of the spreadabout its mean; that is, the total count fractions above and below the meanare likely to be at 0.5. However, we will do just the unconstrained versionhere.The cost function just shown is then minimized using various values ofl to obtain estimates of the count fraction function. A plot of the fit erroragainst values of l in a log scale is shown in Figure 8.5. Note that the fiterror remains relatively constant, close to zero for small values of l. Forthese small values of l, the fit error dominates the cost function. After a particular threshold value, further increases in l are accompanied by fit error increases. Let us call this threshold value the heel of the curve. This is the pointafter which the regularization cost measure takes control. Increases in l lead
12
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FIGURE 8.5
0Log (λ)
Lamda Plot.
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to a deterioration of fit in favor of increased smoothness. Then again, pastanother value of l, the fit error becomes insensitive to the l value. At thispoint the function is more or less a straight line at the mean of all the points(since this is the line that would lead to the lowest cost for the smoothingfunction.Our choice for l is right at the heel of the curve. If we move l to the leftof the point, then the fit error dominates the cost function. Moving to theright of the point results in the smoothing term dominating the cost function. Choosing the value of l to be at the heel of the curve achieves a finebalance between the two cost measures and is the value that we choose.The function values for this value of l are evaluated and plotted in Figure 8.6. The profit profile is then computed from the regularized curve andis as shown in Figure 8.7. The maximum is now easily read off from thisgraph and determines where we place the threshold for our trading signal.The maximum occurs at 0.75 times the standard deviation, which is also thetheoretical value, thus validating our approach.
0.5Count Adjusted for MonotonicityRegularized Curve0.4
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FIGURE 8.6
Regularized Plot of Counts.
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Regularized Profit ProfileRaw Profit Profile
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0.000.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 1.95 2.10Amount of Sigma Away from Mean
FIGURE 8.7
Profit Profile Estimation.
TYING UP LOOSE ENDSIn this section we highlight some issues of importance that were not coveredin the earlier discussions.
Multiple Thresholds DesignAs mentioned earlier, the need for multiple threshold design arises when thespread is generated using a mixture density white noise process. We could dothis by segmenting the time series sample based on periods or time of day.The thresholds are then estimated for each sample. Subsequently, they arechecked to see if the difference between the values corresponding to the datasets is significant. If it is, then we have multiple thresholds. Otherwise, westick to the single threshold approach. Note that in the case of multiplethresholds there are inventory limits to be decided at each level, and theproblem is far from being solved in a conclusive manner.An approach to circumvent multiple threshold design is to do thethreshold estimation in a dynamic fashion. The levels to apply currently are
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determined based on the data in the immediate past. The process could berepeated at regular time intervals and the spread levels determined on a continuous basis.
Sharpe Ratio CalculationsWe will desist from calculating the Sharpe ratio for a single pair because theportfolio of pairs traded determines in a great way the eventual Sharpe ratio.
Time-Based StopsAs noted in Chapter 6, the mere passage of time adds to the risk of holdingan unconverged spread. This is because of the residual common factor exposure resulting in the phenomenon of mean drift. With the passage of time,the SNR ratio deteriorates. We therefore need to calibrate the SNR ratio atwhich we would end up unwinding the position regardless of convergence.In some cases it may be worthwhile looking at unwinding at the meanvalue or zero spread instead of waiting for the spread to swing to the opposite direction.
SUMMARYWhen trading the spread, it is desirable to trade at threshold levels thatyield the maximum profits.A large threshold value trades infrequently for a large profit, and a smallthreshold value trades frequently for a small profit.The optimal value for the threshold is between the extremes.Finding the optimal value for the threshold is easier done using nonparametric methods rather than parametric methods.The profit function to be maximized can be constructed from sampledata using a two-step process of ensuring monotonicity of the crossoverdistribution followed by Tikhonov-Miller regularization.The abcissa for the maximum value of the profit function is the desiredthreshold value.
FURTHER READING MATERIALHidden Markov ModelsMacDonald, Iain L., W. Zuchinni and W. Zucchii. Hidden Markov and other Models for Discrete Valued Time Series. (Boca Raton, Florida: CRC Press, 1997).Ergodic TheoryWalters, Peter. An Introduction to Ergodic Theory. (New York: Springer Verlag,2000).
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determined based on the data in the immediate past. The process could berepeated at regular time intervals and the spread levels determined on a continuous basis.
Sharpe Ratio CalculationsWe will desist from calculating the Sharpe ratio for a single pair because theportfolio of pairs traded determines in a great way the eventual Sharpe ratio.
Time-Based StopsAs noted in Chapter 6, the mere passage of time adds to the risk of holdingan unconverged spread. This is because of the residual common factor exposure resulting in the phenomenon of mean drift. With the passage of time,the SNR ratio deteriorates. We therefore need to calibrate the SNR ratio atwhich we would end up unwinding the position regardless of convergence.In some cases it may be worthwhile looking at unwinding at the meanvalue or zero spread instead of waiting for the spread to swing to the opposite direction.
SUMMARYWhen trading the spread, it is desirable to trade at threshold levels thatyield the maximum profits.A large threshold value trades infrequently for a large profit, and a smallthreshold value trades frequently for a small profit.The optimal value for the threshold is between the extremes.Finding the optimal value for the threshold is easier done using nonparametric methods rather than parametric methods.The profit function to be maximized can be constructed from sampledata using a two-step process of ensuring monotonicity of the crossoverdistribution followed by Tikhonov-Miller regularization.The abcissa for the maximum value of the profit function is the desiredthreshold value.
FURTHER READING MATERIALHidden Markov ModelsMacDonald, Iain L., W. Zuchinni and W. Zucchii. Hidden Markov and other Models for Discrete Valued Time Series. (Boca Raton, Florida: CRC Press, 1997).Ergodic TheoryWalters, Peter. An Introduction to Ergodic Theory. (New York: Springer Verlag,2000).
PART
Three
Risk Arbitrage Pairs
CHAPTER
9
Risk Arbitrage Mechanics
INTRODUCTIONRisk arbitrage in its general connotation relates to trading around corporateevents that alter the capital structure of a firm. Note that we have introducedtwo terms, capital structure and corporate events. Let us briefly describewhat they mean, starting with capital structure. When a business wants toraise initial capital to finance its operation, it can generally do it two ways.One way is to borrow money from lenders and pay interest on the borrowedcapital. This approach is also called issuing bonds or issuing debt. The otherapproach is to promise a percentage ownership in the business commensurate with the fraction of capital invested. This is known as issuing equity. Afirm may choose to go entirely one route or alternately issue equity for someportion of the capital and issue debt for the remainder. The percentage ofdebt and equity that comprise the firm’s capital is termed capital structure.It is important to note that the study of capital structure and its impact onfirm value is a vast subject area in itself. However, for our purpose, the simplistic definition will suffice. We now move on to Corporate event. A corporate event may be described as an action undertaken by a company thataffects its shareholders and/or bondholders. Typical corporate events couldbe paying of dividends, stock splits, tender offers, mergers, exchange offers,spinoffs, and recapitalizations. Of these events, the paying of dividends andstock splits do not affect capital structure. The others, however, could potentially alter the capital structure of the firm. So, in the context of risk arbitrage we are interested only in those.Let us briefly describe each of the corporate events of interest. Recapitalizations are situations in which companies deliberately decide to altertheir capital structure. The current shareholders or bondholders would receive securities or a combination of cash and new securities in exchange fortheir shares. Spinoffs occur when a firm splits its business into separateunits. Current shareholders receive new shares in the spun-off entity in addition to the current shares owned by them. A tender offer is a situation that139
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occurs when a company decides to acquire another company. Shareholdersof the acquired company receive cash in exchange for their shares. In exchange offers and mergers, shareholders of the acquired company receiveshares in the acquiring company in exchange for their shares. Sometimes, inthe case of mergers and tender offers, the acquired company shares are exchanged for a combination of cash and shares. In all of these situations thecommonality lies in the fact that they involve an exchange of one security foranother on a scheduled date in the future. Trading on the price disparity between the two exchanged securities is termed risk arbitrage.How does pairs trading figure in all of this? The answer is quite straightforward. Of the two securities involved in the exchange, we buy the lowerpriced security, sell the higher-priced security, and lock in the pricedifference for our profit. Note that this is possible only when both the securities in question are traded currently in the open market. In the case of recapitalizations and spinoffs, one side of the exchanged securities is issuedafresh and cannot be traded before issue. Keeping with the theme of pairstrading, we therefore focus on mergers and exchange offers.1
HISTORYRisk arbitrage in America is more than 100 years old. In the 1890s, therewas a five-year depression and about one quarter of the railroad industryfaced bankruptcy and was reorganized. In the reorganization, old debt wasexchanged for new debt plus equity consisting of both preferred and common stock. The new securities often represented more value than the oldones. As a result, arbitrageurs could buy the debt and sell the new securitiesafter a time for a profit.Another occasion conducive for the practice of risk arbitrage was whenthe large processing industry trusts were converted to corporations. Trustswere vehicles for interstate commercial activity when holding companieswere not permitted by state law. Subsequently, the New Jersey HoldingCompany Act permitted the trusts to be reorganized. In these reorganizations, arbitrageurs could buy trust certificates in the market and exchangethem for new preferred and common stock that the market bid at a pre1
Quite a large percentage of risk arbitrage practitioners focus primarily on mergersand acquisitions. Some of the practitioners trading around recapitalizations andbankruptcies call their practice distressed security investing, or, more colorfully, asvulture investing. It is also true that the two differ significantly in their method of security analysis, not to mention the even greater difference in the legal aspects surrounding them. We are thus here in this murky area of nomenclature, and whatexactly falls under the umbrella of risk arbitrage could be debated.
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mium. For Bernard Baruch and other players of the day, this presented agood arbitrage opportunity.America experienced its first merger boom in the early 1900s. Shares ofone company could be exchanged for shares of another for the first time.There were plenty of deals. Subsequent merger booms occurred in the 1920s,1960s, 1980s, and late 1990s. Risk arbitrage was practiced in some form oranother during these periods with the arbitrageurs acting as market makersfor investors.The practice of risk arbitrage in its modern form probably started a little after World War II. In the early days, it was done by an exclusive club ofstock traders. Some of the individuals in the group were Gustave Levy, latersenior partner at Goldman Sachs; Salim Lewis of Bear Stearns; Harry Cohnof L.F. Rothschild; Joseph Gruss of Gruss & Co; and Eugene Wyser-Pratt ofBache. The group was secretive about their trading methods, and the practice was shrouded in mystery. The trading was, however, based on publiclyavailable information.Then, in the mid 1980s Ivan Boesky wrote a book2 on risk arbitrage,and a lot of the details on the practice of risk arbitrage came to be knownpublicly. Later, when the SEC charged him with insider trading offenses, thepractice of risk arbitrage took a big hit. It took a while for the erroneous perceptions to come to terms with the fact that risk arbitrage can indeed bepracticed based on information gathered from public sources. Since then, thebusiness has rebounded and competition in the marketplace has steadily increased. There are now quite a few mutual funds and hedge funds specializing in risk arbitrage.
THE DEAL PROCESSOne of the crucial components in the practice of risk arbitrage is the understanding of the deal process. Any discussion on the subject matter withouttalking about the deal process would be incomplete. Let us therefore brieflyoutline various steps involved in a deal. The term deal is used generically torefer to both mergers or exchange offers. In the ensuing discussion we willhighlight both the similarities and the differences between them.The typical chain of events leading to a merger is as follows. First, thetwo companies do their due diligence on each other’s business and siftthrough the nitty-gritty. The attorneys for both the companies then draft acontract known as a definitive agreement. The two companies then makethe announcement through a joint press release. In some instances, the2I. F. Boesky, Merger Mania-Arbitrage: Wall Street’s Best Kept Money Making Secret. New York: Holt, Rinehart and Winston, 1985.
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announcement is made prior to the drafting of the definitive agreement. Insuch cases, the announcement would be construed as an agreement in principle. Subsequently, a registration statement is filed with the Securities andExchange Commission (SEC). The SEC looks at the statement in the contextof various legal statutes causing rounds of amendments based on its comments. Subsequently, the registration statement is declared “effective,” andthe document is mailed to the shareholders for their approval. The shareholders vote then takes place and is followed by deal closing.An exchange offer is somewhat of a hybrid between a merger and a tender offer. It is an unsolicited bid like in the case of a tender offer. However,unlike the tender offer, the bid is made in terms of the acquirer’s stock as opposed to cash. Thus, in this aspect, it is similar to mergers. The formal exchange offer is made though advertisements in the Wall Street Journal andlocal newspapers. Since this involves the issuance of new stock, it goesthrough the same registration process with the SEC as is required for mergers. In this case, however, the completion of the transaction does not requirea shareholders’ vote.Also note that mergers and exchange offers both have quite a bit in common with regards to their transaction terms. This is the topic of discussionin the next section on transaction terms and unless specifically mentioned,our discussions apply to both of them.
TRANSACTION TERMSTransaction terms, in the case of mergers, are usually contained in the proxystatement that is part of the merger agreement document. In the case of anexchange offer, the transaction terms are available in the exchange offeradvertisement.It is useful at this point to specify the convention used in the ensuing discussion. The two companies involved in the transaction will be referred to asthe bidder (B) and target (T). In all of our references involving exchange ofshares, the convention we will follow is as follows. The shares of the targetfirm are given up and exchanged for shares of the bidder firm. With that, weare now ready to discuss transaction terms.The valuation of the target firm during the due diligence process is typically in dollar terms, leading to a specific dollar amount for the target stock.While this dollar amount may be agreeable to both the parties involved ina transaction, note that it is a little hard to pay the exact specific dollaramount. This is because the payment for the target stock is made in terms ofthe bidder stock, and the price of the bidder stock varies on a day-to-daybasis. Therefore, the key challenge in structuring the transaction is in finding an approach to pay a specific dollar amount for the target stock that isequitable to both the bidder and target companies.
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The preceding description of the merger process may seem cut anddried. Nothing, however, could be further from the truth. The duediligence is intense, and negotiations are complex. To get an idea of theextremes to which the process can play out, we strongly suggest reading “Barbarians at the Gate: The Fall of RJR Nabisco,”3 a narrative onthe RJR Nabisco leveraged buyout.As for the arbitrageur, the process is fraught with uncertainty (insome cases more than the others). Antitrust issues, actions of other potential bidders, second thoughts on part of the principals involved, financing problems, defensive merger tactics (poison pills), problemsduring the shareholder voting process, a general collapse of the overall market, and reaction of market participants to merger announcement (the laundry list goes on) are all issues that the arbitrageur needsto consider while placing the bet. In fact, the practice of risk arbitragecould require one to be a renaissance man, knowledgeable in analyzing financial statements, well versed in legal procedures, and be atrader, all at the same time.
3
Brian Burrough and John Helyar, “Barbarians at the Gate: The Fall of RJRNabisco.” HarperCollins (Jan. 1991).
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The role of the transaction terms in a deal is to mitigate some of the variability of the price of the bidder stock and try to achieve the objective ofpaying a specific dollar amount for the target stock. We now list typicaltransaction terms that one is likely to encounter.
Fixed Ratio Stock ExchangeThe simplest form of a merger transaction is the stock for stock transaction.Shares of the target are exchanged for shares of the bidder in a fixed exchange ratio. This exchange ratio is determined by the accountants and analysts of the merging companies based on the valuation of the target firm. Itis, however, self-evident that in such fixed exchanges the price paid for thetarget varies with the stock price of the bidder.
Fixed Value Stock ExchangeThe fixed value stock exchange may be considered an attempt to do a better job at mitigating the variability. The transaction structure also helps toprevent excessive shorting of the bidder stock on the eve of deal announcements. In such transactions, the dollar value of the target stock is fixed. Theexchange ratio is then determined based on the stock price of the bidder during what is known as the pricing period. A commonly used approach is touse the average closing price of the bidder stock in the pricing period to determine the exchange ratio. For example, if the price of the target stock isfixed at $10 and the average of the closing prices of the bidder stock duringthe pricing period is $20, then the exchange ratio is 0.5; that is, two sharesof the target stock is good for one share of the bidder stock. Sometimes, toprevent manipulation of the closing price by arbitrageurs, the volumeweighted average price during the pricing period is used to determine theratio. Thus, in the fixed value approach, the exchange ratio is gradually revealed as the prices unfold over the pricing period.Although, on a rare occasion, the average of the closing prices on a predetermined number of randomly chosen days in the pricing period has alsobeen used to determine the exchange ratio, the typical approach is to usesome sort of average. The length of a typical pricing period is 20 tradingdays or less. The length is fixed more or less to mitigate the effects of volatility on the bidder stock. Although this approach works slightly better thanthe fixed ratio approach, it is still subject to some extent on the volatility ofthe bidder stock.
Stock and Cash ExchangeThe effect of bidder stock volatility on the exchange ratio may be furthermitigated by paying for the target stock with a combination of cash and securities. Generally speaking, though, the exchange ratio in this case could be
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determined using either the fixed ratio or the fixed value method. We illustrate the transaction terms with an example.Bidder B pays for target T, 70 percent in stock, 30 percent in cash. Theshare exchange ratio is 0.5; that is, 1/2 a share of B for one share of T. Cashamount paid for the remaining 30 percent is $20. Based on the precedingspecification, let us now compute the exchange on a per target stock basis.Share amount:
share percentage × ratio = 0.7 × 0.5 = .35
Cash amount:
cash percentage × cash value = 0.3 × $20 = $6.0
Thus, for each share of the target the holder of record would receive 0.35share of the bidder and $6 in cash.Note that while these sets of transaction terms reduce the dependence ofthe exchange ratio on the price of the bidder stock, paying partly in cash reduces the amount of equity issued and has an effect on the capital structureof the new entity.
CollarsAn explicit attempt to reduce the dependence of the exchange ratio to thevolatility of the bidder stock without resorting to cash payments can be seenin the case of collars. Collars are transaction terms that are contingent on theprice of the bidder. They come primarily in two flavors, the fixed exchangecollar and the fixed value collar.In a fixed exchange collar, a constant exchange ratio is specified over arange of the Bidder’s stock price. This ratio is subject to adjustment; that is,to a maximum or minimum value if the bidder’s stock price falls out of therange. In a fixed value collar, a constant dollar value is set for each share ofthe target stock. The ratio is then determined using the pricing period approach. However, the terms would be adjusted to a maximum or minimumexchange ratio if the bidder’s stock price falls out of the range.Also included in some cases is a walk away right, which grants the target company an option to terminate the deal in case the price of the bidderfalls below a specified level. In other cases, the bidder is also granted a termination option if the stock price of the target experiences a very steep increase after announcement.
THE DEAL SPREADThe transaction terms in a merger or exchange offer create a strict parity relationship between the bidder and target stocks. Violations of this parityrelationship can be measured based on the stock prices of both the stocksand is called the spread. To see how the spread is calculated, consider the
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fact that each target share is exchanged for a fixed number of bidder sharesor bidder shares plus cash, the value of which we shall call the exchangevalue of a target share. This can be calculated exactly with the knowledgeof the transaction terms and the current price of the bidder stock. Let us saythat we also know the current price of the target stock. The spread is nowgiven asSpread = exchange value of a target share – current price of target shareThe magnitude of the spread is indicative of the disparity and is therefore representative of the profit potential in dollar terms on a per-target share basis.Prior to the date of deal completion, the target shares almost alwaystrade at a discount to their exchange value. This implies that the value of thespread as calculated above is usually positive. Let us see why that is. If thespread were negative—that is, the exchange value of a target share is lessthan the current price of the target share—then there is not much reason tohold the target stock. One could sell the target stock right away and makemore money than waiting until deal close. If the spread is zero, then also itmakes no sense to wait until deal completion date, as it is better to cash inright now than wait till deal completion to get the same amount of money.In either of the cases, market participants would begin to sell the targetstock until the value of the spread is no longer negative or zero. Thus, onecan expect the spread to be positive.It is also useful at this point to remind ourselves that there is the risk ofdeal break. Now, it is natural for market participants to expect a greater reward for greater deal break risk and a lesser reward for taking moderaterisks. This is also reflected in the spread with large spreads, implying highrisk and vice versa. Thus, the spread is indicative not only of the return butalso the perceived risk of deal break. Naturally, it is a key market variablethat the risk arbitrageur relates to.As an illustration, let us look at a real-life merger between Intel Corporation (INTC) and Level Communications (LEVL). The deal was announcedon March 4, 1999. INTC was the bidder, and LEVL was the target. The exchange ratio was 0.86; that is, every share of LEVL was exchanged for 0.86share of INTC. The deal was completed on August 10, 1999.Figure 9.1a is a plot of the prices of LEVL and INTC. The price of INTCis adjusted for the exchange ratio. The first set of points in Figure 9.1a is theclosing price a day prior to the announcement of the merger. We can see thatLEVL registers a big jump in price on the day of the announcement, narrowing the spread to about $4.26 on a close-to-close basis. In fact, it is common on the day of announcement to see bidder shares fall in price and targetshares rise. As the day of deal completion approaches, the uncertainty in thedeal decreases, the spread narrows and approaches zero. This can be seen inFigure 9.1b.
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70605040INTC (adjust close)LEVL (close)
30200
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40
FIGURE 9.1A
60
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Price Dynamics (INTC–LEVL).
5
Spread
432100
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40
FIGURE 9.1B
60
80
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Spread (INTC–LEVL).
So, how do we go about trading on this disparity? This is indeed thetopic for the next section, Trading Strategy.
TRADING STRATEGYAn arbitrageur may take on a position any time during the course of themerger process. This would, however, be dictated by the risk and returncharacteristics of the deal. The typical trade executed is to short the biddershares (sell high), buy the target shares (buy low), and pocket the spread.When the deal is completed, the target shares owned are exchanged for bidder shares. These bidder shares are then applied to cover the short position.
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Alternately, when it is close to deal completion (the spread is usually closeto zero) the position is reversed; that is, cover short position on bidder sharesand sell target shares.In the example of INTC and LEVL, trade would be to short 0.86 shareof INTC against every share of LEVL that we buy. This is done on the dayof deal announcement. We would thus pocket about $4.26. When the dealis complete, the positions are reversed. This could be done in one of twoways. One way to do it is to exchange the target shares for the bidder sharesand use those to close out the short position on the bidder. Alternately, wecould unwind both the positions individually; that is, we sell the shares ofLEVL and cover the short position on INTC. Given that the spread valuearound deal completion is typically zero, we can expect the proceeds fromthe sale of one share of LEVL would be equal to the price of 0.86 shares ofINTC. These proceeds are now used to buy the shares of INTC to cover theshort position. Thus, for every share of LEVL we were long, we would havepocketed about $4.26.The following list is an example of a typical rate of return calculationfor a deal.
ExampleDeal DetailsThe spread on the deal is calculated as follows:spread = price of B × 0.35 + 6.0 – price of TAnnouncement Terms:1/2 a share of B to be exchanged for everyshare of T, that is, ratio = 0.5.Dividends:No dividends will be paid by B or TCurrent stock prices:Spread:Estimated Time to Close:
price(B) = $20.0, price(T) = $8.00.5 × price(B) – price(T) = (0.5 × 20.0) – 8.0= $2.03 – 6 months. (average = 135 days)
The Strategy1. Purchase two shares of T2. Sell Short one share of B3. Lose Dividend on B,receive dividend on T, net4. Rebate on short position5. Net proceeds (2 + 3 + 4)6. Gross Profit (5 – 1)7. Return = 4.21/16.0 × 165/360
($16.0)$20.0$ 0.0$0.21________$20.21$ 4.2112.05 %
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This return is usually compared with 2 × 90 day treasury bill rate as thebenchmark to determine if the trade is viable. Note that the return calculation here is slightly different from the return calculation used in the statistical arbitrage case. The return is calculated on the dollars invested in the longposition alone. This is more to adhere to conventions followed in practice.
QUANTITATIVE ASPECTSIn the following chapters, we focus on some quantitative aspects of risk arbitrage. The focus will be in the areas of trade execution and risk measurement.When trading spreads in size, it is not possible to trade the entire quantity in one go due to liquidity constraints. The position is therefore filled onan incremental basis. Note, however, that the trades need to be executed inwhat we shall call a paired transaction. We need to ensure that both legs ofthe pair be filled in a manner so as to satisfy the ratio constraint and also inthe process manage to capture a spread that is greater than a specified value.We shall discuss issues relating to the unambiguous specification and verification of such trades.We will then address executions in the fixed value exchange case. In thefixed value exchange transaction, we noted earlier, the exchange ratio is revealed gradually as the prices unfold during the pricing period. The exact exchange ratio is known only at the end of pricing period. However, it ispossible to trade during the pricing period and still manage to be perfectlyhedged. We will illustrate how trading may be done during the pricing period when the exchange ratio is unclear.Next, we demonstrate how market implied deal break probability canbe evaluated using the time series of the spread. This is followed by a discussion on how the measured probability may be applied to evaluate thevalue at risk in risk arbitrage deals. It turns out that the evaluation of stableestimates of the deal break probability requires us to work on a smoothedversion of the spread. We delineate an approach for smoothing. Thesmoothing process leads us to a methodology that could be useful in timingthe unwinding of the spread position.
SUMMARYRisk arbitrage relates to trading around corporate events, especiallymergers and acquisitions.The practice of risk arbitrage has