Efficient Market Management - naked Credit Default swaps (CDS), and investor Risk

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1 Noname manuscript No. (will be inserted by the editor) Will banning naked CDS impact bond prices? Agostino Capponi Martin Larsson Received: date / Accepted: date Abstract We develop a tractable partial equilibrium model to analyze the impact on the bond market generated by a ban on naked credit default swaps. We demonstrate that such a ban will have a negligible impact on the borrowing costs if CDS speculators are risk averse and take positions which are small relatively to the amount of debt outstanding. We find that the ban only excludes from the market moderately pessimistic investors, and induces the most pessimistic to implement their strategy on the short side of the bond market. Despite the sovereign debtor benefits from the reduced yields on the issued bonds, he will suffer from a diminished borrowing capacity after the ban. Such findings suggest that regulators should consider other measures to reduce instability arising from excessive speculation in derivatives markets. Keywords Partial Equilibrium sovereign debt credit default swaps trading restrictions JEL Classification D5 E4 E5 Agostino Capponi gratefully acknowledges the Institute for New Economic Thinking (INET), which supported his research under the grant number IN Martin Larsson gratefully acknowledges funding from the European Research Council under the European Union s Seventh Framework Programme (FP/ ) / ERC Grant Agreement n POLYTE. A. Capponi Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, acappon1@jhu.edu M. Larsson Swiss Finance Institute, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland martin.larsson@epfl.ch

2 2 Agostino Capponi, Martin Larsson 1 Introduction The recent financial crisis has shown that derivative instruments, while allowing for efficient hedging and risk transfer, can also pose serious threats to financial stability. Governmental and regulator authorities have blamed derivatives, especially over-the-counter credit derivatives, to have contributed to making the financial system more complex and opaque, see Gordy and Willemann (2010), and exacerbated the market panic during the crisis. Most recently, a number of policy makers and regulator authorities have expressed concerns regarding sophisticated market players entering uncovered or naked positions in credit default swaps (CDS). Such portfolio positions, where an investor purchases CDS protection without holding the commensurate amount of the underlying bond instrument, is viewed by critics as a way to speculate on the health of the particular entity. This has been claimed to be responsible for raising the borrowing costs of many issuers of government debt, such as Greece, and worried regulators that naked CDS could pose a systemic risk to the overall economy. For this reason, a decision to reduce, and in some extreme cases ban outright the use of this practice, has been taken. 1 This also initiated a debate among economists regarding whether a ban on naked sovereign CDS should be imposed. Portes (2010) and Soros (2009) argued in favor of it, claiming that naked speculation causes a rise in CDS prices, and consequently increases the cost of funding of the underlying sovereign entity. On the other hand, influential researchers such as Duffie (2010) and Jarrow (2010) argued against the ban, claiming that it reduces market liquidity and raises trading execution costs, thereby lowering the quality of information contained in CDS prices, see also Gorton (2010). Our paper belongs to the stream of literature studying the impact of credit default swaps on financial stability. Sovereign CDS trading may affect bond prices in two ways. Firstly, it may introduce feedback effects affecting the fundamentals of the reference country and consequently impact the primary market for bond issuance. This aspect has been partly addressed by existing literature, some of which surveyed next. Secondly, it has been questioned whether CDS trading restrictions can affect the price of sovereign bonds in the secondary market. This question has recently received significant attention, given the expressed concerns that short positions on credit risk through naked CDS can be detrimental for market stability. Our goal is to make a first step towards providing a complete analysis to answer such a question. To this purpose, we develop a tractable partial equilibrium model, where investors with heterogenous default beliefs and mean-variance preferences can trade two credit risky securities, namely a bond and a credit default swap, both referenc- 1 The German Chancellor Angela Merkel effectuated a ban on naked short-selling of Euro bonds on May 25, 2010, as well as prohibited naked CDS on bonds of European governments and short-selling of shares in ten major German financial institutions. A similar action was taken by the EU, who banned naked CDS on sovereign debt with a regulation adopted on November 15, 2011.

3 Will banning naked CDS impact bond prices? 3 ing the same entity. It is partial because we do not consider the interaction of the CDS market with other market activities, such as the cross-hedging use of CDS by investors holding other risky assets characterized by various degrees of correlation with the risk of sovereign debt. Our results indicate that naked CDS speculation would only have a negligible impact on the underlying bond price if CDS manipulators take positions not exceeding significantly the amount of outstanding debt. Notice that this is indeed the case for the financially weaker Eurozone sovereigns, where the aggregate net CDS positions represent small fractions of their respective amounts of debt outstanding. 2 Such conclusions are also in line with empirical findings of Duffie (2010), who does not see any significant relationship between amounts of CDS referencing Greece, Italy, Ireland, Spain, Portugal and the borrowing costs of these sovereigns. Moreover, they are consistent with the evidence reported by Ashcraft et al (2009), showing that on average bond prices are not impacted by CDS trading activities. Although the ban has been introduced to remedy the instability caused by excessive speculation in financially distressed periods, we demonstrate that this is not the case if investors are pessimistic and risk averse. As we illustrate later in the paper, the ban would only exclude some moderately pessimistic investors from the market, those for which the expected gain from a short bond position is now swamped by the larger transaction cost. Highly pessimistic investors will still implement their activity on the short side of the bond market. The prohibition of naked CDS may therefore have a marginal effect on changes in bond prices, and at the same time can impact the liquidity of the CDS market by reducing the number of participants. Moreover, despite the fact that after the ban the sovereign debtor benefits from reduced yields on the issued bonds, he would still suffer from a diminished borrowing capacity, and ultimately see his expected wealth decrease. Previous research by Bolton and Oehmke (2011) demonstrates that CDS have important ex-ante commitment benefits because they increase investment and eliminate strategic default, but also show that creditors generally tend to overinsure, thus resulting in an empty creditor problem, see also Hu and Black (2008) for related analysis, and Subrahmanyam et al (2012) for an empirical illustration. Ashcraft et al (2009) show that, perhaps counter-intuitively, the firms that have benefited the most from CDSs are the safest and less exposed to default risk. Indeed, for such firms, the CDS is more liquid and they tend to pay lower interest once CDSs are written on their bonds. Che and Sethi (2012) show that naked credit default swaps can have a significant effect on borrowing costs and likelihood of default. They develop a mechanism according to which pessimists about future prospects of the borrower would buy naked protection 2 In the case of Greece, the aggregate of the net CDS positions held in the entire market has remained well under 3% of the total amount of Greek debt outstanding. Furthermore, Duffie (2010) documents that since 2008 there was never an increase in aggregate CDS positions on Greece that was more than 0.18% of Greek debt outstanding.

4 4 Agostino Capponi, Martin Larsson from those who are more optimistic, asking them to post cash collateral to eliminate counterparty risk. Such collateral is financed by a third class of investors who shift the terms of financing against the borrower, and for a given borrowing requirement ask for larger bond issue and smaller bond price. When only covered credit default swaps are allowed, such investors become less pessimistic and consequently the terms of lending are shifted in favor of the borrower, thus resulting in a smaller bond issue and larger bond price. Our model is designed to capture empirical features, which are typical of sovereign CDS markets. According to statistics reported in (IMF 2010, Annex 1.2), dealers in the sovereign CDS market are net sellers of credit protection and do so for hedging purposes, while investors (real money and hedge funds) are net buyers of protection. Although dealers represents about 90% of the sovereign CDS markets, with investors only accounting for the remaining 10%, we consider the case where they have a equal share of the CDS market, in order to assess the impact of CDS speculation on the bond market under scenarios of extreme speculation. More specifically, we assume that investors can only buy credit default swaps, while market makers can only sell CDS. An investor who wishes to go long credit risk may do so by purchasing the bond security, while an investor who wants to short credit can do so by either buying (possibly naked) CDS or by short-selling the bond security. We remark, however, that the model we develop is general and can be used to analyze the price impact that trading restrictions in credit derivatives market would have on (not necessarily sovereign) bond markets. Both bond and CDS trades require the intermediation of market makers, whose main function is to provide liquidity to the market. In the sovereign bond market, most market participants have a dual role as market makers and investors (see Table 1 in Cheung et al (2005)). In our model, we therefore let bond trades happen directly across investors, without explicitly modeling the intermediation. In contrast, there are few market participants willing to meet the high demand for sovereigns CDS, as argued above. Moreover, Austin and Miller (2011) demonstrate that traders in the CDS market generally do not have market making responsibilities. To this purpose, we consider the stylized situation where there is only one market maker in the CDS market. CDS market makers typically hedge their exposure. As documented in Boone et al (2010), in the sovereign case this is generally done using either government bonds, or offsetting CDS transactions. However, in times of crisis, the rarity of market makers or investors willing to sell protection can make the bond a more attractive hedging instrument. 3 Our model aims to capture this by requiring the CDS market maker to maintain a riskless position by perfectly hedging his CDS exposure using sovereign bonds. We additionally assume that he has a competitive advantage deriving from the possibility to short-sell bonds at low cost, and strategically set the CDS premium to maximize his wealth. 3 We also remark that, although the equity options market can be used to hedge corporate credit exposure approximately, this is much harder to do for sovereign credit exposure.

5 Will banning naked CDS impact bond prices? 5 The rest of the paper is organized as follows. Section 2 introduces the model and provides some preliminary results. Section 3 recovers the optimal strategies of the investors, taking prices as fixed. Section 4 analyzes the market maker activity, while Section 5 recovers the equilibrium prices of the securities in the model. We perform a comparative statics analysis in Section 6 to assess the economic implications of our findings and the impact of the ban on the bond market. 2 The model We consider an economy where three assets trade. There are two credit sensitive assets, and one risk-free asset. The risky assets are the defaultable bond, and a credit default swap written on that bond. The risk-free asset is used as the numeraire, meaning the unit in which all payoffs will be measured. We assume that the risk-free asset pays no interest. There are two time periods, t = 0 and t = 1. The following table gives the prices and payoff structure of the two risky assets, whose prices will be determined endogenously in our partial equilibrium model. Asset Price at t = 0 Payoff at t = 1 Defaultable bond P b 1 {no default} CDS P c 1 {default} There is clearly a redundancy in this system of payoffs: in the absence of trading restrictions, simultaneous holding of one unit of the defaultable bond and of the CDS is equivalent to holding one unit of the risk-free asset. However, we will also assume that there is a cost associated with short-selling the defaultable bond. This will ultimately induce pessimistic market participants (i.e., agents perceiving a high default probability) to bet on default by investing in the CDS. This short-selling cost will be described momentarily. There are n + 1 agents in the economy, each endowed with an initial wealth w i. Agents 1,..., n are investors who maximize mean-variance utility over the absolute return on their portfolios. The investors are heterogeneous both regarding their attitude towards risk, and their beliefs concerning the probability of default. They can invest in all the assets, but are not allowed to hold a short position in the CDS. This is the function of the market maker, which is the zeroth agent i = 0. The market maker is required to maintain a riskless position (so in particular there will be no counterparty risk in this model), as well as meeting the demand for CDS. Therefore, we have market

6 6 Agostino Capponi, Martin Larsson segmentation with respect to the CDS asset: investors are only allowed to stay on the long side, while the market maker can only stay on the short side of the market. In our market, therefore, there are two ways for the investors of being short credit risk: (1) short-selling the bond security, and (2) buying the credit default swap. Since the market maker s portfolio is riskless, his wealth at t = 1 is deterministic. Due to his monopoly power with the respect to CDS asset, he can set the price P c of the CDS in such a way to maximize his wealth, while taking into account how his choice will affect investor demand. In this sense, the market maker has some market power, because he correctly accounts for the impact his choice of P c has on CDS demand, but he acts as a price taker with respect to the defaultable bond (that is, he regards P b as fixed). As already mentioned, there is a short-selling cost associated with the defaultable bond. Specifically, the market maker pays an amount δ for every bond he sells short. For the investors, the per-bond cost is δ. In our model, the short-selling costs are exogenous deadweight losses, as they only dissipate part of the total welfare without causing wealth redistribution. We refer the reader to Duffie (1996) for endogenous models of short-selling costs, and to Bongaerts et al (2011) for the impact of short-selling costs on asset liquidity, in equilibrium models with zero net supply assets. We assume that 0 < δ < δ, and the fact that it is cheaper for the market maker to short-sell bonds is what allows him to profit on the CDS market: investors who wish to bet on default will try to avoid the higher cost δ by instead purchasing the CDS. This will remain beneficial even if the market maker charges a premium strictly higher than δ (but no higher than δ.) Finally, since our goal is to study the impact of banning naked CDS positions, we will also consider this additional constraint. More precisely, we look at two different versions of the model: one that is as described above; and one where we impose the additional restriction that any long position in the CDS must be accompanied by a long position of at least equal size in the bond. Before proceeding further, we introduce some notation. We denote by q i = (qi b, qc i, qr i ) the vector of quantities held by the i-th agent. Here, qb i and qc i are the quantities held in the defaultable bond and CDS, while qi r is the quantity held in the risk-free asset. Further, we write P = (P b, P c, 1), where P b and P c are the prices of a single unit of defaultable bond and CDS, respectively, which will be determined in equilibrium, while the third component is the price of a single unit of the numeraire asset (which is one by definition). We now describe the preferences and beliefs of the investors, i = 1,..., n, as well as the optimization problem they face. Investor i seeks to determine a vector q i of quantities that he wishes to hold for each of the assets. His aim is

7 Will banning naked CDS impact bond prices? 7 to solve min q γ i 2 q Σ i q µ i q s.t. P q + δ (q b ) = w i q c 0, where x = max( x, 0) denotes the negative part, and Σ i and µ i denote the covariance matrix and mean vector of the payoffs. 4 If we let π i = P i [default] be the i th investor s subjective probability that the reference entity defaults, the covariance matrix and expected payoffs are given by π i (1 π i ) π i (1 π i ) Σ i = π i (1 π i ) π i (1 π i ) 0 = π i (1 π i ) and µ i = 1 π i π i. 1 In particular, the subjective default probabilities π i and the risk-aversion parameters γ i completely characterize the preferences of the investors. It will be convenient to label them from the most optimistic to the most pessimistic, so that 0 < π 1 π 2 π n < 1. When the ban on naked CDS positions is in place, the investors have to adhere to the following additional constraint: (q b ) + q c (naked CDS ban), where x + = max(x, 0) denotes positive part. The market maker s budget constraint is of the same form as the investors, except that his short-selling cost for the bond is δ. If his portfolio quantities are denoted q 0 = (q b 0, q c 0, q r 0), the budget constraint is P q 0 + δ(q b 0) = w 0. Moreover, the requirement that his portfolio be riskless and that he meet the CDS demand yields the restrictions n q0 b = q0 c and q0 c = qi c. 4 We can work with payoffs instead of absolute returns. Indeed, the latter are given by the payoffs minus the expected payoffs, hence have the same covariances. Moreover, since the budget constraint is binding, the total expected absolute return is µ i q w i, and we change nothing by dropping the constant term w i from the objective function. The budget constraint is binding due to the existence of the risk-free asset. i=1

8 8 Agostino Capponi, Martin Larsson The defaultable bond is in fixed positive supply, denoted by α, while the credit default swap asset is available in zero net supply. The supply of the numeraire asset, β, is endogenously given by aggregate feasibility: the aggregate expenditure plus the short-selling costs incurred must equal the aggregate initial wealth netted of the position in the numeraire asset. Thus, the market clearing conditions are given by αp b + δ(q b 0) + δ n i=1 n qi b = α (1) i=0 (q b i ) + β = w, (2) where w = w 0 + w w n is the aggregate wealth. We make the following mild assumptions. Market with deep pockets. The market is sufficiently rich, i.e. it has enough initial aggregate wealth w that all transactions can be financed using the wealth inside the system. In other words, the aggregate net supply β of the numeraire asset is positive. Sufficient heterogeneity. The market is sufficiently heterogeneous in terms of beliefs and risk preferences that trades are triggered, and the system is well functioning. One of our goals is to provide conditions on the model primitives under which these conditions are satisfied. To this end, we develop a number of preliminary results, which will be used later to characterize the behavior of investors in equilibrium. We start with the following Lemma, whose proof is reported in Appendix A. Lemma 1 The market maker s final wealth is given by w 0 + ( P b + P c 1 δ ) n qi c. (3) Before computing optimal quantities and prices, it is useful to characterize the participation constraints. Indeed, trading will only take place if both sides of the market are able to extract positive surplus: for the market maker, this means that the price of the CDS has to be large enough to compensate the short-selling costs that he will incur for perfectly hedging his exposure via short-sale of bonds. For the investor, the price of the CDS cannot be too high, otherwise he could achieve the same objective of going short credit risk by short-selling bonds even if this comes with the short-selling cost δ. The next lemma fully characterizes the participation constraints. Lemma 2 (i) If P b + P c 1 > δ, then no investor buys the CDS. i=1

9 Will banning naked CDS impact bond prices? 9 (ii) In equilibrium, the market maker may restrict his choice of P c to satisfy δ P b + P c 1 δ. Any other choice of P c results in a final wealth less than or equal to w 0. (iii) If P b + P c 1 < δ and naked CDS positions are allowed, then no investor has a short position in the bond. The proof of this lemma is reported in Appendix A. The following is a trivial but useful bound on the bond price P b, whose proof is reported in Appendix A. Lemma 3 In any equilibrium, P b < 1. In view of Lemma 2, it is reasonable to restrict attention to equilibria in which the participation constraint (condition of Lemma 2(ii)) is satisfied. Moreover, when naked CDS positions are allowed, we conjecture that no investor takes a short position in the bond. The above condition does not directly guarantee this because, when P b +P c 1 = δ, the investors are indifferent between shortselling the bond and purchasing the CDS. Notice that investors can profit both from activities in the bond and CDS market, while the market maker profits exclusively from selling CDS protection. Consequently, we naturally allow for a tie-breaking benefit offered by the market maker to induce investors wishing to short credit to purchase CDS protection. Hence, Ansatz. In equilibrium, the market maker chooses P c so that δ P b + P c 1 δ. (4) Moreover, when naked CDS positions are allowed, all investors have nonnegative bond positions: qi b 0, i = 1,..., n. We notice that deviation from this Ansatz would result in the market maker setting the CDS price so as to decrease the level of his initial wealth after implementing his protection selling activity (see Lemma 2). As he knows that market participants will always prefer the CDS security over a short bond position if P b +P c 1 δ, then it is natural to restrict attention to equilibria where this Ansatz is satisfied. 3 Optimal investor quantities This section derives the optimal quantities of securities held by investors, taking prices as fixed. We solve the investor optimization problem when naked protection is allowed in Section 3.1, and when the ban is imposed in Section 3.2.

10 10 Agostino Capponi, Martin Larsson 3.1 Naked CDS allowed Under our Ansatz, all bond positions are nonnegative in the case where naked CDS positions are allowed. No short-selling costs are therefore incurred, so the budget constraint for the i th investor simply becomes q P = w i. Substituting for q r in the objective function yields the new objective f i (q b, q c ) = γ i 2 π i(1 π i )(q b q c ) 2 + π i (q b q c ) q b ( w i q b P b q c P c), which is minimized subject to q b 0, q c 0. We expect that typical speculators would compare their beliefs about the default probability with the market estimates resulting from bond and CDS prices. They would then decide to implement their speculation activity in the bond or in the CDS market, taking into account the outcome of this comparison and the banning restriction, when the latter is in place. The following result shows that this is indeed the case by giving the optimal quantities, and indicating how they depend on the relation between the prevailing market prices and the investor beliefs. Proposition 1 Under the assumption of no short bond positions, no short CDS positions, and P b + P c > 1, the optimal quantities q i of investor i are unique, and given as follows: (i) If π i < P c and 1 π i P b, then q b i = (ii) If π i P c, then 1 ( 1 πi P b), qi c = 0. (5) γ i π i (1 π i ) q b i = 0, q c i = (iii) If π i < P c and 1 π i < P b, then 1 γ i π i (1 π i ) (π i P c ). (6) q b i = 0, q c i = 0. (7) The optimal holding in the risk-free asset, qi r, is then given via the budget constraint. The proof of Proposition 1 is reported in Appendix B. The three investor types corresponding to case (i) (iii) in Proposition 1 can be interpreted, respectively, as optimistic, pessimistic, and what one may call bearish. The pessimistic investor (type (ii)) believes that default is more likely than recognized by the CDS market, and thus finds the CDS attractive. From the point of view of the

11 Will banning naked CDS impact bond prices? 11 optimistic investor (type (i)), the CDS is overpriced, but the survival probability is larger than priced by the bond market. As such, he finds the bond an attractive instrument. The bearish investor (type (iii)) views both the bond and the CDS as overpriced, and therefore does not invest in either. Instead he puts all his wealth in the risk-free asset. To simplify terminology, when we refer to an investor as optimistic, it will be understood that he holds a strictly positive quantity of bonds. Similarly, pessimistic investors hold strictly positive quantities of CDS. 3.2 Naked CDS banned When naked CDS positions are no longer allowed, the conclusion of Lemma 2(iii) is no longer valid. The budget constraint of investor i is therefore This leads to the objective function P q + δ (q b ) = w i. f i (q b, q c ) = γ i 2 π i(1 π i )(q b q c ) 2 +π i (q b q c ) q b ( w i q b P b q c P c δ (q b ) ). We can now characterize the optimal quantities in this case. Proposition 2 Under the assumption of no short CDS positions, no naked CDS positions, and P b + P c > 1, the optimal quantities q i of investor i are unique, and given as follows: (i) If 1 π i P b, then q b i = 1 ( 1 πi P b), qi c = 0. (8) γ i π i (1 π i ) (ii) If P b δ < 1 π i < P b, then q b i = 0, q c i = 0. (9) (iii) If 1 π i P b δ, then qi b 1 ( = P b δ ) 1 + π i, γ i π i (1 π i ) q c i = 0. (10) The optimal holding in the risk-free asset, q i r, is then given via the budget constraint.

12 12 Agostino Capponi, Martin Larsson The proof of Proposition 2 is reported in Appendix B. Remark. When the naked CDS ban is imposed, the investor types change. The condition in (i) is now enough to yield an optimistic investor who invests in the defaultable bond. If the condition is violated, the investor will no longer go long the bond, and as a consequence of the naked CDS ban, will not invest in the CDS. However, he may decide to short sell the bond. This happens when he perceives default to be so likely that the short selling cost does not deter him from going short. If the investor decides to hold zero quantity of the defaultable bond (case (ii) in Proposition 2), we call him moderately pessimistic. If, instead, he decides to go short in the defaultable bond (case (iii) in Proposition 2), we call him highly pessimistic. 4 The market maker The market maker chooses a portfolio q 0 = (q b 0, q c 0, q r 0) and the CDS premium P c, such that his final wealth is maximized, his portfolio is riskless, his budget constraint is satisfied, and he meets the investors demand for CDS. As already mentioned, he acts as a price taker with respect to the defaultable bond, and therefore treats P b as fixed. We first consider the setup where naked CDS positions are allowed, and then discuss the case when the ban is introduced. We start with the general case of n investors, and then specialize it to the case n = Naked CDS allowed Suppose there is no ban on naked CDS. By Proposition 1, we have q c i = 1 γ i π i (1 π i ) (π i P c ) + for i = 1,..., n. Substituting this into the expression for the final wealth of the market maker, see (3), and recalling Lemma 2(ii), we see that the market maker chooses P c as a solution to max x 0 { (x + P b 1 δ ) n i=1 (π i x) + γ i π i (1 π i ) : δ x + P b 1 δ }. (11) The optimal value is the profit, i.e. the amount by which the market maker s final wealth exceeds w 0. In order for the market maker to be active, there must be at least one investor who buys the bond (long credit), and another investor who buys the CDS (short credit). Indeed, if there were no investors buying protection, the market

13 Will banning naked CDS impact bond prices? 13 maker would not have any function and simply deposit all his wealth in the risk-free asset. If, instead, there were no investors buying the bond security, then the market maker would not be able to maintain his riskless position, as no investor would buy the bonds that he is short-selling to hedge his credit exposure. The following lemma establishes this intuitive result, and thus allows one to reduce the number of scenarios to consider with respect to investor types (optimistic, pessimistic, or bearish). The proof is reported in Appendix C. Lemma 4 Suppose that, in equilibrium, the market maker acts in such a way that his final wealth is strictly greater than w 0. Then there is necessarily at least one optimistic and at least one pessimistic investor. We now focus attention on the case of two investors, n = 2, with π 1 < π 2. In this setting, the market maker s optimization problem becomes tractable and can be solved explicitly. By Lemma 4 we know that there must be one optimistic and one pessimistic investor in equilibrium. It is then clear that i = 1 will be the optimistic investor, and i = 2 the pessimistic one. We then have the following proposition, proven in Appendix C. Proposition 3 Let n = 2, assume that naked CDS positions are allowed, and suppose that 1 π 2 < P b δ < 1 + π 2. (12) Then, in equilibrium, the market maker s optimal choice of P c is strictly positive and given by ( δ P P c b ) + (1 + π 2 ) = min, δ P b + 1. (13) 2 If (12) fails, the market maker s final wealth is necessarily at most w 0. Proposition 3 indicates that in determining the CDS price the market maker will give equal weight 1 2 to (1) the minimum selling price 1 P b + δ needed to ensure that he profits from his protection selling activity, and (2) the subjective probability π 2 of the pessimistic investor. When the latter probability is high, the pessimistic investor will have higher incentive to purchase protection and hence the market maker can increase the price further. 4.2 Naked CDS banned Let us now consider what happens when the CDS ban is introduced. According to Proposition 2, there will be no CDS holding whenever P b + P c > 1. On the other hand, as shown in Section 2, choosing P c so that this inequality fails is worse for the market maker than simply investing all his wealth in the

14 14 Agostino Capponi, Martin Larsson risk-free asset. This will indeed be the case, given the market maker s budget constraint and the requirement that he hold a riskless portfolio. Therefore, the price of the CDS will not be uniquely determined when the naked CDS ban is in force. We only know that it certainly could not be low enough for the CDS to become an attractive investment; this would at the very least require P b + P c 1, which would be inconsistent with the optimal choice on the side of the market maker. 5 Equilibrium market prices We compute the equilibrium prices of the securities, along with the endogenous supply of the numeraire asset. When naked positions are allowed, this amounts to computing the price of the defaultable bond and of the credit default swap, whereas in the case where naked protection is banned, the price of the CDS will not be uniquely determined, given that no trade takes place (see Proposition 2). Throughout the section, we assume two investors and one market maker. 5.1 Naked CDS allowed The following result gives necessary and sufficient conditions on (P b, P c ), in terms of the model primitives, guaranteeing that they correspond to a wellbehaved equilibrium. Proposition 4 In order for strictly positive prices (P b, P c ) to generate an equilibrium where the Ansatz of Section 2 is satisfied and the market maker has a final wealth strictly greater than w 0, the following conditions are necessary and sufficient: as well as δ < P b + P c 1 δ, (14) π 1 < P c, 1 π 1 > P b, π 2 > P c, (15) P b γ 1 π 1 (1 π 1 ) 1 π 2 < P b δ < 1 + π 2, (16) P c γ 2 π 2 (1 π 2 ) = 1 γ 1 π 1 1 γ 2 (1 π 2 ) α (17) and ( δ P P c b ) + (1 + π 2 ) = min, δ P b + 1. (18) 2

15 Will banning naked CDS impact bond prices? 15 The proof of Proposition 4 is reported in Appendix D. From Eq. (15) we obtain that 1 π 1 + π 2 > P b + P c. Using Eq. (14), we deduce that π 2 π 1 > δ. (19) Eq. (17) states that the price of the bond measured in units of risk weighted variance of the default belief of the optimistic investor equals the price of the CDS measured in units of risk weighted variance of the default belief of the pessimistic investor plus an adjustment factor. Such a factor is decreasing in the risk weighted default probabilities perceived by the optimistic and pessimistic investor, and obviously decreasing in the bond supply, given that higher supply contributes to decreasing the price of each bond unit. Remark 1 Eq. (17) shows that P c is an affine function of P b. This function is unbounded and strictly increasing, whereas the right side of Eq. (18) is an unbounded strictly decreasing function of P b. The system (17) (18) thus has a solution with P b > 0 if and only if ( ) 1 δ γ 2 π 2 (1 π 2 ) min π2, δ > α + 2 γ 2 (1 π 2 ) 1. γ 1 π 1 This solution is unique. Through elementary but cumbersome verifications of the conditions in Proposition 4 (which may be done with the help of software packages such as Mathematica), it can be checked that, if there is sufficient heterogeneity in the sense that π 2 π 1 > δ and 0 < α < ᾱ := π 2 π 1 δ γ 1 π 1 (1 π 1 ), then the equilibrium prices are positive and given as follows. Define the relative risk-weighted variance as ν = γ 2π 2 (1 π 2 ) γ 1 π 1 (1 π 1 ), (20) and set Then we have δ = δ 1 + ν 1 + 2ν + ν ( ) π 2 π 1 αγ 1 π 1 (1 π 1 ) ν (i) If δ > δ and α < α, then P b = 1 [ ( 1 δ + γ 2 π 2 (1 π 2 ) + 2 ) ] 2α 1 + 2ν γ 2 π 2 γ 1 π 1 (21) P c = 1 [ ] π 2 + αγ 2 π 2 (1 π 2 ) + ν(π 1 + π 2 + δ). (22) 1 + 2ν

16 16 Agostino Capponi, Martin Larsson (ii) If δ < δ δ and α > 0, then P b = ν [ δ + γ 2 π 2 (1 π 2 ) ( ) ] α γ 2 π 2 γ 1 π 1 (23) P c = δ + 1 P b (24) In both cases, the endogenous supply β of the numeraire asset is given by β = w αp b δ ( π 2 P c). γ 2 π 2 (1 π 2 ) If the aggregate wealth w is large enough to make β > 0, then the market with deep pockets assumption made in Section 2 is satisfied. Remark 2 Eq. (6) indicates that the number of CDS units purchased by the pessimistic investor decreases as the price of the CDS increases. From (24), we can see that as the short selling costs incurred by the investors get higher, the price of the CDS rises. All this suggests that in an environment where transaction costs are high, investors will be more reluctant to purchase the CDS, resulting in a diminished speculation activity. This is in the same spirit as the Tobin tax, introduced in the foreign exchange market on all spot conversions of one currency into another, to reduce volatility induced by exchange rate speculation. We notice that the bond price and CDS premium are continuous in δ. Indeed, at δ = δ the bond prices given by Eq. (21) and (23) coincide, and the CDS prices given by Eq. (22) and (24) coincide. 5.2 Naked CDS banned The following result is the analog of Proposition 4 for the case with naked CDS ban. We continue to consider the case of one market maker and two investors. Proposition 5 Assume naked CDS positions are banned. In order for strictly positive prices (P b, P c ) to generate an equilibrium where the Ansatz of Section 2 is satisfied and where there is one optimistic and one moderately pessimistic investor, the following conditions are necessary and sufficient: P b = (1 π 1 )(1 αγ 1 π 1 ) and α < 1 γ 1, (25) and P b δ < 1 π 2 < P b, (26) δ P b + P c 1 δ. (27)

17 Will banning naked CDS impact bond prices? 17 The proof of Proposition 5 is reported in Appendix D. Proposition 6 Assume naked CDS positions are banned. In order for strictly positive prices (P b, P c ) to generate an equilibrium where the Ansatz of Section 2 is satisfied and where there is one optimistic and one highly pessimistic investor, the following conditions are necessary and sufficient: ( ) P b 1 γ 1 π 1 (1 π 1 ) + 1 γ 2 π 2 (1 π 2 ) and = δ + γ 1 π 1 γ 2 π 2 γ 2 π 2 (1 π 2 ) α > 0, (28) 1 π 1 > P b, 1 π 2 < P b δ, (29) δ P b + P c 1 δ. (30) The proof of Proposition 6 is reported in Appendix D. Remark 3 Note that the CDS price P c is not uniquely determined in the case with naked CDS ban. We only get the bounds in (27) respectively (30). Similarly to Section 5.1, one can verify the conditions in Propositions 5 and 6, under the following sufficient heterogeneity condition: π 2 π 1 > δ and 0 < α < ᾱ := π 2 π 1 γ 1 π 1 (1 π 1 ). Defining the relative risk-weighted variance ν as in (20), and δ = π 2 π 1 αγ 1 π 1 (1 π 1 ), we have that the equilibrium price of the defaultable security is then positive and given as follows. (i) If α < α and δ δ, then P b = (1 π 1 )(1 αγ 1 π 1 ) (31) and the endogenous supply β of the numeraire asset is given by β = w + α(1 π 1 ) (αγ 1 π 1 1) (ii) If δ < δ < δ and α > 0, then P b = ν [ δ + γ 2 π 2 (1 π 2 ) ( ) ] α γ 2 π 2 γ 1 π 1 (32) and the endogenous supply β of the numeraire asset is given by β = w αp b δ ) (P b δ 1 + π 2. γ 2 π 2 (1 π 2 )

18 18 Agostino Capponi, Martin Larsson In addition, if the aggregate wealth w is large enough to make β > 0, the market with deep pockets assumption made in Section 2 is satisfied. As expected the bond price P b is continuous with respect to δ. Indeed, it can be checked that at δ, where the investor switches from highly to moderately pessimistic, the bond prices from Eq. (31) and (32) coincide. 6 Comparative Statics Analysis We evaluate the economic implications of the results derived in the previous sections. We remark that the quantities of bonds purchased by the optimist equal the amount of bonds issued plus the magnitude of the market-maker s short position: some optimistic investors own real bonds issued by the sovereign debtor, and some are counterparties to the market maker. Results are presented under two different scenarios. A normal scenario where π 1 = 0.01 and π 2 = 0.1, and a crisis scenario, where π 1 = 0.05 and π 2 = Denote by W1 N and W2 N the optimal wealth of the optimistic and pessimistic investor, respectively, at the end of the period in the regime where naked CDS are allowed. Then, the subjective expected wealth positions of the two investors, given their own beliefs, are given by E [ π1 W1 N ] = w1 + q1( b 1 π 1 P b), E π2 [ W N 2 ] = w2 + q c 2( π 2 P c), where q1 b and q2 c are given, respectively, by Eq. (5) and (6), and P b and P c are given by Eq. (21) (22), or Eq. (23) (24), depending on the values of δ, δ, and α. Similarly, let W1 NN and W2 NN denote the optimal wealths when naked CDS positions are banned. Then E [ π1 W1 NN ] = w1 + q1( b 1 π 1 P b), E [ π2 W2 NN ] = w2 + q2( b 1 π 2 P b) δ (q2) b, where q b 1 and q b 2 are now given by Eq. (8) (10), as determined in Proposition 2, and P b is given by Eq. (31) or (32). Next, we provide a numerical study to measure the predictive power of the model in the case when a ban on naked CDS purchases is introduced. Some of our findings lead to testable implications which may be validated further through a detailed empirical study. We leave this for future research. Testable Implication 1: The ban has small impact if the size of net outstanding CDS contracts is smaller than the bond supply.

19 Will banning naked CDS impact bond prices? Normal Crisis CDS positions / bond supply Percentage Change Normal Crisis Bond supply α Fig. 1 The left panel shows the ratio of CDS positions purchased by the pessimist to the total bond supply. The right panel shows the percentage change in bond price after the ban. The parameter settings are γ 1 = 0.2, γ 2 = 0.1, δ = 0.05, δ = 0.2, α = 5. Figure 1 demonstrates that the ban is most effective when the number of CDS units purchased by the pessimist is significantly higher than the bond supply. As the CDS market becomes thinner, the bond price is only mildly affected by the ban, and it experiences percentages increase of only 1% when the purchased CDS units amount to 10% of the total bond supply. The figure also shows that the ban induces the largest bond price appreciation during crisis regimes, when a higher number of naked CDS units are traded. If we consider that among the twenty largest sovereign CDS markets, the share of net notional CDS outstanding to risky government debt averages to about 2%, without exceeding 7% in any country (see also Figure 1.38 in IMF (2010)), then we can safely conclude that such a ban would not achieve his presumed objective of reducing the borrowing costs of sovereign entities. Testable Implication 2: A ban on naked CDS does not reduce credit speculation. Figure 2 shows that naked CDS positions would be replaced by short bond positions, unless the short-selling costs become too high. Indeed, if the investor is highly pessimistic, he will not be refrained from taking short positions in the bond market when the ban is imposed. As he moves from moderately to highly pessimistic, the short-selling costs will become of minor concern to him, and he will start shorting bonds. The number of shorted units becomes approximately equal to his holdings of naked CDS before the ban, when bond short selling costs are not prohibitive (see left panel of Figure 2), or his default likelihood is large enough (right panel of Figure 2). The figure also reveals that the speculative impact of naked CDS holdings is larger during crisis periods. During these times, the pessimistic investor will tolerate higher short-selling costs before reducing the size of his short bond position, initially equal to the size of naked CDS holdings. However, when the cost becomes too high relative to his default belief, extra increases in short-selling costs will be coupled with a reduction in his bond position (such a reduction will occur at a slower rate than in a normal regime).

20 20 Agostino Capponi, Martin Larsson 3 8 δ 1 = δ 1 = Naked CDS Short Bond Positions Normal Crisis Naked CDS Short Bond Positions δ π 2 Fig. 2 The left panel shows the difference between naked CDS positions (before ban) and short bond positions after the ban as a function of bond short-selling costs. The right panel shows how the same quantity varies with respect to π 2, when π 1 = The parameter settings are γ 1 = 0.1, γ 2 = 0.1, δ = 0.05, α = Expected wealth change Pessimist Normal Pessimist Crisis Optimist Normal Optimist Crisis Expected wealth Change Optimist Normal Optimist Crisis Pessimist Normal Pessimist Crisis δ γ 2 Fig. 3 The subjective expected wealths of optimistic and pessimistic investors. The left panel reports these quantities with respect to δ 1, γ 2 = The right panel reports the same quantities with respect to γ 2, δ = 0.1. The other parameters are set to γ 1 = 0.01, δ = 0.05, α = 5. Testable Implication 3: Both investors experience a trading loss after the ban. As the ban shuts moderately pessimistic investors out of the market, the sovereign debtor would benefit from it by paying lower interest rates when he sells bonds. However, he would also suffer a diminished capacity to borrow, as he needs to borrow the security in the repo market which yields him a cost. This appears evidently from Figure 3, where it is shown that during crisis regimes, the pessimistic investor wants to be short on credit after the ban, and thus takes positions in the bond market. However, the incurred short-selling costs dominate the advantageous increase of interest rate implied by the ban, and yield him a trading loss. The optimistic investor will also be affected by the ban, because he is forced to sell bonds at lower yields. Consistently with Figure 4, the ban has no impact on trading profits of investors, if the pessimist is highly risk averse. As explained earlier, he would replace his naked CDS holdings in full with short bond positions after the ban. Hence, the same

21 Will banning naked CDS impact bond prices? Percentage change Normal Crisis Naked CDS Short Bond Positions Normal Crisis γ γ 2 Fig. 4 The left panel reports the percentage change in bond price with respect to γ 2. The right panel reports the difference between naked CDS positions (before ban) and short bond positions after the ban as a function of γ 2. The parameter settings are γ 1 = 0.1, δ 1 = 0.1, δ = 0.05, α = 5. downward pressure on the bond price will be exerted before and after the ban, and the profits realized by the investors would not be altered. We conclude our study by analyzing the sensitivity of the ban to the risk aversion levels of the investors. Next, we list and discuss the two main findings. The ban has no impact if the pessimist is risk averse. Figure 4 shows that pessimistic investors who are less risk averse will be exposed to higher levels of default risk after the ban, i.e. they will take a smaller number of short bond positions relatively to the number of naked CDS holdings held before the ban. Hence, the amount of speculation on credit will experience the highest drop after the ban, and consequently the bond price will exhibit the highest appreciation. Notice, however, that such an appreciation never exceeds 2%. As the investor becomes more risk averse, he increases his level of short credit exposure on the bond market till shorting a number of bonds equal to the number of naked CDS units before the ban. When this happens, the same pressure on the bond price will be exerted under both regimes, and the ban will have no effect. We further notice that the largest percentage increase occurs during a crisis regime, even if the investor is only roughly four units long on credit risk as opposed to being over twelve units long as in the normal regime. This shows that the price impact caused by the ban is non-linear and higher when the levels of naked CDS speculations are larger. The ban has the largest impact if the optimist is sufficiently risk averse. When the optimist is more risk tolerant, then his strong appetite for risk will induce him to buy bonds at higher prices. Hence, the market maker can use his competitive advantage, and set up smaller CDS prices so to induce the pessimist to purchase a high number of naked CDS units. However, this will not have a noticeable downward impact on the bond price, as the less riskaverse optimist will be willing to purchase the bonds shorted by the market

22 22 Agostino Capponi, Martin Larsson Percentage change Normal Crisis Naked CDS Short Bond Positions Normal Crisis γ γ Normal Crisis CDSPrice γ 1 Fig. 5 The left panel reports the percentage change in bond price with respect to γ 1. The right panel reports the difference between naked CDS positions (before ban) and short bond positions after the ban as a function of γ 1. The parameter settings are γ 2 = 0.1, δ 1 = 0.1, δ = 0.05, α = 5. maker at higher prices. As he becomes more risk averse, even if the pessimist reduces his level of naked CDS speculation, the bonds shorted by the market maker will be purchased at lower prices. Therefore, after the ban, the bond will experience a higher appreciation. Clearly, when the optimistic becomes highly risk averse, the market maker will have to increase the price of the CDS, as each shorted bond will be paid less. Consequently, there will be a very small number of naked CDS positions to hedge, and the net amount of long bond units will be approximately the same before and after the ban. When this occurs, the impact of the ban becomes negligible. All these effects are visible in Figure 5. 7 Conclusions We have proposed a partial equilibrium model to evaluate the impact of a ban on naked credit default swaps. Our model identifies two types of investors, pessimistic and optimistic, trading bonds bilaterally, and CDS with a market maker who maintains a riskless position. We have solved for the optimal strategies of the investors as well as for the equilibrium prices, both in the regime where naked protection is allowed and in the regime where it is not.

23 Will banning naked CDS impact bond prices? 23 We find that if the CDS market has a smaller size than the bond market, then the ban will have a minor impact on the price of the reference bond. This suggests that manipulation through demand-based price pressure is potentially viable only for corporate or sovereign issuers, where the CDS market is significantly larger than the bond market. Since this is not the case for the ten largest Euro-zone sovereign entities, our analysis suggests that the ban would not achieve its intended objective of reducing borrowing costs. We have shown that if short-selling costs are not too prohibitive, risk averse pessimists would not be discouraged from the ban and implement their short-credit strategy with short bond positions after the ban. We also find that the sovereign debtor would not have a net benefit from the ban. Although he would obtain a gain by paying lower interest rates when selling bonds, he would also suffer a diminished capacity to borrow, which ultimately lowers his trading profits. Considering the important role played by CDS prices in leading bond prices, see Fontana and Scheicher (2010), and the evidence that spread variation is linked to fundamentals, see IMF (2010), our analysis suggests that such a restriction in the CDS market would not be beneficial. Regulators should thus explore alternative avenues to improve financial stability in derivatives market, for example imposing high collateral requirements to risky entities, or requiring central clearing houses for OTC transactions, see also Duffie et al (2010) and Duffie and Zhu (2011). Acknowledgements The authors would like to thank Gabriele Camera for very useful discussions improving the quality of this manuscript. A Proofs related to Section 2 Proof (Lemma 1) Since the market maker s portfolio is riskless, and since he meets the investors demand for CDS (and the CDS is available in zero net supply), the final wealth of the market maker is n q01 b {no default} + q01 c {default} + q0 r = q0 r qi c. Moreover, noting that qi c constraint gives 0 (i = 1,..., n) since investors cannot sell CDS, the budget i=1 w 0 = q0 b P b + δ q0 b 1 {q0 b<0} + qc 0 P c + q0 r ( n ) ( = qi c δ P b P c) + q0 r i=1 ( n ) ( = q0 r n ) ( ) qi c + qi c δ P b P c + 1. i=1 i=1 The final wealth can thus be written as n q0 r ( ) qi c = w n 0 δ P b P c + 1 qi c, i=1 i=1