110. A trader who owns the 100/105 box will exercisehis 100 calls in order to be able to tender stock. If he tenders 1000 shares, he mightexpect half, or 500 shares, to be accepted at the tender price of 110. But after the tenderis completed, because it was for only half the outstanding shares, the stock price willreturn to its pre-tender price of 100. The trader’s remaining 500 shares will still betrading at 100. In other words, prior to the tender offer the 100 calls are worth 10, butafter the tender the 105 puts are worth a minimum of five. Of course, one cannot saythat the box is worth 15, because only half the outstanding shares are worth 110. Theother half are worth 100. Still, the 100/105 box is likely to trade for significantly morethan its usual price of five.
The possibility of early exercise can also affect arbitrage strategies in the futuresoption market, but the difference between American versus European values is muchless than in the stock option market because there is less interest considerationassociated with futures options. For practical purposes, unless the options are verydeeply in-the-money or are very long-term, the difference between American andEarly Exercise of American Options 255
European vaiues for futures options is negligible. Some sample option and arbitragevalues are shown below (we assume that all options are subject to stock-type settle-ment):
Futures price = 100; Volatility = 25%; Interest rate = 8.00%;Time to expiration: March = 3 months, June = 6 months
European European American American
Option Value Delta Value DeltaMarch 95 call 7,60 66.9 7.67 67.5March 95 put 2.70 ~31.2 2.73 -31.3March 100 call 4,88 51.5 4.88 51.8March 100 put 4.88 —46.6 4.88 46.9June 95 call 9,27 62.2 9.34 63.5June 95 put 4.46 -33.9 4,48 -34,2June 100 call 6.76 51.4 6.80 52.2June 100 put 6.76 -44.7 6.80 ~45.4European European American AmericanStrategy Value Delta Value DeltaMarch 95 conversion 4.90 +1.9 4.94 +1.2June 95 conversion 4.81 4+3.9 4.86 +23March 100 conversion 0 +19 0 +13June 100 conversion 0 +3.9 0 +2.4March 95/100 box 4.90 0 4.94 +.1June 95/100 box 4.81 0 4.86 +.1March/June 95 jelly roll 1.85 0 1.71 +11March/June 100 jelly roll 1.95 0 171 +1.6
Many new traders seem to worry unduly about the possibility of early exercise: “IfIsell an option, what happens if] am suddenly assigned?” It is true that sometimes earlyassignment can cause a loss. But there are many factors which can cause a trader to losemoney; early exercise is only one such factor. A trader should be prepared to deal withthe possibility of early exercise just as he should be prepared to deal with the possibilityof movement in the price of the underlying contract, or the possibility of changes involatility. Margin requirements established by the clearing houses usually require atrader to keep sufficient funds in his account to cover the possibility of early assignment.But this is not always true. If he is short deeply inthe-money options, an earlyassignment notice may cause a cash squeeze. If this happens, he will need sufficientCapital to cover the situation. Otherwise, he may be forced to liquidate some or all ofthe remaining position. And forced liquidations are invariably losing propositions.256 Chapter 12
An experienced trader should be able to foresee the likelihood of early exercise. Heneed only ask himself: “If! owned this option, would I logically exercise it now?” If theanswer is “yes,” then the trader ought to be prepared for assignment. Early exerciserarely comes as a surprise. If it does, it is probably good for the trader who was assigned.If an option is exercised too early, someone has erroneously abandoned the time valueor protective characteristics associated with the option. When that happens, the traderwho is assigned will find that he has just received an unexpected gift.a 13 %
Hedging with Options
The original justification for introducing options and futures was to enable marketparticipants to transfer part or all of the risk associated with holding a position in theunderlying instrument from one party to another. Options and futures act as insurancecontracts. Unlike a futures contract, an option transfers only part of the risk from oneparty to another. In this respect an option acts much more like a traditional insurancepolicy than a futures contract.
Even though options were originally intended to function as insurance policies,option markets have evolved to the point where, in most markets, hedgers (thosewanting to protect an existing position), make up only a small portion of marketparticipants. Other traders, including arbitrageurs, speculators, and spreaders, typicallyoutnumber true hedgers. Nevertheless, hedgers still represent an important force in themarketplace, and any sophisticated market participant must be aware of the strategieshedgers use to protect their positions.
Many hedgers come to the marketplace as either natural longs or natural shorts.Through the course of normal business activities they will profit from either a rise or fallin the price of some underlying instrument. The producer of a commodity (grain, oil,precious metals) is a natural long; if the price of the commodity rises, he will receivemore when he goes to sell it in the marketplace. The user of a commodity is a naturalshort; if the price of the commodity falls, he will have to pay less for it when he goes tobuy it in the marketplace. In the same way, lenders and borrowers in the financialmarkets are natural longs and shorts in terms of interest rates. A rise in interest rateswill help lenders and hurt borrowers. A decline in interest rates will have the oppositeeffect.
Other potential hedgers come to the marketplace because they have voluntarilychosen to take long or short positions, and now wish to lay off part or all of the risk ofthat position. A speculator may have taken a long or short position in a certain contract,but wishes to temporarily reduce the risk associated with an outright long or shortposition. For example, a portfolio manager is usually paid to choose which stocks to buyfor a portfolio. He is a voluntary long. If he wants to continue to hold certain stocks, butbelieves that the stocks might decline in the short term, he may find it cheaper totemporarily hedge the stocks with options or futures than to sell the stocks and buy themback ata later date.
As with insurance, there is a cost to hedging. The cost may be immediately apparentbecause it requires an immediate cash outlay. But the cost may also be more subtle,either in terms of lost profit opportunity, or in terms of additional risk under some
257258 Chapter 13
circumstances. Every hedging decision is a tradeoff: what is the hedger willing to giveup under one set of market conditions in order to protect himself under another set? Ahedger with a long position who wants to protect his downside will almost certainlyhave to give up something on the upside; a hedger with a short position trying to protecthis upside will have to give up something on the downside.
PROTECTIVE CALLS AND PUTS
The simplest way to hedge an underlying position using options is to purchase either acall to protect a short position, or a put to protect a long position. In each case, if themarket moves adversely, the hedger is insulated from any loss beyond the exercise price.The difference between the exercise price and the current price of the underlying isanalogous to the deductible portion of an insurance policy. And the price of the optionis analogous to the premium which one has to pay for the insurance policy.
For example, suppose an American firm expects to take delivery of DM 1,000,000worth of German goods in six months. If the contract requires payment in Deutsche-marks at the time of delivery, the American firm has automatically acquired a shortposition in Deutschemarks against U.S. dollars. If the Deutschemark rises against thedollar, the goods will cost more in dollars; if the Deutschemark falls against the dollar,the goods will cost less. If the Deutschemark is currently trading at .60 (60¢ per DM)and remains there for the next six months, the cost to the American firm will be$600,000. If, however, at delivery the Deutschemark has risen to .70 (70¢ per dollar),the cost to the American firm will be $700,000.
The American firm can offset the risk it has acquired by purchasing a call option onDeutschemarks, for example a .64 call. If the firm wants an exact hedge, the underlyingcontract will be DM 1,000,000, and the option will have an expiration date correspond-ing to the date of delivery of the goods. If the value of the Deutschemark begins to riseagainst the U.S. dollar, the firm will have to pay a higher price than expected when ittakes delivery of the goods in six months. But the price it will have to pay forDeutschemarks can never be greater than .64. If the price of Deutschemarks is greaterthan .64 at expiration, the firm will simply exercise its call, effectively purchasingDeutschemarks at .64. If the price of Deutschemarks is less than .64 at expiration, thefirm will let the option expire worthless since it will be cheaper to purchase Deutsche-marks in the open market.
A hedger who chooses to purchase a call to protect a short position, or a put toprotect a long position, has risk limited by the exercise price of the option. At the sametime, the hedger still maintains open-ended profit potential. If the underlying marketmoves in the hedger’s favor, he can let the option expire and take advantage of theenhanced value of his position in the open market. If, inour example, the Deutschemarkwere to fall to .55 at the time of delivery, the firm would simply let the .64 call expireunexercised. At the same time, the firm would purchase DM 1,000,000 for $550,000,resulting in an unexpected windfall of $150,000.
There is a cost involved in buying insurance in the form of a protective call or put,namely the price of the option. The cost of the insurance is commensurate with theHedging with Options 259
amount of protection afforded by the option. If the price of a six-month .64 call is .0075(Yc), the firm will pay an extra $7,500 (.0075 x 1,000,000) no matter what happens.A call option with a higher exercise price will cost less, but it also offers less protectionin the form of an additional deductible amount. If the firm chooses to purchase a .66call trading at .0025 (ae), the cost for this insurance will only be $2,500 {.0025 x1,000,000). But the firm will have to bear any loss up to a Deutschemark price of .66.Only above .66 is the firm fully protected. In the same way, a lower exercise price callwill offer additional protection, but at a higher price. A .62 call will protect the firmagainst any rise above .62, but if the price of the call is .015 (1 2¢), the purchase of thisprotection will add an additional $15,000 (.015 x 1,000,000) to the final costs.
The cost of purchasing a protective option and the insurance afforded by thestrategy is shown in Figures 13-1 (protective call] and 13-2 (protective put}. Since eachstrategy combines an underlying position with an option position, it follows fromChapter 11 that the resulting protected position is a synthetic option:
short underlying + long call = long putlong underlying + long put = long call
Ahedger who buys a call to protect a short underlying position has effectively purchaseda put at the same exercise price. A hedger who buys a put to protect a long underlyingposition has effectively purchased a call. In our example, if the firm purchases a .64 callto protect a short Deutschemark position, the result is identical to owning a .64 put.
Figure 13-1: A Protective Call
 
A
      
short an underlying position
ra buy a protective call aNA ”
 
combined position NN
 
 
 
 260 Chapter 13
Figure 13-2: A Protective Put
 
 
 
~oNoe ~ — ia~<a >, buy a protective puta“oO combined position“4
 
 
 
Which protective option should a hedger buy? That depends on the amount of riskthe hedger is willing to bear, something which each market participant must determineindividually. One thing is certain. There will always be a cost associated with thepurchase of a protective option. If the insurance afforded by the option enables thehedger to protect his financial position, the cost may be worthwhile.
COVERED WRITES
While the purchase of a protective option offers a limited and known risk, a hedgermight be willing to accept more risk in return for some other advantage. Instead ofpurchasing an option to protect an existing position, a hedger might consider selling, orwriting, an option against the position. This strategy does not offer the limited riskafforded by the purchase of a protective option but, in its favor, the strategy results in acash credit rather than a debit. This credit offers some, though not unlimited, protectionagainst an adverse move in the underlying market.
Consider a fund manager whose portfolio consists of a long position in a certainstock. If the manager is worried about a decline in the price of the stock, he mightchoose to sell a call option against the stock. The amount of protection the manager isseeking, as well as the potential upside appreciation, will determine which call is sold,whether in-the-money, at-the-money, or out-of-the-money. Selling an in-the-money calloffers a high degree of protection, Dut will eliminate most upside profit potential. SellingHedging with Options 261
an out-of-the-money call offers less protection but leaves room for large upside profitpotential.
For example, if the stock is currently trading at 100, the fund manager might sella 95 call. If the 95 call is trading at 614, the sale of the call will offer a high degree ofprotection against a decline in the price of the stock. As long as the stock declines by nomore than 61, to 9314, the manager will do no worse than break even. Unfortunately,if the stock begins to rise, the fund manager will not participate in the rise since thestock will be called away when the manager is assigned on the 95 call. Still, even if thestock rises, the manager will get to keep the 1 4-point time premium he received fromthe sale of the 95 call.
If the manager wants to participate in upside movement in the stock, and is alsowilling to accept less protection on the downside, he might sell a 105 call. If the 105call is trading at two, the sale of this option will only protect him against a two-pointfall, to 98, in the price of the stock. If, however, the stock begins to rise, he willparticipate point for point in this advance up to a price of 105. Above 105, he can expectthe stock to be called away.
Which option should the manager sell? Again, that is a subjective decision basedon how much risk the manager is willing to take as well as the amount of upsideappreciation in which he wants to participate. Many covered write programs involveselling at-the-money options. Such options offer less protection than in-the-money calls,and less profit potential than out-of-the-money options. An at-the-money option is puretime premium. If the market sits still, a portfolio of at-the-money options have been soldagainst an existing position in the underlying instrument will show the greatestappreciation in value.
The value of typical covered writes, also known as overwrites, are shown in Figures13-3 (covered call] and 13-4 (covered put). When the underlying is purchased and callsare simultaneously sold against the position, a covered write is also referred to as abuy/write. As with the purchase of a protective option, a covered write consists of aposition in the underlying and an option. It can therefore be expressed as a syntheticposition:
long underlying + short call = short putshort underlying + short put = short call
A hedger who sells a call to protect a long underlying position has effectively sold a putat the same exercise price. A hedger who sells a put to protect a short underlyingposition has effectively sold a call. In our example, if the portfolio manager sells a 100call to protect a long stock position, the result is identical to selling a 100 put.
The purchase of a protective option and the sale of a covered option are the twomost common hedging strategies involving options. If given a choice between theseStrategies, which one should a hedger choose? In theory, the hedger ought to base hisdecision on the same criteria as the pure trader: price versus value. In general, if thePrice of an option is lower than its value, then the purchase of a protective call or putmakes the most sense. If the price of an option is higher than its value, then the sale ofa covered call or put makes the most sense. By comparing implied volatility to a forecast262 Chapter {3
Figure 13-3: A Covered Call
 
combined position
sell a covered call <
 
 
a“ long an underlying position &
 
 
 
Figure 13-4: A Covered Put
 
_— combined position
N\ sell a covered put
 
 
ia short an underlying position a
 
 
 Hedging with Options 263
volatility, a hedger ought to be able to make a sensible determination as to whether hewants to buy or sell options. Of course, he is still left with the choice of exercise prices.This will depend on the amount of adverse or favorable movement the hedger foresees,as well as the risk he is willing to accept if he is wrong.
While theoretical considerations will often play a role in a hedger’s decision,practical considerations may also be important. Ifa hedger knows that he will be forcedout of business if the underlying contract moves beyond a certain price, then thepurchase of a protective option at that exercise price may be the most sensible strategy,regardless of whether the option is theoretically overpriced.!
Most hedgers find, however, that there is a psychological barrier to buying options.“Why should I lay out money for an option when J will probably lose the premium?”The hedger is right. Most of the time he will lose the premium because most of the timethe option will expire worthless. But the same hedger who refuses to purchase an optionto protect his business dealings would never consider leaving his home or familyuninsured. Like options, the great majority of insurance policies expire without claimsever being made against them. Houses do not burn down; people live on; cars are notstolen. That is the reason insurance companies make a profit. But the purchaser of aninsurance policy is willing to accept the fact that insurance companies make profits.Unlike the insurance company, which is selling the insurance contract with the clearintent of making a profit, the purchaser of an insurance contract is not purchasing thecontract to make money. He is purchasing it for the peace of mind which it affords. Thesame philosophy should apply to the purchase of options. If a hedger needs theprotection afforded by the option, the purchase makes sense in spite of the fact that itis likely to be a losing proposition in the long run.
FENCES
Suppose a hedger desires the limited risk afforded by the purchase ofa protective option,but also wants to avoid the cash outlay associated with such a strategy. What can hedo? A popular strategy, known as a fence, is to simultaneously combine the purchaseof a protective option with the sale ofa covered option. For example, with an underlyingcontract at 50, a hedger with a long position might choose to simultaneously sell a 55call and purchase a 45 put. The hedger is insulated from any fall in price below 45, sincehe can then exercise his put. At the same time, he can participate in any upward moveto 55, at which point his underlying position will be called away. The cost of thecomplete hedge depends on the prices of the 45 put and 55 call. If they are the same,the cost of the hedge is zero. If the price of the 55 call is greater than the price of the45 put, the hedge can be established for a credit which the hedger gets to keep no matterwhere the underlying finishes. With prices of 1.25 and 1.75 for the 45 put and 55 call,the hedger will receive a credit of .50. This will reduce his break-even price on theunderlying position to 49.50.
 
1. Ofcourse, if the options are wildly overpriced, a hedger might be reluctant to buy a protective option.But this is unlikely to happen. If option prices are high, there is usually a valid reason.264 Chapter 13
Recalling the basic synthetic relationships, we find that a long fence (long theunderlying, short a covered call, long a protective put) is simply a bull vertical spreaddone synthetically. Suppose in our example fence consisting of a long 45 put and short55 call, we replace the 45 put with its synthetic equivalent:
long 45 put = short underlying + long 45 call45/55 fence = long underlying + long 45 put + short 55 call= long underlying + (short underlying + long 45 call) + short 55 call= long 45 call + short 55 call
A short fence consists of a short underlying position together with the purchase ofa protective call and sale of a covered put. Since we can always rewrite one of theoptions with its synthetic equivalent, this is simply a bear vertical spread. Long and shortfences are shown in Figures 13-5 and 13-6,
Fences are popular hedging tools because they offer known protection at a low cost,or even a credit. At the same time, they still allow a hedger to participate, at leastpartially, in favorable market movement. Fences go by a variety of names: rangeforwards, tunnels, cylinders; among floor traders they are sometimes known as spilt-price conversions and reversals.
As the borrowing and lending of funds has become more widespread, and sincesuch activity is often based on a floating interest rate, the use of options to hedge longor short interest rate positions has become especially popular. A firm which borrows
Figure 13-5: A Long Fence
 
 
 
aaVv. sell a covered callSS combined positionN ’“ ut“a~<a. -- ~ - a C _‘ aa La ——A LA~<—_- LE ee ~ 7 oer nn - _a ‘,| “ “ ,“ | *a“ \ buy a protective put .“4 long an underlying position -
 
 
 
 Hedging with Options 265
Figure 13-6: A Short Fence
 
 
 
 
 
me | ¢bined position Ycom p SN sel] a covered put aNs a_~a ~ — =TO ~ ~ a NY “= ~|~™~_-- - eens AS >uo buy a protective call ~> ~NNshort an underlying position x
 
 
 
 
 
funds at a variable interest rate has a short interest rate position. A fall in interest rateswill reduce its cost of borrowing, while a rise in interest rates will increase its costs. Tocap the upside risk, the firm can purchase an interest rate call, thereby establishing amaximum amount it will have to pay for borrowed funds. No matter how high interestTates rise, the borrower will never have to pay more than the cap’s exercise price.
An institution which lends funds at a variable interest rate has a long interest rateposition. A rise in interest rates will increase its returns, while a fall in interest rates willreduce its returns, To set a floor on its downside risk, the institution can purchase aninterest rate put, thereby establishing a minimum amount it will receive for loanedfunds, no matter how low interest rates fall. No matter how low interest rates fall, thelender will never receive less than the floor’s exercise price.
If a borrower (lender) simultaneously buys a cap (floor) and sells a floor (cap), hehas established a collar to his position. This is simply a fence done in the interest ratemarket.
COMPLEX HEDGING STRATEGIES
Since most hedgers are not professional option traders, and have neither the time nordesire to carefully analyze option prices, simple hedging strategies involving the pur-chase or sale of single options are the most widely used. However, if one is willing todo a more detailed analysis of options, it is possible to construct a wide variety of hedging266 Chapter 13
strategies which involve both volatility and directional considerations. In order to dothis, a hedger must be familiar with both volatility and its impact on option values, andthe delta as a measure of directional risk. The hedger can then combine his knowledgeof options with the practical considerations of hedging.
Asa first step in choosing a strategy, a hedger might consider the following:
1. Does the hedge need to offer protection against a “worst case” scenario?2. How much of the current directional risk should the hedge eliminate?3. What additional risks is the hedger willing to accept?
A hedger who needs disaster insurance in order to protect against a “worst case”Scenario only has a choice of which option(s) to buy. Even so, he still needs to decidewhich exercise price to purchase and how many options. For example, suppose a hedgerhas a long position in a certain underlying contract currently trading at 100. Supposealso that the hedger decides to buy a put because he needs to limit the downside risk tosome known and fixed amount. Which put should he buy?
Suppose the hedger has determined that options are generally overpriced (impliedvolatility is too high). Clearly, any option purchase will be to the hedgers’ disadvantage.If his sole purpose is to hedge his downside risk without regard to upside profit potential,he ought to avoid options and hedge himself in the futures or forward market. If,however, he still wants upside profit potential, he must ask himself how much of a longposition he wants to retain. For example, he might be willing to retain 50% of his currentlong position. This simply means that he has to purchase puts with a total delta of -50.He can do this by purchasing one at-the-money put with a delta of —50, or severalout-of-the-money puts whose deltas add up to —50. But in a high implied volatilitymarket one usually tries to buy as few options as possible and sell as many options aspossible, (This is analogous to constructing a ratio vertical spread.) Hence, purchasingone put with a delta of —5O will be less costly, theoretically, than purchasing severaloptions with a total delta of -50. If the hedger wants to eliminate even more of thedirectional risk, say 75%, under these circumstances he would be better off purchasingone put with a delta of -75.
All other factors being equal, in a high implied volatility market a hedgershould buy as few options as possible and sell as many options as possible.Conversely, in a low implied volatility market a hedger should buy as manyoptions as possible and sell as few options as possible.
This means that if all options are overpriced (implied volatility is too high}, and thehedger decides he is willing to accept the unlimited downside risk that goes with thesale of a covered call, in theory he ought to sell as many calls as possible to reach hishedging objectives. If he is trying to hedge 50% of his long underlying position, he cando a ratio write by selling several out-of-the-money calls with a total delta of 50, ratherthan selling a single at-the-money call with a delta of 50.
There is an additional problem if one sells several calls against a single longunderlying position. Now the hedger not only has the unlimited downside risk whichHedging with Options 267
goes with a covered call position, but he also has unlimited upside risk because he hassold more calls than he can cover with the underlying. If the market moves up farenough, he will be assigned on all the calls. Most hedgers want to restrict their unlimitedrisk to one direction, usually the direction of their natural position. A hedger with a longunderlying position may be willing to accept unlimited downside risk, but he is probablyunwilling to accept unlimited upside risk. A hedger with a short underlying position maybe willing to accept unlimited upside risk, but he is probably unwilling to acceptunlimited downside risk. A hedger who constructs a position with unlimited risk ineither direction is presumably taking a volatility position. There is nothing wrong withthis, since volatility trading can be highly profitable. But a true hedger ought not losesight of what his ultimate goal is: to protect an existing position, and to keep the cost ofthis protection as low as possible.
A hedger can also protect a position by constructing one-to-one volatility spreadswith deltas which yield the desired amount of protection. For example, a hedger whowants to protect 50% of a short underlying position can buy or sell time spreads orbutterflies with a total delta of +50. Such spreads offer partial protection within a range.The position still has unlimited upside risk, while it also retains unlimited downsideprofit potential. Such volatility spreads also give the hedger the choice of buying orselling volatility. If implied volatility is generally low, with the underlying marketcurrently at 100, the hedger might protect a short position by purchasing a 110 call timespread (purchase a long-term 110 call, sell a short-term 110 call). This spread has apositive delta and, at the same time, is theoretically attractive since the low impliedvolatility makes a long time spread relatively inexpensive. If the 110 call time spreadhas a delta of +25, to hedge 50% of his directional risk the hedger can buy two timespreads for each short underlying position. Conversely, if implied volatility is high, thehedger can consider selling time spreads. Now he will have to choose a lower exerciseprice to achieve a positive delta. If he sells the 90 call time spread (purchase a short-term90 call, sell a long-term 90 call), he will have a position with a positive delta and apositive theoretical edge. If he wants to protect 75% of his position, and the spread hasa delta of +25, he can sell the spread three times for each underlying position.
In the same way, a hedger can also buy or sell vertical spreads to achieve a desiredamount of protection. Depending on whether options are generally underpriced orOverpriced (implied volatility is too low or too high), the hedger will work around theat-the-money option. With the underlying market currently at 100, the hedger whoWants to protect a long position can sell a vertical call spread with a negative delta (sellthe lower exercise price call, buy the higher exercise price call). If implied volatility ishigh, he will prefer to sell an at-the-money option and buy an out-of-the-money option.If implied volatility is low, he will prefer to sell an in-the-money option and buy anat-the-money option. Each spread will have a negative delta, but will also have a positivetheoretical edge since