904The average price lower bound for the five days, lowerAvg = 74.7976The final price upper bound,pupper =
(T − N) avg c + N.upperAvg= 80.6476T
The final price lower bound,plower =
(T − N) avg c + N.lowerAvg= 74.7976T
Upper bound on number of target sharespupper= nBptgt80.647660.0= 134413 shares (with 95% confidence)= 100000 ×
Lower bound on number of target sharesp= n B lowerptgt74.797660.0= 129,917 shares (with 95% confidence)= 100000 ×
Note that in the preceding example, a volatility assumption is made forthe bidder stock. Therefore, a realistic value for the volatility leads to goodestimates. A reasonable approach to determining the volatility to use is tolook at the implied volatility of the options on the bidder stock and choosethe volatility corresponding to the contract with the most appropriate strikeand expiration.
SHORT SELLINGAs discussed in the introduction, the uptick rule makes it harder to short thestock. Until some time ago, a trading technique called married puts allowedthe traders to sidestep the rules that prevented short sales when a stock’sprice was falling steadily. The strategy involved the following steps. First,buy a deep-in-the-money put with the closest maturity date. Then, tender for
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the shares. Now we are theoretically long the stock and can therefore sell itwithout waiting for the uptick. This loophole has, however, been detectedand is now disallowed by the SEC.In another approach to circumvent the uptick rule, one may also enterinto a stock swap, receiving the acquirer’s stock in exchange for cash. Nowwe are long the stock and can therefore sell the stock without the uptick rulerestriction. To settle, we buy the acquirer’s stock in the market and return itor pay the stock returns. In turn, we would receive LIBOR with a haircut.Notice that this is similar to short selling except that we pay a premium forthe stock.
SUMMARYThe arbitrageur normally executes the paired transaction through abroker.Verifying the executions by pairing them is an approach fraught with inconsistencies. It makes sense for the arbitrageur to insist on a firm average spread or better.It is possible to put on a spread position during the pricing period andbe perfectly hedged. In such situations, however, the exact position sizeof the target stock is gradually revealed.Putting on a spread position typically involves a short sale and must beexecuted in accordance with the uptick rule.
FURTHER READING MATERIALOn Linear Programming and Network FlowsChavatal, Vasek. Linear Programming. (New York: Freeman, 1983).
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APPENDIXDINIC’S ALGORITHM FOR MAXIMUM FLOW INA NETWORKIn this section, we describe Dininc’s algorithm for maximum flow on a network. The following are some additional concepts used in the description ofthe algorithm.
Blocking FlowA flow on a network is a blocking flow if every path from source to sinkcontains a saturated edge (edge with full capacity flow through it). An example of a blocking flow is shown in Figure 10.4. Also note that the flow isnot maximum. The optimal flow is shown in Figure 10.4.
Residual GraphAssociated with every feasible flow through the network is a weighted digraph called the residual graph. This is constructed in the following manner.Let C be the capacity of the edge connecting vertices A and B. Let F be theflow from A to B. The residual graph has a directed edge from A to B withcapacity C – F and another from B to A with capacity F. All edges with zerocapacity are removed from the graph thus constructed. The resultant graphis the residual graph. An example of the residual graph corresponding to theproblem and the blocking flow is shown in Figure 10.4.
Flow Augmenting PathsA flow augmenting path is a path on the residual graph from source to sink.The maximum flow that can be sent through this path is limited by capacityof the edges in the path. It is equal to the smallest capacity of an edge alongthe path.
Dinic’s AlgorithmThe steps in Dinic’s Algorithm are as follows:1. Find a blocking flow for a given network.2. Construct the residual graph corresponding to the current flow.3. Find the shortest length augmenting path on the residual graph. (Thelength of the path is equal to the number of edges in it.) If such a pathdoes not exist, then the current flow is the maximum flow.4. If such a path exists, then augment the flow along the path. (Make thecorrections to the flow graph and the residual graph.) Go to Step 3.
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3units
2
2units
3
2
Src
Sink
31
31unit
1
3units
The Max-Flow Problem
3units
2units
3Src
Sink
331unit
3unitsBlocking Flow
3units
2
2units
3
2
Src
Sink
31
31unit
1
3units
Residual Graph
3units
1
Src
2unitsSink
21unit
1
3units
Optimal Solution
FIGURE 10.4
Max-Flow Algorithm.
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LAZY ALLOCATION ALGORITHMDinic’s algorithm is required for a general network. We hasten to add thatthe matching problem we are required to solve has some additional properties. To recognize that, we organize the trades on the stock being sold in ascending order (group A) and the trades on the stock being bought indescending order (Group B). Thus, each trade has a rank associated with it.Also, if there is a feasible edge between the ith vertex in group A and the jthvertex in group B, then there is a feasible edge between the ith vertex and allthe vertices in group B with rank greater than j. Similarly, there is a feasibleedge between the jth vertex in group B and all the vertices in group A withrank less than i. This is illustrated in the ordering of the nodes in Figure 10.2.To perform lazy allocation, we start from the vertex with the lowestrank in group A and saturate the edge connecting it to the lowest possiblerank in group B. When the entire trade quantity from the vertex is accountedfor, we move on to the next vertex and do the same and work our way to thelast vertex. The consequence of the lazy allocation is that on a graph withthis structure, there is no flow augmenting path on the associated residualgraph, and the allocation is optimal. An example of such an allocationmethod is demonstrated in Figure 10.2, configuration B.Although there may not be any flow augmenting paths, there may be analternate flow routing on the residual graph, such that the flow along the directed edge is zero. This implies that there could potentially be redundantoptimal routings.
CHAPTER
11
The Market ImpliedMerger Probability
INTRODUCTIONThe spread value in a merger deal is a measure of the profit potential of atrade. Knowledgeable players in the marketplace are likely to carefully assessthe profit potential and inherent risks and put on a position according totheir judgements. If the value of the spread is large and the risks inherent in171
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the successful completion of the merger are small, then we can expect thatthe players would put on a large position causing the spread to narrow. If,however, the value of the spread narrows disproportionate to the risks, thenwe can expect some profit taking causing the spread to widen. If there aresufficient players in the marketplace engaging in risk arbitrage, the spreadwould then in some sense represent the consensus estimate of the risk involved in the deal and thereby the odds of successful completion of themerger. Therefore, it makes sense to construct models to estimate the oddsof merger success taking the value of the spread into account.In this chapter, we discuss a method to assess the probabilities of mergeras reflected by the spread between the stock prices of the merging companies. The method is based firmly on the classical results of the Arrow-Debreutheory of contingent claims. The probabilities derived are called risk neutralprobabilities.Risk neutral probabilities calculated for a merger are in many ways similar to the implied volatility parameter encountered in option pricing. Forpurposes of option pricing, stock price movement is modeled as a lognormal process. One of the important parameters of the log-normal processis its standard deviation. This standard deviation is commonly termedvolatility. The volatility of the underlying stock plays a crucial role in thepricing of an option. Conversely, the price of an option quoted in the marketplace implies a certain volatility that when plugged into the option pricing model results in the quoted option price. This volatility implied by theoption price is called the implied volatility.In an analogous fashion, the probability of merger success as implied bythe observed spread may be calculated. Note that both implied volatility andthe merger probability specify probability distributions. Furthermore, theyare both calculated from values observed in the marketplace, and therein liesthe similarity between the two constructs. A key question, however, pertainsto the premise of the chapter. What is the value in knowing the market implied probability of merger? Well, if one has a view based on independent research that the true chance of merger success is better than the impliedprobability, then entering into a trade may have a good risk–reward profile.Thus, knowing the probability of merger can provide value in making investment decisions.The implied probability of deal success can also prove useful in the areaof risk management. The risk manager is typically charged with assessing therisk in multiple portfolios running multiple strategies. Unlike the risk arbitrage traders, risk managers are not disposed to know the intimate details ofevery deal in the risk arbitrage portfolio. Using the market-implied probabilities, the risk manager will be able to design meaningful value at risk(VAR) measures for such deals. The VAR plays a key role in determining the
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reserve capital requirements for a bank. The requirements for reserve capital are designed to help markets survive extreme conditions by making sure(to some extent) that the counterparties in a trade have enough reserve capital to meet their obligations. Ideally, we would like the reserve capital to bethe proverbial Goldilocks value, not too little and not too much. If the riskmeasurement is too conservative, then we will need to post more reserves,leading to underutilization of capital. If it is too aggressive, then we may nothave enough reserves to meet obligations during extreme moves. We willpropose a practical value at risk measurement approach for risk arbitragedeals, based on the merger probabilities.The chapter is organized as follows. First, we discuss briefly the ArrowDebreu theory, which forms the basis for our probability measure. We thendescribe the single-step model for measuring merger probability success. Thesingle-step model is then extended to multiple steps. Subsequently, we reconcile between the proposed theory and practice. This is followed by an example application to risk management.
IMPLIED PROBABILITIES ANDARROW-DEBREU THEORYThe purpose of this section is not so much to provide a formal descriptionof the Arrow-Debreu theory as much as to provide a flavor for it. Let us consider the scenario that involves placing bets on a set of outcomes. Examplesof such events could be a boxing match or a horse race. In these cases, theset of outcomes is finite and well defined. We will use the horse race example for purposes of illustration. Important to the discussion is the notion ofbetting. If, for example, the bet is placed in favor of a horse and it wins therace, then the reward is the payoff from the bet. If it happens to lose, thenhere, too, the reward is the payoff from the bet, except that the payoff isprobably zero dollars. Thus, a bet is completely defined when we specify thepayoff for every possible outcome. To place a bet, one has to put up thestake money. This is specified by the bookie.The Arrow-Debreu theory states that the full and complete specificationof bets with the stake money and the payoff for each outcome automaticallyimplies a probability for a particular outcome.1 Additionally, the stake
1
The reasoning stems from the maximization of a linear utility function resulting ina linear program with constraints. The weights happen to be the values of the dualvariables in the solution of the linear program. This work by Arrow and Debreu wasawarded the Nobel Prize in Economics.
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money must be the probability weighted payoff (also called the expectedpayoff) across all outcomes. Also, if it so happens that a single set of probability weights is not able to account for the entire set of bets, then arbitrageopportunities exist.According to the theory, the probabilities are derived such that any twobets with the same expected payoff have the same current value. It may bethat one of the two bets yields the expected payoff almost certainly and therisk associated with it is minimal when compared to the other bet. Thisscheme, however, treats both bets on an equal footing; that is, we are neutral to risk. For this reason, the set of probabilities that are implied by thedefinition of the bets are called risk neutral probabilities.Continuing with the horse race example, let us say that the odds givenby the bookie for the horse race is as follows: 3 to 5 in favor of NiceAndEasy, 2 to 3 in favor of WindSlicer, and 1 to 2 in favor of ButterBiscuit. Forinstance, a successful bet of one dollar on ButterBiscuit returns the stake plustwo dollars, which is a total of three dollars. The terms of the bets are presented in Table 11.1.Note that the first scenario described in Table 11.1 is the risk-free scenario where a deposit of x dollars with the bookie results in a payoff of onedollar no matter which horse wins. According to Arrow-Debreu theory, thebet amount must be a weighted combination of the payoffs. If the probability weights for each of the three horses winning are denoted as pne, pws, pbb,then the following equations as given in matrix form must hold.1800
TABLE 11.1
x1 1  pne   0 0 3⋅  pws  =  5 0 2  pbb   0 3 1
Terms of the Bet.Payoff from the Bets
Bet scenarioRisk-Free ScenarioBet On NiceAndEasyBet on WindSlicerBet on ButterBiscuit
Betamount
NiceAndEasywins
WindSlicerwins
ButterBiscuitwins
x321
1800
1050
1003
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175
or specificallypne + pws + pbb = xpne = 3/8 = 0.375pws = 2/5 = 0.4pbb = 1/3 = 0.333The value of x from the preceding equations is therefore pne + pws + pbb= 1.108. So, if we deposit close to a dollar and 11 cents with the bookie, wewill get back a dollar. The loss for the bettor in this enterprise is therefore100 × (0.108/1.108), which is approximately 9.7 percent. In other words, onaverage the bookie gets to keep 10 cents on every dollar that is deposited asstakes. Normalizing the probability weights to add up to one, we now haveProbability of NiceAndEasy winning = 0.375/1.108 = .338Probability of WindSlicer winning = 0.4/1.108 = .361Probability of ButterBiscuit winning = 0.333/1.108 = .301Thus, WindSlicer is favored to win the race, with ButterBiscuit being theunderdog.Note that we started our example with the specification of the bets andtheir odds and have now derived the probabilities from it. Are the probabilities the true probabilities for the outcome of the race? Maybe, then againmaybe not. This is, however, where the bookie will allow the bet to bemade. In the case of the markets, unlike the example here, the price/stakeamount of a bet is decided by the auctioning process. The price, therefore,represents the consenus opinion of the participants. In such situations itmay be argued that the risk neutral probabilities represent the consensus ofthe market.
THE SINGLE-STEP MODELWe are now ready to formulate the single-step model to calculate the probabilities implied by the spread. It is, however, good to remind ourselves ofthe implicit assumptions we make during the modeling process. First, putting on a spread involves a short position and we need to borrow stock. Attimes it may turn out that a particular stock is unavailable for borrow. However, for purposes of our model, we will assume that the stock is alwaysavailable to borrow. Additionally, we also assume that there are no liquidity issues and that it is possible to put on a spread position in size at thecurrent spread level observed in the market. This would imply that the transaction costs if any are negligible and may be assumed to be zero.
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To construct our model, let us now go through the mechanics of a riskarbitrage trade. Initially, a spread position is created. We do this by establishing a short position in the target and a long position in the bidder,yielding a dollar value equal to the spread. The proceeds from the trades arereinvested at the risk-free rate. Let us call this spread at time zero S0. Uponsuccessful completion of the deal, the spread would converge to zero. Theposition is reversed for no additional cost at time T, and the reward earnedby the investor will be erT S0, with r being the interest rate. In case there issome cash paid out for the target shares, the payoff will be erT S0 + cash.However, if the deal ends in a failure, the spread will not converge to zeroas expected. Instead, it inflates to a value of, say, ST. The position will stillneed to be reversed, and a cost equal to the spread at time T, ST, is incurred.The net payoff in the event of deal failure is therefore erT S0 – ST. The discussion of the scenarios here is illustrated in a state diagram as shown inFigure 11.1.
Payoff onSuccessertS0+cash
πsuccess
CurrentSpread: S0
πfailure
Payoff onDeal BreakerTS0−ST
FIGURE 11.1
Single-Step Model.
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177
The state diagram in Figure 11.1 comes with a few caveats. Very often,the fact that the deal is going to be unsuccessful is known much before thetime it would take for successful completion, in which case, treating both thepayoffs as if they are known at the same time would not be appropriate. Forour purposes, however, we choose to reflect this difference in timing in thevalue of ST while keeping T a constant. Later on in the multistep model formulation, we will eliminate the need to estimate ST. Therefore, keeping T thesame for both the success and failure scenarios does not affect the outcomeof the model.Recall that the initial cost of setting up the trade was zero. By the no arbitrage condition the expected payoff is also zero. Writing out the equations,we have
π success (e rT S0 + cash) + π failure (e rT S0 − ST ) = 0
(11.1)
psuccess + pfailure = 1
(11.2)
where psuccess and pfailure are the risk-neutral probabilities of successful mergerand failed merger, respectively. Solving the two equations, we have
π failure = e rT (S0 + e − rT cash) / (ST + cash)
(11.3)
Note here that in the derivation of the one-step model we apply theArrow-Debreu ideas to the spread. For the curious mind, the same resultsmay be obtained by applying the Arrow-Debreu ideas to the individualstocks in question, also. The derivation of the formulas for the model usingthe individual stocks is presented in the appendix.In Equation 11.3 for failure probability, all the values are known or observable except for ST, the spread in the future if the deal happens to break.The value for this spread is anybody’s guess. Unless a reasonable value forthe deal break spread is known, the one-step model as it stands is of notmuch use. We deal with this issue in the multistep model.
THE MULTISTEP MODELThe multistep model relates the changes in the risk-neutral probability to thedynamics of the spread movement. In this case, it is crucial to have an estimate of deal break probability on the eve of the announcement. If we areable to assess the probability of successful merger on the eve of the announcement, the value may be updated periodically based on the spread dynamics. This eliminates the need to guess at the deal break spread and thusmitigates the problem of the one-step model. We now proceed to describethe model.
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Spread=0
π t1success
π t2success
0
T
St1
St2
π t1failure
π t2failure
ST
FIGURE 11.2
Multistep Model.
Let t1 and t2 be points in the interval [0,T] and t2 > t1. We denote thet1t2, π failureprobabilities of failed merger at times t1 and t2 by, π failureand thespreads by St1 , St2, respectively. See Figure 11.2.Applying the one-step model, we havet1π failure= e r(T − t1) (St1 + e − r(T − t1) cash) / (ST + cash)
(11.4)
t2π failure= e r(T − t2 ) (St2 + e − r(T − t2 ) cash) / (ST + cash)
(11.5)
Dividing one by the other, we havet1t2π failure/ π failure= e r(t2 − t1) (St1 + e − r(T − t1) cash) / (St2 + e − r(T − t2 ) cash) (11.6)
To get a better feel for what the equation signifies, let us consider the situation of a pure stock-for-stock deal without a cash component. Making thecash to zero, the equation in this case reduces to
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179
t2t1π failureπ failure= e ( 1 1) St1 St1r t −t
(11.7)
Taking logarithms on both sides, we now have
(
t
)
(
t
)
(
)
( )
( ) (11.8)
21log π failure− log π failure= − r t 2 − t1 + log St2 − log St1
Let us see how this equation may be interpreted. First, when t2 – t1 is asmall quantity, the above becomes a difference equation. The left-hand sideis the difference in the logarithms of the probabilities. The negative logarithm of probabilities can be interpreted as a measure of information content. Therefore, the left-hand side may be interpreted as the change ininformation content between the times t1 and t2. The right-hand side has aterm consisting of the difference in the logarithm of the spreads. This is, infact, the unrealized profit per target share. The other term on the right-handside represents the risk-free return. Therefore, the equation may be interpreted as saying that any return in excess of the risk-free rate is equal to thechange in the information content, that is, the reduction in the uncertaintyof deal break.0Once we have the initial probability value, that is, π failure, all the probabilities in the interval [0,T] can be evaluated using the difference Equation11.8. We have thus eliminated the requirement to make a guess at the spreadvalue in case of deal break. This is replaced with the assessment of the initial deal break probability. As a practical matter, the risk-free rate values inEquations 11.7 and 11.8 are rather negligible. We can therefore set r = 0 inour calculations.
LOGARITHMS AND INFORMATION THEORYThe interpretation of the negative logarithm of probabilities as the information content was originally proposed by Claude E. Shannon inhis groundbreaking article “A Mathematical Theory of Communication,” published in 1948.2 Shannon was then working at Bell Laboratories. The powerful ideas describing the ways to measure rates ofinformation flow very soon became a discipline in its own right calledinformation theory. As a matter of fact, the word bit (as in bits per second) that is so commonly used today is attributed to Shannon.2
“A Mathematical Theory of Communication,’’ Bell System Technical Journal27 (July and October 1948): 379–423; 623–656.
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The initial deal break probability can be naively estimated using a number of robust statistical methods based on past data relating to deal announcements and successful completions. Alternately, an assessment of thefundamentals of the two merging companies can be used to arrive at the dealbreak probability. Yet another approach could be to base it simply on the instantaneous reduction of the spread on deal announcement. The average ofthe logarithm of the spread for the 10 days just prior to deal announcementcould be used as the logarithm of the spread corresponding to a failure probability of 1.
RECONCILING THEORY AND PRACTICETo see how the theory bears out in practice, let us look at some implicationsof the preceding model. According to the theory, the expectation is that thespread will converge to zero in the case of successful completion of merger.Alternately, we expect it blow up or widen in case the merger does not gothrough. We present both examples and counterexamples. In cases wherethe model might not hold, we also provide possible reasons as to why thatwould be the case. All the examples presented are taken from the mergerboom period of the late 1990s.Also, the key value in the theory relates to the logarithm of the spread.If the logarithm of the spread goes down in a linear fashion, then the spreadgoes down in an exponential fashion. As a matter of fact, in the case ofmergers with no glitches along the way, it is not unreasonable to expect thelogarithm of the spread to decrease in a linear fashion. We could thereforeexpect the spread to exponentially approach zero as the merger date nears.Now when the spread goes to zero, the logarithm of the spread goes tonegative infinity. In practice, we may assume a lower bound to the value ofthe spread between the two stocks to be at least equal to or greater than thebid-ask spread between them. At best, the spread could go to a penny. Thelogarithm of the spread for a penny is –4.6. That would be about the lowestvalue that we could expect for the logarithm of the spread.With that said, let us look at some real-life deals. We start by looking atsome mergers where the spread behavior follows the model. Figure 11.3shows examples where the theory holds. Figure 11.3a is a plot of the spreadfor a successful merger. The bidder in this case was Newell Company, andthe target was Rubbermaid Corporation. The ratio for exchange was 0.7883.The deal was announced on October 21, 1998, and completed March 24,1999. Notice from the figure that it is conceivable that we could fit an exponential to the overall profile of the spread.Figure 11.3b is a plot of the spread of an unsuccessful merger. The bidder was American Home Products (AHP), and the target was Monsanto
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181
Spread on Close
2.21.81.41.00.60.2–0.20
Spread on Close
FIGURE 11.3A
60
40
20
80
100
Successful Merger (NWL-RBD).
161412108642010
FIGURE 11.3B
30
50
70
90
Unsuccessful Merger (AHP-MTC).
Corp. (MTC). The exchange ratio was 1.15. The deal was announced June 1,1998. It became obvious that the deal was a failed deal around October 13,1998. The plot of the spread in the graph is for these dates. Notice that thespread does not converge at all. This is indicative of the risk involved in thisparticular merger. When it becomes obvious that the deal will not gothrough, the spread in fact blows up.We also present cases where the theory does not bear out. Figure 11.4ais a plot of an inverted spread. The bidder in this case was Venetor Group(Z). The target was Sports Authority (TSA). The exchange ratio was 0.8.The deal was announced May 7, 1998, and it became apparent that the dealwas unsuccessful around September 10, 1998. Note that the spread in thiscase takes on negative values. The logarithm of the spread in this case isundefined. It is not possible to apply the theory here. Such a situation is
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Spread on Close
210–1–210
50
30
FIGURE 11.4A
90
70
Inverted Spread (Z-TSA).
Spread on Close
242016128400
20
FIGURE 11.4B
40
60
80
100
120
140
Unconverged Spread (BRK-GRN).
possible if there are not enough arbitrageurs participating in the merger. Asa consequence, the spread dynamics do not imply anything.Figure 11.4b is a plot of an unconverged spread. The bidder in this casewas Berkshire Hathaway (whose chairman was the famous Warren Buffet).The target was General Re, a reinsurance company. The deal was announced June 19, 1998, and completed December 21, 1998. It is interestingto note that the spread on close was about $ 4.00; that is, the convergenceto zero did not happen. To see why that would be the case, note the fact thatBerkshire stocks were priced very high, and the exchange ratio was about0.0035; that is, for every thousand shares of General Re we would need 3.5shares of Berkshire to be perfectly hedged. This awkward exchange ratio ishard to achieve given the average per-trade volume of General Re and Berk-
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183
shire. Thus, the liquidity of the stocks involved increases the execution riskand caused the inefficiency.We now apply the theory to the INTC–LEVL example discussed inChapter 10. Figures 11.5a and b are plots of the spread and the probabilityimplied by the spread as of the close. The initial probability is determined asthe percentage by which the spread narrows on deal announcement with respect to the spread calculated on the 10-day average prices of the securities.In this case it turns out to be close to 30 percent; that is, the spread narrowedby about 70 percent on deal announcement. Note that the probabilities calculated follow the spread values very closely. This results in a very noisypicture. It is conceivable that the probabilities do not vary as much. It istherefore useful to use some smoothing function on the spread before evaluating the implied probabilities. We will discuss this in greater detail inChapter 12.
5
Spread
432100
20
40
FIGURE 11.5A
60
80
100
120
100
120
Spread (INTC-LEVL