our position.
Can we do any better with the second choice, exercising the option? If we exercisethe option, we will pay the exercise price of 90 for the stock, and we will discard the10-point value of the option, effectively purchasing the stock at 100. When the stockgoes ex-dividend we will lose two points when it opens two points lower the next day,but we will also receive the dividend since we now own the stock. We will break evenbecause the two-point loss on the stock price will be offset by the two-point dividendwe receive. Clearly, we are better off exercising the option than holding it, not becausewe will show some additional profit, but because we will avoid a two-point loss. Wemust exercise the option to ensure that we break even.
What about the third choice, selling the option and buying stock ourselves? Thisseems to be very similar to early exercise. In both cases we are replacing the option withthe stock. If the option is trading at parity, in this case 10, there is no difference betweenexercising the option, or selling the option and buying the stock. In each case the cashflow will be identical and we will own the stock when it goes ex-dividend. But supposethe 90 call is trading at a price other than 10. Suppose, for example, it is trading at91, Clearly, this price is not possible in a market where the options are American andcan be exercised at any time. Purchasing a 90 call for 91% is equivalent to purchasingthe stock for 991. If the stock is trading at 100, this means the call is trading at lessthan parity. If this were indeed true, all traders would buy the call, sell the stock, andimmediately exercise the call. This would result in a riskless profit of 14 point, theamount by which the option was below parity. In a market where options are subjectto early exercise, no option ought to be trading at less than parity.
What if the 90 call is trading for more than parity, say 1014? Now if we sell theoption and purchase the stock we will still receive the dividend, since we will own thestock. But we will end up with an additional 14 point which we would not have
2. On the day a stock gives up its dividend, prices are typically disseminated with the dividend alreadydeducted from the stock’s price. Hence, a 100-point stock which gives up a 2-point dividend overnightand opens the next day at 98 would be deemed to have opened unchanged since the 2-point drop inits price results from the dividend payout, not from any change in investor demand,Early Exercise of American Options 245
collected through exercise of the call. Therefore, our third choice, selling the call andbuying the stock, is the optimum choice.
Since the only reason a trader would ever consider exercising astock option callearly is to receive the dividend, ifa stock pays no dividend there is no reason to exercisea call early. If the stock does pay a dividend, the only time a trader ought to considerearly exercise is the day before the stock goes ex-dividend. At no other time in its life isa stock option call an early exercise candidate.
We noted that in an American (exercise) market an option should never trade atjess than parity. Is the same true in a European (exercise) market? In our example ofthe 90 call and stock at 100, with an expected two-point dividend, the call price willclearly drop when the stock goes ex-dividend. If the option cannot be turned into stockthrough exercise prior to the ex-dividend date, the option is clearly going to lose value.If it has very little time value, the amount it will lose when the dividend is given up maymake the option worth less than parity. In our example, if the option expires in twoweeks with interest rates at 8% and a volatility of 20%, the European value of the 90call is approximately 81/4. This is 154 under parity, due to the fact that a call option, nomatter how deeply in-the-money, does not carry with it the right to the dividend. Onlyownership of the stock carries that right.
Early Exercise of Puts for Interest
As we did with a stock option call, we can express a stock option put’s value in termsof its components:
put value = intrinsic value — interest rate value + volatility value + dividend value
In the case of a put, the only component that affects its value negatively is the interestrate component. If the negative effects of interest rates are greater than the positiveeffects of volatility and dividends, it might be possible for a put, if it is European, to beworth less than parity. Consider this situation:
Stock price = 100, Time to expiration = 8 weeks, Volatility = 20%Interest rate = 8.00%, Dividend = 0
With these assumptions, the value of a 110 call is approximately .70. Using the put/callparity relationship, we can calculate the value of the 110 put:
put value = call value + exercise price — stock price — carrying costsThe carrying costs on the exercise price of 110 are:110 x 56/365 x 8% = 1.35The European put value is therefore:.70 + 110-100 - 1.35 = 9.35
Since the 110 put is only worth 9.35 if we hold it to expiration, but is worth 10 pointsif we exercise it today, the option is apparentiy worth more dead (exercised) than alive246 Chapter 12
(unexercised). When we exercise the 100 put we get to sell stock at 110, and we canearn interest on this 110-point credit to expiration.
Whereas a stock option call can only be an early exercise candidate on the day priorto the stock’s ex-dividend date, a stock option put can become an early exercisecandidate anytime the interest which can be earned through the sale of the stock at theexercise price is sufficiently large. Determining exactly when this happens is a difficultproblem, but if the stock pays a dividend it is most likely to occur on the day after thestock goes ex-dividend. Since a put is a substitute for a short stock position, one of theadvantages of holding a put is to avoid paying the dividend. Hence, a trader will almostalways want to hold the put through the ex-dividend date. Then, if the interestconsiderations are sufficient, the trader will exercise his put.
In an American (early exercise) market no put option can be trading at less thanparity. Otherwise there is an immediate arbitrage opportunity available by purchasingthe stock, purchasing the put, and immediately exercising the put. The profit will beequal to the amount by which the put is trading at less than parity. This is not necessarilytrue in a European (no early exercise} market, where the option must be carried toexpiration. In our example of the 110 put where the theoretical value derived from thecall price was 9.35 (.65 less than parity), the market in the European 110 put might be914-914. A market maker is willing to sell the option at 914 (14 less than parity) becausehe can hedge himself by selling stock, and he will be confident in the knowledge thathe will not be required to buy back the stock until expiration. The interest he earns onthe stock to expiration will more than offset the amount he will lose by selling the putbelow parity.
Conditions for Early Exercise
If given the chance, there are clearly reasons why a trader might want to exercise anAmerican option early. In the case of futures options, when the option is subject tostock-type settlement, the trader is trying to capture the interest on the option’s intrinsicvalue. In the case of stock option calls, he is trying to capture the dividend which thestock pays. In the case of stock option puts, he is trying to capture the interest on theproceeds from the sale of the stock at the exercise price. From the foregoing discussionwe can infer two conditions which are necessary before a trader considers exercising anoption early to capture this additional profit:
1. The option must be trading at parity.2. The option must have a delta close to 100.
If the option is trading at more than parity, the trader will always do better sellingthe option and taking a position in the underlying market himself. Determining whetheran option is trading at parity is simply a matter of finding out what the market in thatoption is. In the great majority of cases, if an option is deeply enough in-the- money tobe an early exercise candidate, the bid/ask spread is probably so wide that for practicalpurposes the option is trading at parity.Early Exercise af American Options 247
Why should the delta be close to 100? If we exercise an option early, we are tradingin the option for a position in the underlying contract. We therefore want to bereasonably certain that the option and the underlying contract have the same charac-teristics. In other words, we want to be certain that the option has no additionalinsurance value left, so that when we exercise the option we are not throwing awaythis insurance value.
For example, suppose there is an 80 call available with the underlying contracttrading at 100. If we want to take a long position we can purchase either the 80 call orthe underlying contract. If we feel there is no chance that the market can fall below 80prior to expiration, there is no difference between holding the 80 call or holding theunderlying contract. But suppose we feel that there is some chance that the marketcould fall below 80. In that case we will prefer to hold the 80 call. If the market wereto fall below 80, holding the call would limit our potential loss to the option’s premium.If, however, we hold a long underlying position and the market drops below 80, ourpotential loss is unlimited.
Recail that one interpretation of the delta is the probability that the option will finishin-the-money. A delta close to 100 means that the underlying market has almost nochance of going through the exercise price, and the option therefore has almost nochance of going out-of-the-money.
What degree of certainty justifies early exercise? With a delta of 100 one certainlywould consider early exercise. What about 99? 96? 90? With a delta above 95, manytraders will at least consider early exercise. How much interest will be earned, or whatdividend will be collected, through early exercise? With a delta of 95 or lower, the riskis probably too great to consider early exercise. There is at least a 5% chance that theunderlying market will go through the exercise price. If that happens, a trader will regrethaving exercised an option early, regardless of the additional benefits to be gainedthrough early exercise.
By ensuring that the delta is close to 100, we are also ensuring that we do not throwaway any insurance value that the option might still retain. As an option goes moredeeply into-the-money, its time value (insurance value) begins to disappear. With a deltaof 100, in theory the option has no time value left. However, as the delta declines, thetime value increases. If the delta were significantly less than 100, the option would stillretain a significant amount of time value. If we were to exercise such an option, wewould be throwing away this insurance value.
Since the desirability of early exercise is dependent, at least in part, on the option’sdelta, the accuracy of the delta is important. But there are many factors which affect thedelta, and we might incorrectly assess any one of these factors. If we feel that the correctvolatility is 15%, an option might have a delta of 98. But if we decide to raise ourvolatility to 17%, the delta of the option will decline. Under the new conditions its deltamight only be 93. At a 15% volatility a trader might consider early exercise, while at a17% volatility he might not. For this reason, options in low-volatility markets areexercised eatly much more often than options in high-volatility markets.248 Chapter 12
In the same way, an option with three months remaining to expiration might havea delta of 92, and would therefore not be an early exercise candidate. But if two monthspass, and the price of the underlying contract is unchanged, the delta will rise, and thenew delta might be 99. With three months remaining the option was not an earlyexercise candidate, but with one month remaining the option might be early exercisecandidate.
Of course, we ought to ask why we were considering early exercise in the firstplace. In many cases, we exercise to earn interest over the life of the option. We caneal more interest over three months than over one month, so from this point of viewa three-month option is more likely to be an early exercise candidate than a one-monthoption. We can see that many factors affect the desirability of early exercise, and thesefactors can pull in different directions, with some factors dictating for early exercise andsome dictating against.
Thus far we have only addressed the question of when a trader might find itdesirable to exercise an option early. There should be an additional value associatedwith an American option over an identical European option because the Americanoption carries with it additional rights. Even an out-of-the-money option, which no onewould consider exercising today, might someday go deeply enough into-the-money sothat it could become an early exercise candidate. This possibility should always makean American option more valuable than a European option.
How much more should a trader be willing to pay for an American option over anidentical European option? The Black-Scholes model makes no attempt to answer thisquestion since it is a European pricing model. In spite of the fact that Black-Scholesgenerated values are likely to show some inaccuracies in an American market, for manyyears traders continued to use the model because no model with equal ease of useexisted to evaluate American options. The problem of early exercise was dealt withthrough intuition, or through slight adjustments to Black-Scholes generated values. Forexample, when a stock is expected to pay a dividend, an American call value can beapproximated by comparing the Black-Scholes value of the call option under twocircumstances:
1. The call expires the day before the stock goes ex-dividend.
2. The call expires on its customary date, but the underlying stock price used toevaluate the call is the current price less the expected dividend.
Whichever value is greater is the pseudo-American call value.
In the case of options on futures, or of put options on stock, traders usedBlack-Scholes generated values, but raised any option with a theoretical value less thanparity to exactly parity.
Models eventually were developed to more accurately evaluate American options.The most widely used of these are the Cox-Ross-Rubenstein model, developed by JohnCox, Stephen Ross, and Mark Rubenstein; and the Whaley model, developed byGiovanni Barone-Adesi and Robert Whaley. Unlike the Black-Scholes model, neither ofEarly Exercise of American Options 249
these models is closed form. One can’t simply sit down and add up all the numbers toget the correct value. Rather, both models are algorithms, or loops, and each time theuser makes a pass through the loop, the closer he comes to the true American value ofthe option. While the Cox-Ross-Rubenstein model is quite easy to understand, bothintuitively and mathematically, it may require numerous passes through the loop togenerate an acceptable value. On the other hand, the Whaley model is more complexmathematically, but converges to an acceptable value much more quickly. The Cox-Ross-Rubenstein model may require 40 or 50 passes through the loop to reach the samedegree of accuracy as the Whaley model achieves with four or five passes through theloop. In addition to evaluating American options, both models also determine when anoption should be exercised early. We were somewhat vague on this point in our earlierdiscussion, stating that the delta of an option should be close to 100 to justify earlyexercise. When using a true American model, an option is optimally exercised earlywhen its theoretical value is exactly parity and its delta is exactly 100.
While the Cox-Ross-Rubenstein and Whaley models in most cases generate similarvalues, more and more computer services are using the Whaley model to evaluateAmerican options because of the speed with which it can achieve results, However, theCox-Ross-Rubenstein model, while slower, is still very widely used because it offerssome advantages which are not available in the Whaley model. For example, theWhaley model handles the dividend payout of a stock as if it were a continuous interestpayment over the life of the option. In reality dividends are paid all at once, and theCox-Ross-Rubenstein model more accurately reflects the impact of this one-time pay-ment on 4 call option on the stock. Unlike the Whaley model, the Cox-Ross-Rubensteinmodel can also be used to evaluate some of the recently introduced, exotic options. Suchoptions are path dependent since their value depends not only on the likely pricedistribution of the underlying contract at expiration, but also on the various paths whichthe underlying price might take to achieve that distribution. Both these models arediscussed further in Appendix B.
Regardless of the model a trader chooses, the accuracy of model-generated valueswill depend at least as much on the inputs into the model as the theoretical accuracy ofthe model itself. If a trader evaluates an American option using an incorrect volatility,or an incorrect interest rate, or an incorrect underlying price, the fact that he deriveshis values from an American rather than a European model is likely to make littledifference. Both models will generate incorrect values because the inputs are incorrect.The American model may produce less error, but that will be small consolation if theincorrect inputs lead to a large trading loss.
American option values are most important when there is a significant differencebetween the carrying cost associated with the option and the carrying cost associatedwith the underlying contract. The greater the difference between the two carrying costs,the greater the value of early exercise. When the underlying contract is a futurescontract and the option on the contract is subject to futures-type settlement, the cost ofcarrying either position is, in theory, zero. This is the same as assuming an effective250 Chapter 12
interest rate of zero. If we use a zero interest rate, there is no difference betweenEuropean and American option values.
When the options on a futures contract are subject to stock-type settlement, thereis a small difference between the carrying cost on the option and the carrying cost onthe underlying contract. Although che option is subject to stock-type settlement, theprice of the option is only a small fraction of the price of the underlying contract. Theadditional value for early exercise is therefore small, and is only likely to show up indeeply in-the-money options. Even in this case, the difference between European andAmerican values is often less than the minimum price increment. In such a market, atrader is unlikely to profit significantly because he is using an American pricing modelwhile other traders are using a European pricing model. Practical considerations, suchas the accuracy of the trader’s volatility estimate, his ability to anticipate directionaltrends in the underlying market, and his ability to control risk through effectivespreading strategies, will far outweigh any small advantage gained by using an Americanrather than a European model.
The importance of early exercise is greatest when the underlying contract is a stockor physical commodity.3 In such a case there is a significant difference between thecarrying cost on an option and the carrying cost on an underlying position. Thisdifference will especially affect the difference between European and American putvalues, since early exercise will allow the trader to earn interest on the proceeds fromthe sale at the exercise price. An option trader in either the stock or physical commoditymarket will find that the additional accuracy offered by an American model, such as theCox-Ross-Rubenstein or Whaley models, will indeed be worthwhile. The significance ofthis difference is shown in Figure 12-1.
THE EFFECT OF EARLY EXERCISEON TRADING STRATEGIES
The delta of an American option will always be greater than the delta of an equivalentEuropean option. The extent to which it is greater will depend on how deeplyin-the-money the option is, as well as the interest considerations resulting from earlyexercise. In most cases the delta of an American option will differ only slightly from aEuropean option, and the possibility of early exercise is therefore unlikely to have asignificant impact on volatility or directional strategies, In the former case, a trader mayhave to adjust the ratio of a strategy slightly if he wants to remain close to delta neutral.In the latter case, a trader may find that he is a few deltas longer or shorter than hewould be if employing the same strategy with European options.
Because early exercise of an option is not automatic, there are strategies whichdepend on someone making an error and not exercising an option early when he oughtto do so. For example, a stock option trader might try to execute a dividend play. Thisstrategy consists of buying stock and selling deeply in-the-money calls as the ex-dividend
3. Early exercise can also be an important consideration in the foreign exchange market if the interestrates associated with the domestic currency (the currency in which the option is settled) and foreigncurrency (the currency to be delivered in the event of exercise) are significantly different.Early Exercise of American Options 251
 
Figure 12-1European vs. American Put ValuesStock price = 100.00 Volatility = 25%; interest rate = 8.00%; Dividend = 0
4 weeks to expiration:
Black-Scholes Black-Scholes Cox-Ross Cox-RossValue Delta Value Delta80 put 00 0 .00 090 put 15 -5.0 15 -6.1100 put 2.46 45.1 2.51 ~46.4110 put 9.67 ~89,5 10.03 95.3120 put 19.28 -99.4 20.00 —100.0
13 weeks to expiration:
Black-Scholes Black-Scholes Cox-Ross Cox-RossValue Delta Value Delta80 put At 2.2 aa 2.390 put 97 -14.3 1.00 -14.9100 put 4.00 ~41.2 4.20 44.0110 put 10.00 -70.6 10.71 78.2120 put 18.25 -89.2 20.00 —100,0
26 weeks to expiration: 4
Black-Scholes Black-Scholes Cox-Ross Cox-RossValue Delta Value Delta80 put 46 -67 50 6A90 put 1.89 -18.1 2.02 ~19.6100 put 5.11 37,7 5.55 ~42.1110 put 10.43 58.9 11.58 ~68.8120 put 17.56 -76.4 20.04 -94.8
 
date for the stock approaches. If the trader is not assigned on the calls, he will breakeven on the stock (the stock price will fall, but he will collect the dividend). At the sametime he will profit when the deeply in-the money calls which he has sold fall by theamount of the dividend. Of course, if he is assigned on the calls, as he ought to be, hewill only break even. But for each call which goes unexercised the trader will profit bythe amount of the dividend. Dividend plays were much more common in the early daysof option trading when the market was less sophisticated and many options whichshould have been exercised were not. As markets have become more efficient, only aprofessional trader, with very low transaction costs, can afford to take advantage of sucha possibility. Even then, he may find that he is assigned on the great majority of the callshe has sold.252 Chapter 12
A trader can execute a similar type of interest play by selling stock and simultane-ously selling deeply in-the-money puts. Now, instead of profiting by the amount of thedividend, the trader will profit by the amount of the interest he can earn on the exerciseprice (the proceeds of the stock sale and the put sale combined). This profit will continueto accrue as long as the puts remain unexercised. If the puts are exercised, the traderdoes no worse than break even. Again, only a professional trader, with his lowtransaction costs, is likely to employ such a strategy.
If options are subject to stock-type settlement, an interest play can also be done inthe futures option market by either purchasing a futures contract and simultaneouslyselling a deeply in-the money call, or selling a futures contract and simultaneously sellinga deeply in-the-money put. If the option is deeply enough in-the-money, it ought to beexercised early. If, however, the option remains unexercised, the trader will continueto earn interest on the proceeds from the option sale. Since the amount on which thetrader will earn interest is the difference between the exercise price and futures price,this will not be as profitable as a similar strategy in the stock option market, where thetrader will earn interest on the exercise price. Still, if the transaction costs are lowenough, it may be worth doing.
A variation on dividend and interest plays can also be executed using deeplyin-the-money vertical spreads. For example, suppose a stock is trading at 100 with theex-dividend date approaching If both the 80 and 85 calls are deeply enough in-the-money to be early exercise candidates, a trader might try to purchase the 80/85 callvertical spread for five points. If he does so, he will exercise his 80 call in order to collectthe dividend, and at the same time hope not to be assigned on the 85 call. Oddly, if boththe 80 and 85 calls are early exercise candidates, the trader should also be willing tosell the 80/85 call vertical for five points. If he does so, he will exercise his 85 call inorder to collect the dividend, and at the same time hope not to be assigned on the 80call. In other words, a professional trader might logically make a market under thesecircumstances of 5 bid/5 ask. He is willing to buy or sell the 80/85 call vertical at fivepoints. He intends to immediately exercise whichever call he buys, while hoping not beassigned on the call which he sells.
The possibility of early exercise can also have an impact on arbitrage strategies. Forexample, suppose a stock option trader executes a reverse conversion:
buy a callsell a putsell stock
If he executes this strategy at what he believes to be profitable prices, part of his profitwill come from the interest he expects to earn on the sale of the stock. But what willhappen if the stock price begins to fall, and in fact falls so far that the trader is assignedon the put? This will eliminate the interest earnings, since he will be required to buyback the stock. Of course, he can still sell the call and take in some cash. But if the valueof the call is insufficient to offset the interest loss, the trader may find that his profitablereversal has in fact become unprofitable.Early Exercise af American Options 253
The trader in our example is worried about being assigned on his put. If the marketdrops there will be a greater likelihood of assignment; if the market rises there will bea lesser likelihood of assignment. Since the trader prefers the market to rise, he must bedelta long. This is confirmed by the following delta values:
Stock price = 100; Time to expiration = 3 months; Volatility = 25%Interest rate = 8.00%; Dividend = 0
European European American AmericanOption Value Delta Value Deita100 call 5.97 58.8 5.97 58.8100 put 4.00 41.2 4.20 ~44.0
If these are European options, the total delta of the reversal is:+58.8 +41.2 -100 =0If, however, the options are American, the total delta is:+58.8 +44.0 ~100 = +2.8
The positive delta of 2.8 reflects a slight preference for the market to rise so that thetrader will avoid being assigned on the put.
For a similar reason, if the trader executes a conversion (sell call, buy put, buystock), he is 2.8 deltas short. He wants the stock to fall so that he can exercise his putearly and thereby avoid the interest costs of carrying a long stock position.
Because the desirability of early exercise, and therefore the likelihood of earlyexercise, can increase or decrease as the underlying market rises or falls, conversionsand reversals using American options are not delta neutral. While these strategies maybe unbalanced by only two or three deltas, the fact that they are often done in large sizecan result in an additional risk that the trader cannot afford to ignore. This also appliesto boxes and jelly rolls, which are simply combinations of conversions and reversals.Consider these option values:
Stock price = 100; Volatility = 25%; Interest rate = 8.00%; Dividend = 0Time to expiration: March = 3 months, June = 6 months
European European American AmericanOption Value Deita Value DeltaMarch 95 call 9.00 73.7 9.00 73.7March 95 put 2.13 -26.3 2.24 27.8March 100 call 5.97 58.8 5.97 58.8March 100 put 4.00 -41.2 4.20 —44.0June 95 call 11.96 72.7 11.96 72.7June 95 put 3.24 27.3 3.49 -29.9June 100 call 9.03 62.3 9.03 62.3
June 100 put 5.11 -37.7 5.55 -42.1254 Chapter 12
We can see how the possibility of early exercise affects the values and deltas of variousarbitrages:
European European American American
Strategy Value Delta Value DeltaMarch 95/100 box 4.90 0 4.99 -1.3June 95/100 box 4.80 0 4.99 -1.8March/June 95 jelly roll 1.85 0 1.71 +14March/June 100 jelly roll 1.95 0 L71 +1.6
Boxes become more valuable using American options because a trader who is longthe box owns a put with a higher exercise price. If a trader buys the 95/100 box, he islonga 95 calland a 100 put, and short a 95 put and 100 call. The 100 put will becomean early exercise candidate before the 95 put, making the American box more valuablethan the same European box. The delta of the box is negative with American optionsbecause a trader who owns the box would like the market to decline so that he will beable to exercise the 100 put as quickly as possible.
Jelly rolls become less valuable using American options because a trader who ownsthe jelly roll has sold a long-term put. If a trader buys the March/June 00 jelly roll, heis longa March 100 put and June 100 call, and is short a March 100 call and June 100put. The June 100 put, being a longer term option, has more potential for early exercisethan the March 100 put. The delta of the jelly roll is therefore positive because there isa greater danger of being assigned on the June 100 put. Hence, the trader who ownsthe jelly roll would like the market to rise so that he is less likely to be assigned on theJune 100 put.
A unique, early exercise situation affecting the value of boxes can also arise in astock option market if there is a tender offer to buy a portion of the stock in a company.For example, with a stock trading at 100, one would expect to see the 100/105 boxtrading at close to five points. But suppose that a tender offer is made to buy half theoutstanding shares at a price of