may force market makers to accept conversion positions that produce a proﬁt less than the target proﬁt. In the preceding example, for instance, with the stock being offered at $57.70 andthe 50 Put being offered at 1.45, a market maker seeking a 5-centper-share proﬁt would offer the 50 Call at 4.69. However, if this callis offered at 4.67 by other traders, then the market maker mustchoose between doing some business at a lower rate of proﬁt or foregoing that opportunity. Of course, there may be options at otherstrike prices that offer higher-yielding conversions, and there might
Arbitrage Strategies • 173
be other stocks with better opportunities. Deciding which conversion opportunities are “acceptable” is part of the art of being a market maker.
Pricing a Conversion with DividendsA dividend is a payment made by a company to its shareholders. Notall companies with publicly traded stocks pay dividends, but those thatdo pay dividends typically pay them on a regular and predictable basis.Many companies pay dividends quarterly, but some pay dividendssemiannually or annually, and some pay a relatively small quarterlydividend and then an extra dividend at year end if earnings are goodenough to support such a payment.Dividends have an impact on the pricing of conversions because adividend is extra income for the stockowner. Tables 6-6 through 6-8show the three-part process of pricing a conversion when dividendsare included.Table 6-6 makes the same assumptions for stock price, put price,borrowing rate, etc. as stated in Table 6-3 with one addition of a22-cent dividend. Tables 6-6 through 6-8 demonstrate that the addition of a 22-cent dividend reduces the price of the 55 Call by 22 cents,from 4.69 to 4.47.Line 8 of Table 6-6 calculates the DPV of the strike price plus dividend rather than just the strike price, as in Table 6-3, because the dividend revenue to the stock owner affects the price of a conversionposition. Although there is a timing difference between receipt of thestrike price (at expiration) and receipt of the dividend (generally later),the thinking is that getting $55 at expiration and getting 22 cents abouta month later is almost the same thing as getting $55.22 today. Tradersalso generally ignore the lost interest on the dividend, 22 cents in thisexample, because it usually amounts to less than 1 cent.Table 6-7 shows how the sale price of the 55 Call is calculated usingthe DPV of the strike price plus dividend from line 8 in Table 6-6, andTable 6-8 analyzes the cash ﬂow, net proﬁt, and time values.
174 • Trading Options As a Professional
Table 6-6
Pricing a Conversion with Dividends—Part 1: Stating theAssumptions
Assumptions:
Question:
1 Strike price55.002 Stock price57.703 Price of 55 Put1.454 Price of 55 Call?5 Borrowing rate5%6 Dividend (ex-date beforeexpiration) 0.227 Days to expiration608 DPV of strike price  dividend (strike  div)  [1 (0.05  60/365)]54.779 Stock cost0.01 per share10 Option cost (call and put) 0.02 per share11 Exercise/assignment cost 0.01 per share12 Target proﬁt0.05 per share
Transactioncosts 0.04
What is the sale price of the 55 Call?
Table 6-7
Pricing a Conversion with Dividends—Part 2: Calculate the SalePrice of the 55 Call
Step 1:
Calculate the net investment per share (NI)NI  DPV of strike  dividend minus the sum of costs plustarget proﬁtNI  $54.77  (0.04  0.05)  $54.68
Step 2:
Sell the 55 Call at a price so that the net cost of the three-partposition (long stock, long 55 Put, and short 55 Call) equalsthe NI in step 1.2-1 If  stock  55 Put  55 Call   NI2-2 Then  55 Call   stock  55 Put  NI2-3 Therefore,  55 Call  57.70  1.45  54.68  4.47
The conclusion of pricing conversions with dividends is stated atthe bottom of Table 6-8. For a conversion when there is a dividend,the difference between the time value of the call and the time valueof the put equals the gross proﬁt. The net investment, however, isreduced by the dividend.
Arbitrage Strategies • 175
Table 6-8
Pricing a Conversion with Dividends—Part 3: Analysis of CashFlows, Net Proﬁt, and Time Values
Revenue  amount received at expiration strike price Cost  net debit paid for position (NI) Gross proﬁt Borrowing costs  54.46  (0.05  60/365) Loss before transaction costs Transaction costs Loss before dividend Dividend Net proﬁt
= 55.0054.68 0.32 0.45 0.13 0.04 0.17 0.22 0.05
Analysis of Time Values10Time value of 55 Call price – intrinsic  4.47  2.7011Time value of 55 Put  price – intrinsic  1.45  0.0012Time value of call – time value of put
 1.77 1.45 0.32
123456789
Conclusion: For a conversion when there is a dividend, the difference between thetime value of the call and the time value of the put (12) equals the gross proﬁt(3). The net investment, however, is reduced by the dividend.
The Ex-Dividend DateTiming is important when dividends are involved because there arespeciﬁc dates that are signiﬁcant in the dividend-paying process. Theex-dividend date is the ﬁrst day that a new stock purchaser will notreceive the next dividend. On this day, the stock is trading withoutthe dividend, or ex-dividend. In Table 6-6, the ex-dividend date, orex-date, is prior to option expiration.The ex-date is crucial because of its relationship to another datethat follows called the record date. The record date is the day that astock purchaser must be a shareholder on the company’s books to qualify for the dividend payment. Stock transactions typically are processedin a manner known as T  3, which means “transaction date plusthree business days.” The transaction date is the day of the stock trade.The settlement date is three days after the transaction date, whencash and ownership certiﬁcates are exchanged. In order to receive a
176 • Trading Options As a Professional
dividend, the settlement date must be on or before the record date.A trade made two days before the record date will not be settled untilthe day after the record date. That day, therefore, is the ex-date.For example, assume that the record date is Friday, May 7. Stock mustbe owned on this date in order to receive the dividend. If stock is purchased on Tuesday, May 4, the transaction date, the settlement date willbe on Friday, May 7, which is T  3. The new owner receives the dividend as the owner on the record date. Shares purchased on Wednesday,May 5, however, will not be settled until Monday, May 10, and will notbe eligible to receive the dividend. May 5 is the ex-date in this example.
Pricing Conversions by Strike PriceThe strike price affects the conversion-pricing relationship becausethe DPV changes as the strike price changes. As a result, the amountby which the time value of the call must exceed the time value of theput also changes. Table 6-9 uses the same assumptions as Tables 6-3through 6-5. The stock price is 57.70, there are 60 days to expiration,the borrowing rate is 5 percent, and there are no dividends.Table 6-9
Pricing Conversions by Strike PriceCol 1
Col 2
Row
StrikePrice
DPV ofStrike
12345
4550556062
44.6349.5954.5559.5164.47
Col 3
Col 4
StrikeCosts PlusMinus DPVTargetof StrikeProﬁt0.370.410.450.490.53
0.090.090.090.090.09
Col 5Time Value ofCall MinusTimeValue of Put0.460.500.540.580.62
DPV  discounted present value.Assumptions: 60 days to expiration, interest rate of 5%, no dividends, costs of 0.04, and target proﬁtof 0.05.Sample DPV calculation: 55  [1  (0.05  60/365)]  $54.55.
Arbitrage Strategies • 177
In row 1 of Table 6-9, the strike price is 45 (column 1), the DPV ofthe strike is 44.63 (column 2), the difference between the strike priceand the DPV is 0.37 (column 3), and the costs and target proﬁt are 9cents per share (column 4). In order to make a proﬁt in this example,the time value of the 45 Call therefore must be greater than the timevalue of the 45 Put by 46 cents (column 5). In row 2, the strike priceis 50, and the difference of time value of the call minus time value ofthe put is 50 cents.As the strike price rises by $5.00 in each row in Table 6-9, the difference between call and put time values increases by 4 cents. Sincethe DPV of the strike price increases as the strike price increases, amarket maker must borrow more money to ﬁnance a conversion position. Thus borrowing costs increase. The difference between call timevalue and put time value increases to cover these additional costs.
The Concept of Relative PricingThe essence of the conversion relationship is that given a strike price,costs, and a target proﬁt, a constant difference between the time valueof the call and the time value of the put always will exist. Relative pricing means that the price of a call, a put, or a stock can be calculatedif prices of the other two are known. In Table 6-3, the time value ofthe 55 Call had to be 69 cents greater than the time value of the55 Put. Therefore, if the prices of the stock and the 55 Put are known,then the price of the 55 Call can be calculated. Also, if the prices ofthe stock and the 55 Call are known, then the price of the 55 Put canbe calculated, and if the prices of the 55 Call and 55 Put are known,then the price of the stock can be calculated.In the days when option trading was conducted in open outcry,market makers would start their days by calculating the relative pricesof calls to puts by strike price. If trading the 60-strike options in row 4of Table 6-9, for example, a market maker would note that the timevalue of the 60 Call must be 58 cents greater than the time value ofthe 60 Put in order to make a 5-cent-per-share proﬁt on a conversion.
178 • Trading Options As a Professional
Then, during the day, if 60 Puts were offered at 3.10 when stock wasoffered at 60.47, the market maker would offer 60 Calls at 4.15. Ifsomeone were to buy the calls for 4.15 (and the market maker soldthem), the market maker would then simply buy the stock at 60.47and buy the puts at 3.10 to complete the conversion and thereby lockin a 5-cent-per-share proﬁt.As stock prices changed and as option orders entered the pit, themarket maker would constantly use the 58-cent difference to calculate the relative prices of 60 Calls and 60 Puts and make bids and offersaccordingly.In today’s electronic trading environment, computers handle thepricing of options—and therefore the pricing of conversions. Nevertheless, the same concepts apply, and today’s option traders mustunderstand these concepts for those occasions when they need to override the computer or adjust its assumptions.
The Reverse ConversionAs its name implies, the reverse conversion or, simply, the reversal isthe opposite of the conversion. It involves the purchase of syntheticstock and the sale of real stock. A reverse conversion is a three-part strategy consisting of short stock, short puts, and long calls on a share-forshare basis. The calls and puts have the same strike price and sameexpiration date. A reverse conversion is established for a net credit,which is invested at the risk-free rate. To be proﬁtable, the interestearned from the net credit must be greater than transaction costs plusthe difference of call time value minus put time value.Table 6-10 and Figure 6-2 illustrate a reverse conversion that yieldsa gross proﬁt of 75 cents per share before transaction costs and interest. The three-part position consists of one share of stock sold short at$102, one short 100 Put sold at 5.25, and one long 100 Call purchasedfor 6.50. As column 5 in Table 6-10 and the solid line in Figure 6-2show, the ﬁnal outcome at expiration, 75 cents per share proﬁt, is thesame regardless of the stock price.
Arbitrage Strategies • 179
Table 6-10
The Reverse Conversion: Short Stock at 102, Short 100 Put at5.25, and Long 100 Call at 6.50Col 1
Col 2
Col 3
Col 4
Col 5
Row
Stock Priceat Expiration
Short Stock@ 102
Short 100Put @ 5.25
Long 100Call @ 6.50
CombinedP/(L)
12345
9095100105110
12.007.002.003.008.00
4.750.255.255.255.25
6.506.506.501.503.50
0.750.750.750.750.75
100Stock Price
105
Profit/(Loss)
151050–5–1095
Short Stock @ 102Short100 Put @ 5.25Figure 6-2
Long100 Call @ 6.50Reverse Conversion
The Reverse Conversion
This example assumes that the net credit proceeds from the reverseconversion are invested. Consequently, the net proﬁt or loss of areverse conversion depends on the amount of interest earned.
Reverse Conversion—Outcomes at ExpirationAs with synthetic positions and conversions, there are three possibleoutcomes at expiration for reverse conversions. Table 6-11 summarizesthe possibilities. If the stock price closes below the strike at expiration,
180 • Trading Options As a Professional
outcome 1, then the short put is in the money. It therefore will beassigned, and the stock will be purchased at the strike price. If the stockprice is above the strike price at expiration, outcome 3, then the longcall is in the money. It therefore will be exercised, and the stock willbe purchased at the strike price.It does not matter if the stock price closes above or below the strikeprice at expiration; the stock will be purchased, and the reverse conversion position will be closed. This result is the desired outcomebecause a proﬁt will be realized, and the position will be closed. Thenthere will be no open position with the attendant risk that couldreduce the proﬁt or create a loss.
Reverse Conversion—Pin RiskPin risk poses as much of a problem for the reverse conversion as itdoes for the conversion. When the stock price closes exactly at thestrike price at expiration, the trader must make a difﬁcult decision.
Table 6-11
The Reverse Conversion at Expiration
Long 1 XYZ 100 Call @ 6.50 Long stock (synthetically) @ 101.25Short 1 XYZ 100 Put @ 5.25Short XYZ stock @ 102.00  short stock @ 102.00Combined position at expiration  no position (gross proﬁt 0.75)Three Possible Outcomes at Expiration1 XYZ  Strike
2 XYZ  Strike
3 XYZ  Strike
Call expiresworthlessPut assigned(buy stock)Shares bought forstrike priceResult: No position
Call expires worthless
Call exercised (buy stock)Put expires worthless
Put expires worthlessShort shares are keptResult: Short stock
Shares bought forstrike priceResult: No position
Arbitrage Strategies • 181
The three-part position of short stock, short puts, and long calls produces uncertainty about how many puts, if any, will be assigned. If allthe calls are exercised but only some of the puts are assigned, then along stock position is created.Similarly, if the calls are not exercised, and if some of the puts arenot assigned, then a short stock position remains. In either case, a stockposition must be carried over the weekend with all the related risk.Just as with a conversion, there is no solution that eliminates this overthe-weekend risk. One common practice is to exercise half the callsand hope that only half the puts are assigned. Whatever stock positionremains is then closed at the open on Monday.
Pricing a Reverse ConversionGiven the high probability that the stock price will be above or belowthe strike price at expiration, traders can reasonably expect that thecall will be exercised or the put will be assigned. Consequently,the reverse conversion strategy will result in a cash payment equal tothe strike price at expiration. Unlike the conversion, which resemblesa Treasury bill, the reverse conversion can be compared to borrowingmoney that is repaid with a maturing investment. When a reverse conversion is established, stock is sold short. The proceeds from that shortsale are invested at the risk-free rate. Subsequently, at option expiration, the money-market funds, including interest earned, are used torepurchase the stock at the strike price.For example, a trader creating a reverse conversion might sell stockshort at $34.55, sell a 55-day 35 Put for $2.10, and purchase a 55-day35 Call for $1.75. The net credit available to invest is $34.90 (34.55 2.10  1.75  34.90). This position results in a loss before interest of0.10 (35.00 paid at expiration minus 34.90 net credit received for establishing the position). Assuming 4 percent interest for 55 days on $34.90,the interest income is approximately 21 cents (34.90  0.04  55/365).Therefore, the loss before interest of 0.10 becomes a net proﬁt afterinterest of 0.11, or 11 cents per share, before transaction costs.
182 • Trading Options As a Professional
The proﬁtability of a reverse conversion depends on three factors:transaction costs, the lending rate, and the competitive environment.Tables 6-12 through 6-14 show, in three parts, how a reverse conversion might be priced. Table 6-12 makes 11 initial assumptions. Thestrike price (1) is 55. The stock price (2) is $57.70. The price ofthe 55 Put (3) is 1.45, and the price of the 55 Call (4) is unknown.The lending rate (5) of 4 percent and the days to expiration (6) of 60lead to the DPV of the strike price (7) of 54.64. Trading costs (8–10)consist of 1 cent per share to trade stock, to trade each option, and foroption exercise or assignment. These transaction costs lead to totalcosts of 4 cents per share for opening the position (i.e., short stock,short put, and buy call) and for closing the position (i.e., either thecall is exercised or the put is assigned). Finally, the target proﬁt (11)is 5 cents per share in this example.The 10 known assumptions provide all the information needed tosolve for the unknown price of the 55 Call. The question is, “What isthe purchase price of the 55 Call that yields the target proﬁt?”Part 2 of pricing a reverse conversion in Table 6-13 shows that thepurchase price of the 55 Call can be calculated in two steps. The ﬁrstTable 6-12
Assumptions:
Question:
Pricing a Reverse Conversion—Part 1: Stating the Assumptions1 Strike price2 Stock price3 Price of 55 Put4 Price of 55 Call5 Lending rate6 Days to expiration7 DPV of strike price strike  [1 (0.04  60/365)]8 Stock cost9 Option cost (call and put)10 Exercise/assignment cost11 Target proﬁt
55.0057.701.45?4%60
54.640.01 per share0.02 per share0.01 per share0.05 per share
What is the purchase price of the 55 Call?
Transactioncosts0.04
Arbitrage Strategies • 183
step calculates the funds required to invest, or the net credit, which isthe DPV of the strike price plus costs plus the target proﬁt. From line7 in Table 6-12, the DPV of 55, using the lending rate of 4 percent, is$54.64. Adding costs of 0.04 and the target proﬁt of 0.05 yields a netcredit required of 54.73. Note that this formula uses the strike pricefor the DPV calculation because it is the strike price that will be paidat expiration.Step 2 in Table 6-13 uses basic algebra to ﬁnd that the price of the55 Call should be 4.42 to make the target proﬁt.As with the conversion, this formula for reverse conversions calculates costs and target proﬁt on a per-share basis because a tradercannot know in advance how many options might come into the marketplace and how many shares need be traded. A per-share estimatemakes it possible to trade in small or large quantities and still get thetarget proﬁt.The third and ﬁnal part of pricing a reverse conversion involvesanalyzing the cash ﬂows, net proﬁt, and time values, as shown inTable 6-14. The revenue (1) is 54.73, which is the net credit receivedwhen the reverse conversion is established, as calculated in step 1 ofTable 6-13. The cost (2) is the amount paid at expiration, which isthe strike price of 55, and the gross loss (3) of 0.27 is the differencebetween cost and revenue. The interest income (4) of 0.36 is basedTable 6-13
Pricing a Reverse Conversion—Part 2: Calculate the PurchasePrice of the 55 Call
Step 1: Calculate the net credit required per share (NC)NC  DPV of strike plus transaction costs plus target proﬁt marginNC  $54.64  0.04  0.05  $54.73Step 2: Buy the 55 Call at a price so that the NC for the three-partposition (short stock, short 55 Put, and long 55 Call)equals the NC in step 1.2-1 If  stock  55 Put  55 Call  NC2-2 Then  55 Call   stock  55 Put  NC2-3 Therefore,  55 Call  57.70  1.45  54.73  4.42
184 • Trading Options As a Professional
on the net credit and the lending rate. The proﬁt before transactioncosts (5) of 0.09 is the difference between the interest income andthe gross loss. Subtracting transaction costs of 4 cents per share(6) results in the net proﬁt (7) of 5 cents per share.Lines 8 through 10 in Table 6-14 analyze the relationship of thetime value of the 55 Call and the time value of the 55 Put. In this case,the time value of the 55 Call (8) is 1.72, and the time value of the 55Put (9) is 1.45, for a difference (10) of 0.27.The conclusion of this exercise is stated at the bottom of Table6-14. For a reverse conversion, the difference between the time valueof the call and the time value of the put equals the absolute value ofthe gross loss.
Competition and Reverse ConversionsCompetition may force market makers to accept reverse conversionpositions that produce a proﬁt less than the target proﬁt. In the
Table 6-14
Pricing a Reverse Conversion—Part 3: Analysis of Cash Flows,Net Proﬁt, and Time Values
1 Revenue  net credit received forestablishing position2 Cost  amount paid at expiration strike price3  Gross loss4  Interest income  54.73  (0.04  60/365)5  Proﬁt before transaction costs6  Transaction costs7  Net proﬁt
54.73= 55.00 (0.27) 0.36 0.090.04 0.05
Analysis of Time Values8 Time value of 55 Call  price  intrinsic  4.42  2.709 Time value of 55 Put  price  intrinsic  1.45  0.0010 Time value of call  time value of put
Conclusion: For a reverse conversion, the difference between the timevalue of the call and the time value of the put (10) equals the absolutevalue of the gross loss (3).
 1.72 1.45 0.27
Arbitrage Strategies • 185
preceding example, for instance, when the bid price for the stock is$57.70 and the bid price for the 55 Put is 1.45, a market maker seeking a 5-cent-per-share proﬁt would bid 4.42 for the 55 call. However,if other market makers are bidding 4.44 for this call, then the marketmaker must choose between doing some business at a lower rate ofproﬁt and doing no business. Of course, there may be options at otherstrike prices that offer higher-yielding reverse conversions, and theremay be other stocks with better opportunities. Deciding which opportunities are “acceptable” is part of the art of being a market maker.
Pricing a Reverse Conversion with DividendsWhen stock that pays a dividend is sold short, the borrower of the stockmust pay the dividend to the lender, or owner, of the stock. Dividendstherefore increase costs for a reverse conversion position rather thanreducing costs, as they do for conversion positions.Tables 6-15 through 6-17 expand the pricing calculation for reverseconversions to include dividends. Table 6-15 states the same assumptions for stock price, put price, borrowing rate, etc. as statedin Table 6-12 with one difference, the addition of a 22-cent dividendthat reduces the call price by 22 cents, from 4.42 (Table 6-13,line 2–3) to 4.20.In Table 6-15, line 8, the DPV calculation begins with “Strike dividend” rather than “Strike,” as in Table 6-12, because the dividendis an extra cost to the short seller of stock. Although there is a timingdifference between payment of the strike price (at expiration) and payment of the dividend, the thinking is that paying $55 today and paying22 cents in about a month is almost the same thing as paying $55.22today. Traders generally ignore the interest on the 22 cents because itusually amounts to less than 1 cent.Table 6-16 shows how to calculate the purchase price of the 55 Callusing the DPV of the strike price plus dividend from line 8 inTable 6-15, and Table 6-17 analyzes the cash ﬂow, net proﬁt, and timevalues. The conclusion of pricing reverse conversions with dividends
186 • Trading Options As a Professional
Table 6-15
Assumptions:
Pricing a Reverse Conversion with Dividends—Part 1: Stating theAssumptions1 Strike price55.002 Stock price57.703 Price of 55 Put1.454 Price of the 55 Call?5 Lending rate4%6 Dividend (ex-date before expiration)0.227 Days to expiration608 DPV of strike price plus dividend (strike  div)  [1 (0.04  60/365)]54.869 Stock cost0.01 per share10 Option cost (call and put)0.02 per share11 Exercise/assignment cost0.01 per share12 Target proﬁt0.05 per share
Transactioncosts 0.04
Question: What is the purchase price of the 55 Call?
Table 6-16
Pricing a Reverse Conversion with Dividends—Part 2: Calculatethe Purchase Price of the 55 Call
Step 1:
Calculate the net credit required per share (NC)NC  DPV of strike plus transaction costs plus target proﬁt marginNC  $54.86  0.04  0.05  $54.95
Step 2:
Buy the 55 Call at a price so that the NC for the three-partposition (short stock, short 55 Put, and long 55 Call) equalsNC in step 1.2-1 If  stock  55 Put  55 Call   NC2-2 Then  55 Call   stock  55 Put  NC2-3 Therefore,  55 Call  57.70  1.45  54.95  4.20
is stated at the bottom of Table 6-17. For a reverse conversion, the difference between the time value of the call and the time value of theput equals the absolute value of the gross loss. The net credit, however,is increased by the dividend.
Arbitrage Strategies • 187
Table 6-17
Pricing a Reverse Conversion with Dividends—Part 3: Analysis ofCash Flows, Net Proﬁt, and Time Values
Revenue  net credit received for establishing position Cost  amount