interest rate of 8% and a time to expiration of threemonths, since 3/12 x .08 = .02. As the amount of time to expiration changes, or asinterest rates change, the delta of the futures contract will also change.
We can now express in delta terms the amount by which a program trade isunbalanced. If a trader buys 50 futures contracts, each with a delta of 102, and sells anappropriate amount of the underlying stocks, his delta total position is
(50 x 102} - (50 x 100) = +100
He is long as if he had purchased stock outright corresponding to one index contract. Ifhe wants to be exactly delta neutral, he must sell a basket of stocks corresponding toone index contract.
A change in interest rates will not only affect the delta of the futures contract, butcan also affect the profitability of a program trade. If a trader institutes a buy program(buy the stocks, sell the futures), he is effectively borrowing cash in order to purchasethe stocks. If the cost of funds is tied to a floating interest rate, any increase in rates willhurt his position and any decrease will help. If he institutes a sell program (sell thestocks, buy the futures) he is effectively lending cash. Now any increase in rates willhelp his position and any decrease will hurt. If the change in interest rates is sufficientlylarge, an initially profitable program trade might become unprofitable. This can beespecially true if the program trade consists of very long-term futures contracts. In sucha case, the interest considerations are magnified because of the increased costs ofborrowing or lending over extended periods. In the same way, because of reducedinterest considerations, changes in interest rates are unlikely to affect program tradesconsisting of short-term futures.
We have thus far assumed that the dividend payout of all the stocks in an indexremains constant, but this is not necessarily true. Companies can have good years andbad years, and their dividend policies can change accordingly. In a buy program (buythe stocks, sell the futures), any increase in dividends will help the position, and anydecrease will hurt. In a sell program (sell the stocks, buy the futures), the effect is justthe opposite. In a broadly based index consisting of hundreds of stocks it is unlikely thata change in the dividend policy of any one company, or even several companies, willhave a significant impact on the profitability of a program trade. But in a narrowly basedindex consisting of only a few stocks, a change in the expected dividend payout of evenone firm can alter the potential profitability of the trade. In sucha case, the trader mustcarefully consider beforehand the possibility of a dividend change in the companiesmaking up the index.Stock Index Futures and Options 313
INDEX OPTIONS
There are really two types of stock index options, those where the underlying is a stockindex futures contract, and those where the underlying is the index itself. Although theyare alike in many respects, they also have unique characteristics which set them apartfrom each other. For this reason, we will look at each market separately.
Options on Stock Index Futures
Since an option on a stock index futures contract is often evaluated in the same way asa traditional futures option, a trader in such a market has all the usual problems whichgo with futures option evaluation. He must pick 4 suitable pricing model, determine thecorrect inputs into the model, choose appropriate strategies, and consider the risks thatgo with these strategies. Since exercise or assignment of a futures option contract willresult in the trader taking a long or short position in the stock index futures contract,the trader might logically use the current futures price as the underlying price input intohis theoretical pricing model. But a futures option trader is often faced with a newproblem. Should he also use the current price of the futures contract as his price inputif the futures contract is clearly mispriced and trading at other than fair value?
For example, if a stock index futures contract is trading at 424.00 but an optiontrader believes that the fair value is 425.00, which value should the trader feed into histheoretical pricing model? If the trader really believes that the futures contract ought tobe trading at 425.00, and also believes that all the assumptions upon which thetheoretical pricing model is based are valid, then he might very well choose to use425.00 as the underlying price in his model. He can then take option positions, whetheroutright or spread positions, by purchasing options which he believes are underpricedand selling those which he believes are overpriced.
Notice, however, that with the futures contract trading at 424.00, he cannot use425.00 as his price input and still use the futures contract to hedge his option position.If he were to do so, whatever positive theoretical edge he achieved because of his optiontrades would automatically be offset by the negative theoretical edge resulting from hisfutures trades. This becomes obvious when we recall the basic synthetic relationship forfutures options from Chapter 11:
call price — put price = futures price — exercise price
If the trader believes the underlying futures contract ought to be trading at 425.00, thenhe also believes that the 425 call and the 425 put ought to be trading at an identicalPrice. The difference between the presumed futures price and the exercise price is zero,so the difference between the call price and put price ought to also be zero. But withthe underlying futures market actually trading at 424.00, the call price will be approxi-mately 1.00 less than the put price. If the 425 call is trading for 8.50 and the 425 putis trading for 9.50, the call will appear too cheap to the trader and the put will appeartoo expensive. Unfortunately, if the trader buys the call for 8.50 and sells the put for9.50, he has no way to lock in his presumed profit of 1.00. If he attempts to completethe reverse conversion (buy the call, sell the put, sell the underlying), he will have to314 Chapter 15
do so by selling the futures contract at 424.00. What he makes on the options he willgive back on the futures contract.
If a trader simply wants to execute volatility or directional spreads based on apresumed underlying price of 425.00, then it might very well make sense to use thatprice as the input into the theoretical pricing model. But professional option traders relyheavily on arbitrage relationships for many of their trading strategies. If a price otherthan the actual underlying price is used, these arbitrage relationships will no longer bevalid. Because the choice of underlying price may vary from trader to trader, manyon-line option evaluation services give traders the choice of using either the actualfutures price or the theoretical futures price as the basis for theoretical evaluation.
Although the ultimate decision about the underlying price is the trader’s, in a stockindex futures option market a trader should be very careful about using an underlyingfutures price different from the quoted price. As we have already seen, the theoreticalvalue of a stock index futures contract depends on information which may not be readilyavailable to the trader. If he is wrong about the price at which the index is actuallytrading because the individual stock prices do not reflect the true market, his theoreticalevaluation of the futures contract will be incorrect. Most traders have learned throughexperience that the apparent mispricing of a stock index futures contract is oftenillusory, or at least smaller than it appears.
Options on a Cash Index
The exercise or assignment of a stock option requires the buyer or seller to take deliveryor make delivery of the underlying stock. In theory there is no reason why the exerciseor assignment of a stock index option could not be treated the same way. If a traderwere to exercise such an option, he would take delivery or make delivery of all thestocks in the index, in their correct proportion, at a price equivalent to the value of theindex. The trader who was assigned would take an opposing position at the same price.
While this sounds reasonable, in practice it presents the same problems that wouldarise if we were to settle a stock index futures contract through the physical delivery ofall the stocks. The delivery of many different stocks, in their appropriate amounts (whichmight include fractional shares), would overwhelm the clearing organization. As withfutures, the popular solution is to settle the contract in cash rather than through physicaldelivery. If a trader holds a 440 call and exercises it with the index at 450, he is creditedwith a cash amount equivalent to 10 index points. If each index point has a value of$100, his account is credited with $1,000. The account of the trader who is assignedon the same call is debited $1,000. In the same way, ifa trader exercises a 475 put withthe index at 450, his account will be credited with $2,500. The account of the traderwho is assigned will be debited $2,500.
In theory, if the index option is American, carrying with it the right of earlyexercise, the cash settlement should be based on the index value at the time the traderexercises his option. But this is impractical since exercise is not an instantaneoustransaction, An exercise notice must be filled out by the trader, submitted to the clearingfirm, and then submitted to the clearing house. Even if one could pinpoint the precisetime at which an option was exercised, it might be difficult to ascertain the exact valueStock Index Futures and Options 315
of the index. For all these reasons, the exercise or assignment value for an index optionis calculated from the index value at the end of the trading day.3 Whether an exercisenotice is submitted at 10:00 AM or 3:00 PM, both traders will be credited with cash equalto the difference between the exercise price and the index value at the close of tradingon that day.4
Evaluating a Cash Index Option
The most common method for evaluating cash options on a stock index is based on theassumption that the underlying contract for a cash index option is a basket of stockswhich, taken together, have the characteristics of one individual stock. Therefore, atrader might logically choose to evaluate options on an index using a traditional stockoption model, the Black-Scholes model for European options or the Cox-Ross-Ruben-stein model for American options. In either case, it will be necessary to feed into themodel the usual information: time to expiration, exercise price, underlying price,interest rates, volatility, and dividend payout. Unfortunately, the dividend payoutpresents a problem which we have already touched upon in the discussion of stockindex futures. If the options are European, with no possibility of early exercise, the totalamount of dividend payout expected to expiration is sufficient for theoretical evaluation.But if the options are American, where early exercise is a possibility, we need to knowthe exact date of the dividend payout. There is, however, no one exact date sincedividends on different stocks are paid at different times. In order to accurately evaluateAmerican options, we need to create a matrix of dividend payouts with the dates andamounts, and then feed this into our theoretical pricing model. Ifa trader wants an exactevaluation of a stock index option, he will need software which is capable of exactdividend calculations.
A less accurate method, but one which might be practical for many traders, is toassume a constant dividend payout. A trader can attempt to determine the averagepercent dividend paid by each stock, deduct this from the interest rate component, anduse the result as the interest rate input. For example, if the average dividend payout is4%, and interest rates are 6%, the trader can ignore the dividend input and simply usean interest rate of 2%. Some models allow the user to enter the interest rate and thedividend payout (as a percent) individually, and from these inputs the model calculatesthe value of an American option. In both cases, however, the value of early exercise willbe distorted since stock option calls are often exercised just prior to a dividend payout,and puts are often exercised just after a dividend payout. When no exact dividend datehas been entered into the model, optimum early exercise cannot be calculated. Thedistortion can sometimes be quite large if many stocks in the index happen to pay a large
 
3. Because of the order imbalances which can occur in individual stocks when a large number of stockindex arbitrages are liquidated at the same time, some exchanges have chosen to settle stock Indexoptions and futures on expiration day based on the opening index price. This opening price {fsdetermined by the opening prices of each stock in the index on that day. Since stocks can open atdifferent times, the index value may not represent simultaneous stock trades.
4. There is usually a cut-off time after which an index option may not be exercised on a given day. In theUnited States this is usually within an hour after the index has been fixed at the end of the trading day.316 Chapter 15
dividend on the same date, or pay no dividend for an extended period. Figure 15-1shows that this is often the case for the S&P 500.
A Phantom Variable
In some markets, stocks or their derivative products continue to trade after the closingindex price has been fixed for the day. In the United States, most stock indices are fixedat 4:00 PM Eastern Time when trading ceases on the New York Stock Exchange.However, many of the stocks in the index continue to trade on regional stockexchanges, and futures and options on the indices may also continue to trade for someperiod after the close. The fact that the underlying index may be fixed while optionsand futures on the index continue to trade results in an anomaly concerning earlyexercise.
Suppose a trader owns a stock which settles at a price of $50 when trading ceasesat 4:00 PM on the primary stock exchange. Suppose also that the trader has the right tosell that stock to someone at the settlement price of $50 until 12:00 midnight the sameday. If the trader’s intention is to hold onto the stock, their is no reason for him toexercise this right. But suppose some negative news on the stock comes out at 4:30 PM,and he finds that the stock is now trading on other exchanges at $48. What will he do?If he decided that he wanted to sell the stock because of the bad news, the trader wouldcertainly exercise his right to sell at $50 rather than selling it at $48 on some otherexchange. Even if he wanted to retain ownership of the stock, he would immediatelyrealize that he could show an extra $2 profit by exercising his right to sell at $50, andbuying the stock back at $48. Indeed, he would be foolish not to do so since if he didnot, he would be throwing away $2.
A similar situation occurs in stock index options where early exercise is a possibil-ity, and where alternative instruments are available on the index after the index hasbeen fixed at the close of the trading day. To see what can happen, consider an optionmarket on a very simple, non-dividend paying index which has been fixed at 400.00when the stock exchange ceases trading at 4:00 PM. If there are 30 days remaining toexpiration, with implied volatility at 14%, interest rates at 6%, and no dividendexpected, option prices and implied deltas might be the following:
Index = 400.00Time to June expiration = 30 days —_— Implied volatility = 14%Interest rates = 6% Dividend = 0
Exercise Call Call Put PutPrice Price Delta Price Delta380 22.44 92 59 -8390 13.98 78 2.12 -23400 7.42 56 5.62 —46410 3.24 32 11.71 -72
420 1.14 14 20.12 -94Stock Index Futures and Options 317
Suppose that at 4:05 PM some very negative economic news comes out. Stocks startto fall on other exchanges which are still open, and traders in the index futures andoption markets, which continue to trade after 4:00 PM, react as if the index will open10 points lower the next morning. If the option market does open the next day at390.00, option prices and deltas might then look like the following:
Index = 390.00Time to June expiration = 29 days —_ Implied volatility = 14%Interest rates=6% Dividend =0
Exercise Cali Call Put PutPrice Price Delta Price Deita380 13.70 79 1.95 -22390 7.10 55 5.40 -46400 2.98 31 11.55 ~-73410 98 13 20.08 —95420 22 4 30.00 -100
Suppose a trader originally owned ten 380 calls. What should he do when thenegative news hits the market? If he holds his calls, and the market does open at 390.00the next morning, his calls will be trading at 13.70, He will show a loss of 22.44 —13.70, or 8.74 per call. But if he exercises his calls today, he will receive 20.00 points,the difference between the exercise price of 380 and that day’s index settlement priceof 400.00. If he exercises, his loss will only be 22.44 — 20.00, or 2.44 per option. If hefails to exercise, he will automatically show an additional loss of 6.30.
Anytime after the index value has been fixed for the day, a trader who finds thatthe next day’s presumed theoretical value is less than parity will be better off exercisingan index option early rather than holding it. In this scenario the 390 calls will alsobecome early exercise candidates, although the savings will not be as great as with the380 calls. Prior to the news, the 390 calls might be trading for 13.98. But if the marketdrops to 390.00 the next morning, the 390 calls will only be trading for 7.10. If thetrader owns 390 calls and doesn’t exercise, he will lose 13.98 - 7.10, or 6.88 per call.{f he does exercise, he will lose only 13.98 — 10.00, or 3.98. He will save 2.90 per callthrough early exercise.
Of course, implied volatility might increase following such a violent drop in theMarket. But even if implied volatility jumps to 18%, the prices of the 380 and 390 callsthe next morning will only be 15.06 and 8.84, respectively. This is still less than parity,So a trader holding these calls would still want to exercise them early.
Although it does seem that stock index markets are more susceptible to bad newsthan good, puts might also become early exercise candidates if the market were to jumpfollowing unusually bullish news. If, in our example, there was a perception that theMarket would open the next morning 10 points higher, at 410.00, the option pricesMight look like this:318 Chapter 15
Index = 410.00Time to June expiration = 29 days Implied volatility = 14%Interest rates=6% Dividend =0
Exercise Call Call Put PutPrice Price Delta Price Delta380 31.92 98 12 ~2390 22.45 92 61 -8400 14.01 78 2.16 23410 7.46 35 5.67 -46420 3.28 32 11.75 -72
Now the 410 and 420 puts have become early exercise candidates. Based on theclosing index price of 400.00, their parity values are 10.00 and 20.00. But with theindex perceived to be at 410.00 their prices are only 5.67 and 11.75. Even if impliedvolatility were to rise to 18%, their prices would only be 7.49 and 13.37, respectively.This is still well below parity, so a trader would want to exercise the puts rather thanhold them.
In a market where early exercise is a possibility, options should never trade for lessthan parity. If a stock index option market remains open after the index has been fixedfor the day (as it does for 15 minutes in the United States), even if traders begin to tradethe options as if the market will open the next morning at a price significantly differentthan that day’s closing index price, all options must trade at a price of at least parity. Ifthis were not true, there would be an immediate arbitrage opportunity. If the 390 callwere really trading at 7.04 as a result of the negative news, a trader would buy as manycalls as possible and immediately submit exercise notices, in the process collecting 10points per option. He would realize an immediate profit of 2.96 per option.
What would happen if the marketplace were wrong about where the index wouldopen the next morning? Suppose that traders believe the market will open the nextmorning 10 points lower, but in fact the market opens unchanged, or even severalpoints higher. Wouldn’t a trader who exercised the 380 or 390 calls regret his actions?Regardless of what happens the following morning, a trader will always be better offexercising as long as he takes action to offset the effects of the exercise.
For example, suppose the trader initiated the following delta neutral ratio spreadbefore the negative news reached the market:
long 10 June 390 calls at 13.98 delta = 78short 25 June 410 calls at 3.24 delta = 32
If the trader holds the position, including his 390 calls, and the market opens the nextmorning at 390, he will show a loss of 12.75, since:
-10 x (13.98 — 7.10) +25 x (3.24 - .98) = -12.30
But suppose the trader decides to exercise his 390 calls. He still needs to balance hisdelta position by taking in the 780 deltas which disappeared when he exercised his calls.Stock Index Futures and Options 319
One way to do this is to simply close out the spread by buying back the 410 calls at theirnew price of .98. If he does this, he will show a profit of 16.85, since:
-10 x (13.98 - 10.00) +25 x (3.24 — .98) = +16.70
If the trader does not want to liquidate his short position in the 410 calls, he can lookfor another way of taking in 780 deltas. If other stock exchanges are still open and thetrader can purchase the stocks making up the index at the new, lower prices and in thecorrect proportion, he might consider buying stock equivalent to eight index contracts(800 deltas) to make up his loss of 780 deltas. Or, if a futures contract is available onthe index, and the futures market, in reaction to the negative news, is now trading 10points lower, he might buy eight futures contracts since each contract will haveapproximately 100 index deltas.
It may seem odd, but in fact it doesn’t matter whether the index opens the nextmorning at a higher price, lower price, or unchanged. What matters is that themarketplace believes that the market will change, and that all contracts are pricedaccordingly. In such a case, the trader must exercise those options which, given theperceived change in the underlying price, now have a value less than parity, and replacethem with other contracts which are not limited by parity constraints.
There is clearly an additional early exercise value associated with cash indexoptions, a value which does not result from the usual early exercise considerations ofdividends or interest. This value arises because under some circumstances a trader canexercise index options and replace them with other instruments at more favorableprices. How much is this replacement value worth? The answer depends on theprobability of some significant event occurring between the time the index is fixed andthe close of trading in other instruments linked to the index: stocks, futures, and options.Itis doubtful that anyone really knows how to quantify that probability, and traders tendto regard the possibility of a significant event occurring as a phantom variable. No oneknows how to evaluate it, but everyone agrees that it exists.
The extra early exercise value resulting from this phantom variable is most evidentin the trading of boxes in an index option market. Suppose we are interested in buyinga 420/430 box (buy the 420 call and 430 put, sell the 420 put and 430 call). If the boxis made up of European options where early exercise is not a possibility, the value of thebox is simply the 10-point difference between exercise prices less the carrying cost on10 points to expiration. If there are four weeks to expiration and interest rates are 6%,the value of the box is approximately:
10.00 — (28/365 x 6% x 10.00) = 9.95
If these are American options there will be some early exercise value attached to thembecause of dividend and interest considerations. The extra value will be greatest for the420 call and 430 put, so the value of the box will be slightly greater than 9.95. Using abinomial model, and given a normal dividend payout, we might find that a 10-point boxin the S&P 100 (OEX) market has a value of approximately 10.05. Yet such boxes in the
5. For simplicity, we ignore the likelihood that the trader who continues to hold a spread position willmake adjustments to his delta position as a result of the movement in the underlying Index.
 320 Chapter 15
S&P 100 market often trade for as much as 1014 to 1034. The additional amount overexpected theoretical value is the value the marketplace places on the possibility of earlyexercise after the index has been fixed at the end of the day.
If the right of early exercise can represent an additional reward to the holder of astock index option, then the possibility of early assignment must represent an additionalrisk to the seller of such an option. Indeed, the risk to the seller of an index option mayseem even greater than the reward to the buyer. The seller is not only at the mercy ofthe buyer in terms of when the option will be exercised; he may not even find out thathe has been assigned until it is too late to take protective action.
The problem for the seller is that index options are settied in cash, and cash doesnot carry a delta value. When a trader is assigned in a normal stock option market, theassignment may result in a loss to the trader in terms of dividends or interest costs, butat least it does not upset his delta position. If the stock option is deeply enoughin-the-money to be exercised, it probably has a delta very close to 100. When the optionis exercised, the 100 deltas associated with the option disappear, but they are immedi-ately replaced with the 100 deltas associated with the stock contract. But when a traderis assigned on an in-the-money stock index option, not only does assignment cause anegative cash flow, but the deltas associated with the assigned options completelydisappear. If the trader knows immediately that he has been assigned, he might be ableto do something to rebalance his delta position. Unfortunately, exercise notices areprocessed overnight, so the trader will not find out that he has been assigned until thenext trading day. If he is short deeply in-the-money index options and is assigned, thefollowing day he may find himself with a big positive or negative delta position, perhapsmuch bigger than he would like. If the market moves substantially on the opening, thetrader could find himself with severe losses before he ever has a chance to take anydefensive action.
Synthetic RelationshipsSuppose a stock is trading at 100. With two months to expiration, interest rates at 6%,and no dividend expected prior to expiration, what should be the relationship betweenthe 100 call and the 100 put? If we ignore the possibility of early exercise, we knowfrom the synthetic relationship in Chapter 11 that the call price should exceed the putprice by approximately the carrying costs on the exercise price of 100:
call price — put price = exercise price — stock price + carrying costsSince the carrying costs on the exercise price are100 x 6% x 2/12 = 1.00
we know that the call should be 1.00 more expensive than the put. If the put is tradingfor 4.00, the call should be trading for 5.00.
Suppose that the stock is trading for 100 and the put 4.00, but the call is actuallytrading for 4.25. What should a trader do? Regardless of the trader’s opinion aboutvolatility, it’s clear that the call is .75 too cheap with respect to the put. A trader familiarwith synthetic relationships will attempt to take advantage of this relative mispricing byStock Index Futures and Options 321
executing a reverse conversion: buy the call for 4.25, sell the put for 4.00, sell the stockfor 100. The result will be a profit at expiration of .75 when the trader closes out theposition by either exercising the put or being assigned on the call.
True arbitrage relationships are unlikely to be violated in an option market wherethere is a well defined and freely traded underlying contract. If such a relationship isviolated, a trader can always trade the underlying instrument against the mispricedarbitrage relationship and essentially lock in a profit equal to the mispricing. Thecumulative effect of all traders attempting to profit from a mispriced relationship is thatthe mispricing quickly disappears.
Now consider a trader who encounters a similar mispricing ina stock index optionmarket where the options are settled in cash. Suppose that the index is at 400 and that,according to the synthetic relationship, the 400 call should be 3.00 greater than the 400put. With the 400 put trading at 6.00, the trader would expect to see the 400 calltrading at 9.00. What should he do if he finds that the 400 call is actually trading at7.50?
The trader would like to take the same action in the index that he would take if hewere dealing with options in an individual stock: buy the call for 7.50, sell the put for6.00, and sell the index for 400.00. This ought to result in a profit of 1.50, the amountby which the call and put are mispriced relative to each other. Is this a practical strategyin the index option market?
Certainly the trader can buy the 400 call and sell the 400 put. But when the tradergoes to sell the index he may find that it is not as easy as he originally thought.Depending on how broadly based the index is, the trader may have to sell severalhundred stocks in various proportions. It might be possible to do so, but it will certainlynot be as easy as selling one stock.
Even if he is able to sell all the stocks in the index in the right proportion, has heexecuted a true reverse conversion? A true reverse conversion consists of being long acall and short a put, and short the underlying instrument, where the underlyinginstrument is the instrument to be delivered when an option is exercised. If the basketof stocks were the true underlying instrument, the trader would make delivery of, ortake delivery of, the basket if the 400 call or 400 put were exercised. But this is notwhat happens in an index market. If the trader exercises the call, or is assigned on theput, there is a cash transfer. But the trader is still left holding the basket of stocks.
The fact that the index is settled in cash can make a conversion or reverseconversion a much riskier strategy in an index option market than in an individual stockoption market, especially if the options are American. Moreover, arbitrage strategies aretypically done in large size because of their narrow profit margin. If