ABSTRACTThis article presents a comprehensive framework for valuing financial instruments subject to credit risk. In particular, we focus on the impact of default dependence on asset pricing, as correlated default risk is one of the most pervasive threats in financial markets. We analyze how swap rates are affected by bilateral counterparty credit risk, and how CDS spreads depend on the trilateral credit risk of the buyer, seller, and reference entity in a contract. Moreover, we study the effect of collateralization on valuation, since the majority of OTC derivatives are collateralized. The model shows that a fully collateralized swap is risk-free, whereas a fully collateralized CDS is not equivalent to a risk-free one.Key Words : asset pricing; credit risk modeling; unilateral, bilateral, multilateral credit risk; collateralization; comvariance; comrelation; correlation.IntroductionA broad range of financial instruments bear credit risk. Credit risk may be unilateral, bilateral, or multilateral. Some instruments such as, loans, bonds, etc, by nature contain only unilateral credit risk because only the default risk of one party appears to be relevant, whereas some other instruments, such as, over the counter (OTC) derivatives, securities financing transactions (SFT), and credit derivatives, bear bilateral or multilateral credit risk because two or more parties are susceptible to default risk. This paper mainly discusses bilateral and multilateral credit risk modeling, with a particular focus on default dependency, as correlated credit risk is one of the greatest threats to global financial markets.There are two primary types of models that attempt to describe default processes in the literature: structural models and reduced-form (or intensity) models. Many practitioners in the credit trading arena have tended to gravitate toward the reduced-from models given their mathematical tractability. They can be made consistent with the risk-neutral probabilities of default backed out from corporate bond prices or credit default swap (CDS) spreads/premia.Central to the reduced-form models is the assumption that multiple defaults are independent conditional on the state of the economy. In reality, however, the default of one party might affect the default probabilities of other parties. Collin-Dufresne et al. (2003) and Zhang and Jorion (2007) find that a major credit event at one firm is associated with significant increases in the credit spreads of other firms. Giesecke (2004), Das et al. (2006), and Lando and Nielsen (2010) find that a defaulting firm can weaken the firms in its network of business links. These findings have important implications for the management of credit risk portfolios, where default relationships need to be explicitly modeled.The main drawback of the conditionally independent assumption or the reduced-form models is that the range of default correlations that can be achieved is typically too low when compared with empirical default correlations (see Das et al. (2007)). The responses to correct this weakness can be generally classified into two categories: endogenous default relationship approaches and exogenous default relationship approaches.The endogenous approaches include the contagion (or infectious) models and frailty models. The frailty models (see Duffie et al. (2009), Koopman et al. (2011), etc) describe default clustering based on some unobservable explanatory variables. In variations of contagion or infectious type models (see and Lo (2001), Jarrow and Yu (2001), etc.), the assumption of conditional independence is relaxed and default intensities are made to depend on default events of other entities. Contagion and frailty models fill an important gap but at the cost of analytic tractability. They can be especially difficult to implement for large portfolios.The exogenous approaches (see Li (2000), Laurent and Gregory (2005), and White (2004), Brigo et al. (2011), etc) attempt to link marginal default probability distributions to the joint default probability distribution through some external functions. Due to their simplicity in use, the exogenous approaches become very popular in practice.Collateralization is one of the most important and widespread credit risk mitigation techniques used in derivatives transactions. According the ISDA (2012), 71% of all OTC derivatives transactions are subject to collateral agreements. The use of collateral in the financial markets has increased sharply over the past decade, yet the research on collateralized valuation is relatively sparse. Previous studies seem to turn away from direct and detailed modeling of collateralization (see Fuijii and Takahahsi (2012)). For example, Johannes and Sundaresan (2007), and Fuijii and Takahahsi (2012) characterize collateralization via a cost-of-collateral instantaneous rate (or stochastic dividend or convenience yield). Piterbarg (2010) regards collateral as a regular asset in a portfolio and uses the replication approach to price collateralized contracts.This paper presents a new framework for valuing defaultable financial instruments with or without collateral arrangements. The framework characterizes default dependencies exogenously, and models collateral processes directly based on the fundamental principals of collateral agreements. Some well-known risky valuation models in the markets, e.g., the CDS model, the risky interest rate swap (IRS) model (Duffie and Huang (1996)), can be viewed as special cases of this framework, when the default dependencies are ignored.IRSs and CDSs are two of the largest segments of the OTC derivatives market, collectively accounting for around two-thirds of both the notional amount and market value of all outstanding derivatives. Given this framework, we are able to analyze the value of IRSs with bilateral credit risk and look at how swap rates are affected by correlated default risk. Our study shows that counterparty default correlations have a relatively small impact on swap rates. Furthermore, we find that the value of a fully collateralized IRS is equal to the risk-free value. This conclusion is consistent with the current market practice in which market participants commonly assume fully collateralized swaps are risk-free.We also study the value of CDS contracts with trilateral credit risk and assess how spreads depend on the risk of the buyer, seller, and reference entity in a CDS contract. In general, a CDS contract is used to transfer the credit risk of a reference entity from one party to another. The risk circularity that transfers one type of risk (reference credit risk) into another (counterparty credit risk) within the CDS market is a concern for financial stability. Some people claim that the CDS market has increased financial contagion or even propose an outright ban on these instruments.The standard CDS pricing model in the market assumes that there is no counterparty risk. Although this oversimplified model may be accepted in normal market conditions, its reliability in times of distress has recently been questioned. In fact, counterparty risk has become one of the most dangerous threats to the CDS market. For some time now it has been realized that, in order to value a CDS properly, counterparty effects have to be taken into account (see ECB (2009)).We bring the concept of comvariance into the area of credit risk modeling to capture the statistical relationship among three or more random variables. Comvariance was first introduced to economics by Deardorff (1982), who used this measurement to correlate three factors in international trading. Furthermore, we define a new statistics,comrelation , as a scaled version of comvariance. Accounting for default correlations and comrelations becomes important in determining CDS premia, especially during the credit crisis. Our analysis shows that the effect of default dependencies on CDS premia from large to small is i) the default correlation between the protection seller and the reference entity, ii) the default comrelation, iii) the default correlation between the protection buyer and the reference entity, and iv) the default correlation between the protection buyer and the protection seller. In particular, we find that the default comvariance/comrelation has substantial effects on the asset pricing and risk management, which have never been documented.There is a significant increase in the use of collateral for CDS after the recent financial crises. Many people believe that, if a CDS is fully collateralized, there is no risk of failure to pay. Collateral posting regimes are originally designed and utilized for bilateral risk products, e.g., IRS, but there are many reasons to be concerned about the success of collateral posting in offsetting the risk of CDS contracts. First, the value of CDS contracts tends to move very suddenly with big jumps, whereas the price movement of IRS contracts is far smoother and less volatile than CDS . Second, CDS spreads can widen very rapidly. Third, CDS contracts have many more risk factors than IRS contracts. In fact, our model shows that full collateralization cannot eliminate counterparty risk completely for a CDS.This article also shows that the pricing process of a defaultable instrument normally has a backward recursive nature if the payoff can be positive or negative. Accordingly, we propose a backward induction approach for risky valuation. In contrast to the popular recursive integral solution (see Duffie and Huang (1996)), our backward induction method significantly simplifies the implementation. One can make use of the well-established algorithms, such as lattice/tree and regression-based , to price a defaultable instrument.The rest of this paper is organized as follows: Pricing bilateral defaultable instruments is elaborated on in Section 2; valuing multilateral defaultable instruments is discussed in Section 3; the conclusions are presented in Section 4. All proofs and some detailed derivations are contained in the appendices.Pricing Financial Instruments Subject to Bilateral Credit RiskWe consider a filtered probability space (,, , ) satisfying the usual conditions, where denotes a sample space, denotes a -algebra, denotes a probability measure, and denotes a filtration.In the reduced-form approach, the stopping (or default) time of firmi is modeled as a Cox arrival process (also known as a doubly stochastic Poisson process) whose first jump occurs at default and is defined by,(1)where or denotes the stochastic hazard rate or arrival intensity dependent on an exogenous common state , and is a unit exponential random variable independent of .Dependence between the default times is only introduced by the dependence of the intensity on a common process . Consequently, conditional on the path of , defaults are independent, which is the reason why this setup is also often called the conditional independence setup.It is well-known that the survival probability from time t tos in this framework is defined by(2a)The default probability for the period (t, s ) in this framework is given by(2b)Three different recovery models exist in the literature. The default payoff is either i) a fraction of par (Madan and Unal (1998)), ii) a fraction of an equivalent default-free bond (Jarrow and Turnbull (1995)), or iii) a fraction of market value (Duffie and Singleton (1999)). The whole course of the recovery proceedings under the Bankruptcy and Insolvency act is a complex process that typically involves extensive negotiation and litigation. No model can fully capture all aspects of this process so, in practice, all models involve trade-offs between different perspectives and views. In general, the choice for a certain recovery assumption is based on the legal structure of an instrument to be priced. For example, the recovery of market value (RMV) assumption is well matched to the legal structure of an IRS contract where, upon default close-out, valuation will in many circumstances reflect the replacement cost of the transaction, whereas the best default recovery assumption for a CDS is that the claim made in the event of the reference default equals a fraction of the face value of the underlying bond.There is ample evidence that corporate defaults are correlated. The default of a firm’s counterparty might affect its own default probability. Thus, default correlation/dependence arises due to the counterparty relations.Two counterparties are denoted as A and B . The binomial default rule considers only two possible states: default or survival. Therefore, the default indicator for party j (j=A, B ) follows a Bernoulli distribution, which takes value 1 with default probability , and value 0 with survival probability , i.e., and . The marginal default distributions can be determined by the reduced-form models. The joint distributions of a multivariate Bernoulli variable can be easily obtained via the marginal distributions by introducing extra correlations.Consider a pair of random variables (,) that has a bivariate Bernoulli distribution. The joint probability representations are given by(3a)(3b)(3c)(3d)where ,, and where denotes the default correlation coefficient, and denotes the default covariance.A critical ingredient of the pricing of a bilateral defaultable instrument is the default settlement rules. There are two rules in the market. The one-way payment rule was specified by the early International Swap Dealers Association (ISDA) master agreement. The non-defaulting party is not obligated to compensate the defaulting party if the remaining market value of the instrument is positive for the defaulting party. The two-way payment rule is based on current ISDA documentation. The non-defaulting party will pay the full market value of the instrument to the defaulting party if the contract has positive value to the defaulting party.1.1 Risky valuation without collateralizationConsider a defaultable instrument that promises to pay a from partyB to party A at maturity date T , and nothing before date T . The payoff may be positive or negative, i.e. the instrument may be either an asset or a liability to each party. All calculations are from the perspective of party A .We divide the time period (t, T ) into n very small time intervals () and use the approximation provided that y is very small. The survival and the default probabilities for the period (t , ) are given by(4a)(4b)Suppose that the value of the instrument at time is that can be an asset or a liability. There are a total of four () possible states shown in Table 1.The risky value of the instrument at time t is the discounted expectation of all the payoffs and is given by(5a)where(5b)(5c)(5d)where is an indicator function that is equal to one if Y is true and zero otherwise, is the expectation conditional on the , is the risk-free short rate, and is the recovery rate.The pricing equation above keeps terms of order . All higher order terms of are omitted. Similarly, we have(6)Note that is -measurable. By definition, an -measurable random variable is a random variable whose value is known at time . Based on thetaking out what is known and towe r properties of conditional expectation, we have(7)By recursively deriving from t forward over T where and taking the limit as approaches zero, we obtain(8)We may think of as the bilateral risk-adjusted discount factor and as the bilateral risk-adjusted short rate. Equation (8) has a general form that applies in a particular situation where we assume that partiesA and B have independent default risks, i.e. and . Thus, we have:(9a)where(9b)(9c)(9d)
ABSTRACTThis paper attempts to assess the economic significance and implications of collateralization in different financial markets, which is essentially a matter of theoretical justification and empirical verification. We present a comprehensive theoretical framework that allows for collateralization adhering to bankruptcy laws. As such, the model can back out differences in asset prices due to collateralized counterparty risk. This framework is very useful for pricing outstanding defaultable financial contracts. By using a unique data set, we are able to achieve a clean decomposition of prices into their credit risk factors. We find empirical evidence that counterparty risk is not overly important in credit-related spreads. Only the joint effects of collateralization and credit risk can sufficiently explain unsecured credit costs. This finding suggests that failure to properly account for collateralization may result in significant mispricing of financial contracts. We also analyze the difference between cleared and OTC markets.Key words : unilateral/bilateral collateralization, partial/full/over collateralization, asset pricing, plumbing of the financial system, swap premium spread, OTC/cleared/listed financial markets.JEL Classification: E44, G21, G12, G24, G32, G33, G18, G28IntroductionCollateralization is an essential element in the so-called plumbing of the financial system that is the Achilles’ heel of global financial markets. It allows financial institutions to reduce economic capital and credit risk, free up lines of credit, and expand the range of counterparties. All of these factors contribute to the growth of financial markets. The benefits are broadly acknowledged and affect dealers and end users, as well as the financial system generally.The reason collateralization of financial derivatives and repos has become one of the most important and widespread credit risk mitigation techniques is that the Bankruptcy Code contains a series of “safe harbor” provisions to exempt these contracts from the “automatic stay”. The automatic stay prohibits the creditors from undertaking any act that threatens the debtor’s asset, while the safe harbor, a luxury, permits the creditors to terminate derivative and repo contracts with the debtor in bankruptcy and to seize the underlying collateral. This paper focuses on safe harbor contracts (e.g., derivatives and repos), but many of the points made are equally applicable to automatic stay contracts.Financial derivatives can be categorized into three types. The first category is over-the-counter (OTC) derivatives, which are customized bilateral agreements. The second group is cleared derivatives, which are negotiated bilaterally but booked with a clearinghouse. Finally, the third type is exchange-traded/listed derivatives, which are executed over an exchange. The differences between the three types are described in detail on the International Swap Dealers Association (ISDA) website (see ISDA (2013)).Under the new regulations (e.g., Dodd-Frank Wall Street Reform Act), certain ‘eligible’ OTC derivatives must be cleared with central counterparties (CCPs) (see Heller and Vause (2012), Pirrong (2011), etc.). (2011) further recommends mandatory CCP clearing of all OTC derivatives. Meanwhile, Duffie and Zhu (2011) suggest a move toward the joint clearing of interest rate swaps and credit default swaps (CDS) in the same clearinghouse.The posting of collateral is regulated by the Credit Support Annex (CSA). The CSA was originally designed for OTC derivatives, but more recently has been updated for cleared/listed derivatives. For this reason people in the financial industry often refer to collateralized contracts as CSA contracts and non-collateralized contracts as non-CSA contracts.There are three types of collateralization: full, partial or over. Full-collateralization is a process where the posting of collateral is equal to the current mark-to-market (MTM) value. Partial/under-collateralization is a process where the posting of collateral is less than the current MTM value. Over-collateralization is a process where the posting of collateral is greater than the current MTM value.From the perspective of collateral obligations, collateral arrangements can be unilateral or bilateral. In a unilateral arrangement, only one predefined counterparty has the right to call for collateral. Unilateral agreements are generally used when the other counterparty is much less creditworthy. In a bilateral arrangement, on the other hand, both counterparties have the right to call for collateral.Upon default and early termination, the values due under the ISDA Master Agreement are determined. These amounts are then netted and a single net payment is made. All of the collateral on hand would be available to satisfy this total amount, up to the full value of that collateral. In other words, the collateral to be posted is calculated on the basis of the aggregated value of the portfolio, but not on the basis of any individual transaction.The use of collateral in financial markets has increased sharply over the past decade, yet analytical and empirical research on collateralization is relatively sparse. The effect of collateralization on financial contracts is an understudied area. Collateral management is often carried out in an ad-hoc manner, without reference to an analytical framework. Comparatively little research has been done to analytically and empirically assess the economic significance and implications of collateralization. Such a quantitative and empirical analysis is the primary contribution of this paper.Due to the complexity of quantifying collateralization, previous studies seem to turn away from direct and detailed modeling of collateralization (see Fuijii and Takahahsi (2012)). For example, Johannes and Sundaresan (2007), and Fuijii and Takahahsi (2012) characterize collateralization via a cost-of-collateral instantaneous rate (or stochastic dividend or convenience yield). Piterbarg (2010) regards collateral as a regular asset in a portfolio and uses the replication approach to price collateralized contracts. All of the previous works focus on full-collateralization only.We obtain the CSA data from two investment banks. The data show that only 8.92% of CSA counterparties are subject to unilateral collateralization, while the remaining 91.08% are bilaterally collateralized. The data also reveal that all contracts in OTC markets are partially collateralized due to the mechanics of CSAs, which allow for the existence of limited unsecured exposures and set minimum transfer amounts (MTAs), whereas all contracts in cleared/listed markets are over-collateralized as all CCPs/Exchanges require initial margins. Therefore, full-collateralization does not exist in the real world11Singh (2010) and ECB (2009) come to a similar conclusion, although they do not provide any data to justify their statements.. The reason for the popularity of full-collateralization is its mathematical simplicity and tractability.This article makes a theoretical and empirical contribution to the study of collateralization by addressing several essential questions concerning the posting of collateral. First, how does collateralization affect expected asset prices? To answer this question, we develop a comprehensive analytical framework for pricing financial contracts under different (partial/full/over and unilateral /bilateral) collateral arrangements in different (OTC/cleared/listed) markets.In contrast to other collateralization models in current literature, we characterize a collateral process directly based on the fundamental principal and legal structure of the CSA agreement. A model is devised that allows for collateralization adhering to bankruptcy laws. As such, the model can back out differences in prices due to counterparty risk. This framework is very useful for valuing off-the-run or outstanding financial contracts subject to credit risk and collateralization, where the price quotes are not available. Given this model, we are able to explain credit-related spreads and provide an important tool for credit value adjustment (CVA).Our theoretical analysis shows that collateralization can always improve recovery and reduce credit risk. If a contract is over-collateralized (e.g., a repo or cleared contract), its value is equal to the risk-free value. If a contract is partially collateralized (e.g., an OTC derivatives), its CSA value is less than the risk-free value but greater than the non-CSA risky value.Second, how can the model be empirically verified? To achieve the verification goal, this paper empirically measures the effect of collateralization on pricing and compares it with model-implied prices. This calls for data on financial contracts that have different collateral arrangements but are similar otherwise. We use a unique interest rate swap contract data set from an investment bank for the empirical study, as interest rate swaps collectively account for around two-thirds of both the notional and market value of all outstanding derivatives.ISDA mid-market swap rates quoted in the market are based on hypothetical counterparties of AA-rated quality or better. Dealers use this market rate as a reference when quoting an actual swap rate to a client and make adjustments based on many factors, such as credit risk, liquidity risk, funding cost, operational cost and expected profit, etc. Unlike most other studies, this study mainly concentrates the analysis on swap adjustments/premia related to credit risk and collateralization, which are to be made to the mid-market swap rates for real counterparties.Prior research has primarily focused on the generic mid-market swap rates and results appear puzzling. Sorensen and Bollier (1994) believe that swap spreads (i.e., the difference between swap rates and par yields on similar maturity Treasuries) are partially determined by counterparty default risk. Whereas Duffie and Huang (1996), Hentschel and Smith (1997), Minton (1997) and Grinblatt (2001) find weak or no evidence of the impact of counterparty credit risk on swap spreads. Collin-Dufresne and Solnik (2001) and He (2001) further argue that many credit enhancement devices, e.g., collateralization, have essentially rendered swap contracts risk-free. Meanwhile, Duffie and Singleton (1997), and Liu, Longstaff and Mandell (2006) conclude that both credit and liquidity risks have an impact on swap spreads. Moreover, Feldhütter and Lando (2008) find that the liquidity factor is the largest component of swap spreads. It seems that there is no clear-cut answer yet regarding the relative contribution of the liquidity and credit factors. Maybe, the recently revealed LIBOR scandal can partially explain these conflicting findings.Unlike the generic mid-market swap rates, swap premia are determined in a competitive market according to the basic principles of supply and demand. A client who wants to enter a swap contract first contacts a number of swap dealers and asks for a swap rate. After comparing all quotations, the client chooses the most competitive rate. A swap premium is supposed to cover operational, liquidity, funding, and credit costs as well as a profit margin. If the premium is too low, the dealer may lose money. If the premium is too high, the dealer may lose the competitive advantage.Unfortunately, we do not know the detailed allocation of a swap premium, i.e., what percentage of the adjustment is charged for each factor. Thus, a direct empirical verification is impossible.To circumvent this difficulty, this article uses an indirect process to verify the model empirically. We define a swap premium spread as the difference between the swap premia of two collateralized swap contracts that have exactly the same terms and conditions but are traded with different counterparties under different collateral agreements. We reasonably believe that if two contracts are identical except counterparties, the premium spread should reflect the difference between two counterparties’ unsecured credit risks only, as all other risks and costs are identical.Empirically, we find quite a number of CSA swap pairs in the data, where the two contracts in each pair have different counterparties but are otherwise the same. The test results demonstrate that the model-implied swap premium spreads are very close to the market swap premium spreads, indicating that the model is quite accurate.Third, what are the effects of counterparty risk and collateralization, alone or combined, on swap premium spreads? We first study the marginal impact of counterparty risk on spreads. CDS premia (the prices of insuring against a firm’s default) theoretically reflect the credit risk of the firm. Presumably, differences in CDS premia should be largely attributable to differences in counterparty risk. We estimate a regression model where market swap premium spreads are used as the dependent variable and differences between CDS premia as the explanatory variable. The estimation results show that the adjusted is 0.7472, implying that approximately 75% of market premium spreads can be explained by CDS spreads. In other words, counterparty risk alone can provide a good but not overwhelming prediction on spreads.Next, we assess the joint effect of both counterparty risk and collateralization on premium spreads. Since the model-generated swap premium spreads take into account both counterparty risk and collateralization, we present another regression model where the market swap premium spreads are regressed on the implied swap premium spreads. The estimation results show that the constant term is insignificantly different from zero; the slope coefficient is close to 1 and the adjusted is very high. This suggests that the implied premium spreads explains nearly all of the market premium spreads.Finally, how does collateralization impact pricing in different markets? Our proprietary data reveal that cleared swaps have dramatically increased since 2011, reflecting the financial institutions’ compliance to regulatory requirements. We find evidence that the economical determination of swap rates in cleared markets is the same as that in OTC markets, as all clearinghouses claim that cleared derivatives would replicate OTC derivatives, and promise that transactions through the clearinghouses would be economically equivalent to similar transactions handled in OTC markets.Although the practice recommended by CCPs is popular in the market, in which derivatives are continuously negotiated over-the-counter as usual but cleared and settled through clearinghouses, some market participants cast doubt on CCPs’ economic equivalence claim. They find that cleared contracts have actually significant differences when compared with OTC trades. Some firms even file legal action against the clearinghouses, and accuse them of fraudulently inducing the firms to enter into cleared derivatives on the premise the contracts would be economically equivalent to OTC contracts (see Pengelly (2011)).In fact, our study shows that there are many differences between cleared markets and OTC markets. After much discussion, we come to the conclusion that cleared derivatives are not economically equivalent to their OTC counterparts. These discussions may be of interest to regulators, academics and practitioners.The remainder of this paper is organized as follows: Section 2 discusses unilateral collateralization. Section 3 elaborates bilateral collateralization. Section 4 presents empirical evidence. The conclusions and discussion are provided in Section 5. All proofs and some detailed derivations are contained in the appendices.Unilateral CollateralizationA unilateral collateral arrangement is sometimes used when a higher-rated counterparty deals with a lower-rated counterparty, in which only one party, normally the lower-rated one, is required to deliver collateral to guarantee performance under the agreement. Typical examples of unilaterally collateralized contracts include: repos, cleared/listed derivatives, sovereign derivatives (see AFME-ICMA-ISDA (2011)), and some OTC derivatives22Our CSA data from two investment banks show that 8.92% of CSA counterparties are subject to unilateral collateralization.We consider a filtered probability space (,, , ) satisfying the usual conditions, where denotes a sample space, denotes a -algebra, denotes a probability measure, and denotes a filtration.Since the only reason for taking collateral is to reduce/eliminate credit risk, collateralization analysis is closely related to credit risk modeling. There are two primary types of models that attempt to describe default processes in the literature: structural models and reduced-form models. Many people in the market have tended to gravitate toward the reduced-from models given their mathematical tractability and market consistency. In the reduced-form framework, the stopping (or default) time of a firm is modeled as a Cox arrival process (also known as a doubly stochastic Poisson process) whose first jump occurs at default and is defined by,(1)where or denotes the stochastic hazard rate or arrival intensity dependent on an exogenous common state , and is a unit exponential random variable independent of .It is well-known that the survival probability from time t tos in this framework is defined by(2a)The default probability for the period (t, s ) in this framework is given by(2b)In order to assess the impact of collateralization on pricing, we study valuation with and without collateralization respectively.Valuation without collateralizationLet valuation date be t . Consider a financial contract that promises to pay a from party B to party A at maturity dateT , and nothing before date T . We suppose that partyA and party B do not have a CSA agreement. All calculations are from the perspective of party A. The risk free value of the financial contract is given by(3a)where(3b)where is the expectation conditional on the , denotes the risk-free discount factor at time t for the maturity T and denotes the risk-free short rate at time u ().Next, we discuss risky valuation. In a unilateral credit risk case, we assume that party A is default-free and party B is defaultable. We divide the time period (t, T ) into n very small time intervals (). In our derivation, we use the approximation provided that y is very small. The survival and default probabilities of party B for the period (t , ) are given by(4a)(4b)The binomial default rule considers only two possible states: default or survival. For the one-period (t, ) economy, at time the contract either defaults with the default probability or survives with the survival probability . The survival payoff is equal to the market value and the default payoff is a fraction of the market value33Here we use the recovery of market value (RMV) assumption.: , where is the recovery rate. The non-CSA value of the contract at time t is the discounted expectation of all possible payoffs and is given by(5)where denotes the (short) risky rate and denotes the (short) credit spread.Similarly, we have(6)Note that is -measurable. By definition, an -measurable random variable is a random variable whose value is known at time . Based on thetaking out what is known and towe r properties of conditional expectation, we have(7)By recursively deriving from t forward over T where and taking the limit as approaches zero, the non-CSA value of the contract can be obtained as(8)We may think of as the risk-adjusted short rate. Equation (8) is the same as Equation (10) in Duffie and Singleton (1999), which is the market model for pricing risky bonds.In theory, a default may happen at any time, i.e., a risky contract is continuously defaultable. This Continuous Time Risky Valuation Model is accurate but sometimes complex and expensive. For simplicity, people sometimes prefer the Discrete Time Risky Valuation Model that assumes that a default may only happen at some discrete times. A natural selection is to assume that a default may occur only on the payment dates. Fortunately, the level of accuracy for this discrete approximation is well inside the typical bid-ask spread for most applications (see O’Kane and Turnbull (2003)). From now on, we will focus on the discrete setting only, but many of the points we make are equally applicable to the continuous setting.If we assume that a default may occur only on the payment date, the non-CSA value of the instrument in the discrete-time setting is given by(9)where can be regarded as a risk-adjusted discount factor.The difference between the risk-free value and the risky value is known as the credit value adjustment (CVA). CVA is required by regulators, such as, International Accounting Standard Board (IASB) and committee. The CVA reflects the market value of counterparty risk or the cost of protection required to hedge counterparty risk and is given by(10)Since the recovery rate is always less than 1, we have or . In other words, the risky value is always less than the risk-free value. An intuitive explanation is that credit risk makes a financial contract less valuable.Valuation with collateralizationThe posting of collateral is regulated by the CSA that specifies a variety of terms including the threshold, the independent amount, and the minimum transfer amount (MTA), etc. The threshold is the unsecured credit exposure that a party is willing to bear. The MTA is used to avoid the workload associated with a frequent transfer of insignificant amounts of collateral. The independent amount plays the same role as the initial margin (or haircut).We define effective collateral threshold as the threshold plus the MTA. The collateral is called as soon as the mark-to-market (MTM) value rises above the effective threshold. A positive effective threshold corresponds to partial/under-collateralization where the posting of collateral is less than the MTM value. A negative effective threshold represents over-collateralization where the posting of collateral is greater than the MTM value. A zero-value effective threshold equates with full-collateralization where the posting of collateral is equal to the MTM value.Suppose that there is a unilateral CSA agreement between partiesA and B in which only party B is required to deliver collateral when the mark-to-market (MTM) value arises over the collateral threshold H .The choice of modeling assumptions for collateralization should be based on the legal structure of collateral agreements. According to the Bankruptcy Law, if the collateral value is greater than the default claim, creditors can only have a claim on the collateral up to the full amount of their default demand. Any excess collateral is returned to the estate of the failed institution for the payment of unsecured creditors. If the demand for default payment exceeds the collateral value, the balance of the demand will be treated as an unsecured claim and subject to its pro rate distribution under the Bankruptcy Code’s priority scheme (see Garlson (1992), Routh and Douglas (2005), and Edwards and Morrison (2005)). The default payment under a collateral agreement, therefore, can be mathematically expressed as(11a)or(11b)where is an indicator function that is equal to one if Y is true and zero otherwise, and C(T) is the collateral amount at timeT .It is worth noting that the default payment in equation (11) is always greater than the original recovery, i.e., , since is always less than 1. Said differently, the default payoff of a CSA contract is always greater than the default payoff of the same contract without a CSA agreement. That is why the major benefit of collateralization should be viewed as an improved recovery in the event of a default.Let us consider repo/cleared/listed markets first. Contracts in these markets are always over collateralized, as the parties with collateral obligation are required to deposit initial margins and are also charged variation margins in response to changes in the market values. The total collateral (initial margin plus variation margin) posted at timet is given by(12)where is the CSA value of the contract at time t and is the effective threshold. Note that for over-collateralization, the effective threshold is negative, i.e., , which equals the negative initial margin. The collateral in equation (12) is a linear function of the asset value.In general, initial margins are set very conservatively so that they are sufficient to cover losses under all scenarios considered. Also, the initial margins can be adjusted in response to elevated price volatility. Moreover, daily marking-to-market and variation margin settlement can further eliminate the risk that a loss exceeds the collateral amount. Thus, it is reasonable to believe that under over-collateralization the collateral amount is always greater than the default claim, i.e., .At time T , if the contract survives with probability , the survival value is the promised payoff and the collateral taker returns the collateral to the collateral provider. If the contract defaults with probability , the collateral taker has recourse to the collateral and obtains a default payment up to the full value of the promised payoff . The remaining collateral returns to the collateral provider. The CSA value of the over collateralized contract is the discounted expectation of the payoffs and is given by(13)Equation (13) tells us that the CSA value of an over-collateralized contract is equal to the risk-free value . This result is consistent with the market practice in which market participants commonly assume that repos and cleared contracts are virtually free of default risk because of the implicit guarantee of the contracts provided by the clearinghouse and backup collateral.It is worth keeping in mind that clearing does not eliminate any risk. It has no effect on the counterparty’s default probability and does not improve the counterparty’s credit rating. Instead, it uses some mitigation tools, e.g., collateralization, to perfectly hedge the credit risk, making a contract appear to be risk-free.Next, we turn to OTC markets. We obtain the CSA data from two investment banks. The data show that 61.32% of CSA counterparties have a zero threshold, and the remaining 38.68% use a positive threshold ranging from 25,000 to 750,000,000. Moreover, all CSA counterparties in the data maintain a positive MTA ranging from 500 to 60,000,000, which means that the effective thresholds are always greater than zero, i.e., . If the value of the contract is less than the effective threshold , no collateral is posted; otherwise, the required collateral is equal to the difference between the contract value and the effective threshold. The collateral amount posted at time t can be expressed mathematically as(14a)or(14b)In contrast to repo/cleared markets, collateral posted in OTC markets is a nonlinear function of daily market value changes. In fact, this discontinuous and state-dependent indicator function is the root cause of the complexity of collateralized valuation in OTC markets.Since all CSA derivatives in OTC markets are partially collateralized, the default claim is almost certainly greater than the collateral amount. For a discrete one-period (t, T ) economy, the collateral amount posted at time t is defined in (14). At time T , if the contract survives, the survival value is the promised payoff and the collateral taker returns the collateral to the collateral provider. If the contract defaults, the collateral taker possesses the collateral. The portion of the default claim that exceeds the collateral value is treated as an unsecured claim. Thus, the default payment is , where is the future value of the collateral. Since the most predominant form of collateral is cash according to ISDA (2012), it is reasonable to consider the time value of money only for collateral assets. The large use of cash means that collateral is both liquid and not subject to large fluctuations in value. It can be seen from this, that collateral does not have any bearing on survival payoffs; instead, it takes effect on default payments only. The CSA value of the partially collateralized contract is the discounted expectation of all the payoffs and is given by(15)Suppose that default probabilities are uncorrelated with interest rates and payoffs44Moody’s Investor’s Service (2000) presents statistics that suggest that the correlations between interest rates, default probabilities and recovery rates are very small and provides a reasonable comfort level for the uncorrelated assumption.. We have the following proposition after some simple mathematics.
ABSTRACTThis paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.Key Words : jump diffusion, convertible bond, convertible underpricing, convertible arbitrage, default time approach, default probability approach, asset pricing and credit risk modeling.