Are CDS Auctions Biased and Inefficient?

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1 Are CDS Auctions Biased and Inefficient? Songzi Du Simon Fraser University Haoxiang Zhu MIT Sloan and NBER February 9, 015 Abstract We study the design of CDS auctions, which determine the payments by CDS sellers to CDS buyers following the defaults of bonds. Through a simple model, we find that the current two-stage design of CDS auctions leads to biased prices. First, dealers may manipulate the first-stage quotes to profit from their CDS positions. Second, various restrictions imposed on both stages of CDS auctions prevent certain investors from participating in the price-discovery process. The resulting allocations of bonds are also inefficient. As a remedy, we propose a double auction design that delivers more efficient price discovery and allocations. Keywords: credit default swaps, credit event auctions, price bias, manipulation, allocative efficiency, double auction JEL Classifications: G1, G14, D44 First draft: November 010. An earlier version of this paper was distributed under the title Are CDS Auctions Biased? and is available on our websites. For helpful comments, we are grateful to Jeremy Bulow, Mikhail Chernov discussant), Darrell Duffie, Vincent Fardeau discussant), Jacob Goldfield, Alexander Gorbenko discussant), Yesol Huh, Jean Helwege, Ron Kaniel, Ilan Kremer, Yair Livne, Igor Makarov, Andrey Malenko, Lisa Pollack, Michael Ostrovsky, Rajdeep Singh discussant), Ken Singleton, Rangarajan Sundaram, Andy Skrzypacz, Bob Wilson, Anastasia Zakolyukina, Adam Zawadowski discussant), and Ruiling Zeng, as well as seminar participants at Stanford University, the University of South Carolina, the ECB- Bank of England Workshop on Asset Pricing, Econometric Society summer meeting, Financial Intermediary Research Society annual meeting, SIAM, NBER Summer Institute Asset Pricing meeting, European Finance Association annual meeting, Society for the Advancement of Economic Theory meeting, and Western Finance Association annual meeting. We are also grateful to a number of senior executives at ISDA for their comments and insights. Paper URL: Simon Fraser University, Department of Economics, 8888 University Drive, Burnaby, B.C. Canada, V5A 1S6. MIT Sloan School of Management and NBER, 100 Main Street E6-63, Cambridge, MA 014.

2 1 Introduction Credit default swaps CDS) are default insurance contracts between buyers of protection CDS buyer ) and sellers of protection CDS seller ), written against the default of firms or countries. Since the financial crisis, CDS have been one of the financial innovations that received the most policy attention and regulatory actions. 1 As of June 014, global CDS markets have a notional outstanding of $19.5 trillion and a gross market value of $635 billion. This paper studies the design of CDS auctions, a unique and unusual mechanism that determines the post-default recovery value for the purpose of settling CDS. Since the recovery value is a fundamental parameter for the pricing, trading and clearing of CDS contracts, achieving a fair, unbiased auction price is crucial for the proper functioning of CDS markets. Our primary objective is to evaluate, from a theoretical and market-design perspective, whether the current format of CDS auctions can discover efficient recovery values and achieve efficient allocations of defaulted bonds. We show that it cannot. As a remedy, we propose a double auction design that delivers better price discovery and more efficient bond allocations. The current CDS auction mechanism was initially used in 005 and hardwired in 009 as the standard method used for settling CDS contracts after default ISDA 009). From 005 to December 014, CDS auctions have settled 113 defaults of firms such as Fannie Mae, Lehman Brothers, and General Motors) and sovereign countries such as Greece and Argentina). For many firms, separate CDS auctions are held for senior and subordinate debt. As explained in Section, the current mechanism consists of two stages. The first stage of the auction solicits market orders, called physical settlement requests, to buy or sell defaulted bonds. The net market order is called the open interest. Simultaneously, dealers quote prices on the bonds, and a price cap or floor is calculated from these quotes. The second stage is a standard uniform-price auction, subject to the price cap or floor. If the final auction price is p per $1 of face value, the default payment by CDS sellers to CDS buyers is 1 p. To the best of our knowledge, Chernov, Gorbenko, and Makarov 013) CGM) provide the only other theoretical model of CDS auctions. Their model is an extension of the 1 For example, the Dodd-Frank Act of United States has mandated and later implemented mandatory central clearing of standard over-the-counter derivatives, including CDS. In its Financial Markets Infrastructure Regulation, the European Commission states: Derivatives play an important role in the economy but are associated with certain risks. The crisis has highlighted that these risks are not sufficiently mitigated in the over-the-counter OTC) part of the market, especially as regards credit default swaps CDS). See market/financial-markets/derivatives/index en.htm. In 01, European regulators also made a controversial move of banning naked sovereign CDS in the EU. See the semiannual OTC derivatives statistics, Bank for International Settlements, December

3 strategic bidding models of Wilson 1979) and Back and Zender 1993). CGM have an important insight that participants may manipulate the final price in the second stage of the auction in order to profit from their CDS positions. Such manipulation is possible and effective because the second stage of the auction is one-sided, namely only buy orders or only sell orders are allowed. Moreover, CGM s model assumes that certain investors are exogenously constrained such that they cannot buy the defaulted bonds. Those who can buy the bonds would bid strategically. Combining these features, CGM conclude that the final auction price can be either higher or lower than the bond fundamental value. Empirically, CGM find that in 6 CDS auctions on U.S. firms from 006 to 011, CDS auction prices tend to be lower than secondary market bond prices before and after auction dates. This V-shaped price pattern is confirmed by Gupta and Sundaram 01) GS), who also conduct a structural estimation of CDS auctions. In earlier empirical papers with smaller samples, Helwege, Maurer, Sarkar, and Wang 009) find that CDS auction prices and bond prices are close to each other, and Coudert and Gex 010) provide a detailed discussion on the performance of a few large CDS auctions. To the best of our knowledge, these four studies are the only ones in the small literature on CDS auctions. While CGM s model takes a first step in analyzing the mechanism of CDS auctions, it is limited in several respects. First, in their model the fundamental bond value is commonly known after default, that is, CDS auctions provide no additional price discovery. assumption is hardly realistic, since default is an important information event, and the primary function of an auction in general is to provide price discovery. Second, CGM do not model dealers incentive to manipulate the price cap or floor in the first stage of the auction. Because dealers quotes determine the permitted range of the final auction price, understanding dealers incentives is an essential step to better understand the CDS auction design. Lastly, CGM s model does not address allocative efficiency in CDS auctions. In reality, efficient allocation of bonds in CDS auctions is valuable for investors, as trading bonds in CDS auctions incurs zero bid-ask spread, whereas doing so in secondary markets is much more costly. 3 This Our analysis provides novel insights on these dimensions not addressed by CGM s model. We show that CDS auction prices are biased for two reasons. First, dealers may strategically manipulate the price cap or floor in order to profit from their existing CDS positions. Second, various restrictions imposed on the auctions prevent certain investors from participating in 3 For example, Feldhutter, Hotchkiss, and Karakas 013) find that the round-trip bid-ask spread of defaulted bonds are about 1.5% of market value near default dates. The Amihud illiquidity measure also roughly doubles in the week of default, suggesting a higher price impact costs of trading large quantities.

4 the price-discovery process. These two channels are orthogonal to those in CGM s model second-stage manipulation and restrictions on buying bonds). Moreover, we show that CDS auctions deliver inefficient allocations of defaulted bonds. Finally, we propose a double auction design that achieves better price discovery and more efficient allocations. Our analysis is built on a simple model of divisible auctions, which works roughly as follows. A continuum of traders have high or low values for owning the defaulted bonds. The proportion of high-value traders is unobservable and serves as a state variable. Traders also have heterogeneous CDS positions. A trader s utility is the payout from his CDS positions plus the profit of buying or selling bonds in the auction, less a quadratic cost of holding defaulted bonds. Traders select the optimal physical requests in the first stage and the optimal demand schedules in the second stage. Moreover, a finite number of infinitesimal dealers submit first-stage quotes, which determine the second-stage price cap or floor. This infinitesimal-trader model greatly reduces technical complexity in illustrating our insights. 4 Optimal strategies in the two stages are solved in a subgame-perfect equilibrium. The first, manipulation channel for price bias is straightforward. A dealer s benefit of manipulating quotes in the first stage is the more favorable payout from CDS positions, and a major cost is reputation. Without loss of generality let us suppose that dealers are CDS buyers, who benefit from a low final auction price. If dealers CDS positions are sufficiently large, dealers will quote prices below their best estimate of bond value in the first stage, leading to downward biased prices. Conversely, if dealers are CDS sellers with sufficiently large positions, they quote prices above their best estimate of bond value, leading to upward biased prices. We further show that dealers ability to manipulate the quotes is stronger if they are in the money on their CDS positions, namely if they tend to be CDS buyers sellers) when the post-auction bond value is low high). The manipulation incentive in CDS auctions is similar to the manipulation incentive in survey-based financial benchmarks, such as the London Interbank Offered Rate LIBOR). Since LIBOR manipulation is already an established fact, 5 the current CDS auction design causes the same concern. Although the incentive of manipulation does not necessarily imply 4 Realism aside, we believe the only major missed opportunity of our model is that an infinitesimal trader cannot unilaterally affect the second-stage price. But as discussed above, this strategic incentive is already modeled by Chernov, Gorbenko, and Makarov 013). Moreover, the large-trader model of Chernov, Gorbenko, and Makarov 013) does not handle price discovery, first-stage manipulation, or allocative efficiency. Therefore, to make a more interesting contribution to the literature, we opt for a infinitesimal-trader model rather than a large-trader model. 5 See, for example, Market Participants Group on Reference Rate Reform 014) and Official Sector Steering Group 014) for the institutional details and suggested reforms on reference rates such as LIBOR. 3

5 its occurrence, proper designs of auction mechanisms should not give dealers disproportionately large discretion in affecting the final price, such as by setting a price cap or floor. The second, information channel for price bias is slightly more involved. For simplicity, suppose for now that the price cap or floor is dropped altogether in the second stage, so price manipulation does not occur. Without loss of generality, consider a CDS buyer who has a high value for owning the defaulted bonds. For example, this trader may specialize in distressed investment.) This high-value trader wishes to buy defaulted bonds, but the current design of the CDS auction stipulates that only CDS sellers can submit physical requests to buy i.e. market orders to buy) in the first stage. Thus, the demands to buy bonds from high-value CDS buyers are suppressed completely in the first stage of the auction. If the open interest is to buy, the auction protocol also stipulates that only orders in the opposite direction, i.e. sell limit orders, are allowed in the second stage. Thus, the demands of high-value CDS buyers to buy bonds are suppressed in the second stage of the auction as well. Consequently, by completely missing the information from high-value CDS buyers, the auction final price is downward biased if the open interest is to buy. 6 This argument applies symmetrically if the open interest is to sell, but in this case it is the low-value CDS sellers who are prevented from participating in either stage of the auction. Thus, the auction final price is upward biased if the open interest is to sell. The combination of the manipulation channel and the information channel leads to either upward or downward biased prices, depending on dealers CDS positions. Biased prices also lead to inefficient allocations of bonds in CDS auctions, even if the price cap or floor is set correctly. While investors can trade the bonds in the secondary markets, doing so incurs higher transaction costs because CDS auctions allow trading at zero bid-ask spread. Our model generates a number of novel predictions regarding quoting behavior, price biases, and post-auction trading activity. For example, in the first stage, dealers who are CDS buyers quote relatively low prices, whereas dealers who are CDS sellers quote relatively high prices. As a result, the final auction price is downward biased if dealers are net CDS buyers, and is upward biased if dealers are net CDS sellers. Moreover, if the open interest is to buy, low-value CDS traders get too much allocation in the auctions and will sell bonds after the auctions. If the open interest is to sell, high-value CDS traders get too little allocation in the auctions and will buy bonds after the auctions. These predictions are new to the literature. 6 We can show that other types of traders including high-value CDS sellers, low-value CDS buyers, and low-value CDS sellers either fully or partially participate in either stage of the auction. 4

6 A major challenge in testing these predictions is that it requires data on CDS positions by identity or transaction data in the secondary bond markets by identity. These types of data are highly confidential and only available to regulatory agencies. That said, our predictions of price biases are generally consistent with the existing literature. For instance, CGM and GS find that if the open interest is to sell, which is the case for the majority of CDS auctions, the final auction prices are downward biased relative to secondary market prices days before and after the auctions). On the other hand, Vause 011) documents that dealers are net CDS buyers in 010 and 011, with hundreds of billions of dollars in net notional amounts. The combination of these two patterns is consistent with our prediction that if dealers are net CDS buyers, then CDS auctions tend to underprice the bonds. Our analysis suggests that a double auction is a better design to reduce price biases and allocative inefficiency. A double auction is by no means unusual or exotic; everyday open auctions and close auctions on stock exchanges are double auctions. Under a double auction design, limit orders in the second stage can be submitted in both directions, buy and sell, regardless of the open interest. Two-way orders allow broader investor participation in the price-discovery process. We also argue that the price caps or floors should be dropped to mitigate the potential manipulation incentives of dealers. Indeed, we show that in our model with infinitesimal traders, the double auction delivers both the competitive price and the efficient allocation. If the number of auction participants is finite, say n, a double auction still delivers the efficient price, with allocative inefficiency of order O1/n). The double auction design is yet another dimension in which we contribute to the extant literature. For instance, CGM argue that the alternative allocation rule proposed by Kremer and Nyborg 004) would mitigate strategic bidding incentives and reduce underpricing in CDS auctions; they additionally suggest that the price cap should depend on the firststage open interest in a fairly complex fashion. Separately, GS examine Vickery auction and discriminatory auction as alternative formats in the second stage, holding the firststage strategies fixed. Under CGM s proposal and GS s proposal, however, certain investors are still left out of the price-discovery process, and modification to the second stage alone does not neutralize dealers first-stage manipulation incentives. We thus expect that those modifications do not correct price biases caused by the manipulation channel and information channel we have identified. 5

7 The Institutional Arrangements of CDS Auctions This section provides an overview of CDS auctions. Detailed descriptions of the auction mechanism are also provided by Creditex and Markit 010). A CDS auction consists of two stages. In the first stage, the participating dealers 7 submit physical settlement requests on behalf of themselves and their clients. These physical settlement requests indicate if they want to buy or sell the defaulted bonds as well as the quantities of bonds they want to buy or sell. Importantly, only market participants with nonzero CDS positions are allowed to submit physical settlement requests, and these requests must be in the opposite direction of, and not exceeding, their net CDS positions. For example, suppose that bank A has bought CDS protection on $100 million notional of General Motors bonds. Because bank A will deliver defaulted bonds in physical settlement, the bank can only submit a physical sell request with a notional between 0 and $100 million. Similarly, a fund that has sold CDS on $100 million notional of GM bonds is only allowed to submit a physical buy request with a notional between 0 and $100 million. 8 Participants who submit physical requests are obliged to transact at the final price, which is determined in the second stage of the auction and is thus unknown in the first stage. The net of total buy physical request and total sell physical request is called the open interest. Also, in the first stage, but separately from the physical settlement requests, each dealer submits a two-way quote, that is, a bid and an offer. The quotation size say $5 million) and the maximum spread say $0.0 per $1 face value of bonds) are predetermined in each auction. Bids and offers that cross each other are eliminated. The average of the best halves of remaining bids and offers becomes the initial market midpoint IMM), which serves as a benchmark for the second stage of the auction. A penalty called the adjustment amount is imposed on dealers whose quotes are off-market i.e., too far from other dealers quotes). The first stage of the auction concludes with the simultaneous publications of i) the initial market midpoint, ii) the size and direction of the open interest, and iii) adjustment amounts, if any. Figure 1 plots the first-stage quotes left-hand panel) and physical settlement requests 7 For example, between 006 and 010, participating dealers in CDS auctions include ABN Amro, Bank of America Merrill Lynch, Barclays, Bear Stearns, BNP Paribas, Citigroup, Commerzbank, Credit Suisse, Deutsche Bank, Dresdner, Goldman Sachs, HSBC, ING Bank, JP Morgan Chase, Lehman Brothers, Merrill Lynch, Mitsubishi UFJ, Mizuho, Morgan Stanley, Nomura, Royal Bank of Scotland, Société Générale, and UBS. 8 To the best of our knowledge, there are no formal external verifications that one s physical settlement request is consistent with one s net CDS position. 6

8 right-hand panel) of the Lehman Brothers auction in October 008. The bid-ask spread quoted by dealers was fixed at per 100 face value, and the initial market midpoint was One dealer whose bid and ask were on the same side of the IMM paid an adjustment amount. Of the 14 participating dealers, 11 submitted physical sell requests and 3 submitted physical buy requests. The open interest to sell was about $4.9 billion. Figure 1: Lehman Brothers CDS Auction, First Stage Points Per 100 Notional First Stage Quotes Bid Ask IMM BofA Barclays BNP Citi CS DB Dresdner GS HSBC JPM ML MS RBS UBS Billion USD Physical Settlement Requests Sell Buy BNP BofA Citi CS DB GS HSBC ML MS RBS UBS Barclays Dresdner JPM In the second stage of the auction, all dealers and market participants including those without any CDS position can submit limit orders to match the open interest. Nondealers must submit orders through dealers, and there is no restriction regarding the size of limit orders one can submit. If the first-stage open interest is to sell, then bidders must submit limit orders to buy. If the open interest is to buy, then bidders must submit limit orders to sell. Thus, the second stage is a one-sided market. The final price, p, is determined as in a usual uniform-price auction, subject to a price cap or floor. Specifically, for an open sell interest, the final price is set at p = min M +, p b ), 1) where M is the initial market midpoint, is a pre-determined spread that is usually $0.01 or $0.0 per $1 face value, and p b is the limit price of the last limit buy order that is matched. If needed, limit orders with price p are rationed pro-rata. Symmetrically, for an open buy 7

9 interest, the final price is set at p = max M, p s ), ) where p s is the limit price of the last limit sell order that is matched, with pro-rata allocation at p if needed. If the open interest is zero, then the final price is set at the IMM. The announcement of the final price p concludes the auction. After the auction, bond buyers and sellers that are matched in the auction trade the bonds at the price of p ; this is called physical settlement. In addition, CDS sellers pay CDS buyers 1 p per unit notional of their CDS contract; this is called cash settlement. Figure plots the aggregate limit order schedule in the second stage of the Lehman auction. For any given price p, the aggregate limit order at p is the sum of all limit orders to buy at p or above. The sum of all submitted limit orders was over $130 billion, with limit prices ranging from the price cap) to 0.15 per 100 face value. The final auction price was CDS sellers thus pay CDS buyers per 100 notional of CDS contract. Figure : Lehman Brothers CDS Auction, Second Stage 1 Price per 100 Notional) Aggregate Limit Orders Open Interest Final Price Quantity Billion USD) 3 A Model of CDS Auctions In this section we describe the baseline model and solve the optimal strategies in both stages of the auction. We also derive the associated prices and allocations. To isolate the effects of 8

10 various restrictions imposed on the auctions, price caps or floors are not considered in this section. Rather, an extended model with price caps and floors is solved in Section 4. There is a unit mass of infinitesimal traders on [0, 1]. This infinitesimal-trader setting greatly reduces technical complexity at little cost of economics. Each trader i [0, 1] has a CDS position Q i, where {Q i } are independent and uniformly distributed on [ Q, Q], for Q > 0. Each trader i also has a private value v i for holding the defaulted bonds. The private value is a reduced form for heterogeneous information or heterogenous inventory among the traders. There are two states for the defaulted bonds: high H) and low L), with equal ex ante probability. In the high state, a fraction m > 1/ of the traders have value v H, and the rest have value v L. In the low state, a fraction m of the traders have value v L, and the rest have value v H. Value v i is independent of CDS position Q i. Throughout the paper we impose the following parameter restriction: Assumption 1. v H v L Q. 3) This condition generates an interior solution and simplifies the analysis. The qualitative results of this paper does not hinge upon this parameter condition. In this section we study the following two-stage auction game: 1. In stage 1, each trader i [0, 1] submits a physical settlement request r i that satisfies r i Q i 0 and r i Q i. Let R r i di 4) be the open interest in the first stage of the auction.. a) If R < 0, then in stage each trader i [0, 1] submits a demand schedule x i : [0, 1] [0, ) to buy bonds. b) If R > 0, then in stage each trader i [0, 1] submits a supply schedule x i : [0, 1], 0] to sell bonds. The final auction price p is defined by r i + x i p ) di = 0. 5) i 9 i

11 The utility of trader i is: U i = 1 p )Q i + v i p )r i + x i p )) r i + x i p )). 6) The first and second terms of U i are, respectively, the payout from CDS settlement and the profits of trading the defaulted bonds. The last term of U i, with some > 0, is interpreted as the inventory cost of creating the bond position r i + x i in CDS auctions. This inventory cost implies that traders do not wish to accumulate unlimited positions. In this regard it plays a similar role as risk aversion, although it is not risk aversion. This kind of linear-quadratic utility is also used by Vives 011), Rostek and Weretka 01), and Du and Zhu 014), among others. Given the first-stage open interest, the second stage strategy is straightforward. Since traders are infinitesimal, each trader takes the price p in the second stage as given and wants to get as close as possible to his optimal bond allocation, v i p. Lemma 1. In any equilibrium, each trader i submits in the second stage the demand/supply schedule: max r i + v i p x i p) =, 0), if R < 0; min r i + v i p, 0), if R > The competitive equilibrium benchmark As a benchmark, we consider the competitive equilibrium, in which each trader submits an unconstrained demand schedule, taking the price as given. The first-order condition of U i yields the competitive equilibrium strategy: 7) x c i = v i p. 8) Given the market clearing condition i xc idi = 0, the competitive equilibrium prices in the high state and low state are, respectively, p c H = mv H + 1 m)v L, 9) p c L = 1 m)v H + mv L. 10) The allocations in the competitive equilibrium are efficient, so trader i s efficient allocation is v i p c H in the high state and v i p c L in the low state. 10

12 3. Equilibrium of the two-stage auction We now characterize the equilibrium in the two-stage auction. Intuition of the equilibrium are discussed immediately after the results. Proposition 1. There is an equilibrium with the following strategies and properties: 1. In the first stage, every trader i submits: minv H mp B 1 m)p S )/, Q i), if v i = v H, Q i < 0 r i v i, Q i ) = maxv L 1 m)p B mp S )/, Q i), if v i = v L, Q i > 0, 11) 0, otherwise where p B = xv L + 1 x)v H, 1) p S = xv H + 1 x)v L, 13) and where x is a function of m and v H v L Q appendix. that is spelled out in Equation 3) in the. In the second stage, every trader i submits the demand schedule in Equation 7). 3. In the high state, the open interest is to buy, and the final auction price is p B. In the low state, the open interest is to sell, the final auction price is p S. 4. If m 8 1 v H v L Q ) 1, p B p S. If m < 8 1 v H v L Q ) 1, p B < p S. Corollary 1. As Q, the final prices in the equilibrium of Proposition 1 converge to if m /3, and to p B, = p S, = p B, = m/ 1 m/ v H + 1 m 1 m/ v L, 14) p S, = 1 m 1 m/ v H + m/ 1 m/ v L. 15) m m 4m 5m + v 1 m) H + 4m 5m + v L, 16) 1 m) 4m 5m + v H + m m 4m 5m + v L. 17) 11

13 if m < /3. Proposition Price biases). In Proposition 1, the equilibrium price p S in the low state is higher than the competitive price p c L ; and the equilibrium price p B in the high state is lower than the competitive price p c H. The equilibrium allocation of bonds in Proposition 1 is inefficient. 3.3 Intuition of the equilibrium The intuition of the equilibria is best understood by asking: which types of traders are constrained in the two stages? Figure 3 plots the first-stage physical requests and second-stage allocations in equilibrium. We first discuss the case of m 81 v H v L Q ) 1, which corresponds to the three left-hand subplots, and then turn to the case m < 81 v H v L Q ) 1, which corresponds to the three right-hand subplots. The prices biases are illustrated in Figure 4. The kink, where p B and p S intercept, corresponds to the threshold of m, 81 v H v L Q ) 1. The case of m 81 v H v L Q ) 1. The top-left subplot shows that high-value CDS buyers v i = v H, Q i > 0, solid red line) and low-value CDS sellers v i = v L, Q i < 0, dashed blue line) are completely prevented from participating in the first stage. This is because r i must have opposite sign from Q i. Moreover, if R > 0, the middle-left subplot shows that all high-value traders are prevented from participating in the second stage. This is because only sell orders are allowed for R > 0 but high-value traders only wish to buy. Putting these two stages together, we see that if R > 0, high-value CDS buyers trading interests are suppressed completely throughout the auction. We also see that other types of traders can at least partially participate in either stage. Consequently, too few high-value traders participate in the price-discovery process, leading to a downward biased price. Similarly, if R < 0, then the top-left and bottom-left subplots reveal that low-value CDS sellers are completely prevented from participating in both stages. Other types can participate in at least one stage.) The resulting final auction price is upward biased. The price biases are the most transparent in Corollary 1, where we have taken the limit of Q. In the limit the distribution of Q i becomes diffuse, and the cutoff point 81 v H v L Q ) 1 converges to /3. Thus, with probability 1, the max ) or min ) operator in 11) no longer binds. This implies that high-value CDS sellers and low-value CDS buyers fully express their trading interest in the first stage. If R > 0 which happens in the high state), 1

14 low-value CDS sellers also fully express their trading interest in the second stage see the middle-left subplot of Figure 3). The only traders that are constrained in both stages are the high-value CDS buyers. Therefore, the effective ratio of high-value traders to low-value traders is m/, lower than the unbiased ratio of m/1 m). This explains why in the limit, 1 m the final price in the high state is m/ 1 m/ v H + 1 m 1 m/ v L, lower than the competitive price of mv H + 1 m)v L. The intuition for the upward price bias if R < 0 is symmetric. The case of m < 81 v H v L Q ) 1. From the three right-hand subplots of Figure 3, we see that the first-stage strategies and second-stage allocations are similar to the case of m 81 v H v L Q ) 1, with one important difference. Since m is relatively close to 1/, the buy-versus-sell trading interest is more balanced. For instance, in the high state, the open interest is to buy, but it is small in magnitude. Indeed, as m 1/, R 0. Thus, the effective presence of high-value traders in the second stage through open interest is very small. But because only low-value CDS sellers participate substantially in the second stage as can be seen in the middle-right subplot of Figure 3), they dominate the price-discovery process. The resulting price is downward biased. Similarly, in the low state, there is a very small open sell interest, and high-value CDS buyers dominate the price discovery in the second stage, resulting in an upward biased price. The price bias in this case is also the most transparent in Corollary 1, where we take the limit Q. As m 1/, the final auction price converges to v L in the high state, but converges to v H in the low state. These biases are large and extreme. Inefficient allocations. It is therefore not a surprise that the equilibrium allocations of bonds are inefficient. Figure 5 depicts the final allocations. In the high state, allocations to low-value traders are uniformly too high. High-value traders either buy too much if Q i is sufficiently negative) or too little if Q i is positive or mildly negative). In the low state, allocations to high-value traders are uniformly too low. Low-value traders either sell too much if Q i is sufficiently positive) or too little if Q i is negative or mildly positive). 13

15 Figure 3: First-stage physical requests and second-stage allocations. Parameters: v H = 1, v L = 0, = 1, Q =. Physical settlement requests m=0.9) r i Physical settlement requests m=0.6) r i Q i Q i High-value trader Low-value trader High-value trader Low-value trader Second stage allocations, open interest to buy m=0.9) * x i Second stage allocations, open interest to buy m=0.6) * x i Q i Q i High-value trader Low-value trader High-value trader Low-value trader Second stage allocations, open interest to sell m=0.9) * x i Second stage allocations, open interest to sell m=0.6) * x i Q i Q i High-value trader Low-value trader High-value trader Low-value trader 14

16 Figure 4: Final auction prices parameters are the same as in Figure 3) Price Second stage prices * High state p B c High state p H * Low state p S c Low state p L m Figure 5: Total allocations in two stages parameters are the same as in Figure 3). Total allocations, high state m=0.9) 0. Total allocations, high state m=0.6) Q i Equilibrium allocation, high-value trader Efficient allocation, high-value trader Equilibrium allocation, low-value trader Efficient allocation, low-value trader Q i Total allocations, low state m=0.9) Total allocations, low state m=0.6) Q i Q i

17 4 Imposing a Price Cap or Price Floor In Section 3 we have shown that the two-stage CDS auctions without price cap or floor lead to biased prices and inefficient allocations. In this section we consider the effect of adding a price cap or a price floor on the second-stage orders, taking the cap or floor as given. We assume that the price cap and floor are known by the traders in the first stage. In Section 5 we study the dealers strategic incentives in quoting prices that eventually determine the price cap or floor. Denote by p the price cap, which applies if R < 0, and denote by p the price floor, which applies if R > 0. We impose the following assumption in this section. Assumption. The price cap and floors are symmetric: p v L = v H p. 18) They also satisfy where p S and p B p < p S and p > p B, 19) are the equilibrium prices from Proposition 1. Condition 18) keeps the model symmetric and hence tractable. Condition 19) makes the price cap and floor binding constraints; if the price cap and floor do not satisfy 19), the equilibria would be unchanged from Proposition 1. By Proposition, Assumption nests an important special case: p = p c L and p = pc H. That is, the cap and floor could be the competitive prices. Proposition 3. Under Assumption, there exists an equilibrium in which every trader i submits in the first stage: minv H mp 1 m)p)/, Q i ) v i = v H, Q i < 0 r i v i, Q i ) = maxv L 1 m)p mp)/, Q i ) v i = v L, Q i > 0, 0) 0 otherwise and submits 7) in the second stage. The equilibrium price is equal to the price cap p given an open interest to sell and is equal to the price floor p given an open interest to buy. For either direction of the open interest, there is strict pro-rata rationing of the second-stage orders at the equilibrium price. The equilibrium allocation with price cap or floor is inefficient. 16

18 Figure 6: Total allocations in two stages with price cap or floor In the high state the price floor is set at p c H, and in the low state the price cap is set at pc L. Other parameters are the same as in Figure 3. Total allocations, high state m=0.9) Total allocations, high state m=0.6) Q i Equilibrium allocation, high-value trader Efficient allocation, high-value trader Equilibrium allocation, low-value trader Efficient allocation, low-value trader Q i Total allocations, low state m=0.9) Total allocations, low state m=0.6) Q i Q i -0.4 Figure 6 depicts the final allocations with a price cap or floor for the equilibrium described in Proposition 3. In the high state the price floor is set at p c H, and in the low state the price cap is set at p c L. This parameter guarantees that the prices are correct. But as shown in Proposition 3, the final allocations are still inefficient. The direction of allocative inefficiency is qualitatively similar to that in Proposition 1 and is caused by the pro-rata rationing in the second stage. In the high state, low-value traders receive too much allocation they do not sell enough). In the low state, high-value traders receive too little allocation they do not buy enough). This allocative inefficiency generally hurts investors because reallocating the defaulted bonds in the secondary markets incurs substantial trading costs. 17

19 5 Manipulation of First-Stage Quotes The last part of our analysis of the current CDS auction mechanism is to add the first-stage quotes by dealers. In practice, these quotes determine the price cap or floor studied in the previous section. The objective of this section is to illustrate that the net CDS positions of dealers can create strong incentives to distort, or manipulate, the first-stage quotes. For this purpose, we will make a few simplifying assumptions, elaborated shortly. We assume that a finite number of infinitesimal traders, denoted j {1,,..., n}, are dealers. Because dealers have measure zero, it is inconsequential whether they are part of the unit mass of traders or outside of it.) In the first stage, each dealer j makes a quote b j [v L, v H ] on the defaulted bonds. Restriction to [v L, v H ] is natural but not necessary for our results.) The average quote is denoted b j b j/n. If R < 0, b is the price cap in the second stage. If R > 0, b is the price floor in the second stage. If R = 0, b is the final auction price, but this does not happen on equilibrium path.) Although the practical determination of price cap or floor involves adding or deducting a spread from the initial market midpoint see Section ), we omit the spread for simplicity here. We denote by p c the competitive price, which depends on the state. The utility of each dealer is given by: 1 maxb, p B π j = ))Q j c E j [p c ] b j, if R > 0, 1) 1 minb, p S ))Q j c E j [p c ] b j, if R < 0. The first term of the utility π j is the payout from dealer j s CDS position Q j, where the final price is either unconstrained p B if R > 0 and p S if R < 0) or constrained by the price cap or floor b. The second term c E j [p c ] b j is a reduced-form proxy for the reputation cost of dealer j to quote a price different from his best estimate of the ex post competitive price, where c > 0 is a given constant. Although a dealer s best estimate of the competitive price is typically a dealer s private knowledge, this information may nonetheless be partially observed by investors through informal communication. In the utility function π j we have effectively assumed that dealers do not submit physical settlement request r j and they do not buy or sell bonds in the second stage. This seemingly strong assumption is actually reasonable because dealers often receive the bid-ask spread in the secondary markets, but in the auction they receive zero spread. By contrast, nondealers would prefer trading in the auction at zero spread to paying bid-ask spreads in the secondary markets.) That is, in our simplified model, the dealers only role is to determine the price 18

20 cap or floor. From 1) it is clear that as long as Q j 0, dealer j may have an incentive to report b j that is different from E j [p c ] in order to profit from his CDS position. Although a full characterization of the equilibrium is complicated, in some special cases the equilibrium can be solved explicitly. In particular, we will focus on cases in which dealers are homogeneous in their CDS positions and the CDS positions are perfectly correlated with the underlying state. These special cases look strong and stark, but they generate very simple predictions that convey the main intuition. Proposition 4. Suppose that conditional on the low state, Q j = Q > 0 for every dealer j; and conditional on the high state, Q j = Q < 0 for every dealer j. 1. In the low state, in equilibrium every dealer j quotes b j = p c L if Q nc and quotes b j = v L if Q > nc. The final auction price is p c L if Q nc and is v L if Q > nc.. In the high state, in equilibrium every dealer j quotes b j = p c H if Q nc and quotes b j = v H if Q > nc. The final auction price is p c H if Q nc and is v H if Q > nc. By assumption, dealers perfectly infer p c from their CDS position Q j. In a sense, dealers are in the money on their CDS contracts because they are CDS buyers in the low state and CDS sellers in the high state. If the CDS positions are sufficiently small in magnitude Q nc), dealers truthfully quote p c L in the low state and pc H in the high state, leading to an unbiased average quote and final price. If, however, dealers CDS positions are sufficiently large Q > nc), they would strategically manipulate the quotes to affect the final price. In the low state, dealers are CDS buyers and profit from a low final price; hence, they quote the lowest price possible, v L, which becomes the price cap. In the high state, dealers are CDS sellers and profit from a high final price; hence, they quote the highest price possible, v H, which becomes the price floor. The final price is downward biased if the open interest is to sell low state), and upward biased if the open interest is to buy high state). The next proposition starts from the opposite assumption from Proposition 4, that is, dealers are out of the money, being CDS sellers in the low state and CDS buyers in the high state. Proposition 5. Suppose that conditional on the low state, Q j = Q < 0 for every dealer j; and conditional on the high state, Q j = Q > 0 for every dealer j. 1. In the low state, in equilibrium every dealer j quotes b j = p c L if Q nc and quotes b j = p S if Q > nc. The final auction price is pc L if Q nc and is p S if Q > nc. 19

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