rue to aconcentrated set of specialists, or broadly to all participants, inwhich case each individual has the perception that the market isindeed ef®cient.
The Clientele Effect
Let us take the reasoning in a slightly different direction and introduce the notion of clientele effect. Different instruments attract different trading populations, and these populations interact with themarket in speci®c ways driving prices, liquidity, and volatility. Theidea of distinct trading populations is very clearly illustrated by thefact that most index derivatives in the world are not listed on thesame exchange as their underlying references.27 Historical as well astechnical reasons explain this situation,28 but it is clear that derivatives were created to address speci®c problems that stocks and``cash'' securities were unable to solveÐhence, the differentexchanges and trading populations. At the extreme are mutualfunds, which by and large are explicitly forbidden to trade derivatives. When a derivative product is involved in the arbitrage, it istherefore not surprising to witness distinct, and unaligned, investorgroups. In the case of risk arbitrage, the two products are similar in27
Note the following examples: the New York Stock Exchange for S&P 500 index; theChicago Mercantile Exchange for S&P 500 futures; the Tokyo Stock Exchange for theNikkei 225 index; the Osaka Stock Exchange for the Nikkei 225 futures and options;the London Stock Exchange for the FTSE 100; the London International FutureExchange for FTSE 100 futures; and the Bourse de Paris for the CAC 40 and theMatif for CAC futures. There are counterexamples: TOPIX (Tokyo Stock Exchange)has its future listed on TSE as well.28For example, the need existed for a central clearinghouse to handle margin requirements for derivative transactions in order to eliminate counterpart risks.
24
THEORY AND TOOLS
nature and are usually listed on the same exchange, but in fact theclientele effect still holds, at a micro level.If in practice it is naturally impossible to identify any set ofinvestors following a particular security, it is important to realizethat the investor population is always fragmented in a multitude ofsmaller groups with distinct interests and expectations.
Equities versus Other Securities
Traditionally equities have not been the preferred instruments of arbitrage. Fixed-income securities and foreign exchange markets haveattracted much more attention and activity in that regard. The reasonsfor this are very natural: These securities exhibit a strong and predictable relationship to macroeconomic aggregates; therefore, a solidunderstanding of economic theory leads to fairly adequate predictionof probable future moves. By contrast, equity valuation is a much moresubtle exercise.For instance, it is dif®cult to envision a country eradicating itsin¯ation overnight, but countless companies have taken ®nancialmarkets by surprise with bad earnings virtually from one quarterto the next. Therefore, macroeconomic aggregates exhibit consistentstability over time, which is critical because it gives plenty of time torevisit basic assumptions and adjust positions if necessary.Obviously the Asian crisis has demonstrated that things can govery bad very quickly, but in general volatility on country macroeconomics is relatively low. Equally important is predictability.Considering that modern and developed governments in generalhave similar goals and tools, it is not extremely dif®cult to preemptfuture moves meant to address known problems. Again, there maybe many variants, but it is not unreasonable to expect, for example,tighter monetary policy against in¯ationary pressure as a generalremedy. Trading based on such reasoning is usually termed macrotrading, and it is generally like arbitrage in nature because it is basedon an expected convergence.However, these same reasons that make ®xed income and foreignexchange so attractive also tend to be drawbacks in practice. If macroeconomic aggregates take time to adjust, arbitrageurs may have tocarry their position for quite a while, and things may get muchworse before they get better. Also if price discrepancies are so largeand fundamental criteria so compelling, chances are good that participants have spotted the same opportunities. Altogether this means that
CHAPTER 1 INTRODUCTION
25
macro arbitrage is not necessarily simpler even if the context is somewhat more attractive.29We would argue that equities in general present a very favorableterrain to arbitrage as well, even for macro arbitrage, although arbitrageurs have to have different goals and methods to be successful.Whereas there are only a few dozen currencies traded worldwideand as many governments susceptible to issue sovereign bonds,several thousand stocks are listed and trade with a decent level ofliquidity in the United States alone. Macro arbitrage targets largepro®ts from infrequent trading and bold mispricings, and these mispricings, as infrequent as they are, can also be exploited from equitypositions chosen wisely. Furthermore, equities offer the additionalopportunity for high-volume±low-pro®t strategies expected to produce results through the accumulation of small successful trades.This is generally the basic premise behind strategies like pair tradingor statistical arbitrage.Consider, for example, the two charts in Figures 1-2 and 1-3 as anillustration of macro arbitrage in the context of an equity strategy. Bothcharts present opportunities resulting from the Asian crisis, respectively in the banking and retailing sectors. The ®rst case is based onthe observation that French banks had a considerably higher exposureto Asia than German banks. After hedging the spread in local marketsÐrespectively versus the CAC and DAX indicesÐthe relative difference between French and German banks exhibits a strikingwidening spread with French banks outperforming, which can beviewed as an anomaly.30 The second situation is also built on thesame type of hypothesisÐthat is, the relative exposure to Asia fromtwo French retailers, Promodes and Carrefour.31 The arbitrage is lessobvious and based on the trend of the Asia ex-Japan index; if it isidenti®ed early, it is possible to speculate that Carrefour/Promodeswill follow a similar path and enter the relative spread accordingly.Overall arbitrage in these two situations is entirely based on aperceived mispricing, and the execution is subject to an additional
29
For a complete discussion of these strategies and numerous illustrations, see, forexample, Gabriel Brustein, Macro Trading and Investment Strategies: MacroeconomicArbitrage in Global Markets (New York: John Wiley & Sons, 1999).30For clari®cation, the trade implied by the chart is the following: short a portfoliocomposed of a spread between French bank hedged against the CAC versus longanother portfolio of German banks hedged against the DAX.31Note that the hedge of local market risk is not necessary because both companies areFrench and traded in Paris.
26
THEORY AND TOOLS_
×10 3
.80
ASIA EX-JAPAN STOCK INDEX IN US$
750
FRENCH BANKS/CAC40GERMAN BANKS/DAX(CURRENCY ADJ)
.70
700650
.60
600
.50
550
.40
500450
.30
400
.20.10
350
SERIOUS MACRO MISPRICING GAP (20%)GIVEN THE LARGER ASIAN LOANS EXPOSURE OF FRENCH BANKS
1992
1993
1994
1995
1996
1997
1998
300
Figure 1-2 Macro Arbitrage, French Banks versus German Banks(Source: Gabriel Burstein, Macro Trading and Investment Strategies: MacroeconomicArbitrage in Global Markets. Copyright # John Wiley & Sons, 1999. Reprinted by permission of John Wiley & Sons, Inc.)
CORRECT PRICING OF ASIAN CRISIS IN AN EQUITY SPREAD
2.40
700
ASIA EX-JAPAN STOCK INDEX IN US$
2.20
600
2.00
550
1.801.60
500
CARREFOUR/PROMODES
450
1.401.20
650
400350
FRENCH RETAILER PROMODES IS MOSTLY DOMESTIC ORIENTEDFRENCH RETAILER CARREFOUR HAS INTERNATIONAL EXPOSURE INCL ASIA
M A M J
J A S O N D J F M A M J
J A S O N D
J F M
Figure 1-3 Macro Arbitrage, Carrefour/Promodes(Source: Gabriel Burstein, Macro Trading and Investment Strategies: MacroeconomicArbitrage in Global Markets. Copyright # John Wiley & Sons, 1999. Reprinted by permission of John Wiley & Sons, Inc.)
CHAPTER 1 INTRODUCTION
27
uncertainty because the speci®c stocks chosen also present idiosyncratic risk related to their general business. In addition, it can beargued that the Asian crisis was certainly an exceptional event andthat opportunities of that type would not necessarily abound undernormal market circumstances. Nevertheless, these are two examplesof astute arbitrage in which equities have been used as proxies forlarger macroeconomic factors, clearly an indication that opportunities exist.
VALUATIONS: MARK TO MARKET VERSUSTHEORETICALAn arbitrage position is built on convergence expectations. It reliesentirely on the comparison between observedÐthat is, tradedÐpricesand a theoretical relationship we believe should prevail in the long run.Therefore, it can always be evaluated at any time from any of these twoangles.A sure convergence can always be used as a foundation for valuation, because whatever the market prices are, we know we can show apro®t if we fully execute each leg at the time of trading. At the otherextreme, in the case of speculative arbitrage, we have to use marketprices because of the inherent uncertainty, and we may have to show aloss, possibly large, if the original divergence widens.Consider, for example, the case of a risk arbitrage situation:Company A is acquiring company B through a 2-for-1 stock swap.Table 1-3 presents their hypothetical prices over a few days. On Day1, we enter a position by buying 2 shares of B and selling one shareof A (we neglect all transaction costs). At the end of the day, thetheoretical valuation is $1: the theoretical spread is 0Ðbecause oneshare of A should be equal exactly to 2 shares of BÐand we realizeda spread of +1 in the market $1  100 2  491/2). However, themark-to-market valuation is 0 because the spread we realized is nobetter than the spread at the close of the market. On Day 2 as a resultof usual market volatility, A and B close with a spread somewhatsmaller than the previous day. The theoretical valuation is unchangedby de®nition, while the mark-to-market shows a pro®t of +1/2.On Day 3, the spread comes back to its previous level. Again thetheoretical valuation does not move while the mark-to-market valuation is losing what it had just made on Day 2, 1/2, for a cumulative®gure back to 0.
28
THEORY AND TOOLS
TABLE 1-3Day
A
B
TheoreticalEvaluation (Cumul.)
Mark-to-MarketEvaluation (Cumul.)
12345
$100$97$98$95$92
$491/ 2$488/ 14$481/ 2$46$431/ 2
11111
01/ 2024
On Days 4 and 5 some bad news accumulates on A and B, alteringsigni®cantly the perspective of the acquisition. Naturally the marketreacts by selling both stocks, and the spread widens signi®cantlybecause the acquisition is not certain anymore. At the extreme, onDay 5, the market simply does not price any convergence anymore.The mark to market turns largely negative, and the theoretical isunchanged.This simple simulation shows that theoretical and mark-tomarket valuations capture different aspects of the position. Whereasone captures the entire pro®t immediately and with 100 percent certainty, the other tracks the actual occurrence of the convergence in themarketplace. Therefore, consistency requires the choice of the valuationmethod be primarily dictated by the nature of the position.Traders in general focus more on the theoretical part because itunambiguously shows an expected pro®t as if it were realized. Riskmanagers,32 on the other hand, tend to focus more on mark to marketbecause these ®gures are unquestionably a re¯ection of what is happening in the market. The difference in methodology can be a source ofmisunderstanding, both ways. For example, for index arbitrage wherethe convergence is certain, virtually all traders value their position ontheoretical grounds. When the futures closes far from its theoreticalvalue, mark-to-market valuation shows huge differences compared toresults declared by front-of®ce valuations. Support for the theoretical®gures is not dif®cult to get, but it requires all parties to understand theunderlying strategy and its structure. Now consider an analogous situation in which the mark-to-market valuation shows huge losses whereastheoretical still indicates pro®ts, but where the convergence is not certain anymore. Risk managers will naturally be as inquisitive as before,32
Risk managers here denotes ®nancial controllers, internal auditors, and more generallyall personnel involved in monitoring risk.
CHAPTER 1 INTRODUCTION
29
and the same argument can be used to demonstrate that mark to marketis temporarily erroneous. But in this situation the demonstration is notlogically acceptable because of uncertainties at maturity, and the markto market is indeed the most accurate valuation. The point in this exercise is to realize that the existence of two different pro®tability measuresfor the same position creates the need for a rigorous reconciliation. Oneof the parties has to give way to the other, for the better or the worst: Ifthe arbitrage argument is used indiscriminately, it may help justifycarrying losing positions long after they should have been liquidatedor at the very least reconsidered.This difference between observed and theoretical valuations willbecome clear as we develop the necessary formalism in the chapterdedicated to valuation techniques. As a safety rule, both should betracked at all times because they are complementary, but more importantly because the reconciliation process to translate one into the otheris generally valuable to all parties involved.
RISKRisk can be segmented according to different criteria. From an analytical standpoint, traders care about parametric risksÐthat is, risks arising from the randomness of market parametersÐand consequently wewill have to develop a framework to understand and manage theserisks later on. For the purpose of our introductory discussion here, weadopt a different segmentation and consider in turn the issues of execution, carry, and nonconvergence.
Execution
Execution risk covers all risk associated with building and unwinding aposition. It would be convenient to associate a high level of con®denceregarding the ®nal convergence to a lower execution risk, on thegrounds that simultaneous execution on highly correlated instrumentsseems a comfortable challenge. Paradoxically, in general, the exactopposite is true, and in fact it is legitimate to turn the hypothesisaround and state that the execution risk becomes relatively minor forspeculative positions.33 Consider, for example, the extreme situation inwhich a particular spread is arbitraged only by one player, everybody33
At expiry, however, it is true that for an absolute or explicit convergence, executionrisk is theoretically nonexistent because of the nature of the position. In practice, itmay persist still, for example, if the position involves a future expiring on an average.
30
THEORY AND TOOLS
else being indifferent to it. This particular player is in no hurry toexecute his or her strategy and indeed should take some time to minimize a possible impact he or she may have on the prices of the securities he or she wants to trade. At the other extreme, an indexarbitrageur always faces a situation in which the whole market perceives an opportunity at the same time he or she does, and he or shehas to compete to be the ®rst to take advantage of it. He or she mayvery well get ``stuck'' in the middle of his or her execution becauseother participants rush to sell the futures when it shows a premium orbuy it when it trades at a discount.This leads us to the notions of timing and hedge in arbitrage trading. Simultaneous trading requires that we carry an equivalent amountof risk on both legs of the arbitrage at all times. Risk here is fairlysimply measured by the traded nominal; therefore every dollar executed on the long side should be matched by the appropriate numberof dollars of executed short and vice versa.34 Therefore, from a riskperspective, synchronous execution on both legs is far more importantthan instantaneous execution of a large nominal, which would be keyto capturing an opportunity. This operating procedure, assuming it isfeasible, is the only guarantee to minimize execution risk by beingperfectly hedged at all time.It is necessary to realize, however, that it also leads to seeminglylost opportunities. If the market has a tendency to form clear trends,even for short periods of time, a trader might very well implement astrategy of trading on the waves. Nonsimultaneous trading on bothlegs in that case may result in higher pro®ts. It is fair to say that thisis still arbitrage, but it is not risk-free arbitrage anymore. This tradercould very well do the exact same thing and capture the same movements using only one of the legsÐthe most liquid and cheapest totrade, for example. This discussion is less trivial than it seems becausenonsimultaneous trading is usually embedded in traders' behavior,and it is very dif®cult to disentangle. Yet it should be clearly separated: Pure arbitrage requires rigorous simultaneous execution, and ifthe only way to make money is to skillfully play the short-termwaves, then the arbitrage activity should be stopped or reducedand resources refocused on pure speculative trading based on waverecognition.
34
Note that we say the ``appropriate number of dollars,'' not ``an equal number ofdollars.'' As we will see shortly, hedging a position may be done in ways otherthan dollar for dollar.
CHAPTER 1 INTRODUCTION
31
Hedge StrategiesFrom a broad perspective we can say that there are two categories ofhedge strategies: static and dynamic. These two broad methodologiescorrespond to different needs and are applicable in speci®c circumstances. A static hedge essentially applies to a position that is consideredappropriately hedged once and for all when it is initiated. Conversely,a dynamically hedged position needs to be continuously monitored andadjusted.Static Hedge: Fixed QuantityA ®xed-quantity hedge rests on the premise that the two legs of aposition are bound by a rigid relationship with respect to theirrespective size. If the position is an arbitrage, this relationship canbe deducted from it, as is the case of an index arbitrage position, forexample. The ratio of the appropriate quantities on each legÐthehedge ratioÐis measured directly on expiration, and the ®xed-quantity approach simply postulates that the same ratio is also valid atany time before. Therefore, once we know the right hedge for apossible transaction, this ratio is kept constant from initiation toexpiry, except for minor adjustments warranted by the speci®cs ofthe position.35 In an index arbitrage, for example, arbitrageurs de®nea strict equation between the number of futures and the stock position they carry, and this equation holds permanently. All executions,including expiry, re¯ect this ratio. In the case of risk arbitrage, forexample, the ratio is part of the terms of the transaction, and once set,it is not expected to change.In general, an absolute or explicit convergence dictates the adequate ratio to be used, in absolute terms. Regardless of who is entering the arbitrage, or at what moment in time, the ratio isconsistently equal to an absolute reference available to all. By contrast, an arbitrage on a more speculative convergence may rely on arelative ratioÐthat is, a ratio deducted from a certain number ofproprietary hypothesesÐfor example, the past behavior of securitiesin the arbitrage.In that last category, a common hedge ratio derived from portfoliomanagement theory is the relative beta of two portfolios or securities.The de®nition of beta is that of the capital asset pricing model, but it isadapted to the needs of the situation. Speci®cally:35
Among such adjustments is the hedging of ®nancing costs for margin calls on futurespositions. See the chapters dedicated to futures and index arbitrage for more details.
32
THEORY AND TOOLS
A=B 
cov rA ; rB  A=B A2BB
where A=B is the beta of A versus B, cov (rA ; rB ) denotes the covariance of returns from A and B over the relevant period of time inthe past, A=B their coef®cient of correlation, and 2A and 2B theirrespective variances. This ratio captures the relative behavior of Aand B: A 1 percent move in the price of B has historically translatedinto a A=B percentage variation in the price of A. Therefore, eachdollar of exposure on A has to be matched with A=B dollars ofexposure on B if the position is to be hedged with regard to pastreturns.36If NA and NB stand respectively for nominal exposures, this lastcondition is equivalent to:NB 
A=B  NA
At all times, the valuation of a spread portfolio between A and B can beexpressed as37:V  NA
NB
This is true because an increase in NA results in a pro®t, whereas anincrease in NB results in a loss. A variation in V captures an overallpro®t or loss from the time we take the position to the time we exit.Alternatively, V can be regarded as the total exposure of the portfoliobecause it gives access to possible losses in the case of unfavorablemarket moves.It follows that:V  NA
NB  NA
A=B  NA
Following a market movement, V varies with NA and NB according tothe following equation:V  NA36
NB
It can be shown that this hedge ratio is the one that minimizes the variance of theposition's pro®t and loss. See, for example, John C. Hull, Introduction to Futures andOptions Markets, 3rd ed. (Englewood Cliffs, N.J.: Prentice Hall, 1998), p. 90.37A spread portfolio is here a portfolio holding a spread position between one orseveral securities. In the simple case above, we are long A and short B.
CHAPTER 1 INTRODUCTION
33
where the traditional operator  indicates a change in the associatedvariable.38 If we de®ne the percentage variations of A and B aspA  NA =NA ; pB  NB =NB , then V becomes:V  pA  NApA
pB  NB  pA  NA
A=B  pB  NA 
A=B  pB   NA
If A and B behave as they have historically, the de®nition of A=Bimplies pA  A=B  pB , which essentially indicates that V  0.Therefore, using a beta as a hedge ratio is clearly motivated byan historical perspective, and the whole position is pro®table onlyif A and B break away from the statistical pattern that betacaptured.Beyond the case of a ratio deducted from a convergenceÐwhichlooks at the futureÐand that of a beta hedgeÐwhich looks at thepastÐthe third option in terms of a ®xed-quantity hedge is to considerthe current market as the most appropriate source of information andderive a ratio from there. Typically, if t  0 indicates today's date, we®x the ratio as of today, to NA =NB t0 , and the valuation of the portfolio at a later date is then:V  NA
NANB
t0
 NB
The clear advantage of this solution is its simplicity: We hedge theposition today and simply keep it static going forward. This is clearly achoice of default, however. If we knew about a likely relationshipbetween A and B, we would use some type of ratio based on it. Inthe absence of such a relationship, the next best alternative is to postulate that historical data capture a large part of the information weneed. Then if this historical perspective is not considered legitimate forsome reason, the only ®xed-quantity hedge left is based on today'smarket.By nature, all ®xed-quantity hedge strategies are static: Once theratio is set, it is unlikely to change. If it does, this change generates apro®t or a loss because the readjustment of the position creates a distortion. This property is convenient in the world of arbitrage because itensures that the carry of the position is easy to implement: There issimply nothing to do.38
Precisely V  Vfinal
Vinitial .
34
THEORY AND TOOLS
It is also important to note that a ®xed-quantity hedge does notusually result in a portfolio being hedged in nominal exposure. Forexample, the two legs of a portfolio hedged in beta are always ofdifferent nominal because V  NAA=B  NA 6 0 unlessA=B  1.This is an important consideration because it shows that the notionof hedge is dependent upon the metrics chosen to evaluate risk.First- and Second-Order ExposureReconsider the above portfolio with two securities, A and B, in thegeneral case where we have a generic spread position, which wewrite V  NA NB . Given the conventions we used earlier, we alreadyknow that:V  NA
NB  pA  NA
pB  NB
If we are hedged in nominal terms, as opposed to hedged innumber of units, NA  NB  N and V  pA pB   N. Therefore, ifpA  pB , the portfolio does not show any pro®t or loss regardless of theamplitude of the absolute market move. Because the outcome is determined by a performance differential, V is what we call a second-orderexposure.Now assume that we carry this position for a while, and at somepoint pA 6 pB resulting in a divergence between NA and NB . If we haveNA 6 NB today with NA  NB  N , the variation in V following amarket move becomes:V  pA  NApA
pB  NB  pA
pB   NB  pA  N 
pB   NA  pB  N
The term V is now the sum of two terms, a similar differential component, (pA pB   NB or (pA pB   NA , and a new quantity pA  N orpB  N .The interpretation of these two terms is very straightforward:pA pB   NB is the result of a performance differential, and can beconsidered a second-order exposure. On the other hand, pA  N isdirectly dependent on the absolute performance of the ®rst security,and as such it is what we call a ®rst-order exposure. N itself representsthe extent to which the two legs of the position are not exposed on thesame nominal.This discussion has the very simple goal of showing that the valuationÐand exposureÐof a spread portfolio can always be separated intwo terms of distinct nature. Whereas this analysis was not relevant toa ®xed-quantity hedgeÐbecause we knew that we could not expect the
CHAPTER 1 INTRODUCTION
35
two legs of the portfolio to be equal anywayÐit is critical to understand a strategy of dynamic hedge.Dynamic Hedge: DeltaA delta hedge39 is based on the premise that we want to get rid of ®rstorder exposure as often as necessary. Typically we trade away the termN in the market to ensure that the overall exposure is only differentialbetween A and B.40This hedging procedure is very common on options, from which itderived its name. It is based on the analysis of the delta of the position,which indicates what is the nominal mismatch between the option andthe rest of the portfolio. This mismatch is generally systematicallytraded away by adjusting the composition of the portfolio with respectto the underlying stock.Delta hedging is very resource intensive in many respects. Itrequires a lot of analysis time to assess the delta of a portfolio.Furthermore, to be ef®cient, trading must be continuous, or at leastvery regular, which may turn out to be costly in the presence of transaction costs. The most important problem with this approach, however, resides in the fact that we are shifting away from true arbitrage,because the position is far from being risk free even after it has beenfully executed.Consider, for example, the common strategy of convertible bondarbitrage. It consists of trading both a convertible bond and its underlying stock, in effect breaking down the bond into a risky straight bondand a warrant with a strike price deducted from the terms of theconversion. It is referred to as arbitrage although in practice it doesnot pass our de®nition for three reasons. First, the warrant imbeddedin the bond is highly dependent on volatility, which is not the case ofthe stock. In other words the two securities do not have a similarbehavior, the volatility exposure remains throughout the life of theposition, and even when delta hedged, this is not a risk-free proposition.41 The second reason is related to the carry of the positionÐthat is,39
For an exhaustive and very practical discussion of dynamic hedging, see NassimTaleb, Dynamic Hedging: Managing Vanilla and Exotic Options (New York: JohnWiley & Sons, 1997):40The choice of trading A or B to get rid of N is based on a risk management perspective. If N > 0; NA > NB , and trading A, for example, will reduce NA to NB .Conversely, trading B will scale the overall nominal of the position up to NA .41It is naturally possible to hedge the convertible bond with options and introducevolatility in both legs of the arbitrage. This would solve the ®rst dif®culty but notthe second because the overall portfolio has to be dynamically hedged anyway.
36
THEORY AND TOOLS
its behavior through time. Delta hedging the option requires continuous trading and is not risk free because if we leave the position unattended, it is susceptible to developing a signi®cant exposure on theoverall market. Finally, the bond portion is exposed to a credit riskthat may be signi®cant and for which we do not necessarily have anysatisfactory hedging procedure.ApplicationIt is certainly useful here to consider a numerical application to understand better the differences between the two approaches ®xed qualityversus delta hedge. Consider two stocks S1 and S2 with the pricesequence given in Table 1-4. The simplest hedge method here is the®xed-quantity approach. Consider, for example, that the price relationship on January 18, S1  2  S2 , is more than a coincidence and isactually the result of a known merger transaction between the twocorporations. If we were to take a position on the spread between S1and S2 , we would naturally adopt the same hedge ratio. Table 1-5summarizes the arbitrage. Assuming all transactions are executed onthe close, we originally buy 1 share of S1 at $100 and sell 2 shares of S2at $51. On January 18 when the convergence is realized, we buy backS2 at $45 and sell S1 back at $90. The total pro®t here is 1  $90 $100 2  $45 $51  $2.Suppose now that we do not factor in the merger transaction but