks inthe portfolio do not remain fixed. When the price of one particular stock in the portfolio risesmore sharply than others, more weight is automatically given to that stock. Some indices areconstructed from a hypothetical portfolio consisting of one of each of a number of stocks. Theweights assigned to the stocks are then proportional to their market prices, with adjustments beingmade when there are stock splits. Other indices are constructed so that weights are proportional tomarket capitalization (stock price x number of shares outstanding). The underlying portfolio isthen automatically adjusted to reflect stock splits, stock dividends, and new equity issues.
Stock Indices
Table 3.2 shows futures prices for contracts on a number of different stock indices as they werereported in the Wall Street Journal of March 16, 2001. The prices refer to the close of trading onMarch 15, 2001.
The Dow Jones Industrial Average is based on a portfolio consisting of 30 blue-chip stocks in theUnited States. The weights given to the stocks are proportional to their prices. One futurescontract, traded on the Chicago Board of Trade, is on $10 times the index.
The Standard & Poor’s 500 (S&P 500) Index is based on a portfolio of 500 different stocks: 400industrials, 40 utilities, 20 transportation companies, and 40 financial institutions. The weights ofthe stocks in the portfolio at any given time are proportional to their market capitalizations. Thisidex accousts fur 80% of ihe inarket capitalization of ail the siovks listed on the New York StockExchange. The Chicago Mercantile Exchange (CME) trades two contracts on the S&P 500. One ison $250 times the index; the other (the Mini S&P 500 contract) is on $50 times the index. TheStandard & Poor’s MidCap 400 Index is similar to the S&P 500, but based on a portfolio of 400stocks that have somewhat lower market capitalizations.
The Nikkei 225 Stock Average is based on a portfolio of 225 of the largest stocks trading on theTokyo Stock Exchan 1Be. Stocks are weighted according to their prices. One futures contract (traded
aa the CMEY: es des deon the CME) is on $5 times the index.
* An exception to this is a total return index. This is calculated by assuming that dividends on the hypotheticalportfolio are reinvested in the portfolio.Determination of Forward and Futures Prices 53
 
Table 3.2 Stock index futures quotes from the Wall Street Journal, March 16, 2001.(Columns show month, open, high, low, settle, change, lifetime high, lifetime low, and openinterest, respectively.)
 
INDEX U.S. Dollar Index (NYBOT)-$1,000 times USDX
Mar” 11440 115.10 11410 11472 + 96 118.72 10804Dd Industrial Average (CBOT) $10 times average Whe Tye Wieas 1185 de? 100 HE tet SDTMar Gre gD Ose. ea 0 Fst vol 3,100; vol Wed 5031: open int 8.133, 4-485.
Hx pri: Hi 115.19: Lo 1138.43: Close 11471, +1.01.
8,02:dune 10170 10200 10060 10105 + 10 11795 9980 24,36;
33
 
   
 
 
  
10260 10260 10160 10200 + 7 11390 10005 204Ef 8 Ase amenx pi ; lose + -S&P 500 Index (CME) $250 times Index ie su te eee i kata 3 - ug wae see toeMar 118000 118200 116900 117330 + © 410 164260 115500 112,346 ~ 160 34500 33030 1,442June 118080 119450 117450 118470 + 390 166560 116550 455531 | pee “ - $2880 = 140 34350 23180 © ‘528119500 120680 119300 119640 + 360 169060 117820 2635 | Ect ya) 28.059: val ‘Wed 15,314: open int 122284, +1388.Dec 120700 21760 120500 120740 + 360 171480 118020 "969 | indey Hi 32639 Lo 30044" Close 40429, 210.wre seas sou post vue + ae ee ‘pour #8 | cac-40 Stock Index (marie Euro 10,00 x IndexEe ee ee ae te sta ene Mar 5140.0 52075 50680 51760 + 500 71020 44890 370,341Eat wd i a 04 arr Ci on ae es Aor 5175.0 5196.5 51000 51950 + 49.0 60225 50020 12,519rt FTIR OF Lo 16671 Close 1170.58, +585. fay 5145.0 5160.5 50705 51670 + 500 55085 49765 75Minl S&P 500 (CME)-$50 times Index June 5119.0 51540 50950 51560 + 51.0 70340 49730 14,742Mar 116950 118: 00 117325 + 400 150000 115400 58,162 Sept 52070 + 510 60135 48040 930Vol Wed 183,181: omen im S076 24. De =, SRBLO + «520 81625 58925 9135&P Midcap 400 (CME)-$500 times Index wre ” "530 + 510 2800Mar 47500 477.00 471.00 471.55 - 1.90 564.00 45050 4,347 Sot Oe BO ‘600jue, HOt 48400 ee om nat Gir 47500 15.5 | Est vol 117.859, vol Wed 94843; open int 407,720, +2i.514val We Open int DAX-30 German Stock index (EUREX)Idx prt Hi 479.23; Lo 471.25; Close 471,25, ~ 2.34 Euro 25 per OAX Index pt.Nikkel 225 Stock Average (cme) -$5 times index. Mar 5850.0 58875 5768.5 P556.0 + 56.0 7699.0 58850 190,163dune 12080, 12175. 12030, 12130. + ie ais. 11255, 16825 | June 58765 59120 57930 98825 + 615 73640 56900 293541Est vol 1,239: vol Wed 2,341; open int 16,846, Sept 5688.0 5959.5 5888.0 5945.5 + 64.0 69525 57680 2,idx prt: Hi 12152.63; Lo 11493.68; Close ‘nee "5002s, ae 125,542; open int 485,548, 49,245.Nasdaq 100 D (CME) $100 times Index Index Hi 5889.95 Lo 5787.06 Ciose 5889.95, +9563.Mar 181400 181600 168000 189250 - 6000 424150 168000 21284 | FrSe' 100 indox (LIFFE)-£10 per index pointJune 178000 184100 170800 171500 ~ 6100 396100 169800 47,728 | pe eS ee eee aso Baa00 54620 71,880Est vol 25457, vol Wed 41,488; open int Be. = 3.889. June 57100 5751.5 5640.0 57080 + 415 6308.0 6180 260,404i 1813.68; Lo 1697.61; Close 1697.92, —47.16. Sept 5749.0 + 39.0 6436.0 59065 6
190 (cme) $20 times ind Index Krenn sere agra) | Est vol 113,700; vol Wed 151,724: open int 339,034, + 10,145,Vor Wet 46,403: open int "104,082, —771, 80.0 3650.0 1676.0 85,740 DJ Euro Stoxx 50 Index {EUREX)-Euro 10,00 x indexcecl ions St heey index Mar 41400 4189.0 4086.0 4160.0 + 630 5536.0 3998.0 340,951GEC! (CME) $260 times INGOT 5000 21440 25g | lume 4135.0 41740 40700 41850 + 820 52020 30840 434081ar 217.00 21700 244. na OO 24. Sept 4195.0 4195.0 4191.0 42020 + 68.0 4913.0 40580 24,311Apr 21690 217,00 214.40 215.50 — 80 237.50 21440 14581 | tel hs ae1.a73 pen int 790.303, 4 44714Est vol 3,241, vol Wed 3,607, q mn int 47,055, 3 Index Hi 4200.08; Lo 4088.84: Close 420008, +76.11.idx pr Hi 217.28; Lo 214,34 215.58, ~ 82. DJ Stoxx 50 Index (EUREX)-Euro 10.00 x indexflussell 2000 (CME oe index sg 44550 aao7_ | Mar 39450 40000 39030 39810 + 51.0 51500 3en30 13.001r - FNe eet ee ib 5 458) 16083 dane 0, $050 | 380, + 520 50500 3611.0 14427Est vol 4,190; vol Wed ed he ae 240, -40 | index Hi 403216; Lo 3908.72; Close 4032.96, +91.22,
ld prt: Hi 457.96; Lo 451.71; Close 462.16, —1.53.
 
Source: Reprinted by permission of Dow Jones, Inc., via copyright Clearance Center, Inc.© 2001 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
The Nasdaq 100 is based on 100 stocks using the National Association of Securities DealersAutomatic Quotations Service. The CME trades two contracts. One is on $100 times the index; theother (the Mini Nasdaq 100 contract) is on $20 times the index.
In the GSCI index futures contract shown in Table 3.2, the underlying asset is the GoldmanSachs Commodity Index. This is not a stock index. It is a broadly based index of commodity prices.All the major commodity groups, such as energy, livestock, grains and oilseeds, food and fiber, andmetals, are represented in the GSCI. Studies by Goldman Sachs have shown that the GSCI isnegatively related to the S&P 500 index, with the correlation being in the range —0.30 to —0.40.
The Russel! 2000 index is an index of small stocks in the United States. The U.S. dollar index is atrade-weighted index of the values of six foreign currencies (the euro, yen, pound, Canadian dollar,Swedish krona, and Swiss franc). The Share Price Index is the All Ordinaries Share Price Index, abroadly based index of Australian stocks. The CAC-40 Index is based on 40 large stocks trading inFrance. The DAX-30 Index is based on 30 stocks trading in Germany. The FT-SE 100 Index isbased on a portfolio of 100 major U.K. stocks listed on the London Stock Exchange. The DJ EuroStoxx 50 Index and the DJ Stoxx 50 Index are two different indices of blue chip European stocks54 CHAPTER 3
compiled by Dow Jones and its European partners. The futures contracts on these indices trade onEurex and are on 10 times the values of the indices measured in euros.
As mentioned in Chapter 2, futures contracts on stock indices are settled in cash, not bydelivery of the underlying asset. All contracts are marked to market at either the opening price orthe closing price of the index on the last trading day, and the positions are then deemed to beclosed. For example, contracts on the S&P 500 are closed out at the opening S&P 500 index onthe third Friday of the delivery month. Trading in the contracts continues until 8:30 a.m. on thatFriday.
Futures Prices of Stock Indices
A stock index can be regarded as the price of an investment asset that pays dividends. Theinvestment asset is the portfolio of stocks underlying the index, and the dividends paid by theinvestment asset ate the dividends that would be received by the holder of this portfolio. It isusually assumed that the dividends provide a known yield rather than a known cash income. If g isthe dividend yield rate, equation (3.7) gives the futures price, Fo, as
Fy = Sye OF (3.12)
Example 3.7. Consider a three-month futures contract on the S&P 500. Suppose that the stocksunderlying the index provide a dividend yield of 1% per annum, that the current value of the index is400, and that the continuously compounded risk-free interest rate is 6% per annum. In this case,
r = 0.06, S) = 400, T = 0.25, and g = 0.01. Hence, the futures price, Fy, is given by
Fy = = 400e .06—0.01)x 0.25 = $405.03
In practice, the dividend yield on the portfolio underlying an index varies week by weekthroughout the year. For example, a large proportion of the dividends on the NYSE stocks arepaid in the first week of February, May, August, and November each year. The chosen value of gshould represent the average annualized dividend yield during the life of the contract. Thedividends used for estimating g should be those for which the ex-dividend date is during the lifeof the futures contract. Looking at Table 3.2, we see that the futures prices for the S&P 500 Indexappear to be increasing with the maturity of the futures contract at about 3.8% per annum. Thiscorresponds to the situation where the risk-free interest rate exceeds the dividend yield by about3.8% per annum.
Index Arbitrage
If Fy > Spe OF profits can be made by buying spot (i-e., for immediate delivery) the stocksunderlying the index and shorting futures contracts. If Fy < Spe""®", profits can be made by doingthe reverse—that is, shorting or selling the stocks underlying the index and taking a long positionin futures contracts. These strategies are known as index arbitrage. When Fo < Sye’~?", indexarbitrage is often done by a pension fund that owns an indexed portfolio of stocks. WhenFy > Soe” ®", it is often done by a corporation holding short-term money market investments.For indices involving many stocks, index arbitrage is sometimes accomplished by trading arelatively small representative sample of stocks whose movements closely mirror those of theindex. Often index arbitrage is implemented through program trading, with a computer systembeing used to generate the trades.Determination of Forward and Futures Prices 55
October 1987
To do index arbitrage, a trader must be able to trade both the index futures contract and theportfolio of stocks underlying the index very quickly at the prices quoted in the market. In normalmarket conditions this is possible using program trading, and Fo is very close to Spe" 9%".Examples of days when the market was anything but normal are October 19 and 20 of 1987.On what is termed “Black Monday”, October 19, 1987, the market fell by more than 20%, and the604 million shares traded on the New York Stock Exchange easily exceeded all previous records.The exchange’s systems were overloaded and if you placed an order to buy or sell a share on thatday there could be a delay of up to two hours before your order was executed. For most of the day,futures prices were at a significant discount to the underlying index. For example, at the close oftrading the S&P 500 Index was at 225.06 (down 57.88 on the day), whereas the futures price forDecember delivery on the S&P 500 was 201.50 (down 80.75 on the day). This was largely becausethe delays in processing orders made index arbitrage impossible. On the next day, Tuesday,October 20, 1987, the New York Stock Exchange placed temporary restrictions on the way inwhich program trading could be done. This also made index arbitrage very difficult, and thebreakdown of the traditional linkage between stock indices and stock index futures continued. Atone point the futures price for the Decemher contract was 18% less than the S&P 500 Index.However, after a few days the market returned to normal, and the activities of arbitrageurs ensuredthat equation (3.12) governed the relationship between futures and spot prices of indices.
The Nikkei Futures Contract
Equation (3.12) does not apply to the futures contract on the Nikkei 225. The reason is quitesubtle. When S is the value of the Nikkei 225 Index, it is the value of a portfolio measured in yen.The variable underlying the CME futures contract on the Nikkei 225 has a dollar value of 5S. Inother words, the futures contract takes a variable that is measured in yen and treats it as though itis dollars. We cannot invest in a portfolio whose valuc will always be 5S dollars. The best we cando is to invest in one that is always worth 5S yen or in one that is always worth 5QS dollars, whereQ is the dollar value of one yen. The arbitrage arguments that have been used in this chapterrequire the spot price underlying the futures price to be the price of an asset that can be traded byinvestors. The arguments are therefore not exactly correct for the Nikkei 225 contract. We willdiscuss this issue further in Section 21.8.
3.11 FORWARD AND FUTURES CONTRACTS ON CURRENCIES
 
 
We now move on to consider forward and futures foreign currency contracts. The underlying assetin such contracts is a certain number of units of the foreign currency. We will, therefore, define thevariable Sy as the current spot price in dollars of one unit of the foreign currency and Fy as theforward or futures price in dollars of one unit of the foreign currency. This is consistent with theway we have defined Sy and Fo for other assets underlying forward and futures contracts. However,as mentioned in Chapter 2, it does not necessarily correspond to the way spot and forwardexchange rates are quoted. For major exchange rates other than the British pound, euro,Australian dollar, and New Zealand dollar, a spot or forward exchange rate is normally quotedas the number of units of the currency that are equivalent to one dollar.
A foreign currency has the property that the holder of the currency can earn interest at the56 CHAPTER 3
risk-free interest rate prevailing in the foreign country. For example, the holder can invest thecurrency in a foreign-denominated bond. We define ry as the value of the foreign risk-free interestrate when money is invested for time T. As before, r is the domestic risk-free rate when money isinvested for this period of time.The relationship between Fo and Sp isFo = Soe? (3.13)
This is the well-known interest rate parity relationship from international finance. To see that itmust be true, we suppose that the two-year interest rates in Australia and the United States are 5%and 7%, respectively, and the spot exchange rate between the Australian dollar (AUD) and theU.S. dollar (USD) is 0.6200 USD per AUD. From equation (3.13) the two-year forward exchangerate should be
0.620020 99%? — 9.6453
Suppose first the two-year forward exchange rate is less than this, say 0.6300. An arbitrageur can:
1. Borrow 1,000 AUD at 5% per annum for two years, convert to 620 USD, and invest theUSD at 7% (both rates are continuously compounded).
2, Enter into a forward contract to buy 1,105.17 AUD for 1, 105.17 x 0.63 = 696.26 USD.
The 620 USD that are invested at 7% grow to 620e°°%? — 713.17 USD in two years. Ofthis, 696.26 USD are used to purchase 1,105.17 AUD under the terms of the forwardcontract. This is exactly enough to repay principal and interest on the 1,000 AUD that areborrowed (1, 000e°°*? = 1,105.17). The strategy therefore gives rise to a riskless profit of713.17 — 696.26 = 16.91 USD. (If this does not sound very exciting, consider following asimilar strategy where you borrow 100 million AUD!)
Suppose next that the two-year forward rate is 0.6600 (greater than the 0.6453 value given byequation (3.13)). An arbitrageur can:
1. Borrow 1,000 USD at 7% per annum for two years, convert to 1, 000/0.6200 = 1, 612.90AUD, and invest the AUD at 5%.
2. Enter into a forward contract to sell 1,782.53 AUD for 1, 782.53 x 0.66 = 1,176.47 USD.
The 1,612.90 AUD that are invested at 5% grow to 1,612.90e°*? = 1,782.53 AUD in two years.The forward contract has the effect of converting this to 1,176.47 USD. The amount needed to payoff the USD borrowings is 1,000e°°7** = 1,150.27 USD. The strategy therefore gives rise to ariskless profit of 1,176.47 — 1,150.27 = 26.20 USD.
Table 3.3 shows futures prices on March 15, 2001, for a variety of different currency futurestrading on the Chicago Mercantile Exchange. In the case of the Japanese yen, prices are expressedas the number of cents per unit of foreign currency. In the case of the other currencies, prices areexpressed as the number of U.S. dollars per unit of foreign currency.
When the foreign interest rate is greater than the domestic interest rate (r; > r), equation (3.13)shows that Fy is always less than Sy and that Fy decreases as the time to maturity of the contract, 7,increases. Similarly, when the domestic interest rate is greater than the foreign interest rate (r > r,),equation (3.13) shows that Fy is always greater than Sy and that Fo increases as T increases. OnMarch 15, 2001, interest rates on the Japanese yen, Canadian dollar, and the euro were lower thanin the interest rate on the U.S. dollar. This corresponds to the r > ry situation and explains whyfutures prices for these currencies increase with maturity in Table 3.3. In Australia, Britain, andDetermination of Forward and Futures Prices 57
 
Table 3.3 Foreign exchange futures quotes from the Wall Street Journal on March 16, 2001.(Columns show month, open, high, low, settle, change, lifetime high, lifetime low, and openinterest, respectively.)
 
CURRENCYJapan Yen (CME, 12.5 milion fen; ¢ yen (.00)ie rn Be) elgg arte Goan ap augMWe #560 Ase8 fos) S200 — ode 10b9 Aibe arp
Sept 8475 8475 845 8363 - .0097 1.0050 Be 579Dec 6450. 8450 8450 8455 - 0009 9880 8450 431Est vol 23,771, vol Wed 95,059; ot nv int 130,445, + 6.474.Beutschemark, om 126, }0 marks; $ per mark
Ma 4852 = O81” 4925 4225 238June 4660 re a ‘ey = 0058 4900 4596 22Est vol 217; vot Wed 161; open int 467, -172.
Canadian | Dollar ee 100, 000 « airs + per Can $Mar 64 7040 401 20,442une bar eau Bins - oe £900 8404 eetSept 8425 6435 ie ‘BA12 — (0018 8906 6405 |” 2.830Dec 6440 6440 6417 BAIS. — 0018 6825 417 1,288Est vol 8574 ) ee ‘on int a, ¥ +998. ‘
ritish Poun s.; $ per p