a long history and is a widely practicedarbitrage technique.
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The mechanics involve putting on a spread position when the deal is announced and unwinding it on deal completion.The spread is a key market variable that the arbitrageur relates to.Its value is roughly equal to the dollar profit per share of target and alsoindicative of the inherent risk of deal break.We also discuss a typical trading strategy.
FURTHER READING MATERIALHistoryNelson, Ralph L. Merger Movements in American History, 1895–1956. (Princeton,New Jersey: Princeton University Press, 1959).Nye, Richard B. and Roy C. Smith, “Event Investing” (working paper, SolomonCenter, Leonard N. Stern School of Business, New York University, WP-S-9527, 1995).Mechanics of Risk Arbitrage & Transaction TermsMoore, Keith M. Risk Arbitrage: An Investor’s Guide. (New York: John Wiley &Sons, Inc., 1999).
CHAPTER
10
Trade Execution
INTRODUCTIONThere is an oft-quoted maxim, “Concentrate on what you do best and delegate the rest.” Going with this idea, the general preference of the arbitrageuris to concentrate on deal analysis and delegate trade execution to a broker.The execution in this case is fairly involved, as the trades need to be done ona paired basis. Not only must the broker ensure that the positions in both thestocks satisfy the particular exchange ratio, but he or she also needs to ensure that the fill prices on the stocks capture the specified spread. Thus, atall times the broker needs to ensure that the ratio and spread constraints aresatisfied. Given that the execution is fairly involved, there are only a selectnumber of brokers who accept orders to be executed on a paired basis. Inany case, there is now a need to specify the order unambiguously to a broker. We will discuss the unambiguous specification of a paired transactionin this chapter. There is also a need to verify that the order was executed tosatisfaction. We will also discuss issues pertaining to execution quality andthe criteria that could be used to measure it.Next, we address trade executions in fixed value exchange transactions.Recall that in such transactions, the exchange ratio is gradually revealed asthe pricing period unfolds, and the exact exchange ratio is known only at theend of the pricing period. This seems to carry with it the implication that thespread position can be put on only after the end of the pricing period andnot any time before that. However, that need not be the case. It is possibleto put on a spread position during the pricing period and still manage to beperfectly hedged. We will discuss how that can be accomplished.Also note that we expect to enter into a short position of considerablesize on the bidder stock when putting on the spread; that is, we need to execute a short sale on the bidder stock. A short sale is the sale of stock that theseller does not own but is committed to repurchasing eventually. The NewYork Stock Exchange and the NASDAQ stock market require that any shortsale must occur when the price is rising; that is, on an uptick. Therefore,151
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there are fewer opportunities to execute a short sale making it relativelyharder to short a stock. In fact, the uptick rule is meant to curb the impactof attempts by short sellers to drive a stock’s price down through aggressiveselling. We will conclude this chapter with a few remarks on short selling.
SPECIFYING THE ORDERThe first step to the successful execution of an intended trade is unambiguous specification. In the case of a single stock execution, the order must at aminimum specify clearly the ticker, the trade direction, and the trade quantity. This is essentially what is known as the simple market order. Additionalconstraints related to the price at which the trade may be executed are addedto this basic instruction and specified in what is well known as limit ordersor stop orders.In the case of a paired transaction, we need to specify the bidder and target tickers, and the trade direction and trade quantity for each of them. Theother constraints specified include the spread to be captured between the
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stocks and the ratio of the bidder to target shares. As in the case of a singleorder, it is also possible to get a partial fill in a paired execution. In this case,the entire specified quantity is not filled. However, the quantity of completedshares of the bidder and the target are expected to satisfy the ratio constraint, and the prices of the fills must be such that the calculated spread isgreater than or less than the specified value as the case may be. Therefore,the two important constraints that distinguish a paired transaction from asingle stock transaction are the ratio and spread constraints.We discuss here the specification of the details of a paired transaction.An arbitrageur specifies an order to the broker by providing the followinginformation.
Bidder and Target TickersTo execute an order, we would have to know the ticker symbols of thestocks involved.
ActionThis is to signal the arbitrageur’s intention. The arbitrageur can either puton or unwind the spread position. Notice that this implicitly sets the direction for trading the bidder and target stocks. Put on implies that the brokeris required to sell the bidder stock and buy the target stock. Unwind refersto the reversal of a spread position; that is, buy the bidder stock and sell thetarget stock.
Ratio and Cash AmountThe ratio and the cash amount are part of the calculation formula used tocalculate the achieved spread. We repeat the formula here for convenience.achieved spread = price paid for bidder stock × ratio ++ cash amount – price paid for target
(10.1)
The Spread ValueThe specification of the spread value by the arbitrageur is similar to the limitorder for a single stock. The broker should aim to match or beat the specified spread value. Now when the spread is put on, the arbitrageur would likefor the spread to be as wide as possible. In this case, the broker should aimto achieve a spread greater than the specified value. The situation is reversedwhen we unwind the spread. Here, a low value for the achieved spread isgood, and the broker should aim to get as low a spread as possible on theexecution.
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Target Stock QuantityThis piece of information in the specification is rather straightforward. Implicit in the target quantity specified is also the share quantity of the bidderstock. The bidder stock quantity may be calculated using the formulabidder quantity = ratio × target stock quantity
(10.2)
Short Sale IndicatorThe short sale indicator can take values yes or no. Putting on a spread position involves selling the bidder stock and buying the target stock. However,if we do not own any of the bidder stock already, the only way we can puton the spread is by selling the bidder stock short. Since this requires a shortsale, we would expect the short sale indicator to be set to yes. Similarly,unwinding the spread involves covering the short position in the bidder witha purchase of bidder stock and selling the target stocks that we already own.This does not require a short sale, and we may expect the short sale indicator to be set to no.In fact, we could even go so far as to say that typically, we can expectthis flag to be set to yes when we put on the spread and to no when we unwind the spread. However, on occasion the arbitrageur may not have aspread position corresponding to a particular merger in his or her portfolio.Additionally, he or she may also happen to take the view that the merger isunlikely to happen and may expect the spread to diverge. To participate insuch moves, the unwind order is placed first before the order to put on thespread. This is sometimes termed as chinesing the spread. The unwindingin these situations involves the sale of target shares without a prior positionand would therefore be accompanied with a short sell indicator of yes.We illustrate the order specification with an example.
ExampleBidder ticker= HWP (Hewlett Packard Company)Target ticker= CPQ (Compaq Computers Corporation)Action= put onRatio= 0.6325Cash amount= 0.0Spread value= $2.30Target stock quantity = 10,000Short sell indicator = yesLet us see what this order means. The action is to put on the spread. Thatmeans we sell the bidder stock (Hewlett Packard) and buy the target stock(Compaq). Also, the short sell indicator is a yes. Therefore, the sale of the bidder stock (Hewlett Packard) has to be a short sale. Now that we have fixed
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the direction of trading for the two stocks, let us determine the specifiedquantities. The quantity of the target stock (Compaq) to be bought is specified as 10,000. The bidder quantity, however, is not specified. We deduce thebidder quantity from the target share quantity and the exchange ratio usingthe formula given. In this case, it is 10,000 × 0.6325 = 6325 shares. Therefore, this is an order to buy 10,000 shares of CPQ and sell short 6,325 sharesof HWP. Are there any additional constraints? The answer is yes, and it isspecified in terms of the spread value. The spread measured on the fill pricesof the two stocks must be greater than or equal to the specified spread valueof $2.30. This spread in this case may be measured using the formula0.6325 × price(HWP) + cash amount – price(CPQ) >= $2.30Thus, the order unambiguously specifies the tickers, the trade direction,trade quantity, the spread constraint, and the ratio constraint that are to besatisfied.
VERIFYING THE EXECUTIONThe paired execution order as specified to the broker is usually for a considerable size in terms of the number of shares involved. Given the large position sizes and the fact that this is a paired transaction, the order is workedby the broker in a series of executions. This is done on a best-effort basis.The broker then delivers to the arbitrageur a list of executions for both thestocks. It is now up to the arbitrageur to evaluate the quality of execution.The quality of executions may be measured two ways, speed and efficacy. In situations when there is news in the market and quick action is required, execution speed is of utmost importance and may very well be thecriterion upon which the execution quality is measured. However, in mostother situations the arbitrageur is interested in capturing as high a spread aspossible when putting on the spread and unwinding at as low a spread aspossible. In such cases, the execution efficacy, characterized by how well theexecutions measure up to the specified spread value, is more important.While one would ideally want to be able to capture as high a spread as possible in the shortest possible time, it is conceivable that a quick executioncould mean that one gives up on efficacy, causing a trade-off between thetwo criteria for measurement of execution quality.We now look at issues related to measuring execution efficacy. Ofcourse, in the course of the discussion we will see how it affects executionspeed. To measure efficacy, one can attempt to pair up the executions andcheck to see if each paired execution satisfies the spread constraint, howmany missed the specified spread value, and by how much. A ratio of thedollar value of the hits versus the misses could very well be the measure ofexecution efficacy. Another option is to compare the specified spread value
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against the average achieved spread calculated using the execution prices ofthe two stocks. We could then use the difference as a measure of executionefficiency. The question therefore is, which measurement approach do wechoose? We recommend going with the second method; that is, the onebased on the average execution prices. We will justify our recommendationby demonstrating that the first idea of pairing up executions to judge efficacy is a process fraught with inconsistencies. To do just that, let us gothrough the exercise of pairing the executions. This exercise in addition toproving our point also provides insight into how we may characterize aggressive and conservative trading.Let us start with the pairing process. The input to the pairing exercise isthe list of executions. Each execution is characterized by the stock ticker, thedirection of execution (long or short),1 the fill price, and the fill quantity.The executions may therefore be partitioned into two sets based on the stockticker. The idea is to pair the executions of one ticker against the executionsof the other ticker such that the spread and ratio constraints are met. We formulate the trade pairing exercise as a network flow problem.
HISTORYOne of the important subject areas of the latter part of the twentiethcentury that has found successful application in a multitude of situations has been the study of network flows. The subject area was pioneered by Ford and Fulkerson. In their seminal paper published in1956, they framed the network flow problem as an integer linear program. Besides detailing an algorithm to solve for the optimal way toroute the flow across a network, they also provided a novel interpretation of the dual of the network flow linear program, now famouslyknown as the max flow–min cut theorem.Since then, network flows have found application in a wide variety of situations. Of relevance to finance, a subclass of the generalmax–flow problem, known as matching, has found application ingame theory and to the theory of auctioning. Other applications of network flows range from something as practical as logistic planning tosomething as abstract as theorem proving,2 making this a fascinatingsubject area for study in its own right.
1
Since these trades are all done to fill the same order, we can expect the trade direction to be uniform across all executions for a given ticker.2For an example, see Umesh Vazirani, “Rapid Mixing Markov Chains,” Proceedingsof the Symposia in Applied Mathematics, Vol. 44, 1991, pp. 99–121.
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As a necessary preprocessing step, we evaluate the equivalent targetshare quantities for the fill quantities on the bidder side trades. This is accomplished by dividing the bidder fill quantities by the exchange ratio. Theseare the share quantities that will be used in the modeling exercise. The network flow problem can now be represented in a pictorial fashion.Every execution is represented as a node. We pick a node from each ofthe two sets (bidder trades and target trades) and determine if the fill priceson the two executions satisfy the spread constraint. If satisfied, we draw anedge or line between the two nodes or executions and assign a weight or capacity to it based on the number of shares that can be matched along thatedge. Thus, each edge represents a potential way to pair the executions. Wewish to match as many shares of the bidder as we possibly can with the target shares. The edges and the nodes form a network called a bipartite graphin graph-theory parlance, and the pairing problem is equivalent to maximizing the flow on this bipartite network. The idea becomes clearer with thefollowing example:
ExampleConsider an order specification with the following data:Exchange ratio = 1.0Cash amount = 0.0Spread value specified = $1.00Action = put onNow, since we are putting on a spread, we sell the bidder and buy the target. The spread constraint is therefore met when the calculated spread isgreater than the specified value of $1. The executions list for the bidder andtarget stocks when putting on the spread position is listed in Table 10.1.The pairing problem can now be posed as a max-flow problem on thenetwork shown in Figure 10.1.
TABLE 10.1
Execution List.Bidder List
Target List
Label
Quantity
Fill Price
RatioAdjustedQuantity
B1B2B3
100100100
19.019.520.0
100100100
Label
Quantity
Fill Price
T1T2T3
50150150
19.018.518.0
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Bidder Trades
Target Trades
100 at$19.0
50 at$19.0
x1Source
100 at$19.5
x4x2
x3
100 at$20.0
FIGURE 10.1
x5
x6
150 at$18.5
Sink
150 at$18.0
The Max-Flow Formulation.
Note that in Figure 10.1 we have explicitly shown the source and thesink to identify the direction of flow, which should be from the source to thesink. Each node represents an execution, and the details of the execution arelisted for each node. The maximum capacity of each edge is not shown in thepicture but is actually the smaller of share quantities of the nodes on eitherend of the edge. The pairing problem is now equivalent to maximizing theflow on this bipartite network. The linear programming formulation3 maybe written asMaximize x1 + x2 + x3 + x4 + x5 + x6sub:x1 ≤ 100x2 + x3 ≤ 100x4 + x5 + x6 ≤ 100x4 ≤ 100x2 + x5 ≤ 150x1 + x3 + x6 ≤ 1503
(1)(2)(3)(4)(5)(6)
The linear programming formulation here is different from the conventional linearprogramming representation of the max-flow problem. The interpretation of thedual in this case is a minimum weighted set cover problem. However, since this is notgermane to the discussion, we will not pursue this in detail.
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x1 ≤ 100x2 ≤ 100x3 ≤ 100x4 ≤ 100x5 ≤ 100x6 ≤ 100x1, x2, . . . , x6 ≥ 0
(7)(8)(9)(10)(11)(12)
Let us examine the formulation shown here. The objective in the linearprogram is the sum of the flows across all the edges, and we are rightfully looking to maximize this quantity. Constraints 1–3 state that the totaloutflow from a bidder node must not be greater than the share quantity ofthat node. Likewise, constraints 4–6 ensure that the inflow into a targetnode must not be greater than the share quantity of the node. The constraints 7–12 fix the maximum capacity of every edge. Thus, maximizingthe flow along this bipartite network leads us to a solution for our pairingproblem.The solution approach to the general max-flow problem has been wellresearched and a number of algorithms have been proposed to solve it. Also,the pairing problem we address has a special structure. On account of this,the solution method that would work for us is fairly straightforward. A detailed description of both the general method and one tailored to our problem can be found in the appendix. The important point, however, is that themodeling process and the solution method provide us with some insightsinto the nature of the pairing problem itself.In fact, the solution approach reveals that the answer to the pairingproblem is not unique, and it is possible to have a number of optimal paringconfigurations for a given set of executions. Looking at Figure 10.2 we cansee that there are at least three possible configurations for the given set of executions in Table 10.1, begging the question as to which configuration weshould use to measure execution efficiency. It is now easy to appreciate thefutility of measuring efficiency by way of pairing. Note, however, that theaverage achieved spread is a constant value regardless of the pairing configuration. We therefore contend that it is this measure that should form thebasis for verifying executions.There is a flip side to looking at just the average spread. The broker maytrade away the edge gained on an earlier execution, of course ensuring thatthe average spread criteria is met. To see what we mean, consider the situation with a target spread of $1.00 where the broker has completed abouthalf of the whole order, achieving a spread of $1.05. To satisfy the averagecriterion, the broker needs to achieve a spread of just $0.95 on the remaining half, which may not be what was intended by the arbitrageur.
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Bidder Trades
Target Trades
100 at$19.0
50 at$19.0100
50
100 at$19.5
100
100 at$20.0
50
150 at$18.5
150 at$18.0
Configuration A:excess 50 shares of target at $18.5
Bidder Trades
Target Trades
Bidder Trades
Target Trades
100 at$19.0
50 at$19.0
100 at$19.0
50 at$19.0100
100100 at$19.5
100 at$20.0
5050100
150 at$18.5
100 at$19.5
150 at$18.0
100 at$20.0
50100
150 at$18.5
50
Configuration B:excess 50 shares of target at $19.0
FIGURE 10.2
150 at$18.0
Configuration C:excess 50 shares of target at $18.0
Redundant Matchings.
In conclusion, it would help to indicate exactly what the evaluation criterion would be to the broker. This will hopefully ensure that the executionwas completed as intended. The possibilities are1. Conservative trading. Ensure that there is at least one way to pair thetrade list such that each pair satisfies the spread criterion.2. Aggressive trading. Ensure that the average spread achieved is the sameas or better than the specified spread value and therefore to the extentpossible the speed of execution matters.
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EXECUTION DURING THE PRICING PERIODIn this section, we focus on trading in the context of fixed value exchangetransactions. In such transactions, the exchange ratio is determined based onthe stock price of the acquiring firm during the pricing period and is gradually revealed as the pricing period unfolds. Since knowledge of the exchangeratio is important to executing a paired transaction, this would imply thatwe can trade only at the end of the pricing period.However, in some cases, this strict constraint may not be ideal for trading. To appreciate the fact, let us consider the situation wherein the end ofthe pricing period is very close to the shareholders’ vote, or even worse, theend of the period is well past the shareholders’ vote (these situations are notuncommon in real life). In such situations, the time interval between the endof the pricing period and the date of deal completion is rather short, leaving very little time to put on a position of substantial size. To further compound the situation, one has to deal with the prospect of rapidly narrowingspreads.Given this, the arbitrageur would prefer to not have to wait until the endof the pricing period to put on a spread position. Therefore, even though theexchange ratio is not known exactly, the arbitrageur would want to begintrading before the end of the pricing period but trade in a way that leaveshim or her perfectly hedged (meet the ratio constraints on the pair) eventually. In this section, we will discuss an approach where we trade during thepricing period and still manage to satisfy the ratio constraint at the end ofthe period.Consider the case where the exchange ratio is computed on a fixedvalue of the target stock pT and the average closing price of the bidderstock in the pricing period. Let the closing prices of the bidder stock bep1,p2, . . . ,pn. starting from day 1 to day n, the last day in the pricing period. Now, according to the terms of the fixed value exchange transactions, the ratio (number of bidder stock exchanged for one target stock) isgiven asr =
npTp1 + p2 + ... + pn
(10.3)
This is the fixed value of target stock divided by the average of the closingprices of the bidder stock in the pricing period. The reciprocal of the exchange ratio r is thereforepp 1 1  p1=  T + T2 + … + Tn r n ppp 
(10.4)
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Interpreting Equation 10.4, the reciprocal of the exchange ratio is an average of the reciprocals of the realized exchange ratio in the pricing period.We will exploit this idea in our approach to trade during the pricingperiod. If we decide to put on a total spread position consisting of nBshares of the acquirer, the number of shares of the target nT to be boughtis given asnBr
(10.5)
p2pn n B  p1++…+n  pT pTpT 
(10.6)
nT =nT =We can write this as
nT = n1T + n2T + … + nnT
(10.7)
whereniT =
n B pinpT
, i = 1, … , n
(10.8)
In other words, the formula tells us the amount of target shares to buyon a daily basis for a fixed number of bidder shares such that we would befully hedged at the end of the pricing period. We can mimic this equationnBduring execution by shortingshares of the acquirer and buyingnnBpi× T shares of the target on the (i + 1)th day. At the end of the pricingnpperiod plus one more day, we will be perfectly hedged. Alternately, we couldstart by holding a naked short position on the bidder stock and buy the corresponding number of target stock on each day in the pricing period. Ofcourse, the captured spread would vary depending on the timing of trades ofthe bidder stock.There is a small caveat in all of this, though. Recall from the introductory chapter that the dollar profit is calculated on a per-target-share basis.However, the total number of target shares that we buy in this case is knownonly at the end of the pricing period. Thus, it would be reasonable to saythat we would know our position size (in target share terms) and expectedprofits only at the end of the pricing period. Hence, by trading in this fashion we will have replaced our uncertainty on the ratio by the uncertainty onthe position size/profits.
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ExampleDays in pricing period n= 20Fixed value of target stock Pr = $60Closing price of bidder stockon ith day, pt= $80.5Total shares of bidderto short nB= 100,000Shares of bidder to short fornBthe next day== 100,000/20 = 5000nShares of target to buy forn B pithe next day== 5000 × (80.5/60.0) = 6708×npT
Bounds on the Position SizeIt was noted in the last section that when trading during the pricing period,an arbitrageur has to be willing to replace the uncertainty in exchange ratiowith an uncertainty in position. However, there is still an interest in evaluating some bounds on the position size of the target shares. In this section,we derive some bounds on the ratio and show how it can be converted tobounds on the position size.To see how we can estimate bounds on the ratio, consider the fact thatthe average of the closing prices of the stock is always between the maximum and the minimum prices. Thus, a very simple bound could be based onthat. If we assume a log-normal process for the price movement of the bidder stock, that is, the logarithm of the prices executes a Brownian motion,the probability distribution for the maximum of a Brownian motion in timet is given as x Fmax = 2φ  − 1, x ≥ 0σ t
(10.9)
where s in the equation is the volatility of the Brownian motion and Φ is thecumulative density function of the normal distribution. The formula represents the probability that the Brownian motion will have a maximum valueless than or equal to x in the time duration t. By symmetry we can expect thedistribution of the minimum to be a flipped version of distribution of themaximum for values less than 0. A plot of the distribution functions for themaximum and minimum is shown in Figure 10.3.Reading the graph, one can say that the maximum value of the Brownianmotion is less than the standardized value of 2.0 with 95-percent accuracy,
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1.00.9Cummulative Probability
0.80.70.6
Distribution of Maximum
Distribution of Minimum
0.50.40.30.20.10.0–3
FIGURE 10.3
–2
–1
01Standardized Variable
2
3
Distribution of the Max and Min of Brownian Motion.
the standardized value being
x
. Thus, we can say with 95-percent conσ tfidence that the average value of the Brownian motion will be betweenx = −2σ t and x.We now combine the preceding information with the fact that the meanof a sample is definitely lower than the maximum of the sample. The maximum can now be used as an upper bound for the mean. In a similar manner,the minimum serves as a lower bound for the ratio. The bounds on the ratiocan quickly be translated into bounds for the target share quantity by multiplying the ratio bounds with the Bidder shares quantity. Thus, an estimateof the maximum and minimum price during the pricing period may be usedto calculate the bounds on the exchange ratio.We, however, believe that we could obtain a tighter bound than the onejust shown. To see that, let us consider the average of n numbers x1,x2,...,xn.Now let the upper bounds for each of the numbers be b1,b2,...,bn. Thenx1 + x2 + … + xn b1 + b2 + … + bn≤nn
(10.10)
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Saying this in words, the upper bound for the average is the average ofthe upper bounds. It is easy to see that the same argument is also true forlower bounds. Exploiting this idea, we compute the bounds on the maximum and minimum for each day in the pricing period and use the averageof the bounds as the bound for the average. This is illustrated in the following example.
ExampleConsider the bidder stock selling at $80, 15 days into the pricing period consisting of a total of 20 days. The realized average price in the 15 days for thebidder stock is $79. The volatility of the bidder stock on an annualized basisis 32 vol. The value of a target share is fixed at $60. We plan on shorting100,000 bidder shares. So, what are the bounds on the number of targetshares that we would end up holding in the end?Step 1: List the given information.The daily volatility σ = 32 252= 2.01%Number of days to end of pricing period, N = 5 days= $ 79Current average price avgc= $80Current closing price pcloseTotal days in pricing period, T= 20 days= 60.0Fixed value of target stock ptgt= 100,000 sharesNumber of bidder shares nBStep 2: We now compute the individual bounds in Table 10.2.TABLE 10.2
Days
n
Bounds on Size of Target Position.
ScalingValue
IncrementalBound
Log ofUpperBound
n
2σ n
log pclose
(
Log ofLowerBound
)
(
log pclose
+2σ n
−2σ n
)
PriceUpperBound
PriceLowerBound
elog(UpBound)
elog(lowBound)
1
1.000
0.0402
4.4222
4.3418
83.2815
76.8478
2
1.414
0.0569
4.4389
4.3252
84.6798
75.5788
3
1.732
0.0696
4.4517
4.3124
85.7687
74.6192
4
2.000
0.0804
4.4624
4.3016
86.6976
73.8198
5
2.236
0.0899
4.4719
4.2921
87.5243
74.7976
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Step 3: CalculationsThe average price upper bound for the five days, upperAvg = 85.5