futures contract, or by trading spreads between futures months ifone of the months is not locked.88 Chapter 5
Transaction costs, on the other hand, can bea very real consideration. If such costsare high, the hedge in Figure 5-1 might not be a viable strategy; all the profits could beeaten up by brokerage fees. In theory, the desirability of a strategy will depend not onlyon the trader’s initial transaction costs, but also on the subsequent costs of makingadjustments. The adjustment cost is a function of a trader’s desire to remain deltaneutral. A trader who wants to remain delta neutral at every moment will have to adjustmore often, and more adjustments mean more transaction costs.
Suppose a trader initiates a hedge but adjusts less frequently, or does not adjust atall. How will this affect the outcome? Since theoretical evaluation of options is basedon the laws of probability, a trader who initiates a theoretically profitable hedge still hasthe odds on his side. Although he may lose on any one individual hedge, if given achance to initiate the same hedge repeatedly at a positive theoretical edge, on averagehe should profit by the amount predicted by the theoretical pricing model, assuming, ofcourse, that his inputs are correct. The adjustment process is simply a way of smoothingout the winning and losing hedges by forcing the trader to make more bets, always atthe same favorable odds. A trader’s disinclination to adjust simply means that there isgreater risk of not realizing a profit on any one hedge. Adjustments do not in themselvesalter the expected return; they simply reduce the possibility of short-term bad luck.
Based on the foregoing discussion, a retail customer and a professional trader arelikely to approach option trading in a slightly different manner, even though bothunderstand and use the values generated by a theoretical pricing model. A professionaltrader, particularly if he is an exchange member, has relatively low transaction costs.Since adjustments cost him practically nothing in relation to the expected theoreticalprofit from a hedge, he is willing to make frequent adjustments. However, a retailcustomer who establishes the same hedge will probably not adjust, or will adjust lessfrequently, because any adjustments are likely to significantly reduce the profitability ofa hedge. But if he understands the laws of probability, the retail! customer will realizethat his position has the same favorable odds as the professional trader’s position. At thesame time he should realize that his position is more sensitive to short-term bad luck.Even though the retail customer may occasionally experience larger losses than theprofessional trader, he will also occasionally experience larger profits. In the long run,on average, both should end up with approximately the same profit.4
Taxes may also be a factor in evaluating an option strategy. When positions areinitiated, when they are liquidated, how the positions overlap, and the relationshipbetween different instruments (options, stock, futures, physical commodities, etc.} mayhave different tax consequences. Such consequences can often have a significant impacton the value of a diversified portfolio, and for this reason portfolio managers must besensitive to the tax ramifications of a strategy. Since each trader has unique taxconsiderations, and this book is intended as a general guide to option evaluation andstrategies, we will simply assume that each trader wishes to maximize his theoreticalpre-tax profits and that he will worry about taxes afterward.
4. This of course ignores the very real advantage the professional trader often has from being able to buyat the bid price and sell at the ask price. A retail customer can never hope to match the profit resultingfrom this advantage, nor should he try to do so.Using an Option’s Theoretical Value 89
Returning to our example in Figure 5-1, note that after the hedge was initiated nosubsequent trades were made in the option market. The trader’s only concern was thevolatility, or price fluctuations, in the underlying market. These price fluctuationsdetermined the size and the frequency of the adjustments, and in the final analysis itwas the adjustments which determined the profitability of the hedge. We might thinkof the hedge as a race between the loss in time value of the June 100 calls and the cashflow resulting from the adjustments, with the theoretical pricing model acting as thejudge. The model says that if options are purchased at less than theoretical value, theadjustments will win the race; if options are purchased at more than theoretical value,the loss in time value will win the race. The conditions of the race are determined bythe inputs into the theoretical pricing model.
While we assumed in our example that the future volatility was known to be18.3%, we might ask what the outcome would have been had volatility been other than18.3%? Suppose, for example, it turned out to be higher than 18.3%. Higher volatilitymeans greater price fluctuations, resulting in more and larger adjustments. In ourexample, more adjustments mean more profit. This is consistent with the principle thatoptions increase in value as volatility increases.
What about the reverse, if volatility had been less than 18.3%? Lower volatilitymeans smaller price fluctuations, resulting in fewer and smaller adjustments. This wouldhave reduced the profit. If the volatility were low enough, the adjustment profit wouldjust be enough to offset the other components, so that the total profit from the hedgewould be exactly zero. This break-even volatility is identical to the option’s impliedvolatility at the trade price. Using the Black model, we find that the implied volatility ofthe June 100 call at a price of 3.25 is 14.6%. At that volatility the race between profitsfrom the adjustments and the loss in the option’s time value will end in an exact tie.Above a volatility of 14.6% we expect the hedge, including adjustments, to show aprofit; below 14.6% we expect the hedge to show a loss,
Since we needed to make adjustments in order to realize a profit, it may seem thatevery profitable hedge requires us to maintain the position until expiration. In practice,however, this may not be necessary. Suppose, for example, that immediately after weestablished the hedge the implied volatility in the option market began to increase.Suppose that it increased from 14.6%, the implied volatility when we bought the June100 calls, to 18.3%, the future volatility over the life of the option. What would happento the price of the June 100 cail? Its price would rise from 3.25 (an implied volatility of14.6%) to 3.88 (an implied volatility of 18.3%}. We could then sell our calls for animmediate profit of .63 per option. Of course, if we wanted to close out the hedge wewould also have to buy back the 57 June futures contracts which we originally sold.What effect would the change in implied volatility have on the price of the futurescontracts? Implied volatility is a characteristic associated with options, not with under-lying contracts. For this reason we would expect to see the underlying futures contractcontinue to trade at its original price of 101.35. By purchasing our 57 outstandingfutures contracts at 101.35 we would realize an immediate total profit from the hedgeof 63.00, exactly the amount predicted by the theoretical pricing model. If we could doall this, there would be no reason to hold the position for the full ten weeks.90 Chapter 5
How likely is an immediate reevaluation in implied volatility from 14.6% to 18.3%?While violent changes in implied volatility are possible, they are the exception ratherthan the rule. Changes usually occur gradually over a period of time, and are the resultof equally gradual changes in the volatility of the underlying contract. As the volatilityof the underlying contract changes, option demand either rises or falls, and this demandis reflected in a corresponding rise or fall in the implied volatility. In our example, ifMarket participants realized that the price of the underlying futures contract werefluctuating at a volatility greater than 14.6%, we would expect implied volatility to beginto rise. If implied volatility ever reached our target of 18.3%, we could sell out our callsand buy in our futures, thereby realizing our expected profit of 63.00 without havingto hold the position for the full ten weeks. However, option prices are subject to a widevariety of market forces, not all of them theoretical. There is no guarantee that impliedvolatility will ever reevaluate up to 18.3%. In that case, we will have to hold the positionfor the full ten weeks and continue to adjust in order to realize our profit.
Every trader hopes for quick reevaluation of implied volatility towards his volatilityforecast. It not only enables him to realize his profits more quickly, but it eliminates therisk of holding a position for an extended period of time. The longer a position is held,the greater the possibility of error from the inputs into the model.
Not only might implied volatility not reevaluate favorably, it might actually moveagainst us, even if the true volatility of the underlying contract moves in our favor.Suppose that after initiating our hedge, implied volatility immediately falls to 13.5%from 14.6%. The price of the June 100 call will fall from 3.25 to 3.06. Now we have apaper loss of 100 x —.19 =--19.00. Does this mean we made a bad trade and shouldclose out the position? Not necessarily. If the volatility forecast of 18.3% turns out to becorrect, the options will still be worth 3.88 by expiration. If we hold the position andadjust, we can eventually expect a profit of 63.00. Realizing this, we ought to maintainthe position as we originally intended. Even though an adverse move in impliedvolatility is unpleasant, it is something with which all traders must learn to cope. Justas a spectilator can rarely hope to pick the exact bottom or top at which to take a longor short position, an option trader can rarely hope to pick the exact bottom or top inimplied volatility. He must try to establish positions when market conditions arefavorable. But he must also realize that conditions might become even more favorable.if they do, his initial trade may show a temporary loss. This is something a trader learnsto accept as a practical aspect of trading.
Let’s look at a somewhat more complex hedge, this time in the form of mispricedstock options. Suppose current market conditions are as follows:
Stock price = 484
Interest rate = 8.00%
Time to March expiration = 10 weeksExpected dividend = 50¢ in 40 days
Note that we now have an additional input, the expected dividend. Of course, westill need the volatility over the life of the option. Again looking into our crystal ball, weUsing an Option’s Theoretical Value 91
see that the volatility over the next 10 weeks will be 32.4%. Again, we decide to lookat a call which is close to at-the-money, specifically the March 50 call. Feeding all of ourinputs into the Black-Scholes model, we find that the March 50 call has a theoreticalvalue of 2.17 and a delta of 46.
Having determined the theoretical value of the call, we still need its price in orderto see if there is any profit opportunity available. In this case, it turns out that the call istrading for 3 (an implied volatility of 42.2%). Since the option is overpriced, we wantto sell it and establish a delta neutral hedge against the underlying contract. We might,for example, sell 100 March 50 calls and simultaneously buy 46 stock contracts.Assuming this option is trading on a U.S. stock options exchange where each underlyingcontract consists of 100 shares of stock, it will be necessary to buy 4600 shares of stock.
Asin allt hedges based on a theoretically mispriced option, it is necessary to maintaina delta neutral position throughout the life of the option. As before, we will make ouradjustments at weekly intervals, but now the underlying contract is shares of stock.Figure 5-2 shows the adjustment process for this hedge. Let’s go through the hedge stepby step and see what the final results are.
Original Hedge: At expiration, with the stock at 523, the 50 calls were worth 234.So on the option we showed a profit of $300 — $237.50, or $62.50. The 4600 sharesof stock we purchased at 48% we were able to sell at 525, for a profit of 37% ($3.875}per share. The total hedge profit was therefore +(100 x $62.50} +(4600 x $3.875) =+$24,075.
Adjustments: The adjustment process forced us to buy and sell stock to remain deltaneutral. At week one we bought 600 shares of stock at 495; at week two we bought1400 shares at 521; at week three we sold 200 shares at 5134; and so on. At weekten we sold out our remaining 900 shares of stock at 524. The result of all these
Figure 5-2Stock Deltaof Total Delta Adjustment Total Share Interest onWeek Price 50call Position inShares Adjustment Cash Flow Cash Flow0 48% 0 0 0 0 0 01 49% 52 -600 buy 600 +600 -23,775 411.142 52% 66 -1400 buy 1,400 +2000 -72,975 —-895.693 51% 64 +200 sell 200 +1800 +10,350 4111.164 50 52 +1200 sell 1,200 +600 +60,000 +552.335 47 28 +2400 sell 2,400 -1800 +112,800 +865.32ex-dividend 50¢6 48% 38 —1000 buy 1,000 —800 ~48,125 —295.347 52 73 -3500 buy 3,500 +2700 182,000 -837.708 52% 78 500 buy 500 +3200 -26,125 ~80.169 50% 55 +2300 sell 2,300 +900 +115,288 +176.88
10 52% sell 900 +47,13892 Chapter S
adjustments was a loss of $13,425, (Again, the reader may wish to confirm this byadding up all the numbers in the “cash flow” column.}
Carrying Costs on the Initial Position: Originally we sold 100 options at $300 each,and bought 4600 shares of stock at $48.50 each. This resulted in a total debit of (100x $300) — (4600 x $48.50} = $193,100. The carrying cost on this debit for ten weeksat an annual rate of 8.00% (our interest rate assumption} was $193,100 x .08 x 70/365= $2,962.63. Note that this is somewhat different than the futures example, where onlythe option contracts entailed a carrying cost. Unlike futures trades, the purchase or saleof stock results in an immediate cash flow, and this cash flow becomes part of thecarrying cost on the initial hedge.
Interest on the Adjustments: Whenever we bought or sold stock to make anadjustment, the trade resulted in a cash flow, either a debit or a credit. We were ableto earn interest on any credit, and we were required fo pay interest on any debit, atarate of 8.00 percent. For example, at week one we bought 600 shares of stock at495, for a total cash outlay of $29,775. The cost of carrying this debit to expiration(nine weeks hence} was -$29,775 x 63/365 x .08 = -$411.14. The total interest costfrom our adjustments is the sum of all the individual interest calculations, or -$81 4.34.
Dividends: We also assumed that 30 days prior to expiration (between week five andsix) the stock paid a dividend of 50¢. What was the dividend earnings or loss to ourposition? We initially bought 4,600 shares of stock. At the end of week five we wereshort 1800 shares of stock as part of the adjustment process. This means that on theex-dividend date we were long a total of 2800 shares of stock. At a dividend of 50¢ pershare, we received a dividend payout $1,400.
Interest on Dividends: We were also able to earn interest on the $1,400 dividendpayout for the remaining 30 days to expiration at a rate of 8.00%. The proceeds fromthis were $1,400 x 30/365 x .08 = $9.21.
Summarizing the results, we have:
Original Interest on Interest on Interest onHedge Adjustments Hedge Adjustments Dividends Dividends+$24,075 -$13,425 -$2,962.63  -$814.34 +$1,400 +$9.21
The total profit on the hedge is therefore:+$24,075 -$13,425 -$2,962.63 -$814.34 +$1,400 +$9.21 = +$8,282.24
versus a theoretical profit of 100 x ($300 - $217) = 100 x $83 = +$8,300.
As in our last example, our profit depended on our knowing the volatility of theunderlying contract over the life of the option. This “known” volatility of 32.4%represents the actual volatility associated with the ten stock price changes in Figure 5-2.
While we continue to assume that markets are frictionless, we noted in our futuresoption example that this is not necessarily true. In a locked futures market it is notalways possible to freely buy and sell the underlying contract. A similar type ofUsing an Option’s Theoretical Value 93
restriction can occur in stock option trading. In our example we established the initialhedge by selling calls and purchasing stock. If, however, the calls had been underpriced,we might have chosen to buy calls and sell stock, stock which we might not necessarilyown. This type of short sale, the sale of stock which is borrowed rather than owned, isprohibited in some markets. This makes it difficult to hedge certain types of optionpositions. As we shall see later, if we choose to sell puts it is necessary to hedge theposition by selling stock. If we are not able to sell stock to hedge the sale of puts, wemight hesitate to sell puts, even at a price greater than the theoretical value. Indeed, inmarkets where short sales are prohibited, puts tend to trade at inflated prices comparedto calls,
The short sale of stock is not totally prohibited in U.S. markets, but it is subject toan up-tick rule. This rule specifies that a short sale is always prohibited at a price lowerthan the price at which the previous trade took place (a down-tick). A short sale isalways permitted at a price higher than the price at which the previous trade took place(an up-tick). Finally, a short sale may take place at the same price at which the previoustrade took place if the previous trade took place on an up-tick (also an up-tick). Theup-tick rule was instituted following the market crash of 1929, with the intent ofpreventing a similar crash by prohibiting the sale, at continuously lower prices, of stocknot actually owned. Below are ten consecutive trade prices (reading from left to right]for a stock with the accompanying ticks (a positive sign for an up-tick; a negative signfor a down-tick}.
4812 +48%% +4854 —4814 -48%% -481/%4 4814 +48% +4826 +4852
Not only might a short sale not be possible because of the up-tick rule, but manybrokerage firms which execute short sales of stock for their customers do not pay fullinterest on the proceeds from a short sale. This can further distort the interestcomponent used in a theoretical pricing model.
Taking one last look at our examples, what enabled us to make a profit approxi-mately equal to that predicted by the theoretical pricing model? A simple way ofinterpreting the results is to realize that, according to the model, the option’s cash flowcan be replicated through the adjustment process. If we know the conditions of anoption contract and the characteristics of the underlying contract, we can replicate thecharacteristics of the option, and therefore replicate the cash flow resulting from aposition in the option, through an adjustment process in the underlying contract. In ourexamples, because we knew the exact market conditions which would prevail over thelife of the options (primarily, this means knowing volatility), we knew we could replicatethe option by continuously calculating the delta and taking an appropriate offsettingposition in the underlying contract. According to the model, at expiration the total cashflow from this dynamic hedge should exactiy equal the value of the option. But in ourexamples we either bought the option at less than theoretical value (our futures optionexample} or sold the option at more than theoretical value (our stock option example).Since the cash flow from the adjustment process exactly replicated the option’stheoretical value, we were left with a profit equal to the difference between the option’sprice and its theoretical value.94 Chapter 5
This type of option replication, using the cash flow from continuous hedging in theunderlying contract, is the basis for many strategies which make use of option charac-teristics but do not actually involve options. We will discuss the best known of thesestrategies, portfolio insurance, in Chapter 13.6 &
Option Values and ChangingMarket Conditions
Every trader who enters the marketplace must balance two opposing considerations,reward and risk. A trader hopes that his analysis of market conditions is correct, andthat this will lead to profitable trading strategies. But no intelligent trader can afford toignore the possibility of error. If he is wrong, and market conditions change in a waythat adversely affects his position, how badly might the trader get hurt? A trader whofails to consider the risks associated with his position is certain to have a short andunhappy career.
Consider the trader who goes into the marketplace and purchases a futurescontract. What is he worried about? Obviously he is worried that the market will fall.Indeed, anyone who takes a position in an underlying contract has only one realconcern, that the market will move in the wrong direction. If a trader has a longposition, he is at risk from a decline in the market; if he has a short position, he is at riskfrom a rise in the market.
Unfortunately, the risks with which an option trader must deal are not one-dimen-sional. A wide variety of forces can affect an option’s value. If a trader uses a theoreticalpricing model to evaluate options, any of the inputs into the model can represent a risksince there is always a chance that the inputs have been incorrectly estimated. Even ifthe trader has correctly estimated current market conditions, it’s possible that over timeconditions will change in a way that will adversely affect the value of his option position.Because of the many forces affecting an option’s value, prices can change in ways whichMay surprise even the most experienced trader. Since decisions must often be madequickly, and sometimes without the aid of a computer, much of an option trader’seducation consists of learning how changing market conditions are likely to change anoption’s value and the risks associated with an option position.
First, let’s consider the general effects of changing market conditions on an option’svalue. These changes are summarized in Figure 6-1.
As the price of the underlying rises or falls, options are more or less likely to finishin-the-money, and their values rise and fall accordingly. As volatility rises, extremeoutcomes in the underlying contract become more likely, and this increases an option’svalue. As volatility falls, or as time to expiration grows shorter, extreme outcomesbecome less likely, and this reduces an option’s value.
9596 Chapter 6
Figure 6-1: The Effect of Changing Market Conditions on an Option’sTheoretical Value
If... Call values will... Put values willl...The price of the underlying contract rises rise fallThe price of the underlying contract falls fall riseVolatility rises rise riseVolatility falls falt faltTime passes tall” fall”
* In some unusual cases it may be possible for an option to rise in value as time passes. Thecircumstances which can cause this will be discussed later.
 
Note that we have not considered the effect of changes in interest rates on optionvalues. Since interest rates affect options differently depending on the underlyinginstrument and the settlement procedure, it is not possible to generalize about the effectof changing interest rates. However, we can logically deduce the effect of changinginterest rates on option values by thinking of a call as a substitute for purchasing theunderlying contract, and a put as a substitute for selling the underlying contract. Forexample, suppose we want to purchase a certain stock. The alternative is to purchase acall option. If interest rates are high, we will prefer the call since the outright purchaseof stock requires a much larger cash outlay, and therefore a greater carrying cost. Onthe other hand, if interest rates are low, the carrying cost associated with stock is not asgreat, and calls become a less attractive alternative to purchasing the stock. Hence,rising interest rates cause stock option calls to rise in value, and falling interest ratescause stock option calls to fall in value.
The effect is just the opposite for stock option puts. If we are considering the saleof stock, the alternative is the purchase of a put. In a high interest rate environment thesale of stock will be more desirable since the cash credit resulting from the sale will earngreater interest. In a low interest rate environment puts become more attractive sincethe earnings on any cash credit resulting from the sale of stock is reduced. Hence, risinginterest rates cause stock option puts to fall in value, and falling interest rates cause stockoption puts to rise in value.
The situation is more complex when it comes to options on foreign currenciesbecause a trader must deal with two interest rates, a domestic rate and a foreign rate.The effect of changes in domestic interest rates will be the same as the effect on stockoptions: a call will avoid the cash outlay required to purchase the currency, and a putwill forego the cash credit resulting from the sale of the currency. However, the effectof a change in the foreign interest rate will be just the opposite. If foreign interest ratesare high, rather than purchase a call one will always prefer to purchase the currencyoutright in order to earn the higher interest rate. Hence, higher (lower) foreign interestrates lead to lower (higher) call values. But if foreign rates are high, a put becomes aOption Values and Changing Market Conditions 97
more desirable alternative to selling the foreign currency since one will want to continueto earn a high rate on the foreign currency. Hence, higher (lower) foreign interest rateslead to higher (lower) put values.
Stock and foreign currency options are always assumed to be subject to stock-typesettlement (immediate cash payment for the option). But the settlement procedure foroptions on futures contracts may vary, depending on the exchange. Sometimes suchoptions are settled like stock and sometimes they are settled like futures. While there isno carrying cost associated with a futures contract, there will be a carrying costassociated with an option if it is subject to stock-type settlement. In such a case, optionvalues will fall when interest rates are high (the option becomes a less desirablealternative to a position in the futures contract} and rise when interest rates are low (theoption becomes a more desirable alternative to a futures position). The effect, however,wil! be small since the value of the option, unless it is very deeply in-the-money, is smallrelative to the value of the underlying contract. Futures options are therefore much lesssensitive to changes in interest rates than options on stocks and foreign currencies.When futures options are subject to futures-type settlement, changes in interest rateshave no effect at all on option values since there is no carrying cost associated with eitherthe underlying contract or the option.
The general effects of a change in interest rates are shown in Figure 6-2. In eachcase, the reader can intuitively arrive at the correct conclusion if he asks himselfwhether a call (put) is a better or worse substitute for the outright purchase (sale) of theunderlying contract.
The dividend which is expected to be paid on stock during the life of a stock optionmay also affect the value of the option. In a simple sense, we can equate a dividendpayment with the foreign interest rate on a currency option. If we own the foreigncurrency we can earn interest on it. In the same way, if we own the stock we will receivewhatever dividend is paid on the stock. If dividends increase, we will prefer to take along position by purchasing the stock rather than purchasing a call. The call thereforebecomes a less desirable alternative to owning the stock. In the case of puts, if dividendsincrease we will prefer to take a short position by purchasing a put rather than sellingthe stock, since the sale of the stock will result in the loss of the dividend. Hence, risingdividends will cause stock option calls (puts) to fall (rise) in value, and falling dividendswill cause stock option calls (puts) to rise (fall) in value. This is summarized in Figure6-3.
We can generalize about the effects of changing market conditions on the value ofoptions, but we must still consider the magnitude of the changes. Will the changes belarge or small, representing either a major or minor risk, or something in between?Fortunately, along with the theoretical value, pricing models also generate several othernumbers which enable a trader to assess not only the direction of the change, but alsothe relative magnitude of the change. While these numbers will not answer all ourquestions concerning changing market conditions, they will help us to better assess therisks associated with both individual and complex option positions.Chapter 6
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