paid at expiration  strike price Gross loss Interest income  54.95  (0.04  60/365) Proﬁt before transaction costs Transaction costs Proﬁt before dividend Dividend Net proﬁt
54.9555.00 (0.05) 0.36 0.310.04 0.270.22 0.05
Analysis of Time Values10 Time value of 55 Call  price  intrinsic  4.20  2.7011 Time value of 55 Put  price  intrinsic  1.45  0.0012 Time value of call  time value of put
 1.50 1.45 0.05
123456789
Conclusion: For a reverse conversion, the difference between the time value of thecall (11) and the time value of the put (12) equals the absolute value of thegross loss (3). The net credit, however, is increased by the dividend.
Pricing Reverse Conversions by Strike PriceTable 6-18 demonstrates the pricing implication of changing strikeprices in reverse-conversion positions. In essence, the net creditrequired changes as the strike price changes, and as a result, theamount by which the time value of the call must exceed the timevalue of the put changes.In row 1 of Table 6-18, the strike price is 45 (column 1), the DPVof the strike is 44.70 (column 2), the difference between the strikeprice and the DPV is 0.30 (column 3), and given costs and a targetproﬁt of 9 cents per share (column 4), the time value of the 45 Callmust be greater than the time value of the 45 Put by 39 cents (column5). Compare this 39-cent difference to the 42-cent difference in column 5 of row 2, where the strike price is 50. Note also that as the strikeprice rises by $5.00 in each row, the difference between call time valueand put time value increases by 3 cents. The difference increases as
188 • Trading Options As a Professional
Table 6-18
Pricing Reverse Conversions by Strike PriceCol 1
Col 2
Col 3
Col 4
Col 5
Row
StrikePrice
DPVof Strike
StrikeMinusDPV ofStrike
Costs PlusTarget Proﬁt
Time Value ofCall MinusTimeValueof Put
12345
4550556065
44.7049.6754.6459.6164.58
0.300.330.360.390.42
0.090.090.090.090.09
0.390.420.450.480.51
DPV  discounted present value.Assumptions: 60 days to expiration, interest rate of 4%, no dividends, costs of 0.04, target proﬁt of 0.05.Sample DPV calculation: 55  [1  (0.04  60/365)]  $54.64.
the strike price increases because interest income increases. With ahigher strike price, more funds are available for investment.
Box SpreadsA box spread is a four-part options-only arbitrage strategy consisting ofa long call and short put at one strike price and a short call and longput at a second strike price. There are two variations of this strategy.A long box spread or, simply, a long box is established for a net costor net debit. A short box spread or, simply, a short box is establishedfor a net credit.
The Long Box SpreadA long box spread consists of a long call and short put at a lower strikeprice and a short call and long put at a higher strike price. A long boxis established for a net debit and is proﬁtable when the differencebetween the strike prices minus the cost of the position is greater thanthe cost of carry. As with a conversion, this strategy assumes that atrader borrows the net cost of a long box.
Arbitrage Strategies • 189
Table 6-19 and Figure 6-3 illustrate a long box spread that yields agross proﬁt of 75 cents per share at expiration before transaction costsand cost of carry. The four-part position consists of one long 90 Callpurchased for 6.50, one short 90 Put sold at 2.00, one short 100 Callsold at 2.25, and one long 100 Put purchased for 7.00.This strategy can be described in two ways. First, it can be describedas the combination of long synthetic stock at the lower strike price and
Table 6-19
The Long Box Spread: Long 90 Call at 6.50, Short 90 Put at2.00, Short 100 Call at 2.25, and Long 100 Put at 7.00
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Row
StockPriceat Exp
Long 90Call @6.50
Short 90Put @2.00
Short 100Call @2.25
Long 100Put @7.00
CombinedP/(L)
1234
8090100110
6.506.503.5013.50
8.002.002.002.00
2.252.252.257.75
13.003.007.007.00
0.750.750.750.75
15
Profit/(Loss)
10
5
090
100
–5
–10Stock Price
Figure 6-3
Long 90 Call @ 6.50
Short 90 Put @ 2.00
Long 100 Put @ 7.00
Long Box Spread
The Long Box Spread
Short 100 Call @ 2.25
190 • Trading Options As a Professional
short synthetic stock at the higher strike price. Second, it can bedescribed as the combination of a bull call spread and a bear putspread with the same strike prices.Matching the calls and puts with the same strike price together, thelong 90 Call and short 90 Put create a synthetic long stock position,and the short 100 Call and long 100 Put create a synthetic short stockposition. Matching the calls together and the puts together, the90 Call and the 100 Call create a bull call spread, and the 100 Putand the 90 Put create a bear put spread.The net proﬁt or loss of a box spread will equal the differencebetween the strike prices minus the cost of creating the position,including transaction and borrowing costs.Just as a conversion is compared with buying a Treasury bill at a discount and receiving full value at maturity, so too does a long box costless than the difference between the strike prices to establish. Thetrader receives that difference at expiration.
Long Box Spread—Outcomes at ExpirationA long box position has ﬁve possible outcomes at expiration. As summarized in Table 6-20, the stock price can be below the lower strikeprice, exactly at the lower strike price, between the two strike prices,exactly at the higher strike price, or above the higher strike price.If the stock price closes below the lower strike price at expiration, outcome 1 in Table 6-20, then both calls are out of the money and expireworthless. Both puts, however, are in the money. The long 100 Put isexercised, creating a stock sale transaction at 100, and the short 90 Putis assigned, creating a stock purchase transaction at 90. The simultaneous purchase and sale of stock incurs two commission costs but doesnot result in a stock position. However, a net amount equal to the difference between the strike prices, or 10 in this example, is received.This amount pays back the borrowed funds, the transaction costs, theborrowing costs, and the net proﬁt, if any. If the amount received is lessthan the loan plus costs, however, then the result is a loss.
Arbitrage Strategies • 191
Table 6-20LongShortShortLong
The Long Box Spread at Expiration
1 XYZ 90 Call @ 6.501 XYZ 90 Put @ 2.001 XYZ 100 Call @ 2.251 XYZ 100 Put @ 7.00
 Long stock (synthetically)@ 94.50 Short stock (synthetically)@ 95.25
Combined position at expiration  no position (gross proﬁt 0.75)Five Possible Outcomes at Expiration#1 XYZ  90 (Lowest Strike)
#2 XYZ  90 (Lower Strike)
90 Call expires worthless100 Call expires worthless90 Put assigned (buy stock)100 Put exercised (sell stock)Result: No position
90 Call expires worthless100 Call expires worthless90 Put expires worthless100 Put exercised (sell stock)Result: Short stock
#3 90  XYZ  100 (Between Strikes)
#4 XYZ  100 (Higher Strike)
90 Call exercised (buy stock)100 Call expires worthless90 Put expires worthless100 Put exercised (sell stock)Result: No position
90 Call exercised (buy stock)100 Call expires worthless90 Put expires worthless100 Put expires worthlessResult: Long stock
#5 XYZ  100 (Highest Strike)90 Call exercised (buy stock)100 Call assigned (sell stock)90 Put expires worthless100 Put expires worthlessResult: No position
If the stock price closes between the strike prices, outcome 3, thenthe short call and short put are out of the money and expire worthless. The long call and long put, however, are in the money and areexercised. Exercising the 90 Call creates a stock purchase transactionat 90, and exercising the 100 Put creates a stock sale transaction at100. As in the preceding outcome, simultaneously buying and sellingstock creates a net result equal to the difference between the strikeprices, or 10.
192 • Trading Options As a Professional
If the stock price rises above the higher strike price, outcome5, then both puts are out of the money and expire worthless. Bothcalls, however, are in the money. The long 90 Call is exercised, creating a stock purchase transaction at 90, and the short 100 Call isassigned, creating a stock sale transaction at 100. Again, the traderwould receive an amount equal to the difference between the strikeprices of 10.Therefore, if the stock price is below the lower strike price, betweenthe strike prices, or above the higher strike price at expiration, thenthe long box position will be closed. As a result, a trader will not beleft with an open stock position and the attendant risk that couldreduce the proﬁt or create a loss.
Long Box—Double Pin RiskA stock price closing exactly at one of the strike prices at expiration,outcomes 2 and 4, creates a pin-risk situation. The out-of-the-moneyoption will expire worthless, and in theory, both the at-the-money calland the at-the-money put will expire worthless. The in-the-moneyoption from the other strike price, however, will create a stock position with pin risk.If the stock price closes exactly at the lower strike price of 90, outcome 2 in Table 6-20, then exercise of the in-the-money 100 Put willcreate a short stock position. If the stock price closes exactly at thehigher strike price of 100, outcome 4, then exercise of the in-themoney 100 Call will create a long stock position. As with pin risk inconversion and reverse conversion positions, it is impossible to predicthow many of the short at-the-money options will be assigned. Consequently, market makers typically respond by exercising half the longat-the-money options, hoping that only half the short options willbe assigned. Undoubtedly, more or less than half the short optionswill be assigned, and a trader will need to close a stock position onMonday. As with conversions and reversals, all a trader can do is hopethat the experience will not be too costly.
Arbitrage Strategies • 193
Pricing a Long Box SpreadGiven the high probability that a box spread will result in a cash payment equal to the difference between the strike prices at expiration,the value of a long box spread is equal to the DPV of the differencebetween the strike prices less costs and a proﬁt margin. The next example discusses a box spread involving 100 and 110 strike prices, not 90and 100 strike prices, as in the preceding example.Tables 6-21 through 6-23 show, in three parts, how a 100–110 boxspread might be priced. Table 6-21 states the assumptions. The price ofthe 100 Call (1) is 9.10. The price of the 100 Put (2) is 2.30. The priceof the 110 Call (3) is unknown. The price of the 110 Put (4) is 6.70. Theborrowing rate (5) of 5 percent and the days to expiration (6) of 60 leadto the DPV of the difference between the strike prices (7) of 9.92. Thereare also trading costs (8 and 9) of 1 cent per share to trade an option andfor option exercise or assignment. The total costs therefore are 6 cents,4 cents for opening the four-part position plus 2 cents for exercise orassignment of the in-the-money options at expiration that close the position. Finally, the target proﬁt (10) is 5 cents per share in this example.Table 6-21
Assumptions:
Pricing a Long 100–110 Box Spread—Part 1: Stating theAssumptions1 Price of 100 Call9.102 Price of 100 Put2.303 Price of 110 Call?4 Price of 110 Put6.705%5 Borrowing rate6 Days to expiration607 DPV of difference between thestrike prices  (110 – 110) (1  0.05  60/365)9.928 Option cost (4 options,0.01/share)0.04 per share9 Exercise/assignment(2 options)0.02 per share10 Target proﬁt0.05 per share
Question: What is the sale price of the 110 Call?
Transactioncosts  0.06
194 • Trading Options As a Professional
Given the nine known assumptions, the unknown price of the110 Call must be determined. The question is, “What is the sale priceof the 110 Call that yields the target proﬁt?”Table 6-22, which contains the second part of pricing a long box,calculates the sale price of the 110 Call in two steps. The ﬁrst steprequires calculating the net investment per share. In the case of a longbox, the net investment per share is the net cost of the position thatyields the target proﬁt if held to expiration. The net investment pershare equals the DPV of the difference between the strike prices (line7 in Table 6-21) minus the sum of costs plus target proﬁt (lines 8–10).The net investment per share is therefore 9.81. As with conversionsand reverse conversions, calculations are made on a per-share basisbecause this method makes the quantity of contracts irrelevant toobtaining the target proﬁt.Step 2 in Table 6-22 uses basic algebra to ﬁnd the price of the110 Call that makes the cost of the long box equal the net investment,and that price is 3.69.The third and ﬁnal part of pricing a long box spread involves analyzing the cash ﬂows, the net proﬁt, and the spread values, as shownin Table 6-23. The revenue (1) is the amount received at expiration,which is the difference between the strike prices, or 10.00 per share
Table 6-22
Pricing a Long 100–110 Box Spread—Part 2: Calculate SalePrice of 110 Call
Step 1: Calculate the net investment per share (NI)NI  DPV of difference between strikes minus sum of costs plus targetproﬁtNI  9.92  (0.06  0.05)  9.81Step 2: Sell the 110 Call at a price so that the net cost of four-part position(long 100 Call, short 100 Put, short 110 Call, long 110 Put) equalsthe NI in step 1.2-1 If  100 Call 100 Put  110 Call  110 Put   NI2-2 Then 110 Call  100 Call  100 Put  110 Put  NI2-3 Therefore,  110 Call  9.10  2.30  6.70  9.81  3.69
Arbitrage Strategies • 195
in this example. The cost of the position (2) is the net investment of9.81, and the difference between revenue and cost is the gross proﬁt(3) of 0.19 per share. The borrowing costs (4) of 0.08 are based on thenet investment and the borrowing rate. The gross proﬁt minus the borrowing costs leaves a proﬁt before transaction costs (5) of 0.11 pershare. Finally, subtracting transaction costs of 6 cents per share (6)results in the net proﬁt (7) of 5 cents per share.Lines 8 through 10 of Table 6-23 analyze the relationship of the values of the two spreads. In this case, the call spread has a value of 5.41(8), and the put spread has a value of 4.40 (9). The sum of the spreadvalues (10) therefore is 9.81, which equals the net investment.The conclusion of this three-part exercise is stated at the bottom ofTable 6-23. For a long box spread, the sum of the debit call spread plusthe debit put spread equals the net investment.
Relative Pricing and Box SpreadsThe concept of relative pricing applies to box spreads just as it does toconversions and reversals. Basically, if the value of the box spread andTable 6-23
Pricing a Long 100–110 Box Spread—Part 3: Analysis of CashFlows, Net Proﬁt, and Spread Values
1 Revenue  amount received at expiration  110 – 1002  Cost  net investment per share paid for position3  Gross proﬁt4  Borrowing costs  9.81  (0.05  60/365)5  Proﬁt before transaction costs6  Transaction costs7  Net proﬁt
10.00 9.81 0.19 0.08 0.11 0.06 0.05
Analysis of Vertical Spread Values8 Value of 100-110 call spread  9.10  3.699 Value of 100-110 put spread  6.70  2.3010 Sum of spread values
 5.41 4.40 9.81
Conclusion: For a long box spread, the sum of the debit call spread plus the debitput spread (10) equals the net investment per share (2).
196 • Trading Options As a Professional
the prices of three of the components are known, then the price of thefourth component can be calculated. Also, if the price of either thecall spread or put spread is known, then the other can be calculated.If, for example, the value of a 50–55 long box were 4.85, and if the50–55 Call spread were offered at 2.65, then a market maker wouldbid 2.20 for the 50–55 put spread. If someone were to sell the putspread at 2.20 so that the market maker bought it, that market makerthen would simply buy the call spread at 2.65 to complete the longbox for a net cost of 4.85 and thereby lock in a proﬁt.
The Short Box SpreadA short box spread consists of a short call and long put at a lower strikeprice and a long call and short put at a higher strike price. A short boxis established for a net credit and is proﬁtable when the credit receivedplus interest earned exceeds the difference between the strike pricesplus costs.Table 6-24 and Figure 6-4 illustrate a short box spread that yields agross proﬁt of 50 cents per share before transaction costs and interestearned. The four-part position consists of one short 90 Call sold at6.50, one long 90 Put purchased for 2.00, one long 100 Call purchased for 1.50, and one short 100 Put sold at 7.50.Table 6-24
The Short Box Spread: Short 90 Call at 6.50, Long 90 Put at2.00, Long 100 Call at 1.50, and Short 100 Put at 7.50Col 1
Col 2
Col 3
Col 4
Col 5
Row
Stock Priceat Exp
Short 90Call @6.50
Long 90Put @2.00
Long 100 Short 100Call @Put @Combined1.507.50P/(L)
1234
8090100110
6.506.503.5013.50
8.002.002.002.00
1.501.501.508.50
12.502.507.507.50
Col 6
0.500.500.500.50
Arbitrage Strategies • 197
10
Profit/(Loss)
5
090
100
–5
–10
–15Stock Price
Figure 6-4
Short 90 Call @ 6.50
Long 90 Put @ 2.00
Short 100 Put @ 7.50
Short Box Spread
Long 100 Call @ 1.50
The Short Box Spread
A short box can be described in two ways. First, it can be describedas the combination of a short synthetic stock at the lower strike priceand a long synthetic stock at the higher strike price. Second, it can bedescribed as the combination of a bear call spread and a bull putspread with the same strike prices.Matching the calls and puts with the same strike prices together,the short 90 Call and the long 90 Put create a synthetic short stockposition, and the long 100 Call and the short 100 Put create a synthetic long stock position. Matching the calls together and the putstogether, the short 90 Call and the long 100 Call create a bear callspread, and the short 100 Put and long 90 Put create a bull put spread.A reverse conversion was compared earlier with making an investment with borrowed funds and then repaying the loan when theinvestment matures. Similarly, a short box spread brings in a creditwhen established and requires a payment when closed at expiration.
Short Box Spread—Outcomes at ExpirationA short box can result in ﬁve possible outcomes at expiration, just likea long box. As summarized in Table 6-25, if the stock price closes
198 • Trading Options As a Professional
below the lower strike price, between the strike prices, or above thehigher strike price at expiration, then the short box spread position willbe closed. This result is the goal because the trader will not be leftwith an open position and the attendant risk that could reduce theproﬁt or create a loss.The stock price closing exactly at one of the strike prices at expiration, outcomes 2 and 4, creates a pin-risk situation. As with the otherarbitrage positions discussed earlier, it is impossible to predict howTable 6-25ShortLongLongShort
The Short Box Spread at Expiration
1 XYZ 90 Call @ 6.501 XYZ 90 Put @ 2.001 XYZ 100 Call @ 1.501 XYZ 100 Put @ 7.50
 Short stock (synthetically)@ 94.50 Long stock (synthetically)@ 94.00
Combined position at expiration  no position (gross proﬁt 0.50)Five Possible Outcomes at Expiration#1 XYZ  90 (Lowest Strike)
#2 XYZ  90 (Lower Strike)
90 Call expires worthless100 Call expires worthless90 Put exercised (sell stock)100 Put assigned (buy stock)Result: No position
90 Call expires worthless100 Call expires worthless90 Put expires worthless100 Put assigned (buy stock)Result: Long stock
#3 90  XYZ  100 (Between Strikes)
#4 XYZ  100 (Higher Strike)
90 Call assigned (sell stock)100 Call expires worthless90 Put expires worthless100 Put assigned (buy stock)Result: No position
90 Call assigned (sell stock)100 Call expires worthless90 Put expires worthless100 Put expires worthlessResult: Short stock
#5 XYZ  100 (Highest Strike)90 Call assigned (sell stock)100 Call exercised (buy stock)90 Put expires worthless100 Put expires worthlessResult: No position
Arbitrage Strategies • 199
many of the short at-the-money options will be assigned. Consequently,market makers typically respond by exercising half the long at-themoney options, hoping that only half the short options will be assigned.
Pricing a Short Box SpreadGiven the high probability that a short box spread will require a cashpayment equal to the difference between the strike prices at expiration, the value of a short box is equal to the DPV of the differencebetween the strike prices plus costs plus a proﬁt margin.Tables 6-26 through 6-28 show, in three parts, how a short 100–110box spread might be priced. Table 6-26 states the assumptions. The priceof the 100 Call (1) is 9.10. The price of the 100 Put (2) is 2.30. The priceof the 110 Call (3) is unknown, and the price of the 110 Put (4) is 6.70.The lending rate (5) of 4 percent and the days to expiration (6) of 60 lead
Table 6-26
Assumptions:
Pricing a Short 100–110 Box Spread—Part 1: Stating theAssumptions1 Price of 100 Call9.102 Price of 100 Put2.303 Price of 110 Call?4 Price of 110 Put6.705 Lending rate4%6 Days to expiration607 DPV of differencebetween the strike prices (110 – 100)  [1 (0.04  60/365)]9.938 Option cost (4 options,0.01/share)0.04 per share9 Exercise/assignment(2 options)0.02 per share10 Target Proﬁt0.05 per share
Question: What is the purchase price of the 110 Call?
Transactioncosts  0.06
200 • Trading Options As a Professional
to the DPV of the difference between the strike prices (7) of 9.93. Trading costs (8 and 9) consist of 1 cent per share to trade an option and foroption exercise or assignment. The total costs therefore are 6 cents,4 cents for opening the four-part position plus 2 cents for exercise orassignment of the in-the-money options at expiration that close the position. Finally, the target proﬁt (10) in this example is 5 cents per share.Given the nine known assumptions, it is possible to solve for theunknown one, the price of the 110 Call. The question is, “What is thepurchase price of the 110 Call that yields the target proﬁt?”Table 6-27 contains the second part of pricing a short box spreadand shows how to determine the purchase price of the 110 Call in twosteps. The ﬁrst step involves calculating the net credit required pershare. Similar to a reverse conversion, the net credit required per share(NC) is the net funds per share received for establishing the positionthat yields the target proﬁt if held to expiration. The net credit requiredper share equals the DPV of the difference between the strike prices(line 7 in Table 6-26) plus the sum of costs plus target proﬁt (lines8–10). The net credit required per share therefore is 10.04. As withconversions and reverse conversions, calculations are made on a pershare basis, allowing a trader to determine a target proﬁt regardless ofthe quantity of option contracts traded.
Table 6-27
Pricing a Short 100–110 Box Spread—Part 2: CalculatePurchase Price of 110 Call
Step 1:
Calculate the net credit required per share (NC)NC  DPV of difference between strikes plus sum of costs plustarget proﬁtNC  9.93  (0.06  0.05)  10.04
Step 2:
Buy the 110 Call at a price so that the net credit received for the fourpart position (short 100 Call, long 100 Put, long 110 Call, short 110Put) equals the NC in step 1.2-1 If  100 Call  100 Put  110 Call  110 Put   NC2-2 Then  110 Call   100 Call  100 Put  110 Put  NC2-3 Therefore,  110 Call  9.10  2.30  6.70  10.04  3.46
Arbitrage Strategies • 201
Step 2 in Table 6-27 uses basic algebra to ﬁnd the price of the 110Call that makes the price of the box spread equal the net fundsrequired, and that price is 3.46.The third and ﬁnal part of pricing a short box involves analyzingthe cash ﬂows, the net proﬁt, and the spread values, as illustrated inTable 6-28. The revenue (1) of 10.04 is the net credit received whenthe short box position was established. The cost (2) of 10.00 is the difference between the strike prices because