read—buy the in-the-money (95) call.
The same principle is equally true for put vertical spreads. We always want to focuson the at-the-money option, buying it when implied volatility is too low, and selling itwhen implied volatility is too high. For example, suppose we want to do a bear putspread with implied volatility too low. In this case we want to buy the at-the- money(100) put. Having done this, we are forced to sell the out-of-the-money (95) put to createour bear spread. We can see from Figure 10-5 that we will pay approximately 1.90 forthis spread, but the spread is worth 2.04, We will end up with a delta position of -20,and a positive theoretical edge of .14.
Of course, a trader is not required to execute any vertical spread by first buying orselling the at-the-money option. Such spreads always involve two options, and a tradercan choose to either execute the complete spread in one transaction, or leg into thespread by trading one option at a time. In the latter case, he may decide to first tradethe in-the-money or out-of-the-money option, and trade the at-the-money option at alater time. This is a practical trading decision he will have to make based on marketconditions and the amount of risk he is willing to accept. Regardless of how the spreadis executed, the trader should focus on the at-the-money option, either buying it whenimplied volatility is too low, or selling it when implied volatility is too high.
In practice it is unlikely that one option will be exactly at-the-money, so that it maybe difficult to decide which option to buy and which to sell. In such a case, it is usuallyBull and Bear Spreads 209
best to focus on the option which is closest to at-the-money. If the underlying market isat 103, with 95, 100, 105, and 110 calls available, it is logical to focus on the 105 callsince it is closest to at-the-money. If implied volatility is too low, a trader will want tobuy the 105 call; if implied volatility is too high, a trader will want to sell the 105 call.He can then trade a different option in order to create a bull or bear vertical spread.
Nor does a trader have to include the option which is closest-to-the- money as partof his spread. A trader who has a strong directional opinion can choose a vertical spreadwhere both options are very far out-of-the-money or very deeply in-the-money. The deltavalues of such spreads will be very low, but a trader can create a highly leveragedposition by executing each spread many times. For example, with the underlying marketat 100, a trader who is strongly bullish might buy the 115/120 call spread (assumingsuch exercise prices are available}. The cost of this spread will be very low since thereis a high probability that the spread will expire worthless. Hence, the trader will be ableto execute the spread many times at a relatively low cost. If he is right and the marketdoes rise above 120, the spread will widen to its maximum value of five points, resultingin very large profits. But regardless of the exercise prices chosen, if implied volatility islow, the trader will attempt to buy the closer-to-the-money option, and if impliedvolatility is high, the trader will attempt to sell the closer-to-the-money option.
The choice of the at-the-money option is slightly different when we move to stockoptions. If we define the at-the-money option as the one whose delta is closest to 50,then we may find that the at-the-money option is not always the one whose exerciseprice is closest to the current price of the underlying contract. This is because the optionwith a delta closest to 50 will be the one whose exercise price is closest to the forwardprice of the underlying contract. In stock options, the forward price is the current priceof the stock, plus carrying costs on the stock, less expected dividends. Assuming anunderlying stock price of 99, six months to expiration, a volatility of 28%, interest ratesof 10%, and no dividend, we can see from Figure 10-6 that the 105 call has a delta of50 and therefore acts like the at-the-money option, even though its exercise price is sixpoints higher than the current underlying price. Therefore, any vertical spread shouldfocus on the 105 call or 105 put.
The approximate prices of vertical spreads with these stock options at impliedvolatilities of 23% and 33% are shown in Figure 10-7.
Figure 10-6Stock price = 99; Time = 6 months; Volatility = 28%; Interest = 8.00%TheoreticalOption Value Delta Gamma Theta Vega95 call 11.97 69 1.8 —.031 .24105 call 6.96 50 2.0 —.030 27115 call 3.74 32 1.8 ~.025 2595 put 4.25 -31 1.8 ~.013 .22105 put 9.20 ~50 2.0 —.011 +26
115 put 15.68 -68 1.8 -.004 .24210 Chapter 10
Figure 10-795 95/105 105 105/415 415Volatility Call Spread Call Spread Call23% 10.77 §.20 §.57 3.05 2.5228% 11.97 (69) §.01 (19) 6.96 (50) 3.22 (18) 3.74 (32)33% 13.21 4.85 8.36 3.34 5.0295 95/105 105 105115 115Volatility Call Spread Call Spread Cail23% 3.05 441 7.46 6.56 14.0228% 4,25 (-31) 4.60 (19) 8.85 (—50) 6.39 (19) 15.24 (68)33% §.49 476 10.25 6.27 16.52
As always, we can never be certain of the delta values of a spread because we cannever be certain that we have the correct inputs into the model, in particular, volatility.If volatility turns out to be lower than our estimate, the deltas will move away from 50.If volatility turns out to be higher than our estimate, the deltas will move towards 50.This will change the delta values of the spreads. However, if we work around theat-the-money option, spreads with approximately the same amount between exerciseprices will have approximately the same delta values. Our primary problem will bedeciding whether implied volatility is too high or too low.
Notice that in every case, whether with futures options or with stock options,whether using a low volatility ora high volatility, if one leg of our spread always involvesan at-the-money option, the vertical spread which includes the in-the-money optionalways has a higher price than the spread which includes the out-of-the-money option.A new trader might take the view that since both spreads have approximately the samedelta values, it must make sense to always purchase the cheaper spread or sell the moreexpensive. This ignores the purpose of option evaluation: to consider not only the initialcost of a strategy, but to compare that to the strategy’s expected return. In our futuresoption example, a trader will always have to pay more for the 95/100 call spread thanthe 100/105 call spread. But if implied volatility is too high, the 95/100 spread willalso return more to the trader. To see why, suppose a trader is choosing between the95/100 bull call spread and the 100/105 bull call spread. Consider three possible cases.In case one, the trader is right and the market moves from 100 to 110. If that happensboth spreads will be winners since they will both widen to their maximum value of fivepoints. In case two, the trader is wrong and the market drops to 90. In that case bothspreads will be losers since they will both collapse to zero. Now consider case three,where the trader is wrong because he expected the underlying market to rise and itdidn’t. But he wasn’t terribly wrong, because it also didn’t fall. It simply remained at100. If that happens the 100/105 spread will collapse to zero, while the 95/100 spreadwill still widen to its maximum value of five points. The 95/100 spread is always morevaluable than the 100/105 spread because it has time on its side. The 100/105 spreadneeds the market to rise to show a profit. The 95/100 spread doesn’t really need theBull and Bear Spreads 211
market to rise; it just needs for the market not to fall. The reader can confirm this bycalculating the gamma and theta positions of the spreads in Figure 10-5. The 100/105spread has a positive gamma and a negative theta, whereas the 95/100 spread hasnegative gamma and a positive theta.
If a trader is considering a Dull vertical spread, how likely is it that the market willrise? That depends on his ability to predict market direction. At the same time, he mustask himself how likely it is that the market will move. That depends on his ability topredict volatility. If a trader believes there is a good chance the market will move (highvolatility), and at the same time believes the movement will be to the upside (bullish ondirection), it makes sense to buy the 100/105 spread. On the other hand, if the traderbelieves the market is unlikely to move significantly (low volatility), but at the same timebelieves that whatever movement does occur is likely to come on the upside (alsobullish), it makes sense to buy the 95/100 spread. In both cases the trader is trying tomaximize the return on his investment when he is right and minimize the loss whenhe is wrong.
Why might a trader with a directional opinion prefer a vertical spread to an outrightlong or short position in the underlying instrument? For one thing, a vertical spread ismuch less risky than an outright position. A trader who wants to take a position whichis 500 deltas long can either buy five underlying contracts or buy 25 vertical call spreadswith a delta of 20 each. The 25 vertical spreads may sound riskier than five underlyingcontracts, until we remember that a vertical spread has limited risk while the positionin the underlying has open-ended risk. Of course, greater risk also means greater reward.A trader with a long or short position in the underlying market can reap huge rewardsif the market makes a large move in his favor. By contrast, the vertical spreader’s profitsare limited, but he will also be much less bloodied if the market makes an unexpectedmove in the wrong direction.
Experienced traders accept the fact that they are human, and that their directionalforecasts will occasionally be wrong. When that happens the vertical spreader has adistinct advantage over the trader with an outright position in the underlying market.By intelligently estimating volatility, an option trader can decide whether he wants timeworking for or against him. When he chooses to have time on his side (positive theta),he can sometimes show a profit when the outright trader either breaks even or showsa loss. When he chooses to have time working against him (positive gamma), the optiontrader’s losses when he is wrong about market direction will often be less than the lossesfrom an outright underlying position.11%
- Option Arbitrage
One important characteristic of options is that they can be combined with other optionsor underlying contracts to create positions with characteristics which are almostidentical to some other contract or combination of contracts. This type of replicationleads to a new category of trading strategies which are unique to the option market.
SYNTHETIC POSITIONS
Suppose a trader has the following position where all options are European (no earlyexercise permitted):
long a June 100 callshort a June 100 put
What will happen to this position at expiration? It may seem that one cannot answerthe question without knowing where the underlying contract will be at expiration.Surprisingly, the price of the underlying contract does not affect the outcome. If theunderlying contract is above 100, the put will expire worthless, but the trader willexercise the 100 call. This results in his buying the underlying contract at 100.Conversely, if the underlying contract is below 100, the call will expire worthless, butthe trader will be assigned on the 100 put. This also results in his buying the underlyingcontract at 100.
Ignoring for the moment the unique case when the underlying contract is right at100, at June expiration the above position will always result in the trader going long theunderlying instrument at the exercise price of 100. He will go long, either by choice(the underlying contract is above 100 and he exercises the 100 call} or by force (theunderlying contract is below 100 and he is assigned on the 100 put). We refer to thisposition as a synthetic long underlying. The position has the same characteristics as theunderlying contract, but won’t actually become an underlying contract until expiration.
If the trader takes the opposite position, selling a June 100 call and purchasing aJune 100 put, he has a synthetic short underlying. This position will always result in hisselling the underlying contract at the exercise price of 100, either by choice (theunderlying contract is below 100 and he exercises the 100 put) or by force (theunderlying contract is above 100 and he is assigned on the 100 call).
213214 Chapter 11
We can express the foregoing relationships as follows:
synthetic long underlying = long call + short putsynthetic short underlying = short call + long put
where all options expire at the same time and have the same exercise price.
A synthetic position acts very much like its real equivalent. For each point theunderlying instrument rises, a synthetic long position will gain approximately one pointin value and a synthetic short position will lose approximately one point in value. Thisleads us to conclude, correctly, that the delta of a synthetic underlying position isapproximately 100. If the delta of the June 100 call is 75, the delta of the June 100 putwill be approximately —25. If the delta of the June 100 put is -60, the delta of the June100 call will be approximately 40. Ignoring the positive sign associated with a call andthe negative sign associated with a put, the deltas of calls and puts with the sameunderlying contract, expiration date, and exercise price will always add up to approxi-mately 100. As we will see, interest rates, as well as the possibility of early exercise, cancause the delta of a synthetic underlying position to be slightly more or less than 100.But for most practical calculations, traders tend to look at a synthetic underlying positionand mentally assign ita delta of 100.
Rearranging the components of a synthetic underling position, we can create fourother synthetic relationships:
synthetic long call = long an underlying contract + long putsynthetic short call = short an underlying contract + short put
synthetic long put = short an underlying contract + long callsynthetic short put = long an underlying contract + short call
Again, all options must expire at the same time and have the same exercise price.Each synthetic position has a delta approximately equal to its real equivalent and willtherefore gain or lose value at approximately the same rate as its real equivalent.
If the underlying instrument in our examples is a June futures contract, we cancreate six synthetic positions:
synthetic long June future = long June 100 call + short June 100 putsynthetic short June future = short June 100 call + Jong June 100 put
synthetic long June 100 call = long June future + long June 100 putsynthetic short June 100 call = short June future + short June 100 put
synthetic long June 100 put = short June future + long June 100 callsynthetic short June 100 put = long June future + short June 100 call
We need not create a synthetic position using the 100 exercise price. We canchoose any available exercise price. Along June 110 call together with a short June 110put is still a synthetic long June futures contract; however, at June expiration the Junefutures contract will be purchased at 110. A short June 95 call together with a long June95 put is a synthetic short June futures contract. But at expiration the June futurescontract will be sold at 95.Option Arbitrage 215
Suppose an underlying contract is trading at 102.00 and we want to take a longposition in the market. We can simply go into the underlying market and buy thecontract at 102.00. But we now have an additional choice. We can take a long syntheticposition by purchasing a June call and selling a June put at the same exercise price.Which of these strategies is best? As with any option strategy, the decision depends onthe prices of the options in the marketplace. Suppose the June 100 call is trading at 5.00and the June 100 put is trading at 3.00. If we buy the June 100 call for 5.00 and sellthe June 100 put for 3.00, we will show an immediate debit of 2.00. If at expiration theunderlying contract is at 110.00, we will show a credit of 10.00 when we exercise ourJune 100 call, for a total profit of 8.00. If we ignore interest considerations, this isidentical to the profit we would have realized had we instead bought the underlyingcontract at 102.00. This is shown in Figure 11-1.
Suppose the underlying contract is still trading at 102.00, but now the June 100call is trading at 4.90 and the June 100 put is trading at 3.05. If we take a synthetic longunderlying position by purchasing the June 100 call and selling the June 100 put, wewill show a debit of 1.85. Now if the underlying contract is at 110.00 at expiration wewill show a total profit of 8.15 (the debit of 1.85 from the option trades, plus the10-point credit when we exercise the 100 call]. This is .15 better than we would do bytaking a long position in the underlying contract at 102.00.
As long as the price of the June 100 call is exactly two points greater than the priceof the June 100 put, the profit or loss resulting from a synthetic position will be identical
 
Figure 11-120:
 
buy a 100 call at 5.00 and sell a 100 put at 3.00184 orbuy an underlying contract at 102.00
 
Profit & Loss9°
 
 
 
 
T —T T T80 85 90 95 100 105 410 415 120Underlying Price216 Chapter 11
to an actual position taken in the underlying contract at a price of 102.00. The differencebetween the call and put price is often referred to as the synthetic market. In the absenceof any interest or dividend considerations, the value of the synthetic market can beexpressed as:
call price — put price = underlying price — exercise price
If this equality holds, there is no difference between taking a position in the underlyingmarket, or taking an equivalent synthetic position in the option market. With the June100 call at 5.00 and the June 100 put at 3.00, we can write:
5.00 — 3.00 = 102.00 —- 100.002.00 = 2.00
There is no difference between the synthetic and its real equivalent.With the June 100 call at 4.90 and the June 100 put at 3.05, we can write:
4.95 — 3.05 # 102.00 — 100.001.90 ¥ 2.00
Here the synthetic side is cheaper, and we therefore prefer to take a long underlyingposition synthetically by purchasing the call and selling the put.
Now suppose the price of the June 100 call is 5.15 and the price of the June 100put is 2.90, with the underlying contract still trading at 102.00. We have:
5.15 = 2.90 102.00 — 100.002.25 # 2.00
Now the real side is cheaper, so we prefer to take a long position by purchasing theactual underlying contract. On the other hand, if we want to take a short underlyingposition, we will prefer to sell the underlying contract synthetically by selling the calland purchasing the put. If we do this, we are selling a contract which is only worth 2.00for 2.25.
The reader may have noted that the synthetic and real markets will he the samewhen the time value of the call and the time value of the put are identical. In ourexample, the synthetic and real markets are identical when the June 100 call and theJune 100 put both have the same amount of time value. If this is not true, there is alwaysa synthetic position which is either too cheap or too expensive with respect to its realequivalent.
The three-sided relationship between a call, a put, and its underlying contractmeans that we can always express the value of any one of these contracts in terms ofthe other two:
underlying price = call price — put price + exercise pricecall price = underlying price + put price — exercise priceput price = call price — underlying price + exercise price
This three-sided relationship is sometimes referred to as put-call parity.Option Arbitrage 217
CONVERSIONS AND REVERSALS
If we take a synthetic long or short underlying position, our primary concern, as withan actual underlying position, is the direction of the market. If the market moves in ourfavor, we expect to show a profit; if it moves against us, we expect to show a loss. If weexecute the synthetic at favorable prices we may gain more or lose less, but it is stillprimarily the direction of the market in which we are interested.
Suppose, as before, the underlying contract is trading at 102.00, the June 100 callis trading at 5.10, and the June 100 put is trading at 2.85. The synthetic market shouldbe 2.00 (underlying price less the exercise price}, but is actually 2.25. If we wanted totake a short position in the underlying market, we would certainly prefer to do sosynthetically (sell the call, buy the put}. Suppose we are not interested in taking adirectional position. Is there any way we can profit from the difference in price betweenthe real and synthetic markets?
When a trader identifies two contracts which are essentially the same but whichare trading at different prices, the natural course is to execute an arbitrage by purchasingthe cheaper contract and selling the more expensive. Since the synthetic position andits real equivalent are essentially the same, and since they are trading at different prices(2.25 vs. 2.00), an option trader might try to purchase the cheaper contract (theunderlying contract} and sell the more expensive (the synthetic equivalent). That is, hemight try to purchase the underlying contract for 102.00, and simultaneously sell thecall for 5.10 and buy the put for 2.85. The cash flows from these transactions are:
Transaction Cash flowunderlying purchase ~102.00call sale +5.10put purchase -2.85exercise or assignment (at 100) at expiration +100.00total 4.25
No matter what happens in the underlying market, the underlying position will doexactly .25 better than the synthetic position, The entire position will therefore show aprofit of .25, regardless of movement in the underlying market. This is shown in Figure11-2.
The foregoing position, where the purchase of an underlying contract is offset bythe sale of a synthetic position, is known as a conversion. The opposite position, wherethe sale of an underlying contract is offset by the purchase of a synthetic position, isknown as a reverse conversion or, more commonly, a reversal. A reversal would beprofitable if the underlying contract were trading at 102.00, and the difference betweenthe prices of the June 100 call and June 100 put were less than 2.00. For example, withthe June 100 call at 4.90 and the June 100 put at 3.05, the synthetic price is 1.85. Bypurchasing the synthetic at 1.85 (buy the call at 4.90, sell the put at 3.05}, and sellingthe underlying contract at 102.00, the reverse conversion would lock in a profit of .15.This is shown in Figure 11-3.218 Chapter I 1
 
Figure 11-23Y vMw short a 100 call at 5.10 oa24 . X long a 100 put at 2.85 a“os ~~“ (synthetic short) ass “* a
combined positions
(conversion) a aS o— .
 
 
 
 
 
Qao6 ~? _of ee .# “ ‘° ~a ws“ NS“14 a“ ~N. . ~sa - ~. “ . ~N long an underlying NN-24 o “ contract at 102.00 _—»+3 T TT T T T99 100 101 102 103 104 105Underlying PriceSummarizing:conversion = long underlying + synthetic short underlying= long underlying + short call + long putreversal = short underlying + synthetic long underlying
= short underlying + long call + short put
As before, we assume that the call and the put have the same exercise price andexpiration date.
Conversions and reversals are classified as arbitrage strategies because of theirsimilarity to traditional arbitrage. Typically, an arbitrageur will attempt to simultane-ously buy and sell the same items in different markets to take advantage of pricediscrepancies between the two markets. He might, for example, buy gold in New Yorkfor $389 per ounce, and sell it in London for $392 per ounce. The profit margin of $3,while small, is secure since the risk of a trade in one market is almost immediately offsetby an identical but opposing trade in another market. Unlike a speculator, who usuallyhopes to make a large profit on a smal! number of trades, an arbitrageur hopes to makea small profit on a large number of trades. An arbitrageur is willing to do much greatersize than the speculator because the risk is much smaller.
Like traditional arbitrages, conversions and reversals involve buying and selling thesame thing in different markets. A conversion involves buying the underlying instru-Option Arbitrage 219
 
 
 
 
 
 
Figure 11-334 SN short an underlying aSS contract at 102.00 “ON “~ -SS -14 combined position ,(reversal) NOg SN . oBS 0 - sane a —= . s5 \o SYsyN17 SN- S“ SN“ ~~ long a 100 call at 4.90 ‘2 short a 100 put at 3.05 ~,-24 eo (synthetic long) NN> , ‘“-3 T T T T T99 100 101 102 103 104 105
Underlying Price
ment, whether a stock or commodity, in the underlying market, and selling theunderlying instrument, synthetically, in the option market. A reversal involves sellingthe underlying instrument in the underlying market, and buying the underlying instru-ment, synthetically, in the option market. The profitability of these strategies is deter-mined by the relationship between the synthetic price and the actual underlying price.Synthetic positions are often used to execute conversions and reversals, so trader'ssometimes refer to the synthetic market (the difference between the call price and putprice) as the conversion/reversal market.
All experienced traders are familiar with the price relationship between a syntheticposition and its underlying contract, so that any imbalance in the conversion/reversalmarket is likely to be short-lived. If the synthetic is overpriced, all traders will want toexecute a conversion (buy the underlying, sell the call, buy the put). If the synthetic isunderpriced, all traders will want to execute a reversal (sell the underlying, buy the call,sell the put). Such activity, where everyone is attempting to do the same thing, willquickly force the synthetic market back to equilibrium. Indeed, imbalances in theconversion/teversal market are usually small and rarely last for more than a fewseconds. When imbalances do exist, an option trader is usually willing to executeconversions and reversals in very large size because of the low risk associated with suchstrategies.220 Chapter 11
We initially broke down a synthetic position into its component cash flows,enabling us to identify the basic relationship between a synthetic and its underlyinginstrument:
call price — put price = underlying price — exercise price
However, when and how the component cash flows occur can alter this relationship.Since there can be different settlement procedures associated with different underlyinginstruments as well as with options, in order to calculate the exact value of a syntheticposition we need to consider how these settlement procedures affect the basic syntheticrelationship.
Futures Option Markets
If the cash flow resulting from an option trade and a trade in the underlying instrumentis identical, the synthetic relationship is simply:
call price — put price = underlying price — exercise price
This will be true if interest rates are zero, or in futures markets where both theunderlying contract and options on that contract are subject to futures-type settlement.In the latter case there is no interest component because no cash changes hands wheneither the futures contract or the options on that contract are traded. This is currentlythe procedure on many non-U.S. exchanges.
In futures markets, where options are subject to stock-type (cash) settlement, whileno cash will change hands when the futures contract is traded, there will be anexchange of cash when options are traded. The interest considerations on the credit ordebit will therefore have to be taken into consideration when calculating the value of asynthetic position.
Using our earlier example, with the June futures contract at 102.00, the June 100call at 5.00, and the June 100 put at 3.00, we have:
call price — put price = underlying price — exercise price5.00 -3.00 = 102.00 - 100.002.00 = 2.00
At these prices there does not appear to be any profit opportunity from either aconversion or reversal. If we do either of these strategies, the credit and debits from thetwo sides of the equation exactly offset each other.
Suppose, however, that we decide to do a reversal by selling the underlying futurescontract for two points (underlying price minus exercise price) and buying the syntheticfor two points (call price minus put price). Suppose also that the options are subject tostock-type (cash) settlement, as is generally true in U.S. futures markets. We will lay out2.00 on the option trades, and we will not get this