money back until expiration whenwe will buy the underlying futures contract, which we originally sold at 102, for 100.If interest rates are currently 8% per annum, and there are three months remaining toexpiration, there will be a 2% carrying cost on the two-point debit resulting from theoption trades. Based on the simple synthetic relationship, we expected to break even onOption Arbitrage 221
the conversion. In fact we will end up losing 2% x 2.00, or .04, because of the cost ofcarrying the debit to expiration. If we really want to break even, we will have to findsome way to offset this interest cost of .04. We might, for example, purchase the callfor .04 less (4.96), or sell the put for .04 more (3.04), or sell the futures contract for .04more {102.04}. Or we might use a combination of these three transactions, for exampleby purchasing the call for 4.98, selling the put for 3.01, and selling the futures contractfor 102.01. The result is a savings of .04, an amount equal to the carrying cost on thedebit resulting from the option trades.
Assuming that all options are European (no early exercise permitted), we can nowexpress the synthetic relationship in futures markets where the options are settle in cashas follows:
call price — put price = futures price — exercise price — carrying costs
where the carrying costs are calculated on either the difference between the futuresprice and the exercise price, or the difference between the call price and put price, bothof which will be approximately the same. In our example, if the call is trading for 4.98,the put for 3.01, and the futures contract for 102.01, the synthetic market is exactlybalanced since:
4.98 - 3.01 = 102.01 - 100.00 - .041.97 = 1.97Using put-call parity, we can always calculate a call, put, or underlying futures pricewhen we know the prices of any two of the other contracts. For example, with a Junefutures contract at 102.00, and the June 100 put at 2.75, we have:
call price = futures price — exercise price + put price — carrying costs102.00 — 100.00 + 2.75 — .04= 471
With the June 100 call at 5.35 and the June futures contract at 101.90, we have:
iH
put price = call price + exercise price — futures price + carrying costs5.35 + 100.00 — 101.90 + .04= 3,49
Finally, with the June 100 call at 3.25 and the June 100 put at 1.25, we have:
futures price = call price— putprice + exercise price + carrying costs3.25 — 1.25 + 100.00 + .04102.04
Stock Option Markets
Consider a stock trading at 103, with aJune 100 call trading at six and a June 100 puttrading at three. If there are no interest or dividend considerations, the synthetic marketseems to be balanced since:222 Chapter 11
call price — put price = stock price — exercise price6-3=103-1003=3
Suppose we decide to do a conversion by purchasing the stock for 103 and sellingthe synthetic for three points (call price minus put price). We will lay out 103 for thestock, take in six for the call, and lay out three for the put. These trades will create atotal debit of 100, a debit which we will have to carry to expiration, at which time wewill sell the stock for 100, either through exercise of the put or assignment of the call.Since interest rates in the real world are not zero, there will be a cost associated withcarrying this debit. If interest rates are currently 8% per annum, and there are threemonths remaining to expiration, there will be a 2% carrying cost on the 100-point debitresulting from the trades. Based on the simple synthetic relationship, we expected tobreak even on the conversion. But in fact we will end up losing 2% x 100, or two points,because of the cost of carrying the debit to expiration. If we really want to break even,we will have to find some way to offset this two-point interest cost. We might do thisby selling the call for two points more (8), or by purchasing the put for two points less(1), or by purchasing the stock for two points less (101). Or we might use a combinationof these three transactions, for example by selling the call for 7, purchasing the put for24, and purchasing the stock for 10214. The result is a savings of two points, anamount equal to the carrying cost on the debit resulting from the trades.
Taking into consideration the interest rate component, we can express the syn-thetic relationship as:
call price — put price = stock price — exercise price + carrying costs
where the carrying costs are calculated on the exercise price.! In our example, if thecall is trading for 7, the put for 21, and the stock for 10214, the synthetic market isexactly balanced since:
7 -214=102'4-100+24,=44
Now suppose that prior to expiration the stock will pay a dividend of 114. When wedid our conversion we purchased stock, so we will receive an extra 114 points whenthe dividend is paid. Given this, we can afford to pay up to 11% points more for theconversion and still do no worse than break even. We could, for example, sell the callfor 51, or buy the put for four, or buy the stock for 104. Or, as before, we mightcombine these three trades by selling the call for six, buying the put for 314, and buyingthe stock for 10314.
Assuming that all options are European (no early exercise permitted), and takinginto consideration both interest rates and dividends, we can now express the full valueof the synthetic relationship in the stock option market as:
1. The exact value depends on the discounted price, or present value, of the exercise price. For a moredetailed explanation, see Appendix B.Option Arbitrage 223
call price - put price = stock price — exercise price + carrying costs — dividends
where the carrying costs are calculated on the exercise price and the dividends are thoseexpected prior to expiration.
Using put-call parity, we can always calculate a call, put, or underlying stock pricewhen we know the prices of any two of the other contracts. For example, with the stockat 102 and the June 100 put at 314, we have:
call price = stock price — exercise price + put price + carrying costs — dividends102 - 100+ 34%4+2-1%= 5%
Ul
With the June 100 call at 61% and the stock at 10114, we have:
put price = call price + exercise price — stock price — carrying costs + dividends614+100-10114%4-2+11%
= 4%
Finally, with the June 100 call at 7 and the June 100 put at 154, we have:
stock price = call price — put price + exercise price — carrying costs + dividends7-—1%4+100-24+1%104%
ll
ARBITRAGE RISK
New option traders are often instructed to go into the market and concentrate onexecuting conversions and reversals because these strategies are riskless. Beware: Thereare no riskless strategies. There are only strategies with greater or lesser risk. The risksof doing conversions or reversals may not be immediately apparent, but they existnonetheless.
Interest Rate Risk
Whenever there is a cash flow associated with a trade, the value of the trade dependson the interest which can be earned on a credit or which must be paid to carry a debitover the life of the trade. This is a function of interest rates, and since interest rates maynot be constant, the interest considerations can change over time. If part of the expectedprofit from a trade depends on our earning 8% ona credit balance, a reduction in interestrates to 6% will certainly reduce our profit, and might even result in a loss. If part of theexpected profit from a trade depends on our paying 7% on a debit balance, an increasein interest rates to 10% will likewise reduce our profit. In practice, large changes ininterest rates over the limited life of most options? are the exception rather than the
2. Weare referring here primarily to exchange traded options with expirations up to approximately ninemonths. Long-term options with expirations of up to several years will obviously be much moresensitive to changes in interest rates.224 Chapter 11
tule. For this reason the interest rate risk associated with conversions and reversals isrelatively small.
Note also that synthetic strategies in the stock option market are much moresensitive to interest rates than the same strategies in the futures option market. A stockoption conversion or reversal includes the cash flow from the underlying stock, while asimilar trade in the futures option market includes no such cash flow. The cash flowfrom a stock trade is always greater than the cash flow from an option trade since theprice of stock is always greater than the price of the options.
Execution Risk
Since no market participant wants to give away money, a trader is unlikely to be offereda profitable conversion or reversal all at one time. Consequently, he will have to executeone or two legs of the strategy, and hope to be able to execute the final leg(s) at a latertime. He may, for example, initially purchase the underlying contract and puts, and laterhope to complete the conversion by selling calls. However, if call prices begin to fall, hemay never be able to profitably complete the conversion. Even a professional trader onan exchange, who would seem to be ina good position to know the prices of all threecontracts, can be fooled. He may purchase a call and sell a put (synthetic longunderlying) at what he thinks are good prices. However, when he tries to sell theunderlying contract to complete the reversal, he may find that the price is much lowerthan what he thought it was. Anytime a strategy is executed one leg at a time, there isalways the risk of an adverse change in prices before the strategy can be completed.
Pin Risk
When we introduced the concept of a synthetic position, we assumed that at expirationeither the underlying market would be above the exercise price, in which case any callwould be exercised, or below the exercise price, in which case any put would beexercised. But what will happen if the underlying market is exactly equal, or pinned,to the exercise price at the moment of expiration?
Suppose a trader has executed a June 100 conversion: he is short a June 100 call,longa June 100 put, and long the underlying contract. If the underlying market is aboveor below 100 at expiration, there is no problem. He wiil either be assigned on the callor will exercise the put. In either case he will offset his tong underlying position so thathe will have no market position on the day following expiration.
Now suppose that at the moment of expiration the underlying market is right at100. The trader would like to be rid of his underlying position. If he doesn’t get assignedon the call, he plans to exercise his put; if he does get assigned on the call, he will letthe put expire worthless. In order to make a decision, he must know whether the callwill be exercised. Unfortunately, he won't know this until the day after expiration,when he either does or does not receive an assignment notice. But then it will be toolate because the call will have expired.
It may seem that an option which is exactly at-the-money at expiration will neverbe exercised since, in theory, it has no value. In fact many at-the-money options do getexercised. Even though such an option has no theoretical value, it does have someOption Arbitrage 225
practical value. For example, suppose the owner of a call which is exactly at-the-moneyat expiration wants to take a long position in the underlying contract. He has twochoices. He can either exercise the call or buy the underlying contract. Since mostexchanges where options are traded include the right of exercise in the originaltransaction cost, it is almost always cheaper to exercise the call. Even if there is a chargefor exercise, it will be less than the cost of trading the underlying contract. Anyoneowning an at-the-money option, and choosing to take a long or short position atexpiration, will find that it is cheaper to exercise the option than buy or sell theunderlying contract.
Clearly, the trader who is short an at-the- money option at expiration has a problem.What can he do?
One course might be to guess whether the trader will be assigned. If the marketappears to be strong on the last trading day, the trader might assume that it will continuehigher on the day following expiration. If the person who holds the call sees the situationsimilarly, it is logical to assume that the call will be exercised. Hence, the trader willchoose not to exercise the put. Unfortunately, if the trader is wrong, and he does notget assigned on the call, he will find himself with a long underlying position which hewould rather not have. Conversely, if the market appears to be weak on the last tradingday, the trader might make the assumption that he will not be assigned on the call. Hewill therefore choose to exercise the put. But again, if he is wrong and does get anassignment notice, he will find himself with an unwanted short underlying position onthe day following expiration.
The risk of a wrong guess can be further compounded by the fact that conversionsand reversals, because of their low risk, are usually done in large size. If the traderguesses wrong, he may find himself naked long or short not one, but several hundredunderlying contracts.
There can be no certain solution to the problem of pin risk. With many, perhapsthousands, of open contracts outstanding, some at-the-money options will be exercisedand some won't. If the trader lets the position go to expiration and relies on luck, he isat the mercy of the fates, and that is a position which an intelligent option trader prefersto avoid. The practical solution is to avoid carrying conversions and reversals toexpiration when there is a real possibility of expiration right at the exercise price. If thetrader has a large number of June 100 conversions or reversals, and expiration isapproaching with the underlying market close to 100, the sensible course is to reducethe pin risk by reducing the size of the position. If the trader doesn’t reduce the size, hemay find that he is under considerable pressure to get out of a large number of riskyPositions at the last moment.
Sometimes even a careful trader will find that he has some at-the-money conver-sions or reversals outstanding as expiration approaches. One way to eliminate the pinrisk which still exists is to liquidate the position at the prevailing market prices.Unfortunately, this is likely to be a losing proposition since the trader will be forced totrade each contract at an unfavorable price, either buying at the offer or selling at thebid. Fortunately, it is often possible to trade out of such a position al! at once at a fairPrice.226 Chapter 11
Since conversions and reversals are common strategies in all option markets, atrader who has an at-the-money conversion and is worried about pin risk can be fairlycertain that there are also traders in the market who have at-the-money reversals andare also worried about pin risk. If the trader with the conversion could finda trader witha reversal and cross positions with him, both traders would eliminate the pin riskassociated with their positions. This is why, on option exchanges, one often finds traderslooking for other traders who want to trade conversions or reversals at even money.This simply means that a trader wants to trade out of his position at a price which is fairto everyone involved, so that everyone can avoid the problem of pin risk. Whateverprofit a trader expected to make from the conversion or reversal presumably resultedfrom the initial trade, not from the closing trade.
Certain options, such as stock indices and Eurodollars, are settled at expiration incash rather than with the delivery of an actual underlying contract. When such anoption expires, the amount of cash which flows into or out of a trader’s account is simplythe amount by which the option is in-the-money, i.e., the difference between theunderlying price and the option’s exercise price. There is no pin risk associated withoptions which settle in this manner because no underlying position results from exerciseor assignment.
Interest rate risk, execution risk, and pin risk are common to all markets, regardlessof the underlying contract. There are, however, certain risks which are unique to amarket depending on the characteristics of the underlying market.
Settlement Risk in the Futures Market
Let’s go back to our original conversion where a trader is short a June 100 call, long aJune 100 put, and long an underlying contract. Suppose the underlying contract is aJune futures contract which is trading at 102.00. If there are three months remainingto June expiration, interest rates are 8%, and all options are subject to stock-type (cash)settlement, the value of the June 100 synthetic market (the difference between the June100 call and the June 100 put) is:
futures price — exercise price — carry on 2 points for 3 months =102 — 100 - (2 x 3/12 x 8%) = 1.96
Suppose a trader is able to sell a June 100 call for 5.00, buy a June 100 put for 3.00,and sell a June futures contract for 102.00. If interest rates do not change, and assumingthere will not be a problem with pin risk, at expiration the trader should realize a profitof .04, since he has done the June 100 conversion at .04 better than its value.
Suppose that shortly after the trader initiates this conversion, the underlying Junefutures contract falls to 98.00. The synthetic (short) side of the position will now showa profit of four points: the short call and long put together will appreciate by four points.But because the options are settled like stock, the profit on the synthetic side will onlybe a paper profit, and will not be fully realized until expiration. On the other hand, thetrader is also long a June futures contract, and this contract, since it is subject tofutures-type settlement, will result in an immediate four point debit when the marketdrops four points. To cover this four point debit, the trader will either have to borrowOption Arbitrage 227
the money or take the money out of an existing, interest-bearing account. In either casethere will be a loss in interest, and this interest loss will not be offset by the paper profitfrom the option position. If the loss in interest is great enough, it may more than offsetthe profit of .04 which the trader originally expected from the position. And in the mostextreme case, where the trader does not have access to the funds required to feed thefutures position, the trader may be forced to liquidate the position. Needless to say,forced liquidations are never profitable.
Of course, this works both ways. A rise in the price of the underlying futurescontract to 106.00 will result in a four point loss to the synthetic side: the short call andlong put together will decline by four points. But the loss on the synthetic side will onlybe a paper loss, and will not be fully realized until expiration. On the other hand, therise in the futures contract will result in an immediate cash credit, a credit on which thetrader can earn interest. This interest will increase his potential profit beyond theoriginal expected profit of .04,
Most futures option traders think of conversions and reversals as being deltaneutral, but in fact this is not necessarily true. With the underlying futures contract at102, the deltas in our example might be:
Option Position Delta Positionshort June 100 call ~60long June 100 put ~38long June futures contact +100total +2
The two extra deltas reflect the fact that the trader would prefer the market to rise ratherthan fall, so that cash will flow into his account from the futures position. The interestfrom this cash flow can represent an unexpected profit or loss, in the case of a declinein the futures price.
Two deltas are really no risk, except when we remember that conversions andreversals, because of their low risk, are often done in very large size. A trader whoexecutes 500 conversions has a risk delta equal to 500 x +2 = +1000. This is the sameas being naked long 10 futures contracts. The risk comes from the interest that can beearned on any cash credit, or which must be paid on any cash debit, resulting frommovement in the underlying futures contract.
The amount by which the delta of a synthetic futures position will differ from 100depends on the interest risk associated with the position. This in turn depends on twofactors: the general level of interest rates and the amount of time remaining toexpiration. The higher the interest rate and the more time remaining to expiration, thegreater the risk. The lower the interest rate and the Jess time remaining to expiration,the less the risk. A 10% interest rate with nine months to expiration represents a muchgreater risk than a 4% interest rate with one month remaining to expiration. In theformer case, the deltas of a synthetic position may add up to 94, while in the latter casethe deltas may add up to 99.
Note that there is no settlement risk when both contracts are subject to the samesettlement procedure. If all contracts are subject to stock-type settlement, no cash flow228 Chapter 1 1
results from fluctuations in the prices of the contracts prior to expiration. If all contractsare subject to futures-type settlement, any credit or debit resulting from changes in theprice of the underlying futures contract will be offset by an equal but opposite cash flowfrom. changes in prices of the option contracts.
Dividend Risk in the Stock Market
Consider a stock trading at 10214, with three months to June expiration, interest ratesat 8%, and a dividend of 11% expected prior to expiration. The value of the June 100synthetic (the difference between the 100 call and the June 100 put) is:
stock price — exercise price + carry on 100 to expiration — expected dividends =10214 - 100 + (100 x 3/12 x 8%) -114=3
Suppose a trader is able to sell a June 100 call for 744, buy a June 100 put for4//, and buy stock for 1021. If interest rates do not change, and assuming there willnot be a problem with pin risk, at expiration the trader should realize a profit of 14,since he has done the June 100 conversion at 1% better than its value.
Since the trader owns the stock, part of his profit comes from the !'% pointdividend which he expects to receive when the stock goes ex-dividend. If the dividendchanges unexpectedly, it could affect the trader’s eventual profit. If, for example, thecompany is doing poorly and decides to cut its dividend in half, to 34, the trader’sconversion will be worth % less, and his profit of '/4 will turn into a loss of 14. Of course,if the company is doing well and decides to increase its dividend to two points, theconversion will be worth 14 more, and the trader’s profit will increase from 1% ta 34.Clearly, the possibility of a change in the expected dividend represents a risk to aconversion or reversal. Moreover, if multiple dividends are expected over the life of thestrategy, the impact of a change in the company’s dividend policy can be greatlymagnified.
BOXES
As we have seen, a conversion or reversal entails risk because these strategies combinea synthetic position in options with a position in the underlying contract. The risk arisesbecause a synthetic position in options and an actual position in the underlying contractcan have different characteristics, either in terms of settlement procedure, as in thefutures option market, or in terms of the dividend payout, as in the stock option market.How might we eliminate this risk?
One way to eliminate this risk is to eliminate the position in the underlyingcontract. Consider a conversion:
short a calllong a putlong an underlying contractOption Arbitrage 229
If we want to maintain this position, but would also like to eliminate the risk that goeswith holding an underlying position, we might replace the long underlying position withsomething that acts like an underlying contract, but which isn’t an underlying contract.For example, we might replace the long underlying position with a deeply in-the-moneycall. Now our position is:
short a calllong a putlong a deeply in-the-money call
If the deeply in-the-money call has a delta of 100, and therefore acts like a longunderlying contract, the position will have the same characteristics as the conversion,
In the same way, instead of replacing the underlying position with a deeplyin-the-money call, we can sell a deeply in-the-money put:
short a calllong a putshort a deeply in-the-money put
If the deeply in-the-money put has a delta of -100, and therefore acts like an underlyingcontract, the position will again have the same characteristics as the conversion.
This type of position, where the underlying instrument in a conversion or reversalis replaced with a deeply inthe-money option, is known as a three-way. Whileeliminating some risks, a three-way is not without its own problems. If a trader sells adeeply in-the-money option to complete his three-way, he still has the risk of the marketgoing through the exercise price. Indeed, as the underlying market moves closer andcloser to the exercise price of the deeply in-the-money option, the option will act lessand less like a position in the underlying contract. The combination of contracts willthen act less and less like a true conversion or reversal.
What else acts like an underlying contract but isn’t an underlying contract? Asynthetic underlying position obviously fills the bill. In addition to a three-way, we mightalso consider replacing an underlying position in a conversion or reversal with asynthetic equivalent. For example, our original position might be a 100 conversion:
short a June 100 calllong a June 100 putlong an underlying contract
Suppose we also execute a reversal at 90:
long a June 90 call short a June 100 callshort a June 90 put long a June 100 putshort an underlying contract long an underlying contract
The long and short underlying contracts cancel out, leaving:
long a June 90 call short a June 100 callshort a June 90 put long a June 100 put230 Chapter 11
We are left with a synthetic long position at the 90 exercise price, and a synthetic shortposition at the 100 exercise price. Or we can think of the position as a reversal at oneexercise price, and a conversion at a different exercise price.
This position, known as a box, is similar to a conversion or reversal, except thatany risk associated with holding a position in the underlying contract has beeneliminated because the underlying position has been replaced with a synthetic underly-ing position at a different exercise price. A trader is long (short) the box when he issynthetically long (short) the lower exercise price and synthetically short (long) thehigher exercise price. The example position is long a June 90/100 box.
At expiration, regardless of the price of the underlying contract, a trader who hasa box will simultaneously buy the underlying contract at one exercise price and sell theunderlying contract at the other exercise price. The value of the box at expiration willtherefore be exactly the amount between exercise prices. In our example, at expirationthe 90/100 box will be worth exactly 10 points because the trader will simultaneouslybuy the underlying contract at 90 and sell itat 100. If itis worth 10 points at expiration,how much is it worth today? If the options are subject to stock-type settlement, the valueof the box today will be the value of the box at expiration less carrying costs. If our10-point box expires in three months with interest rates at 8%, the value today is:
10 - (10 x 3/12 x 8%) = 10—.20 = 9.80
Since a box eliminates the risk associated with carrying a position in the underlyingcontract, boxes are even less risky than conversions and reversals, which are themselveslow-risk strategies. When all options are European (there is no risk of early exercise),and the options are settled in cash rather than through delivery of the underlyingcontract (there is no pin risk), the purchase or sale of a box is identical to lending orborrowing funds over the life of the options. In our example, a trader who sells a10-point box for 9.80 has essentially borrowed funds from the buyer of the box for threemonths at an interest rate of 8%. If a trader were willing to pay a higher rate to borrowfunds, he could sell the box at a lower price, for example 9.70. This corresponds to arate over three months of 12%. When no other method is available, a trading firm mightbe able to raise needed short-term cash by selling boxes. Since the firm will probablyhave to sell the boxes at a price lower than the theoretical value, this will increase thefirm’s borrowing costs. Moreover, if the options can be exercised early, or if there is adanger of pin risk at expiration, this method of borrowing may not be without its ownrisks,
We originally introduced a box as a conversion at one exercise price and a reversalat a different exercise price. Since the long and short underlying positions cancel out,we are left with two synthetic underlying positions. The June 90/100 box was written:
long a June 90 call short a June 100 callshort a June 90 put long a June 100 putThe left side of the box is a synthetic long at 90 and the right side is a synthetic short at
100. Instead of dividing the box into right and left sides, we can also divide it into topand bottom sides. Notice that the top side is a bull vertical call spread (long June 90Option Arbitrage 231
call/short June 100 call}, while the bottom side is a bear vertical put spread (long June100 put/short June 90 put). Since a box is a combination of two vertical spreads, thecombined prices of the vertical spreads must add up to the value of the box.
For example, with three months remaining to expiration and interest rates at 8%,the value of our June 90/100 box is 9.80. Suppose a trader knows that the June 90/100vertical call spread is trading for six points. Without any additional information, thetrader can estimate the fair market price of for the June 90/100 vertical put spread. Heknows 90/100 box is worth 9.80, and that the value ofa call and put vertical spreadmust add up to the value of the box. The price of the put spread must therefore be:
9.80 — 6 = 3.80
If the trader believes he can either buy or sell the call vertical spread at six points, andhe is asked for a market in the put spread, he will probably make his market around anassumed value of 3.80. He might, for example, make a market of 3.50 bid/4.10 ask. Ifhe is able to buy the put vertical for 3.50, he can then try to buy the call vertical for6.00. If he is successful, he will have paid a total of 9.50 for a box with a theoreticalvalue of 9.80. Conversely, if he is able to sell the put vertical for 4.10, he can then tryto sell the call vertical for 6.00. If he is successful, he will have sold a box with atheoretical value of 9.80 at a price of 10.10.
JELLY ROLLS
Another method of eliminating a position in the underlying contract is to take a syntheticposition in a different expiration month, rather than at a different exercise price as witha box. For example, suppose we have executed the following reversal:
long a June 100 callshort a June 100 putshort an underlying