Both retail and institutional investors use theseinstruments to invest in the market as a whole, thereby eliminating the laborious job oftrying to pick individual stocks. Portfolio managers can use index instruments to createa wide variety of risk/reward scenarios for large, diversified stock portfolios. Finally,professional arbitrageurs find that in spite of the highly liquid and usually efficient indexmarkets, pricing disparities occur often enough to warrant close monitoring of thesemarkets. When a disparity does exist, a trader can execute an arbitrage by hedging themispriced index against either other stock indices or against a basket of stocks. Sucharbitrage strategies are commonly referred to as index arbitrage.
For all of the foregoing reasons, no book on options would be complete without atleast some discussion of index options. However, index futures and options are soclosely related, and so many strategies involve both instruments, that it is almostimpossible to discuss one without discussing the other. We will therefore include bothinstruments in our discussion.
WHAT IS AN INDEX?
An index is a number which represents the composite value of a group of items. In thecase of stock indices, the value of the index is determined by the value of a group ofspecified stocks, with the value of the stocks usually being determined by their pricesin the marketplace. As the stocks which make up the index rise in price, the value ofthe index also rises, reflecting the increased value of the stocks; as the stocks fall in price,the value of the index also falls, reflecting the declining value of the stocks. If someStocks in the index rise while others fall, the offsetting changes in stock prices may resultin the index itself remaining unchanged, even though it’s possible that every stock inthe index changed price. The value of the index always reflects the total value of thestocks which make up the index.
301302 Chapter 15
CALCULATING AN INDEX
There ate several different methods of calculating stock index values, but the mostcommon methods entail weighting the stocks either by price or by capitalization. To seehow these methods work, consider an index composed of the following four stocks:
Total Shares TotalStock Price Outstanding Capitalization1 125 10,000 1,250,0002 80 5,000 400,0003 52 25,000 1,300,0004 17 40,000 680,000
Ifan index is price weighted, each stock’s value in the index is directly proportionalto its price. Using this method, the total value of the index is simply the sum of the pricesof all the stocks in the index:
125 + 804524 17 =274The price weighting of each stock is therefore:
stock t: 125/274 = 45.6%stock 2: 80/274 = 29.2%stock 3: 52/274 = 19.0%stock 4; 17/274 = 6.2%
100.0%
Since the importance of each stock is directly proportional to its price, whenmeasured in percent terms movement in higher priced stocks will have a greater impacton the value of the index than equal movement in lower priced stocks. If a trader weremonitoring a price weighted index for possible changes, he would do well to look formovement in the higher priced stocks in the index.
If an index is capitalization weighted, the index value is the total value of all theoutstanding stock of the firms in the index. The total value, or capitalization, of a firmis the price of a share multiplied by the number of shares issued by the firm. The totalCapitalization of the four stocks making up our index is:
(125 x 10,000) + (80 x 5,000) + (52 x 25,000) + (17 x 40,000) =1,250,000 + 400,000 + 1,300,000 + 680,000 = 3,630,000
The capitalization weighting of each stock is therefore:
stock 1: 1,250,000/3,630,000 = 34.5%stock 2: 400,000/3,630,000 = 11.0%stock 3: 1,300,000/3,630,000 = 35.8%stock 4: 680,000/3,630,000 = 18.7%
100.0%Stock Index Futures and Options 303
If an index is capitalization weighted, in percent terms the stocks which will havethe greatest effect on the index are those with the greatest market value. These are oftenstocks which are very widely held, even though they may not be the highest pricedstocks,
For convenience, when an index is first introduced it is common to set the indexvalue equal to some round number, most often 100. In order to achieve this number,the actual value of the index must be either multiplied or divided by some number. Forexample, when we calculated the price weighted value of our four-stock index, thevalue was 274. To transform the index value to 100, we solve:
274/x = 100x=2.74
This number, 2.74, is referred to as the index divisor, and is used in all subsequentindex calculations. Whenever a new value of the index is calculated, that value willalways be divided by 2.74 to yield the published index value. If we wanted to set thevalue of our capitalization weighted index to 100, we solve:
3,630,000/x = 100X = 36,300
In this case, the index divisor is 36,300.
An index divisor is not constant, but can change as the capitalization of the indexchanges. If stocks in the index split or declare special stock dividends, or if stocks arereplaced by other stocks due to buyouts, liquidations, or bankruptcies, the index valuemight go up or down, even though the market prices of the stocks making up the indexTemain unchanged. To offset this effect, when any capitalization change occurs in thestocks making up the index, the divisor is changed to reflect the relative value of index,and to ensure that the index value represents a true picture of price changes in themarketplace.
Regardless of the method used to calculate an index, prices for each of thecomponent stocks must be determined. Ifa stock is traded on only one exchange, thereis never any doubt as to the stock’s price. It is simply the last trade price. But what if astock is traded on more than one exchange? If the last trade for a stock on one exchangeis $52, while the last trade on a different exchange is $5212, what price should be usedin calculating the index? The answer is usually specified in the index prospectus, and isMost commonly the primary marketplace for that stock. In the United States the primarymarketplace for many stocks is the New York Stock Exchange, although this is notalways the case. Some stocks may be primarily traded on the American, Chicago,Pacific, or Philadelphia Stock Exchanges, or in the over-the-counter market. A traderwho follows a stock index should always know the source of the stock prices used tocalculate that index.304 Chapter 15
REPLICATING AN INDEX
Suppose a trader wanted to create a holding of stocks that exactly replicated the valueof the index. What would he have to do?
If each point in our four-stock price weighted index is worth exactly $1, then thetotal value of the index is $274. A trader could simply purchase one share of each stockin the index to exactly replicate the index. Very often, however, for trading purposesthe value of each point is assigned a value other than $1. If, for example, each point inthe index is assigned a value of $100, the total dollar value of the index is:
274 x $100 = $27,400
Now in order to exactly replicate the index, a trader would have to purchase 100 sharesof each stock. The value of his holding would then be:
(125 + 80 + 52 + 17) x 100 = $27,400
Not only must a trader consider the value of each point in an index, he must alsoconsider the effect the divisor will have on the index. Suppose that instead of 274, theindex in our example has been set to 100 by using a divisor of 2.74. If the point valueof the index is still $100, then the total value of the index is:
100.00 (index value) x $100 (point value) = $10,000
Now if we purchase 100 shares of each stock, the dollar value of our stock holding($27,400) greatly exceeds the dollar value of the index ($10,000). Clearly, our stockholding does not replicate the index. Notice, however, that the index value of $10,000is simply our stock value of $27,400 divided by the divisor of 2.74. In other words, ifwe divide each purchase of 100 shares of stock by 2.74, we will have a stock holdingequal to the value of the index. To achieve this, we need to purchase 100/2.74, orapproximately 36/4 shares of each stock.! If we do, the value of our stock holding willreplicate the value of the index since:
(125 + 80 + 52 + 17) x 36.5 = $10,000
The procedure is slightly more complex if we are attempting to replicate acapitalization weighted index. Consider our four-stock capitalization weighted examplewith a divisor of 36,300. Suppose each point in the index has a value of $500, so thatthe dollar value of the index is:
100.00 {index value) x $500 {point value) = $50,000
How can we create a capitalization weighted stock holding with a value of $50,000?In our price weighted example, we purchased shares of stock equal to the point
value divided by the divisor. We can take the same approach with a capitalization
weighted index, with one important change: we must multiply the resulting number by
 
1, Obviously, one cannot trade fractional shares of stock. In the real world, a trader attempting to replicatethe sample index would have to buy either 36 or 37 shares of each stock.Stock index Futures and Options 305
the total number of outstanding shares of the stock. The proper number of shares ofeach stock required to replicate our capitalization index is:
Outstanding Point RequiredShares Value Divisor Sharesstock 1: 10,000 x 500 + 36300 = 137.7stock 2: 5,000 x 500 + 36300 = 68.9stock 3: 25,000 x 500 + 36300 = 344.4stock 4: 40,000 x 500 + 36300 = 551.0
We can see that this replicates the $50,000 value of the index since:(137.7 x 125) + (68.9 x 80) + (344.4 x 52) + (551.0 x 17) = 50,000Summarizing, the number of shares of each stock required to replicate an index
for a price weighted index: point value / index divisorfor a capitalization weighted index: outstanding shares x point value / index divisor
STOCK INDEX FUTURES
In theory one can create a futures contract on a stock index in exactly the same waythat futures contracts have been created on traditional commodities. At expiration, theholder of a long stock index futures position would be required to take delivery of allthe stocks, in their correct proportions, which make up the index. The holder of a shortposition would be required to make delivery of the stocks.
In fact, very few stock index futures contracts are settled through the physicaldelivery of the stocks making up the index. Such a process, requiring the delivery of thecorrect number of shares of many different stocks, would be unmanageable for mostclearing organizations. To be perfectly fair, such a settlement procedure might requirethe delivery of fractional shares of stock, which is impossible.
For all these reasons, exchanges have chosen to settle stock index futures atexpiration in cash. Since futures contracts are settled daily in cash, the final cash transferbetween the buyer and seller of a stock index futures contract is equal to the differencebetween the expiration value of the index and the previous day’s settlement price. Forexample, suppose an index is at 462.50 at the moment of expiration and the previousday’s settlement price for the futures contract was 461.00. If each point in the index isworth $100, the holder of a long position will be credited with a final payment of $150,since (462.50 — 461.00) x $100 = $150. The holder of a short position will be debitedby the same amount.
What should be the fair value of a stock index futures Contract?
Recall from Chapter { that an important difference between a futures contract andStock is the settlement procedure. Whereas the purchase of stock requires the buyer toactually pay cash, the purchase of a futures contract requires only an initial margindeposit. The cash paid for the stock will result in an interest cost, while the margindeposit results in no such interest loss. Moreover, any profits or losses resulting from306 Chapter 15
Stock trading are only paper profits or losses until the stock position is liquidated. Infutures markets, however, profits and losses accrue whenever the price of the futurescontract fluctuates, regardless of whether the position is closed out. These profits orlosses are typically calculated daily from the settlement price of the futures contract,Such profits and losses result in variation, the amount of cash which is credited to ordebited from a trader’s account based on the daily price fluctuation. Variation andmargin requirements are separate and distinct. Margin in theory entails no carryingcosts, while credits or debits resulting from daily variation do indeed earn (in the caseof a credit) or lose (in the case of a debit) interest.
Clearly, the purchase of a futures contract offers one important advantage over thepurchase of the component stocks: no cash outlay is required to purchase a futurescontract. Consequently, there is an interest rate savings equal to the cost of borrowingsufficient cash to purchase all the stocks in the index.
The amount of interest saved by purchasing a stock index futures contract ratherthan the actual stocks in the index can be calculated by multiplying the current indexvalue by the risk-free interest rate, and then multiplying this number by the amount oftime remaining to expiration of the futures contract. By adding the resulting number tothe index value, we have the fair value of the futures contract based on carrying costs.
For example, suppose there are three months remaining to expiration with ourfour-stock index at 274.00. If the risk-free interest rate over this period is 8%, what arethe carrying costs on the index?
8% x 3/12 x 274.00 = 2% x 274.00 = 5.48?
If carrying costs are the only consideration, the fair value for the futures contract willbe the 274.00 value of the index plus the 5.48-point carrying cost to expiration, or279.48,
What other advantages or disadvantages are associated with holding a futurescontract?
If we buy a futures contract, we may have a very good substitute for the stocks inthe index, but we don’t actually own the stocks. Since ownership of a stock is necessaryfor the receipt of a stock dividend, if we buy a futures contract we will have to give upany dividends which the stocks in the index might pay. If the stocks in the index arenon-dividend paying, or will not pay any dividends prior to expiration of the futurescontract, no further calculations are necessary. The fair value will simply be the currentindex value plus carrying costs to expiration. More often, however, some, and perhapsail of the stocks in the index will pay dividends prior to futures expiration. In this casewe must deduct the amount of dividends lost by holding a futures position rather thanthe actual stocks.
Suppose that the stocks in our four-stock index are expected to pay the followingdividends prior to expiration of the futures contract:
2. For simplicity we assume simple interest payments. We can be slightly more accurate by assumingcompound Interest (see Appendix B).Stock Index Futures and Options 307
Stock Price Expected Dividend1 125 $1.802 80 03 52 $.904 17 $.35
If we buy one share of each of the four stocks which make up the index, we willreceive by futures expiration a total dividend payout of:
$1.80 + $.90 + $.35 = $3.05
Jf we buy the futures contract instead of the stocks we will have to forego this 3.05.Therefore, we will want to pay 3.05 less for the futures contract. Taking into considera-tion carrying costs, we showed that with the index at 274.00, the fair value of thefutures contract was 279.48. Deducting the dividends, the fair value of the futurescontract is:
274.00 (index value) + 5.48 (carrying costs) ~ 3.05 (expected dividends) = 276.43
If we own the stocks in the index we not only receive the dividends paid by thestocks, we also have use of the dividend funds. If we own stock 1, we will receive adividend of $1.80 on the ex-dividend date and, in theory, we can also earn interest onthis amount to expiration of the futures contract. If the dividend will be paid two monthsprior to expiration and interest rates are 8%, the interest earnings are:
1.80 x 2/12 x 8% = .02
To calculate an exact futures price, in addition to deducting the dividends themselves,we must also deduct the interest earnings from each dividend payment. This couldbecome a rather complicated exercise, but in practice the interest earnings on dividendsare so small in relation to the actual index that most traders simply deduct the dividendPayments themselves from the index value, and ignore the accompanying interestearnings.
If the index is price weighted, replication of the index requires an equal number ofshares of each stock. So the total dividend payout is simply the sum of all the dividends.If the index is capitalization weighted, replication of the index requires a differentnumber of shares for each stock. In that case, the total dividend payout will be thedividend payout for each stock multiplied by the number of shares required forreplication.
Given the apparently complex dividend calculations, as well as the interest consid-erations on the index and dividends, determining the exact fair value of a futurescontract might seem a daunting task. In practice, however, active trading firms havecomputers which can quickly, and continuously, calculate the fair value for futurescontracts on most commonly traded stock indices,
Even when it is not possible to calculate the exact fair value, many traders find italmost as effective to estimate the fair value by adding the carrying cost to the index andsubtracting the average dividend payout. For example, suppose there are nine weeks308 Chapter 15
(63 days) to expiration on a stock index futures contract with the underlying indexcurrently at 425.00. If interest rates are 8% annually, and the average annual dividendpayout on the stocks making up the index is 6%, we have:
(8% — 6%) x 63/365 x 425.00 = 1.47
Therefore, the approximate fair value of the futures contract is 425.00 + 1.47 = 426.47,The reader should be aware that this is only an estimate. If it turns out that a largenumber of stocks do not pay their regular dividends during this nine-week period, or ifthe dividend payout is unusually large, this estimate could prove to be far from accurate.
The daily dividend payout for the S&P 500 index for one quarterly cycle, fromMarch 1993 to June 1993 expiration, is shown in Figure 15-1, Note the large amountswhich were paid at the beginning of May, compared to the relatively small amountspaid during April. The cumulative effect of this dividend payout, in terms of the total ofdividends expected to June expiration, is shown in Figure 15-2. The line drops onlyslightly during April, but declines sharply during the first two weeks in May. A traderwho intended to initiate some type of arbitrage strategy either just prior to, or just after,a large dividend payout, would need more than a rough estimate of the fair value of anS&P 500 futures contract. If he were not aware of the exact dividend payout, a strategywhich he expected to be profitable might turn out to be just the opposite.
It is also possible to reverse the calculations and ask, given an observed futuresprice, where would the index have to be for the futures to be fairly valued? For example,
Figure 15-1: S&P 500 Index Dividend Payout(March 1993 to June 1993 Expiration)
 
0.25
0.20
 
015
 
 
0.10
Daily Dividends in Index Points
 
 
0.005 30 5 10 15
25 30 5 10.15 20 25 30 5 10 16 20April May dune
MarchStock Index Futures and Options 309
 
Figure 15-2: S&P 500 Dividends to Expiration(March 1993 to June 1993 Expiration)
 
 
 
 
 
 
 
 
 
3503 004-—~ > So2502&Soa.$ 2007—-3££3 1.50coSQZ51.0040.50 \0.00-*25 30 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15March April May June
if the futures price is 432.70, and we assume that it is trading at fair value, what shouldthe index price be if there are 6 weeks (42 days) to expiration, interest rates are 6%, andthe expected dividend payout is 2.23? First we can add the dividend payout of 2.23 tothe futures price of 432.70, giving a value of 434.93. Then we can deduct the carryingcosts on this amount over the next six-weeks:
434.93 x 42/365 x 6% = 3.00The implied index value is therefore 434.93 — 3.00 = 431.93.
INDEX ARBITRAGE
In theory, the price of a futures contract should always reflect the fair value of holdingthe futures contract rather than holding the stocks making up the index. But marketsare not always one hundred percent efficient. If the futures price doesn’t reflect the fairvalue, a trader can execute a profitable arbitrage by purchasing the undervalued asset,either the basket of stocks or the futures contract, and selling the overvalued asset. Ifatrader believes that the fair value of a stock index futures contract is 386.75, and findsthat the futures contract is trading at 387.40, he will attempt to buy all the stocks intheir correct proportion and simultaneously sell the futures contract. If the tradersucceeds, and if the point value of the futures contract is $500, for each futures contracthe executes he will show a theoretical profit of:
(387.40 - 386.75) x $500 = .65 x $500 = $325310 Chapter 15
Of course, this profit will only be fully realized at expiration, when the futures contractis automatically settled at the index value. At the same time, the trader will also haveto liquidate his stock holdings to ensure that at the moment of expiration both hisfutures position and stock position zero out. This can be done by placing a market onclose order, guaranteeing that the last trade price for each stock which determines thefinal index value will be the liquidation price for the trader’s stock holdings.
What would happen if the fair value for the stock index futures contract were386.75, but the futures contract were actually trading at 385.95? Now the trader wouldtake just the opposite position, buying the undervalued futures contract and selling thebasket of stocks. The theoretical profit would be:
(386.75 - 385.95} x $500 = .80 x $500 = $400
Unfortunately, some practical problems might affect the workability of this strategy. Ifthe trader wants to sell stock which he doesn’t own, he will have to borrow it. If nostock is available, or if the market in which he is dealing does not freely permit the shortsale of stock, he may not be able to execute the arbitrage, no matter how mispriced thefutures contract. Finally, a trader might not earn full interest on the short sale of stock.If the actual interest earned is less than the interest rate used to calculate the fair valueof the futures contract, the fair value of the futures contract, for trading purposes, maybe less than the presumed fair value of 386.75.
This type of trading strategy, where one buys or sells a mispriced stock index futurescontract and takes an opposing position in the underlying stocks, is one type of indexarbitrage. Since computers can often be programmed to calculate the fair value of afutures contract, and to exectte the arbitrage when the futures contract is mispriced,such a strategy is also commonly referred to as program trading. A buy program consistsof buying the stocks and selling the futures contract, and a sell program consists of sellingthe stocks and buying the futures contract.
How great a mispricing of a futures contract relative to the price of the underlyingstocks justifies futures arbitrage? If markets are totally frictionless, a trader wouldattempt to execute an arbitrage if there were any mispricing, no matter how small.Unfortunately, markets are not totally frictionless. An arbitrage which in theory isprofitable because of a mispriced futures contract may be unprofitable when transactioncosts are factored in. Ifa trader believes that it will cost approximately 1% index pointto execute all the trades required for an arbitrage, he will not attempt to enter into aprogram trade unless the futures contract is at least 50 too expensive or too cheap. Ifhe also considers the problems associated with the short sale of stock, he may decidethat the futures contract must be .75 too cheap before he will initiate a sell program.With the fair value of the futures contract at 386.75, the trader will initiate a programtrade only if the price of the futures contract is outside the range
386.75 — .75 and 386.75 + .50
or less than 386.00 and more than 387.25.Even if the quoted index price is 386.75, and the futures contract is higher than387.25 or lower than 386.00, the trader cannot be certain that a profit opportunityStock Index Futures and Options 311
exists. In order to execute an arbitrage, it is necessary to trade all the stocks at a priceconsistent with the index value of 386.75. But there is a bid price and an offer price foreach individual stock, and it is likely that the last trade price for some of the stocks inthe index took place at the bid price, some at the ask price, and some at prices inpetween. If a trader wants to immediately execute a buy program, he will have topurchase every stock in the index at the ask price, and this may be at a price well abovethe quoted index value.
Moreover, the quoted index price can sometimes present a distorted picture ofactual market conditions. The index itself may be quoted at 386.75, but does thatrepresent the price at which all the stocks in the index are trading? Perhaps trading insome of the stocks has been halted due to pending news. Or perhaps the last sale forone or more of the stocks took place at a price outside the current market for the stock.The last sale in a stock may have been at 381/4, but if the market in the stock is fallingvery quickly, the offered price for the stock may be 37934. The index value is based onthe last sale price of 381/4, but the actual price is at least as low as 3744, and possiblyeven lower. Ifa trader hopes to profit from a mispriced futures contract he must knowthe true market for all the stocks which underlie that contract.
Assuming the trader can execute all purchases and sales at theoretically profitableprices, is there anything which could go wrong with a program trade consisting of stocksagainst stock index futures? Suppose a trader buys an underpriced futures contract andsimultaneously sells the underlying stocks. If the index starts to fall, and the futurescontract follows, the trader will show a profit on his stock trades and a loss on his futurestrade, and presumably these two components will offset each other. Unfortunately, thestock trades, being subject to stock-type settlement, only result in a paper profit, whilethe futures trade, being subject to futures-type settlement, results in a cash loss. If theindex drops far enough, and the trader must borrow cash to meet the variationrequirements for the futures contract, the interest cost may be so great that the arbitragewhich initially appeared profitable actually turns into a loss. This is similar to thesettlement risk for a conversion or reversal in the futures option market described inChapter 11.
From this, we can see that a program trade always wants the market to move inthe direction of the futures contract so that cash will flow into the trader’s account. Thiswill increase the trader’s potential profit, since he will be able to earn interest on thiscash over the life of the futures contract. In this respect, we can apply the concept ofthe delta to futures contracts in much the same way we do to an option contract. Thedelta is the rate at which a contract will change its theoretical value with respect tomovement in the underlying contract. For example, assume that a stock index is quotedat 300.00. If there are three months to expiration, with interest rates at 8%, and anexpected dividend payout of 4.50, the fair value of a futures contract is approximately:
300.00 + (300.00 x 3/12 x .08) — 4.50 = 301.50
Suppose the index immediately rises 10% to 330.00. The new fair value of the futuresContract is:S312 Chapter 15
330.00 + (330.00 x 3/12 x .08) - 4.50 = 332.10
The index rose 30.00 while the futures contract rose 30.60, The delta of the futurescontract is therefore 102, since 30.60/30.00 = 1.02. The futures contract is changingits value at 102% the rate of the underlying index.
If we assume that the dividend payout remains constant, the delta of a futurescontract is simply a function of interest rates. The extra two deltas associated with thefutures contract results from an