of
156 • Trading Options As a Professional
$100, a 100 Call price of 3, and a 100 Put price of 3. The syntheticstock price is calculated by adding or subtracting the net option premium to or from the strike price. Speciﬁcally, 100 (the strike price)plus 3.00 (the price of the 100 Call) minus 3.00 (the price of the 100Put) equals 100, which is the synthetic stock price.In Table 5-7 and Figure 5-7, the synthetic stock price of $103 alsocan be derived from the call and put prices and the strike price: 100(the strike price) plus 4.50 (the price of the 100 Call) minus 1.50 (theprice of the 100 Put) equals $103 (the stock price). Finally, the stockprice of $97 in Table 5-8 and Figure 5-8 can be calculated by addingthe strike price (100) and the 100 Call price (2.50) and subtractingthe 100 Put price (5.50). Once again, strike price plus call price minusput price equals stock price. The assumptions here are an interestrate of zero, no dividends, and no transaction costs. The next sectionconsiders differences in these factors.
The Role of Interest Rates and DividendsImagine that an investor with $100 cash available to invest wants toinvest in the stock in Table 5-1 and Figure 5-1. Let’s call the stockXYZ, and for simplicity, let’s assume that options cover one share andthat commissions are zero. This investor has two strategy choices. Heor she can buy one share of XYZ for $100, or he or she can buy synthetic stock by simultaneously buying one XYZ 100 Call for 3.00 andselling one XYZ 100 Put for 3.00, for a net cost of zero. An understanding of the role of interest rates and dividends will tell us whetherthe investor will choose the real stock or the synthetic stock. Thisexample also will reveal something about the relationship of optionprices in the real world.First, assume that the interest rate is above zero and that the stockdoes not pay a dividend. If the investor uses the $100 cash to buy thereal stock, then the proﬁt or loss will be determined solely by the priceaction of the stock. If, however, the investor creates a synthetic long
Synthetic Relationships • 157
stock position by buying a 100 Call for 3.00 and selling a 100 Put for3.00 (net cost of zero), then the $100 cash will be available to earninterest. At expiration, if stock XYZ is above or below $100, then thesynthetic stock position will become a real stock position via exerciseof the call or assignment of the put. Regardless of whether the stockprice is up, down, or unchanged, however, the investor will earn interest on the $100 in addition to the proﬁt or loss from the stock priceaction. The conclusion is obvious: If the interest rate is above zero andthere are no dividends, then the investor would choose the syntheticlong stock over the real long stock.Second, assume that the stock pays a dividend and that the interestrate is zero. In this scenario, there is no advantage to holding cash.The proﬁt or loss of synthetic long stock will be determined solely bythe price action of the stock. A real long stock position, however, willreceive the dividend in addition to the proﬁt or loss from the stockprice action. Again, the conclusion is obvious: With positive dividendsand an interest rate of zero, the investor would choose the real longstock over the synthetic long stock.Third, assume that the interest rate is positive and that the stock paysa dividend. In this case, there is no clear choice. If the interest rate ishigher than the dividend yield, then the synthetic long stock is preferred. If dividends are higher, however, then the real stock is preferred.Given that the interest rate is above zero, the conclusion from thepreceding example is that the prices in Table 5-1 and Figure 5-1 cannot exist in the real world. If investors could buy synthetic stock forthe same price as real stock and earn interest during the life of theoptions, then no rational investor would ever buy stock. Rationalinvestors would continue to buy calls and to sell puts as the call pricesrose and the put prices declined until the choice between buying realstock and synthetic stock was equal. Consequently, in the real world,one would expect that the price of the 100 Call would be higher andthat the price of the 100 Put would be lower than Table 5-1 andFigure 5-1 indicate.
158 • Trading Options As a Professional
Figure 5-10 shows a Single Option Pricing screen from Op-Eval Pro,the software that accompanies this book. The assumptions in the leftcolumns are a stock price of $100, a strike price of $100, volatility of 25percent, an interest rate of 5 percent, no dividends, and 30 days to expiration. Given these inputs, two of the outputs in the top rows of thecolumns are the 100 Call price of 3.08 and the 100 Put price of 2.70.These prices verify the thinking presented in the preceding paragraphs,that the time value of the 100 Call is greater than the time value of the100 Put. While the exact monetary difference between the two pricesis 38 cents, one might reasonably ask, “Is this signiﬁcant?” or “Is theresome rule that this follows?” The answer to both questions is, “Yes.”The interest-rate assumption, 5 percent, is the annual rate earnedon short-term investments, such as the rate on 90-day Treasury bills.If one were to invest $100 for 30 days at 5 percent, the interest incomewould be 41 cents (100  0.05  30/365  0.41), which is very closeto the 38-cent difference between the time values of the 100 Call andthe 100 Put. Therefore, the rule to keep in mind is that for calls andOp-Eval Pro: OP-EVAL - Single ViewEQUITY
AMERICAN
CALL
PUT
STOCK PRICE
100.00
VALUE
3.08
2.70
STRIKE PRICE
100.00
DELTA
0.54
–0.47
VOLATILITY %
25.00
GAMMA
0.06
0.06
INTEREST RATE %
5.00
VEGA
0.11
0.11
DIVIDEND
0.00
7-THETA
–0.40
–0.32
DAYS TO EX-DIV
0.00
RHO
0.04
–0.03
DAYS TO EXPIRY
30.00
Figure 5-10
Decimal Places
First Example - Single Option Screen from Op-Eval Pro
2
Synthetic Relationships • 159
puts with the same underlying, the same strike price, and the sameexpiration date, the difference in time values—call time value overput time value—is almost exactly equal to the interest on the strikeprice. The 3-cent difference between the calculated interest of 41cents and the difference in time value of 38 cents stems partly fromrounding and partly from some technical assumptions in the optionpricing formula.The difference in time values—calls over puts—is signiﬁcantbecause it proves there is no theoretical advantage to trading syntheticstock over real stock. Theoretically, an investor should be indifferentbetween trading the two. In the real world, however, factors such astransaction costs and bid-ask spreads lead most nonprofessionalinvestors to trade real stock rather than synthetic stock. Chapter 6 willexplore the role that interest rates and dividends play in arbitragestrategies.To reinforce the concept that the time value of calls is greater thanthe time value of puts by the amount of interest, Figure 5-11presents a second example of a Single Option Pricing screen fromOp-Eval Pro. The assumptions in this example are a stock price of$88, a strike price of $85, volatility of 30 percent, an interest rate of4 percent, no dividends, and 90 days to expiration. Given theseinputs, the price of the 85 Call is 7.27, and the price of the 85 Put is3.48. These prices also verify the theory asserting no theoreticaladvantage to trading synthetic stock. The time value of the 85 Call(4.27) is 79 cents greater than the time value of the 85 Put (4.27 3.48  0.79), and the interest on the strike price is 84 cents (85 0.04  90/365  0.84). Again, the difference between the calculatedinterest of 84 cents and the difference in time value of 79 cents ispartly due to rounding and partly due to technical assumptions in theoption-pricing formula. Nevertheless, a rule of option pricing is thatfor calls and puts with the same underlying, strike price, and expiration, the time value of the call is greater than the time value of theput by the amount of interest.
160 • Trading Options As a Professional
Op-Eval Pro: OP-EVAL - Single ViewEQUITY
AMERICAN
CALL
PUT
STOCK PRICE
88.00
VALUE
7.27
3.48
STRIKE PRICE
85.00
DELTA
0.65
–0.36
VOLATILITY %
30.00
GAMMA
0.03
0.03
INTEREST RATE %
4.00
VEGA
0.16
0.16
DIVIDEND
0.00
7-THETA
–0.23
–0.17
DAYS TO EX-DIV
0.00
RHO
0.12
–0.07
DAYS TO EXPIRY
90.00
Figure 5-11
Decimal Places
2
Second Example - Single Option Screen from Op-Eval Pro
SummaryThere are six real positions—long and short stock, long and short call,and long and short put. Corresponding to these real positions are sixsynthetic positions, each of which consists of positions in the other twoinstruments. The basic put-call parity equation is “Callput  stock” or, in words, “Long call plus short put equal long stock,” assuming that the call and put have the same underlying, the same strikeprice, and the same expiration date. The ﬁve other synthetic relationships logically follow from this basic equation.Whether a stock price lands above, below, or at the strike price atexpiration, a synthetic position results in the same position as a realposition, with the same proﬁt or loss, and with the same effective priceas a real position.If the interest rate is zero and there are no dividends, then thetime value of calls and puts in synthetic positions would be equal.In the real world, however, the relationship of the time values
Synthetic Relationships • 161
depends on the relationship of the interest rate and dividends.Theoretically, investors should be indifferent between trading realpositions and synthetic positions, but factors such as transaction costsand bid-ask spreads affect which types of positions traders actuallychoose.
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Chapter 6ARBITRAGE STRATEGIESrbitrage is skill number four that must be mastered by professional option traders. Knowledge of arbitrage strategies enables atrader to value options on a relative price basis, compare alternativestrategies, and potentially lock in nearly riskless proﬁts. Arbitrage strategies can be complicated, with higher and more numerous transactioncosts. These strategies therefore fall primarily into the realm of professional option traders. Even advanced nonprofessional traders mustbe cautious in this area. This chapter will deﬁne arbitrage and explainthree arbitrage strategies involving options: the conversion, the reverseconversion, and the box spread. Each explanation contains three parts:(1) an explanation of the concept of the strategy, (2) a presentation ofthe mechanics of the strategy, along with proﬁt-and-loss diagrams, and(3) a discussion of pricing considerations.
A
Arbitrage—the ConceptArbitrage is the trading process of buying in one market and selling inanother market with the goal of earning nearly riskless proﬁts. Theclassic example of arbitrage is a situation in which gold for June delivery in New York is trading at $800 per ounce and gold for July delivery in London is trading in $820 per ounce when delivery costs
• 163 •
164 • Trading Options As a Professional
(i.e., shipping, insurance, and ﬁnance charges) are $10 per ounce anddelivery time is less than one month.In such a situation, an arbitrager would, ﬁrst, buy gold for Junedelivery in New York; second, sell gold for July delivery in London;third, take delivery in New York in June; and fourth, ship the gold toLondon in time for July delivery. When the buying, selling, and shipping agreements are ﬁnalized, the arbitrager would feel conﬁdentabout realizing a $10 per-ounce proﬁt. The arbitrager seized theopportunity to “lock in a nearly riskless proﬁt.”Note the proﬁt is not risk-free. It can only be “nearly riskless”because unexpected things can happen to reduce or eliminate it. Ifthe ﬁnancing agreement called for a ﬂoating-rate loan, for example,the interest rate could rise. Also, export or import rules or tariffs couldchange and cause an increase in costs. There also could be shippingdelays owing to a strike- or weather-related port closing. Any numberof unpredictable events could turn opportunity into disaster. Althoughthe risks of such events occurring may be low, they do exist and arewhat create the potential for arbitrage proﬁts. Arbitragers earn profits—or incur losses—by assuming these risks.This example illustrates that an arbitrage pricing relationship existsbetween gold prices in New York and gold prices in London. Theexample also illustrates that the pricing relationship is based on delivery costs. Similarly, arbitrage pricing relationships exist between anytwo markets where delivery between the markets is possible.For option traders an arbitrage pricing relationship exists betweenoptions and stocks. Using synthetic relationships, a trader can buyshares of stock at a stock exchange and then sell those shares synthetically in the options market. A trader also can do the reverse by buyingshares synthetically in the options market and selling them in the stockmarket. The next section explains the process of option-to-stock andoption-to-option arbitrage and how professional option traders attemptto price certain strategies to proﬁt from any arbitrage opportunities thatmight exist.
Arbitrage Strategies • 165
The ConversionThe basic options arbitrage strategy is known as the conversion, and itinvolves the purchase of real stock and the sale of synthetic stock. It isthe fundamental building block of arbitrage techniques: All option-tostock arbitrage techniques are based on the concepts involved in this strategy. A conversion is a three-part strategy consisting of long stock, longputs, and short calls on a share-for-share basis. The calls and puts havethe same strike price and same expiration date. As the following examples show, in order for a conversion to be proﬁtable, the time value ofthe call must be greater than the time value of the put by an amount sufﬁcient to cover transaction costs, the cost of carry, and the desired proﬁt.Table 6-1 and Figure 6-1 illustrate a conversion that yields a grossproﬁt of 75 cents per share before transaction costs and the cost ofcarry. The three-part position consists of one share of long stock purchased for $103, one long 100 Put purchased for 4.50, and one short100 Call sold at 8.25. As column 5 in Table 6-1 and the solid line inFigure 6-1 show, the ﬁnal outcome at expiration, a proﬁt of 75 centsper share, is the same regardless of how high or low the stock pricemight rise or fall.Assuming that the purchase price of the stock is borrowed, the netproﬁt or loss of a conversion depends on the cost of carry. Cost of carry
Table 6-1
The Conversion: Long Stock at 103, Long 100 Put at 4.50, andShort 100 Call at 8.25Col 1
Col 2
Row
Stock Priceat Expiration
Long Stock@ 103
12345
9095100105110
13.008.003.002.007.00
Col 3
Col 4
Long 100 Short 100Put @ 4.50 Call @ 8.255.500.504.504.504.50
8.258.258.253.251.75
Col 5CombinedP/(L)0.750.750.750.750.75
166 • Trading Options As a Professional
10Profit/(Loss)
50–5–10–1595
100
105
Stock PriceLong Stock @ 103Long 100 Put @ 4.50Figure 6-1
Short 100 Call @ 8.25Conversion
The Conversion
consists of borrowing costs and transaction costs associated with ﬁnancing and trading stock and option positions.
Conversion—Outcomes at ExpirationJust as synthetic positions have three possible outcomes at expiration,Table 6-2 summarizes the same three possibilities for conversion positions. If the stock price closes below the strike price at expiration, outcome 1, then the long put is in the money. It therefore will beexercised, and the stock will be sold at the strike price. If the stockprice is above the strike price at expiration, outcome 3, then the shortcall is in the money. It therefore will be assigned, and the stock willbe sold at the strike price.As Table 6-2 demonstrates, if the stock price is above or below thestrike price at expiration, the stock will be sold at the strike price, andthe conversion position will be closed. This result is the desired outcome because a proﬁt will be realized, and no open position will remainwith an attendant risk that could reduce the proﬁt or create a loss.A problem occurs, however, if the stock price closes exactly at thestrike price.
Arbitrage Strategies • 167
Table 6-2
The Conversion at Expiration
Short
1 XYZ 100 Call @
8.25
Long
1 XYZ 100 Put @
4.50
 Short stock (synthetically)at 103.75
Long XYZ stock @ 103.00  long stock at 103.00Combined position at expiration  no position (gross proﬁt 0.75)Three Possible Outcomes at Expiration#1 XYZ  Strike
#2 XYZ  Strike
#3 XYZ  Strike
Call expires worthlessPut exercised(sell stock)Shares sold atstrike priceResult: No position
Call expires worthlessPut expires worthless
Call assigned(sell stock)Put expires worthless
Shares are keptResult: Long stock
Shares sold at strike priceResult: No position
Pin RiskPin risk is the possibility that the stock price closes exactly at the strikeprice at expiration. When this happens, the stock is said to be pinnedto the strike.As outcome 2 in Table 6-2 indicates, if the stock price settles exactlyat the strike price at expiration, then, in theory, both the call and theput expire worthless, leaving the long stock position intact. Exercisingthe puts could eliminate the risk of the long stock position. This actionassumes, however, that none of the short calls will be assigned. In reality, some of the calls will be assigned, but it is impossible to predicthow many.If all the puts are exercised, and if some of the calls are assigned, atrader will be left with a short stock position. This position involves substantial risk because exercise and assignment at expiration always occuron Friday afternoon, but the resulting short stock position cannot beclosed until Monday morning at the earliest. This sequence produceswhat traders call over-the-weekend risk. Similarly, if none of the puts areexercised, and if only some of the calls are assigned, then the trader willbe left with a long stock position that must be carried over the weekend.
168 • Trading Options As a Professional
Given the uncertainty about how many calls, if any, will beassigned, a trader has to decide how many puts to exercise. No solution exists that eliminates the over-the-weekend risk created by pinrisk. However, market makers have developed a common practice todeal with this situation.Marker makers typically respond to pin risk by exercising half thelong puts in their conversion, hoping that only half their short callswill be assigned. Undoubtedly, either more or less than half the callswill be assigned, so there will be a stock position to close on Mondaymorning. All a trader can do is hope that the experience will not betoo costly. You can be sure that pin risk has caused many market makers to endure some sleepless weekends. Pin risk points up one of thereasons arbitrage is best left to professional traders.
Pricing a ConversionOne way to conceptualize how to price conversions is to comparethem with Treasury bills. Anyone can purchase Treasury bills for anamount less than the face value, that is, at a discount, and receive theface value at maturity. A one-year $1,000 Treasury bill, for example,might be purchased for $970. One year later, at maturity, the purchaser gets back a $1,000 payment, $30 of which is interest incomeand $970 of which is return of principal. The interest rate earned inthis example is 30 divided by 970, or approximately 3.1 percent.Conceptually, the value of a conversion is the discounted presentvalue (DPV) of its strike price. The DPV is the investment, like theamount paid for a Treasury bill. The strike price is the amountreceived at expiration, like the Treasury bill maturing at face value.The difference between DPV and strike price is the income earned,like the difference between what was paid for the Treasury bill and theface value received at maturity.Consider, for example, stock purchased for $92.20, a 40-day 90 Putpurchased for $2.30, and a 40-day 90 Call sold at $4.80. This conversion’s net investment totals $89.70 (92.20  2.30  4.80  89.70),
Arbitrage Strategies • 169
and the proﬁt before costs equals 0.30 (90.00 – 89.70). $0.30 incomein 40 days from an investment of $89.70 approximately equates to anannual interest rate of 3 percent (0.30  89.70  365/40).Note for mathematicians: The calculations in the preceding paragraph and the calculations throughout this book are made usingsimple interest rather than compound interest. Simple interest is usedfor ease of presentation. The discussions are intended to be conceptual and accessible to nonmathematicians. For short periods of time,which is typical for arbitrage positions, the difference is insigniﬁcant.After seeing how a conversion is constructed and how its profitability is calculated, you may reasonably ask, “How much proﬁt issufﬁcient to justify a conversion position?” The answer depends onthree factors: costs (including borrowing costs), the target proﬁt, andthe competitive environment.Tables 6-3 through 6-5 provide an example, in three parts, of how aconversion might be priced. Table 6-3 states the 11 initial assumptions.The strike 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. Theborrowing rate (5) of 5 percent and the 60 days to expiration (6) leadto the DPV of the strike price (7) of 54.55. There are also trading costs(8–10) of 1 cent per share to trade stock (buy and sell), to trade eachoption, and for option exercise or assignment. These transaction costslead to total costs of 4 cents per share for opening the position (i.e., buystock, buy put, and sell call) and for closing (i.e., either the put is exercised or the call is assigned). Finally, the target proﬁt (11) is 5 cents pershare in this example.The 10 known assumptions are used to solve for the unknown one,which is the price of the 55 Call (4). The question is, “What sale pricefor the 55 Call will yield the target proﬁt?”Table 6-4 contains the second part of pricing a conversion, whichcalculates the sale price of the 55 Call in two steps. The ﬁrst step ﬁgures the net investment per share. In the case of a conversion, the netinvestment per share is the net cost of the position that yields the target proﬁt if held to expiration. The net investment per share equals
170 • Trading Options As a Professional
Table 6-3
Pricing a Conversion—Part 1: Stating the Assumptions
Assumptions:
Question:
1 Strike price55.002 Stock price57.703 Price of 55 Put1.454 Price of 55 Call?5 Borrowing rate5%6 Days to expiration607 DPV of strike price strike  [1  (0.05  60/365)] 54.558 Stock cost0.01 per share9 Option Cost(call and put)0.02 per share10 Exercise/assignment cost0.01 per share11 Target proﬁt0.05 per share
Transactioncosts 0.04
What is the sale price of the 55 Call?
the DPV of the strike price (line 7 in Table 6-3) minus the sum of costsplus target proﬁt (lines 8–11). The net investment per share, therefore, is $54.46. Calculations are made on a per-share basis because amarket maker cannot know in advance how many options will comeinto the marketplace and, as a result, how many shares will need tobe traded. Calculating a per-share amount facilitates ﬂexible trades—a trader may trade in small or large quantities but use the same pricing system.Remember that the conceptual value of a conversion is the DPVof its strike price. The strike price is used for the DPV calculationbecause the strike price is the amount received at expiration after thecall is assigned or the put is exercised. Regardless of whether the stockprice is above, below, or at the strike price, when a conversion is established, the net investment always will be less than the strike pricebecause stock plus call minus put must equal the DPV of the strikeprice in order for the conversion to be proﬁtable.Step 2 in Table 6-4 uses basic algebra to calculate 4.69 as the priceof the 55 Call that makes the conversion position (stock plus putminus call) equal the net investment of 54.46.
Arbitrage Strategies • 171
Table 6-4
Pricing a Conversion—Part 2: Calculate the Sale Price of the 55 Call
Step 1:
Calculate the net investment per share (NI)NI  DPV of strike minus the sum of transaction costs plustarget proﬁtNI  $54.55  (0.04  0.05)  $54.46
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  54.46  4.69
The third and ﬁnal part of pricing a conversion consists of analyzing cash ﬂows, net proﬁt, and time values, which is accomplished inTable 6-5. The revenue (1) is the amount received at expiration, whichis equal to the strike price of 55.00 per share in this example. The costof the position (2) is the net investment, and the difference betweenrevenue and cost is the gross proﬁt (3), or 0.54 per share in this example. The borrowing costs (4) are calculated by multiplying the netinvestment times the borrowing rate and come to 0.44. The gross proﬁtminus the borrowing costs leaves a proﬁt before transaction costs (5) of10 cents per share. Subtracting transaction costs of 4 cents per share(6) results in the net proﬁt (7) of 6 cents per share. The actual net proﬁtof 6 cents exceeds the target proﬁt of 5 cents owing to rounding.Lines 8 through 10 in Table 6-5 analyze the relationship of the timevalue of the 55 Call and the time value of the 55 Put. In this case, thestock price is 55.70, and the prices of the 55 Call and 55 Put are 4.69and 1.45, respectively. The time value of the 55 Call (8) therefore is1.99, the time value of the 55 Put (9) is 1.45, and the differencebetween them (10) is 0.54. From Table 6-4, 0.54 is the differencebetween the strike price and the net investment (55.00 – 54.46).The conclusion of this three-part exercise is stated at the bottom ofTable 6-5. For a conversion, the difference between the time value ofthe call and the time value of the put equals the gross proﬁt, which isthe difference between the strike price and the net investment.
172 • Trading Options As a Professional
Table 6-5
Pricing a Conversion—Part 3: Analysis of Cash Flows, 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) Proﬁt before transaction costs Transaction costs Net proﬁt
 55.0054.46 0.540.44 0.100.04 0.06
Analysis of Time Values8 Time value of 55 Call  price  intrinsic  4.69  2.709 Time value of 55 Put  price  intrinsic  1.45  0.0010 Time value of call  time value of put
 1.99 1.45 0.54
1234567
Conclusion: For a conversion, the difference between the time value of the calland the time value of the put (10) equals the gross proﬁt (3).
In the speciﬁc example illustrated in Tables 6-3 through 6-5, theconversion position yields a proﬁt of 6 cents per share because thetime value of the 55 Call is 54 cents greater than the time value ofthe 55 Put. The result will differ if the borrowing rate or the transaction costs or both change. Market makers therefore must constantlykeep their interest rate and costs up to date.
The Role of CompetitionCompetition