1.003 Option delta0.55→0.594 Option gamma0.045 Position gamma 4  0.04  0.166 Position delta4  0.55  2.20→2.367 Conclusion: The position gamma of 0.16 estimates that a one-point stockprice rise will cause the position delta to decline by 0.16, and the actualdecline was 0.16 (from 2.20 to 2.36).Assumptions: Days to expiration, 60; volatility, 30%; interest rate, 5%; no dividends.
Short Put Example
Col 1
Col 2
Col 3
1 PositionShort 10 QRS 40 Puts0.81 each2 Stock price41.00→42.003 Option delta0.34→0.244 Option gamma0.115 Position gamma 10  0.11  1.106 Position delta10  0.34  3.40→2.407 Conclusion: The position gamma of 1.10 estimates that a one-point stockprice rise will cause the position delta to fall by 1.10, but the actual decreasewas 1.00 (from 3.40 to 2.40).Assumptions: Days to exp., 40; volatility, 25%; interest rate, 5%; no dividends.
Making less than the amount estimated by the original delta is adisadvantage for the call writer.
Position VegasA position with a positive vega will proﬁt if volatility rises and otherfactors remain constant. Table 4-13 shows that long option positionshave positive vegas. The vega of 0.52 of the long call position inTable 4-13 means that if volatility rises by 1 percent and other factorsremain constant, then the position of long four XYZ 80 Calls willproﬁt by 0.52, or $52, which is exactly equal to the actual increase invalue in row 5.
The Greeks • 127
Table 4-13
Positive Vega Positions—Long Calls and Long Puts
Long Call Example
Col 1
Col 2
Col 3
1 PositionLong 4 XYZ 80 Calls4.20 each →4.332 Volatility30%→31%3 Option vega0.134 Position vega4  0.130.525 Position value4  4.2016.80 debit →17.32 debit6 Conclusion: The position vega of 0.52 estimates that a 1 percent increase involatility will cause the position to proﬁt by $52, and the actual proﬁt was $52(from 16.80 debit to 17.32 debit).Assumptions: Stock price, 80; days to exp., 60; interest rate, 5%; no dividends.
Long Put Example
Col 1
Col 2
Col 3
1 PositionLong 10 QRS 40 Puts 0.81 each→0.862 Volatility25%→26%3 Option vega0.054 Position vega10  0.050.505 Position value 10  0.818.10 debit →8.60 debit6 Conclusion: The position vega of 0.50 estimates that a 1 percent increase involatility will cause the position to proﬁt by $50, and the actual proﬁt was $50(from 8.10 debit to 8.60 debit).Assumptions: Stock price, 41; days to exp., 40; interest rate, 5%; no dividends.
A position with a negative vega will lose if volatility rises and proﬁtif volatility declines, assuming that other factors remain constant.Table 4-14 shows that short option positions have negative vegas. Thevega of the short put position of 0.50 in Table 4-14 means that ifvolatility rises by 1 percent and other factors are the same, then thisposition will lose 0.50, or $50. This estimate exactly equals the loss inrow 5 as the value of the short option position rises from 8.10 to 8.60.The negative vega also estimates that if volatility falls by 1 percent,then the position will proﬁt by $50.
Position ThetasThe theta of a position estimates whether the passing of time will causea position to sustain a proﬁt or a loss. Since option values decay over
128 • Trading Options As a Professional
Table 4-14
Negative Vega Positions—Short Calls and Short Puts
Short Call Example
Col 1
Col 2
Col 3
1 PositionShort 4 XYZ 80 Calls4.20 each→4.332 Volatility30%→31%3 Option vega0.134 Position vega4  0.13 0.525 Position value 4  4.20 16.80 credit →17.32 credit6 Conclusion: The position vega of 0.52 estimates that a 1 percent increase involatility will cause the position value to lose $50, and the actual loss was $50(from 16.80 credit to 17.23 credit).Assumptions: Stock price, 80; days to exp, 60; interest rate, 5%; no dividends.
Short Put Example
Col 1
Col 2
Col 3
1 PositionShort 10 QRS 40 Puts0.81 each→0.862 Volatility25%→26%3 Option vega0.054 Position vega 10  0.05 0.505 Position value 10  0.818.10 credit →8.60 credit6 Conclusion: The position vega of 0.50 estimates that a 1 percent increase involatility will cause the position to lose $50, and the actual loss was $50(from 8.10 credit to 8.60 credit).Assumptions: Stock price, 41; days to exp., 40; interest rate, 5%; no dividends.
time, short option positions proﬁt if time passes, so those positions havepositive thetas. The short call position in Table 4-15 has a position thetaof 0.16 (row 4), which estimates that the position will proﬁt by $16if time to expiration is reduced by “one unit.” In this example, one unitof time is one day, and the theta exactly estimates the change in position value from 16.80 to 16.64 as days to expiration decrease from60 to 59. The short put position has a theta of 0.10 (row 4), whichestimates that the position will proﬁt by $10 in one unit of time.A position with a negative theta will incur a loss if only the time toexpiration changes. Long option positions have negative thetas. InTable 4-16, the theta of the long call position of 0.16 means that ifonly time to expiration is reduced by one day, then the position willlose 0.16, or $16. The theta of 0.10 of the long put position meansthat $10 will be lost if one day passes and other factors remain constant.
The Greeks • 129
Table 4-15
Positive Theta Positions—Short Calls and Short Puts
Short Call Example
Col 1
Col 2
Col 3
1 PositionShort 4 XYZ 80 Calls4.20 each→4.162 Days to expiration60→593 Option theta0.044 Position theta4  0.04  0.165 Position value4 4.20 16.80 credit →16.64 credit6 Conclusion: The position theta of 0.16 estimates that the passing of one daywill cause the position to proﬁt by $16, and the actual proﬁt was $16 (from16.80 credit to 16.64 credit).Assumptions: Stock price, 80; volatility, 30%; interest rate, 5%; no dividends.
Short Put Example
Col 1
Col 2
Col 3
1 PositionShort 10 QRS 40 Puts0.81 each→0.802 Days to expiration40→393 Option theta0.014 Position theta 10  0.01  0.105 Position value 10 0.81 8.10 credit →8.00 credit6 Conclusion: The position theta of 0.10 estimates that the passing of one daywill cause the position to proﬁt by $10, and the actual proﬁt was $10 (from8.10 credit to 8.00 credit).Assumptions: Stock price, 41; volatility, 25%; interest rate, 5%; no dividends.
Position RhosThe rho of a position estimates whether the position will proﬁt orlose as the interest rate changes and other factors remain constant.Long calls and short puts have positive rhos. The long call position inTable 4-17 has a position rho of 0.28 (row 4), which estimates thatthe position will proﬁt by $28 if the interest rate rises by 1 percent.The position rho of the short put example in Table 4-17 is 0.20. Theactual proﬁt of the position of 10 short puts, however, is only $10, asthey decrease in price from 0.81 to 0.80. The difference is due torounding.Short calls and long puts have negative rhos. Table 4-18 shows thatthe short call example has a rho of 0.28, and this exactly estimates theloss as the short call value increases when the interest rate rises from 5to 6 percent. The rho of 0.20 of the long put position in Table 4-18
130 • Trading Options As a Professional
Table 4-16
Negative Theta Positions—Long Calls and Long Puts
Long Call Example
Col 1
Col 2
Col 3
1 PositionLong 4 XYZ 80 Calls4.20 each→4.162 Days to expiration60→593 Option theta0.044 Position theta 4  0.04 0.165 Position value 4 4.2016.80 debit→16.64 debit6 Conclusion: The position theta of 0.16 estimates that the passing of one daywill cause the position to lose $16, and the actual loss was $16 (from 16.80debit to 16.64 debit).Assumptions: Stock price, 80; volatility, 30%; interest rate, 5%; no dividends.
Long Put Example
Col 1
Col 2
Col 3
1 PositionLong 10 QRS 40 Puts0.81 each→0.802 Days to expiration40→393 Option theta0.014 Position theta 10  0.01 0.105 Position value 10 0.818.10 debit→8.00 debit6 Conclusion: The position theta of 0.10 estimates that the passing of one daywill cause the position to lose $10, and the actual loss was $10 (from 8.10debit to 8.00 debit).Assumptions: Stock price, 41; volatility, 25%; interest rate, 5%; no dividends.
over estimates the loss in value as the interest rate rises. In row 5 of thelong put example in Table 4-18, the price of each put decreases from0.81 to 0.80, and this causes the value of 10 long puts to decrease. Theactual loss of 0.10 differs from the estimated loss of 0.20 because ofrounding.
Position Greeks SummarizedTable 4-19 matches long and short options with positive and negativeposition Greeks. Long calls have positive deltas, gammas, and vegasand negative thetas. Short calls have negative deltas, gammas, andvegas and positive thetas. Long puts have negative deltas and thetas,and positive gammas and vegas. Short puts have positive deltas andthetas and negative gammas and vegas.
The Greeks • 131
Table 4-17
Positive Rho Positions—Long Calls and Short Puts
Long Call Example
Col 1
Col 2
Col 3
1 PositionLong 4 XYZ 80 Calls4.20 each→4.272 Interest rates5%→6%3 Option rho0.074 Position rho4  0.07  0.285 Position value 4 4.20  16.80 debit →17.08 debit6 Conclusion: The position rho of 0.28 estimates that a 1 percent increase ininterest rates will cause the position to proﬁt by $28, and the actual proﬁt was$28 (from 16.80 debit to 17.08 debit).Assumptions: Stock price, 80; days to exp., 60; volatility, 30%; no dividends.
Short Put Example
Col 1
Col 2
Col 3
1 PositionShort 10 QRS 40 Puts0.81 each→0.802 Interest rates5%→6%3 Option rho0.024 Position rho10  0.02  0.205 Position value 10 0.81 8.10 credit →8.00 credit6 Conclusion: The position rho of 0.20 estimates that a 1 percent increase ininterest rates will cause the position to proﬁt by $20, but the actual proﬁt was$10 (from 8.10 credit to 8.00 credit) owing to rounding.Assumptions: Stock price, 41; days to exp., 40; volatility, 25%; no dividends.
No two rows in Table 4-19 are the same. Each option position hasits own unique sensitivities to changes in price of the underlying,volatility, and time to expiration, and the sensitivities vary dependingwhether an option is in the money, at the money, or out of the money.Although confusing at ﬁrst, an understanding of position Greeks separates good traders from bad. The Greeks provide an estimate of howa position will change in value as market conditions change. Interpreting such an estimate is the key to selecting appropriate strategies.
SummaryThe Greeks are tools used by option traders to estimate the proﬁt orloss impact of changes in market conditions. Delta is an estimate of
132 • Trading Options As a Professional
Table 4-18
Negative Rho Positions—Short Calls and Long Puts
Short Call Example
Col 1
Col 2
Col 3
1 PositionShort 4 XYZ 80 Calls4.20 each→4.272 Interest rates5%→6%3 Option rho0.074 Position rho4  0.07 0.285 Position value4 4.20 16.80 credit → 17.08 credit6 Conclusion: The position rho of 0.28 estimates that a 1 percent increase ininterest rates will cause the position to lose $28, and the actual loss was $28(from 16.80 credit to 17.08 credit).Assumptions: Stock price, 80; days to exp, 60; volatility, 30%; no dividends.
Long Put Example
Col 1
Col 2
Col 3
1 PositionLong 10 QRS 40 Puts0.81 each→0.802 Interest rates5%→6%3 Option rho0.024 Position rho10  0.02 0.205 Position value 10 0.818.10 debit→8.00 debit6 Conclusion: The position rho of 0.20 estimates that a 1 percent increase ininterest rates will cause the position to lose $20, but the actual loss was $10(from 8.10 credit to 8.20 credit) owing to rounding.Assumptions: Stock price, 41; days to exp., 40; volatility, 25%; no dividends.
the change in option theoretical value given a one-point change inprice of the underlying instrument. Gamma is a measure of changein delta for a one-point change in the price of the underlying. Vega isan estimate of the change in option value resulting from a one percentage point change in volatility, and theta is an estimate of thechange in option value resulting from a one-unit change in time toexpiration. Traders who use computer programs should be sure toknow the deﬁnition of one unit of time used by the program.The Greeks change as market conditions change, and these constant changes complicate the job of estimating how position valueswill behave as market conditions change. The absolute values of deltasof in-the-money options are greater than 0.50 initially and increasetoward 1.00 as expiration approaches. The absolute values of deltas
The Greeks • 133
Table 4-19
Summary of Position Greeks
Position
Delta
Gamma
Vega
Theta
Long CallShort CallLong PutShort Put




 indicates that a position will proﬁt, or beneﬁt, from an increase in an input and incur a loss from, orbe hurt by, a decrease, assuming that other inputs remain constant. indicates that a position will incur a loss from, or be hurt by, an increase in an input and proﬁt, orbeneﬁt, from a decrease, assuming other inputs remain constant.
of at-the-money options remain near 0.50 as expiration approaches,whereas the absolute values of deltas of out-of-the-money options startat less than 0.50 and decrease toward 0 as expiration approaches.Gammas are biggest for at-the-money options and tend to increaseas expiration approaches. Vegas are biggest for at-the-money optionsand decrease as expiration approaches. Thetas are smallest (largestabsolute value) for at-the-money options. The behavior of thetas asexpiration approaches differs for at-the-money options versus in-themoney or out-of-the-money options.Plus signs (“”) and minus signs (“”) indicate “long” or “short”when associated with a quantity of options in a position. They indicate “positive correlation” or “negative correlation” when associatedwith Greeks of individual options, and they indicate “proﬁt” or “loss”when associated with position Greeks.Long calls and short puts have positive deltas. If other factors remainconstant, these positions enjoy a proﬁt with a rise in price of the underlying instrument and suffer a loss with a decline. Short calls and longputs are positions with negative deltas and proﬁt from a stock pricedecline.Positive gamma means that the delta of a position changes in thesame direction as the change in price of the underlying. Stock up,delta up, and stock down, delta down. Long calls and long puts havepositive gammas. Negative gamma means that the delta of a positionchanges in the opposite direction from the change in price of the
134 • Trading Options As a Professional
underlying. Stock up, delta down, and stock down, delta up. Shortcalls and short puts have negative gammas.Positive vega means that a position will proﬁt if volatility rises andlose if volatility declines, assuming that other factors remain constant.Long calls and long puts have positive vegas. Negative vega means thata position will lose if volatility rises and gain if volatility declines. Shortcalls and short puts have negative vegas.Long calls and long puts have negative thetas because they losemoney as time passes toward expiration and other factors remain constant. Short calls and short puts have positive thetas. They proﬁt astime passes toward expiration and other factors remain constant.Knowing how to interpret the Greeks is a valuable skill. It helpsa trader to anticipate how strategies will perform as market conditionschange.
Chapter 5SYNTHETICRELATIONSHIPShe prices of calls, puts, and the underlying stock are linked to eachother by a relationship known as put-call parity. One corollary ofput-call parity is that a position in one instrument (stock, calls, or puts)can be replicated by a two-part position using the other two instruments. These two-part positions are known as synthetic positions.Another corollary is that if the put-call parity relationship is not met,then there will be arbitrage opportunities.This chapter will ﬁrst explain the six basic synthetic positions without taking into consideration interest rates or dividends. It will then discuss the put-call parity equation and ﬁnish with the impact of interestrates and dividends on option prices. The use of synthetic relationshipsto create arbitrage strategies will be discussed in the next chapter.
T
Synthetic RelationshipsThere are six real trading positions: long and short stock, long andshort call, and long and short put. Traders can replicate each of thesereal positions with a synthetic position that consists of the other twoinstruments. For example, a two-part position consisting of a call position and a put position can replicate a stock position. Similarly, a call• 135 •
136 • Trading Options As a Professional
position can be replicated with a combined stock and put position,and a combined stock and call position can replicate a put position.A synthetic position is a two-part position that has the same theoretical risk, the same theoretical break-even point, and the same theoretical proﬁt potential as a real position.In theory, traders should not prefer one type of position to another.In practice, however, there are some differences that do inﬂuence preference. Primarily because synthetic positions have two bid-ask spreadsinstead of one, two transaction costs instead of one, and different margin requirements, the trading of synthetic relationships typically fallsinto the realm of the professional trader. Understanding synthetic relationships is the ﬁrst step in learning about arbitrage strategies.The term effective price means the stock price that takes intoaccount the option premium. Option exercise or assignment createsa stock transaction at the strike price, but the strike price does notaccurately reﬂect the full price to the trader of synthetic positions. If,for example, a 100 Call purchased for 2.00 per share is exercised, thenthe effective price of the resulting long stock position is 102 per share.This term will be used throughout this chapter.The introductory explanation of synthetic relationships that followsmakes four simplifying assumptions. First, it assumes that there is aone-to-one relationship between options and shares of stock. In otherwords, each option covers one share of stock rather than 100 shares.Second, it assumes an interest rate of zero so that time to expirationis irrelevant. Third, no commissions and no dividends are assumed.Fourth and ﬁnally, it assumes an available amount of capital equal tothe stock price. This amount of capital is necessary because, in thereal world, purchasers of stock and options must pay cash, and shortsellers of stock and options must post margin.The assumptions for the examples that follow are a stock price of100, a 100 Call price of 3.00, and a 100 Put price of 3.00. It isalso assumed that $100 of cash is held in reserve to either buy thestock or serve as the margin deposit for a short stock or short optionposition.
Synthetic Relationships • 137
Synthetic Long StockA synthetic long stock position is created with a two-part option position consisting of a long call and a short put, where the call and puthave the same underlying, the same strike price, and the same expiration date. Table 5-1 and Figure 5-1 illustrate that the two-part positionconsisting of a long 100 Call at 3.00 and a short 100 Put at 3.00 is equivalent to long stock at $100. Table 5-1 shows proﬁt-and-loss calculationsfor each component of the synthetic position, for the combined position, and for the corresponding real position at various stock prices. Figure 5-1 graphs the component positions and the combined position.Table 5-1
Synthetic Long Stock: Long 100 Call at 3.00 and Short 100 Putat 3.00 Compared with Long Stock at 100Col 1
Col 2
Col 3
Col 4
Col 5
Short 100Put @ 3.00
CombinedP/(L)
Long Stock@ 100
3.003.003.002.007.00
7.002.003.003.003.00
10.005.00-0 5.0010.00
10.005.00-05.0010.00
95
100
Stock PriceLong 100at Expiration Call @ 3.00Row 1Row 2Row 3Row 4Row 5
9095100105110
10
P/(L)
50–5–10
Long 100 Call@ 3Figure 5-1
Stock PriceShort 100 Put @ 3
105Long Stock @ 100
Synthetic Long Stock = Long Call and Short Put
138 • Trading Options As a Professional
Row 4 in Table 5-1 assumes a stock price of $105 at option expiration (column 1). The 100 Call purchased for 3.00 makes a proﬁt of2.00 per share (column 2), and the 100 Put sold at 3.00 generates aproﬁt of 3.00 per share (column 3). Adding the two proﬁts yields a totalproﬁt of 5.00 per share (column 4), the same proﬁt per share of 5.00that is earned from stock purchased at $100 (column 5). In any rowin Table 5-1, the combined result of the long call and short put in column 4 is the same result as purchasing stock at $100 per share in column 5. The results are the ﬁrst indication that the two-part positionof long call and short put equals a long stock position.
Synthetic Long Stock—Mechanics at ExpirationConsider what happens via exercise and assignment to the option position at expiration. Three outcomes are possible: The closing stockprice at expiration can be above the strike price of 100, below the strikeprice, or exactly at the strike price. Each of these possibilities will beconsidered.If the stock price closes above $100 at expiration, then the short 100Put is out of the money and expires worthless. The long 100 Call, however, is in the money and is exercised. Exercising a call creates a stockpurchase transaction, and it is assumed that the $100 cash reserve isused to pay for the stock. The result is that a long stock position is created. The effective price of the long stock position is $100 per share.The effective stock price is the price of the stock that takes intoaccount the total or net option premiums. In general, an effectivestock price is calculated by adding or subtracting the net option premium from the strike price. In this example, the net option premiumis zero because the call was purchased for 3 per share and the put wassold at 3 per share. Adding zero to the strike price of 100 yields aneffective stock price of $100. Although this calculation yields an effective stock price that is equal to the strike price, the result would be different if either the call price, the put price, or both were different.The conclusion is that with the stock price above the strike price at
Synthetic Relationships • 139
expiration, the synthetic long stock position described in Table 5-1 andFigure 5-1 becomes a real long stock position with exactly the sameproﬁt as buying stock at $100 per share.If the stock price falls below $100 at expiration, then the long 100Call is out of the money and expires worthless. The short 100 Put,however, is in the money and is assigned. Assignment of a short putcreates a stock purchase transaction, and as above, it is assumed thatthe $100 cash reserve is used to pay for the stock. The result is that along stock position is created at an effective price of $100. Thus, withthe stock price below the strike price at expiration, the synthetic longstock position described in Table 5-1 and Figure 5-1 becomes areal long stock position with exactly the same proﬁt as buying stock at$100 per share.The third possible outcome is that the stock price is exactly at $100at expiration. If this happens, then both the 100 Call and 100 Putexpire worthless, and the result is no position with no proﬁt or loss.However, the $100 cash reserve could be used to purchase a real longstock position. If stock were purchased at $100 after both optionsexpired worthless, the result would be a long stock position with noproﬁt or loss, which is the same result as buying real stock originallyat $100 per share.The conclusion is that the two-part position of long call and shortput behaves exactly the same as a real long stock position. It thereforedeserves its name, synthetic long stock.
Synthetic Short StockA synthetic short stock position is created with a short call and a longput. Table 5-2 and Figure 5-2 illustrate that a short 100 Call at 3.00 anda long 100 Put at 3.00 are equivalent to short stock at $100. Table 5-2shows the proﬁt-and-loss calculations for each component of the synthetic position, for the combined position, and for the correspondingreal position at various stock prices. Figure 5-2 graphs the componentsand the combined position.
140 • Trading Options As a Professional
Table 5-2
Synthetic Short Stock: Short 100 Call at 3.00 and Long 100 Put at3.00 Compared with Short Stock at 100Col 1
Col 4
Col 5
Stock Price Short 100Long 100at Expiration Call @ 3.00 Put @ 3.00
CombinedP/(L)
Short Stock@ 100
90
3.00
7.00
10.00
10.00
Row 2
95
3.00
2.00
5.00
5.00
Row 3
100
3.00
3.00
-0-
-0-
Row 1
Col 2
Col 3
Row 4
105
2.00
3.00
5.00
5.00
Row 5
110
7.00
3.00
10.00
10.00
95
100
10
P/(L)
50–5–10105
Stock PriceShort100 Call@ 3Figure 5-2
Long100 Put @ 3
Short Stock @ 100
Synthetic Short Stock = Short Call and Long Put
Row 2 in Table 5-2 assumes a stock price of $95 at expiration (column 1). The 100 Call sold at 3.00 makes a proﬁt of 3.00 per share(column 2), and