a trader executessuch a strategy and finds himself assigned overnight, his position might become horriblyunbalanced. The next day even a small adverse move in the index could provedisastrous.
If a trader could execute all the stocks in the index, and could be certain of holdingthe position to expiration, he would still have to physically liquidate his stock position.Since the option values at expiration are determined by the closing index value, andsince the index value is determined by the closing trades in each individual stock, the322 Chapter 15
trader must ensure that he closes each of his individual stock positions at the closingprice on the exchange from which the index is calculated. In many cases, he canaccomplish this by placing market-on-close (MOC) orders for each individual stock. Buthe must be very careful. If he fails to close out each stock position at the appropriatePrice, his arbitrage may not produce the profit originally expected.
Because it can be difficult to trade a complete and correctly proportioned basket ofstocks, and because there is the additional risk of early exercise after an index arbitragehas been executed, mispriced synthetic relationships are not as easy to exploit in indexoption markets as in other option markets. Not only do such mispricings occur morefrequently in index option markets, they can persist for long periods of time.
Finding a Substitute for the Index
Regardless of the difficulty in executing arbitrage strategies, if a synthetic relationshipappears to be sufficiently mispriced a trader will look for some way to exploit thesituation, even with the additional risks that go with such a strategy. This means eithertrying to execute the basket of stocks underlying the index, or looking for an acceptablesubstitute for the basket.
If the index is narrowly based, with only 20 or 30 stocks, it may not be unreason-able to try and execute a complete basket of stocks. If the index is more broadly based,with several hundred stocks, the difficulties of executing every stock in the rightquantity may make it impractical to execute a complete basket. In such a case, a tradermay question whether a complete basket is really necessary for his purposes. In mostindices not every stock has the same weighting. If an index is very broadly based, pricechanges in the smallest companies may have very little impact on the index. It may bepossible to eliminate these companies and still have a basket which is almost identicalto the index.
In the search for an acceptable tradeoff between ease of execution and correlationwith the index, traders are always experimenting with different pseudo-baskets, aselection of stocks in the index which can be easily bought or sold, and which will reactto changing market conditions in a way which is almost identical to the actual index.The primary question for most traders is one of correlation. What degree of correlationis sufficient for the trader’s purposes? 100 percent? 99%? 95%? A correlation of 99%between a pseudo-basket and the index may sound good. But if the index is at 400.00and a trader executes a conversion with the synthetic market mispriced by 1.00, the99% correlation means he could be wrong by as much as 1% of 400.00, or 4.00. Undersome conditions, his presumed profit of 1.00 might turn into a loss of 3.00. Of course,the laws of probability are still in the trader’s favor. The pseudo-basket’s deviation fromthe true index could also work in the trader’s favor, occasionally giving him a profitgreater than 1.00. Arbitrages tend to be done in large size because they are supposed tobe very low-risk strategies. If the trader executes a large number of conversions orreverse conversions using a pseudo-basket in place of the actual index, he may find thatthe correlation risk has a greater impact than he expected.
Another possible substitute for the index is a futures contract on the index whichunderlies the cash options. If such a contract is available, a trader can use the futuresStock Index Futures and Options 323
contract in place of the underlying basket in his arbitrage strategies. The futures contractis not only easier to execute than a basket, with lower transaction costs, but there willbe a 100% correlation between the futures contract and the actual index. If the futurescontract and the index expire at the same time, the futures contract and the optionvalues will converge to the same price at expiration. Moreover, the trader will not haveto worry about liquidating either his option or futures position. The clearing house willautomatically settle all contracts in cash at expiration.
Providing that the futures contract and the options expire at the same time, afutures contract on the same index which underlies the cash option market is a perfectsubstitute for the basket. Indeed, if the options are European with no chance of earlyexercise, an arbitrage between the futures contract and options is virtually riskless. Oncethe trader has completed his arbitrage, he knows that the futures contract and indexwill correlate perfectly and will converge at expiration. In the United States, traders inthe S&P 500 options market at the Chicago Board Options Exchange often treat the S&P500 futures contract traded at the Chicago Mercantile Exchange as the underlyingcontract in their strategies. Traders take a similar approach on many European andAsian exchanges which trade index options on cash and where a futures contract withthe same underlying index is also available.
If the futures contract offers an acceptable substitute for the index basket, theproblem then becomes one of the prices at which an arbitrage can be executed. Whileit may be easy to buy and sell a futures contract, can it be done at prices which make aconversion or reverse conversion profitable? If the synthetic market is mispriced by 1.00in the index option market, and we can trade the futures contract at exactly its fairvalue, there is no problem. But is that likely to be the case? If it were, everyone in themarketplace would line up to buy or sell a synthetic in the option market, and take anOpposing position in the futures market. This activity would force the markets back tofair value, and any arbitrage opportunity would quickly disappear. Traders find that ifcontracts are mispriced in one index market, they are often mispriced by an equalamount in other index markets. If a synthetic underlying position (long call, short put)is 1.00 too cheap in the option market, a trader usually finds that the futures contractis also 1.00 too cheap. If this were not the case, everyone would take advantage of theSituation by purchasing the cheaper contract, either the synthetic or futures contract,and selling the more expensive. In this respect, markets are typically very efficient. Amispricing in one market usually means a similar mispricing in all related markets.
Tf it can be traded at the right price, a futures contract on an index is perhaps thebest substitute an option trader can find. If such a contract is not available, a trader mightinstead consider a futures contract on a different, but similar, index. For example, thelargest index option market in the United States is the S&P 100 (OEX) market at theChicago Board Options Exchange, Traders in this market are always searching for anacceptable substitute for the underlying basket of 100 stocks. A futures contract on this
 
6. Options on S&P 500 futures are also traded at the CME. Because there is so little early exercise premiumIn futures options, and because the S&P 500 cash options traded at the CBOE are European, it turnsout that the markets for both the futures options and cash options are almost identical.324 Chapter 15
index would be an ideal substitute but, unfortunately, one does not exist,? However,there is a strong correlation between the S&P 100 and S&P 500 indices and, as notedearlier, a futures contract on the latter index is actively traded at the Chicago MercantileExchange.
Using an alternative index futures contract as a substitute for the actual underlyingindex presents several problems. The first is correlation. Just as a pseudo-basket may notcorrelate exactly, not all indices correlate exactly, even those with many of the samestocks. Figure 15-3 shows the spread between two highly correlated indices, the S&P500 and the S&P 100, during 1992 and 1993. Note that the spread had a range of almost20 index points over this period. Moreover, it was not uncommon for the spread tochange by as much as five points over a period of several weeks. This highlights thetrading risk that can arise when one index is used as a substitute for another.
In addition, not all indices expire at the same time. S&P 100 options are settledfrom the closing stock prices on the last trading day, while the S&P 500 futures aresettled from the opening stock prices on the same day. Because of this, holding an S&P100 option versus an S&P 500 futures position to expiration represents a substantial and,for most traders, unacceptable risk. If a trader uses S&P 500 futures as a hedge againstS&P 100 options, he will almost certainly close one side of the position prior toexpiration. Traders have found that using S&P 500 futures contracts as a hedge againstS&P 100 options is a short-term solution at best.
The best solution for most index option traders is to avoid using substitute indexcontracts whenever possible, and to attempt to execute all strategies within the sameindex market. For arbitrage traders, this means focusing on boxes and jelly rolls, and thetime spreads, vertical spreads, and butterflies from which these strategies are con-structed. For example, a trader might buy a call time spread, and then try to sell thecompanion put time spread, thereby creating a jelly roll. Or he might buy a bull verticalcall spread, and then try to buy the companion bear vertical put spread, thereby creatinga box. Or he might look for butterflies which he believes are mispriced, and try tocombine them later with other butterflies to create relatively low risk strategies whichact much like an arbitrage.
But even boxes and jelly rolls can present problems. Both carry the risk of earlyexercise. And a trader who executes a jelly roll still has to decide what to do with hisnear-term synthetic position as near-term expiration approaches. For example, supposea trader is long the March/June 410 jelly roll:
short March 410 call long June 410 calllong March 410 put short June 410 put
At March expiration the March synthetic will be settled in cash, and the trader will beleft with the June synthetic position. If he wants to hold the position and convert it intoa reverse conversion, he will have to sell the underlying basket or a suitable substitute.When he tries to do this he will encounter the same problems we have already
7. In the early 1980s the Chicago Mercantile Exchange did trade an S&P 100 futures contract. But thecontract was delisted due to insufficient activity.325
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discussed. This means that jelly rolls, involving two expiration months, can be mispricedin the same way a simple synthetic relationship can be mispriced within one expirationmonth. If the March 410 synthetic is 1.00 too cheap, while the June 410 synthetic is1.50 too cheap, the March/June 410 jelly roll will be 50 too cheap. But this is only intheory, since to take advantage of the mispricing, a trader would have to be able to carrythe underlying index from March to June expiration. But there is usually no trueunderlying available, only reasonable substitutes.
BIASES IN THE INDEX MARKET
Although there can be widespread mispricing of synthetic relationships in an indexoption market, especially where early exercise is a possibility and options are subject tocash settlement, we might assume that this mispricing is random. Sometimes syntheticmarkets will be too expensive and sometimes they will be too cheap compared to theirtheoretical values. In the real world, however, synthetic markets always seem to beunder pressure. That is, the synthetic market tends to trade at less than theoretical value.If the synthetic market (the difference between a call price and its companion put price)is worth 1.00, the actual market price usually seems to be something less than 1.00.What causes this systematic, downward bias?
Index markets, with their great liquidity, are the most popular of all option markets.They offer all types of traders—directional, volatility, arbitrage—a chance to makedecisions based on general market conditions rather than on the unique factors whichcan affect individual stocks. True or not, most traders believe that the general market isless subject to manipulation than individual stocks, so that index options markets offera more level playing field.
One especially active participant in index markets is the portfolio manager. Aportfolio manager is responsible for going into the market and generating a maximumreturn on capital with a minimum amount of risk. Historically, portfolio managers haveachieved this goal in the equity markets by maintaining a portfolio of stocks which themanager believed would outperform the general market. As the manager identified newstocks which met this criteria he added them to the portfolio, while at the same timeselling off other stocks in the portfolio which had either met his performance goals orceased to perform as expected.
A manager with an equity portfolio might naturally want to protect his holding bytaking advantage of the hedging opportunities in an option market. For example, hemight want to buy puts to protect the stocks in the portfolio, or sell calls to enhance thestocks’ performance. Until the introduction of index options he had to execute thehedging strategy for each individual stock within that stock’s option market. If he owned40 stocks and wanted to execute a uniform hedging strategy for the portfolio, themanager had to go into the individual option markets for the 40 different stocks andexecute each hedging strategy separately. This was not only time consuming, but thetransaction costs tended to reduce the expected benefits of the hedge.
With the introduction of index options, a manager with a broadly based portfoliorealized that his holdings often mimicked an index on which derivative instrumentsStock Index Futures and Options 327
were available. If the manager believed that the characteristics of his portfolio weresufficiently similar to the index, index options now offered a way of hedging the stocksin the portfolio without the time consuming and costly process of executing a hedgingstrategy in each individual stock.
However, the effect of portfolio hedging strategies on stock index derivatives haspeen one-sided. Whereas a professional trader will take either a long or short positionin the underlying instrument, as market conditions dictate, the vast majority of equityportfolio managers take only long positions in equities. Even if a manager believes astock will underperform the market, one rarely hears of a manager selling stock short(selling stock which he doesn’t own) as part of his investment program. Therefore, aportfolio manager is almost always trying to hedge a long position in the market. Thismeans that the manager is either buying protective puts, selling covered calls, orpursuing some combination of these strategies. The result is that there is constant sellingpressure on calls and constant buying pressure on puts.
This downward bias is also evident in stock index futures markets, where aportfolio manager can pursue a dynamic hedging strategy, such as portfolio insurance,by selling futures contracts to protect his equity holdings against adverse marketmovement. The effect is the same as in option markets—there is always downwardpressure on the markets.
If there were a sure way to profit from this downward bias in the market,arbitrageurs would take the opposite position in the underlying index. But we have seenthat executing an exact basket is not always possible. Moreover, when the portfoliomanager protects his long equity position by selling calls, buying puts, or selling futures,a market maker or arbitrageur ends up taking the opposite position. He is long calls,short puts, or long futures. If he wants to hedge his position with an underlying basketof stocks, he must sell stocks short. And selling stocks short is never as easy as buyingstocks. While not totally prohibited in the United States, as it is in many markets, traderswho want to sell short are subject to an uptick rule (see Chapter 5). And there is noguarantee that an uptick will occur in even one stock, let alone several hundred.®Moreover, most traders are subject to different borrowing and lending rates. Even if atrader can sell short all the stocks in the basket, the interest earnings from the shortStock sale may not be the same as the interest cost when stock is purchased.
Given all these factors, the stock index market is not a symmetrical one. ManyMore factors seem to result in downward pressure on synthetics and futures thanupward pressure. This does not mean that such markets can never become inflated,with calls trading for more than expected with respect to puts, or with futures contractstrading at greater than fair value. But this is by far the exception. In stock index marketsaround the world, there tends to be constant downward pressure on derivativeProducts.
 
8. The Chicago Stock Exchange has tried to facilitate the trading of index products by introducing theChicago Basket (CMX), a basket of 20 major U.S. stocks which can be traded at one time, and whichis not subject to the uptick rule. Hence, a trader who wants to be short all 20 stocks can sell the basketwithout worrying about finding an uptick in each individual stock. 
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Finally, there also tends to be a consistent bias in premium levels in index optionmarkets. This can be seen in Figure 15-4, a comparison of the 30-day historical volatilityof the OEX with the OEX implied volatility index from 1989 to 1992.9 Note that theimplied volatility is almost always greater than the historical volatility. If one believesthat the traditional theoretical pricing models are reasonably accurate, it is clear thatthose who purchase index options are consistently overpaying for them.
While a professional trader would not expect to be very successful if he consistentlypaid more for options than their theoretical values, such an approach is not necessarilyunreasonable for a hedger. If one thinks of options as insurance protection, there aredefinite benefits that go with owning options. In particular the owner of an option haslimited risk with unlimited profit potential. Regardless of theoretical value, these arebenefits for which hedgers are very often willing to pay an extra amount. To see why,consider a homeowner who buys insurance to protect his family. Since the insurancecompany is in business to make a profit, the homeowner knows that the premiums beingcharged are more than the policy’s theoretical value. Nonetheless, the homeowner iswilling to pay the extra premium for the privilege of knowing his family is protectedagainst calamity. In the same way, a portfolio manager who believes that his selectionof stocks will outperform the market may be willing to pay something more thantheoretical value to protect the value of the portfolio when the market moves againsthim. If he believes that his portfolio will consistently appreciate by more than the costof the options, he may be willing to overpay since the options will protect him, whileat the same time leaving him ahead in the long run.
 
9. The OEX volatility index (VIX), disseminated by the Chicago Board Options Exchange, represents theimplied volatility of a theoretical at-the-money OEX option with 30 days remaining to expiration.Se 16 &
Intermarket Spreading
While most spreading takes place within one underlying stock, commodity, or index{intramarket spreading), a trader need not limit himself to such a narrow playing field.If a trader can identify a relationship between contracts in two different markets, andfinds that the instruments are mispriced relative to each other, it is possible to spreadone instrument or its derivative products against another instrument or its derivativemarkets (intermarket spreading).
For example, suppose a trader follows two instruments, ABC and XYZ, and decidesthat ABC will be stronger than XYZ over the next several months. Based on thisassessment, the trader might create a spread by purchasing ABC and selling XYZ.
Let’s assume the trader purchases ABC at its current price of 50, and sells XYZ atits current price of 100. If ABC rises to 53 and XYZ falls to 98, the trader will show afive-point profit since he can buy back the spread, which he originally sold for 50, at 45.Note that it is not necessary for ABC to rise and XYZ to fall. If ABC rises to 60 and XYZrises to 105, the trader will still realize a five-point profit. Or if ABC falls to 47 and XYZfalls to 92, the trader will realize the same five-point profit. All that is necessary is thatin total points ABC is stronger, or less weak, than XYZ.
Suppose that after initiating the spread, the trader finds that ABC has risen to 60,while XYZ has risen to 115. The spread seems to have gone against the trader as it haswidened from the original price of 50 to 55. In point terms the trader was wrong sinceXYZ was five points stronger than ABC. But many traders would still contend that ABCwas stronger than XYZ since ABC rose 20% (10/50) while XYZ only rose 15% (15/100)over the same period. If a trader chooses to measure movement in percent terms, hecannot simply buy ABC and sell XYZ. In order to equalize the percent price changes, hisspread must be long and short an equal number of points. In order to accomplish this,for each 100-point XYZ contract the trader sells, he must purchase two 50-point ABCcontracts. Now he is long and short an equal number of points. If ABC rises to 60 (a20% gain) and XYZ rises to 115 (a 15% gain), he will show a profit of:
2 x (60-50) - 1 x (115 - 100) =5
Tf ABC falls 6% to 47 and XYZ falls 10% to 90, the trader will likewise realize a profitSince:
—2 x (50 - 47) + 1 x (100 - 90) = 4
Now in order to make a profit, ABC need only be stronger (or less weak) than XYZ inPercent terms.
331332 Chapter 16
In our example we tried to ensure that we were long and short the same numberof points on each side of our spread. However, the situation can be further complicated ifthe point values of the instruments are unequal. Suppose that each point in ABC is worth$400, while each point in XYZ is worth $100. Now if we buy two ABC contracts and sellone XYZ contract to equalize the point values, we find that the dollar value of our ABCposition is $400 x 2 x 50 = $40,000, while the dollar value of our XYZ position is $100x 1x 100 = $10,000. If ABC falls 6% to 47 and XYZ falls 10% to 90, the result will be:
[-2 x (50 - 47} x $400] + [1 x (100 - 90) x $100] = -$2,400
Even though ABC outperformed XYZ in percent terms, we still ended up losing becausethe dollar values of each point were different.
The trader’s ultimate goal is to equalize the dollar value of each side of his spread.If he wants to achieve this, he must ensure that:
(ABC contracts) x (price of ABC) x (point value of ABC) =(XYZ contracts) x (price of XYZ) x (point value of XYZ)
He must therefore sell two contracts of XYZ for each one contract of ABC which hebuys, since:
1 x 50 x $400 = 2 x 100 x $100
Putting things in slightly different form, we can express the proper ratio of XYZcontracts to ABC contracts required for a balanced spread as:
(price ABC / price XYZ) x (point value of ABC / point value of XYZ)= (50 / 100) x (40/10)= 2 (2:1 ratio)
Now suppose that instead of creating his spread using the underlying instrumentsABC and XYZ, the trader decides to create his spread using options on one or bothinstruments. That is, the trader wants to get long ABC and/or short XYZ by eitherbuying or selling an appropriate number of options. A logical starting point might be toconsider the delta positions of each spread. In traditional terms each underlying contracthas a delta of 100, and if we buy one ABC contract and sell one XYZ contract we seemto have a neutral spread. But we already know that the price and the point value of theinstruments are different. For purposes of intermarket spreading it might be useful toconsider the dollar delta ($delta) of each instrument, the total dollar value of thecontract divided by 100:
$delta = (contract price x point value) / 100
We can interpret the $delta as the dollar value change for a one-percent change inthe price of the underlying contract. This is slightly different than the traditionalinterpretation of the delta, the change in a contract’s value for a one-point change in theprice of the underlying contract. However, the $delta makes it easier to create andevaluate delta positions between instruments trading at different prices and withdifferent point values,Intermarket Spreading 333
Using the $delta calculations, for ABC we have a $delta of SO x $400 / 100 =$200. For XYZ we have 100 x $100 / 100 = $100. For purposes of delta calculationswe can see that each delta of ABC is worth two deltas of XYZ. A balanced spreadtherefore requires us to sell (buy) two deltas of XYZ for each one delta of ABC whichwe buy (sell). This is simply another way of expressing the proper ratio of underlyinginstruments required for a balanced spread.
We can also assign a $delta to an option by multiplying the $delta of the underlyingcontract by the option’s delta, keeping in mind that, even though deltas are oftenexpressed in whole numbers for convenience, they really represent a percent or decimalfraction:
option $delta = underlying $delta x option’s deltaAn option’s $delta represents the dollar change in its theoretical value for a one percentchange in the price of the underlying contract. An ABC call with a delta of 80 has a$delta of:$200 x .80 = $160An XYZ put with a delta of -30 has a $delta of:$100 x -.30 = -$30
Given the $delta relationship, if we want to get long ABC and short XYZ, and wewant to have the same number of dollars at risk on each side of the spread, we need tobe long and short an equal number of $deltas. In other words, in a $delta neutralposition:
number of ABC contracts x contract delta x $delta of ABCmust equalnumber of XYZ contracts x contract delta x $delta of XYZ
Suppose ABC and XYZ are futures contracts with the following options available:
 
ABC XYZExercise Call Put Exercise Call Putprice delta delta price delta delta45 80 20 | 95 70 _ 3050 50 -50 100 50 -5055 20 -80 105 30 -70
Some typical $delta neutral spreads might be:
long 10 ABC futures / short 40 XYZ 100 callslong 10 ABC 45 calls / short 16 XYZ futures
long 10 ABC 50 calls / long 20 XYZ 100 putsshort 30 ABC 45 puts / long 40 XYZ 95 putslong 35 ABC 55 calls / short 20 XYZ 95 calls334 Chapter 16
Each of these spreads is essentially balanced, because our $delta position in ABC optionsis exactly offset by an opposing $delta position in XYZ. Since we are always long ABCdeltas and short XYZ deltas, for small movements in the underlying contracts, eachspread will be profitable as long as ABC is stronger than XYZ.
Why would a trader consider taking a position in options rather than the underlyinginstrument? When a trader considers an option’s theoretical value, in a sense he iscomparing the option’s value to the value of the underlying instrument. That is why thecorrect use of an option’s theoretical value requires us to establish a hedge against theunderlying instrument. If options are underpriced compared to theoretical value, atrader will want to buy options rather than take an equivalent position in the underlyinginstrument; if options are overpriced compared to theoretical value, a trader will wantto sell options rather than take an equivalent position in the underlying instrument.From an option trader’s point of view, the determination of whether options areoverpriced or underpriced depends on one’s opinion of volatility. If implied volatility islow compared to a trader’s forecast, options are underpriced; if implied volatility is high,options are overpriced. An option trader will prefer to take a position in the optionmarket rather than in the underlying instrument if he believes that implied volatility istoo low or too high.
In our example, if a trader is in agreement with the implied volatility in both ABCand XYZ options, there is nothing to be gained by taking an option position rather thana position in the underlying instrument. Suppose, however, that implied volatility inABC options is 22% but the trader believes that 24% is a more reasonable volatility. Nowoptions are underpriced. If the trader wants to take a long position in ABC as part of hisspread, he will be better off doing so by purchasing ABC calls. He will not only have hisintermarket spread, but will, at the same time, be creating a position with a positivetheoretical edge. On the other hand, if the trader believes that 20% is a reasonablevolatility for ABC options, but implied volatility is 22%, the trader will be better offselling ABC puts. The sale of overpriced puts will create a long position in the ABCmarket while also creating a positive theoretical edge.
A trader can take the same approach in the XYZ market. If he wants to take a shortposition in XYZ as part of his intermarket spread, and believes implied volatility in XYZoptions is fairly priced, he can simply sell the underlying XYZ contract. Alternatively, ifhe believes implied volatility is too low, he can buy puts; if he believes implied volatilityis too high, he can sell calls.
What types of markets lend themselves to intermarket spreading? Since a traderwho enters into an intermarket spread makes the assumption that there is a relationshipbetween the markets involved, the important question for most traders is the closenessof that relationship. A trader is much more likely to construct an intermarket spread inmarkets which have similar characteristics, or where the markets are driven by the sameforces, than in those where no apparent similarity exists. While a trader might find itdifficult to identify a relationship between corn and Deutschemarks, he might very wellfind a relationship between corn and soybeans, or between Deutschemarks and SwissFrancs. In the same way, he might not find a relationship between gold and the stockof General Motors Corp., but he might find a relationship between gold and silver, orIntermarket Spreading 335
between