Reduced Form Credit Risk Model Research Articles

Pricing of Credit Risk Derivatives with Stochastic Interest Rate

This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that the stochastic interest rate follows a jump-diffusion Cox–Ingersoll–Ross process. First, we derive the joint Laplace transform of the distribution of the vector process (rt,Rt) by applying piecewise deterministic Markov process theory and martingale theory. Then, using the joint Laplace transform, we obtain the explicit pricing of defaultable bonds and a credit default swap. Lastly, numerical examples are presented to illustrate the dynamic relationships between defaultable securities (defaultable bonds, credit default swap) and the maturity date.

A Practical Guide to the Valuation of Coupon-Bearing Fixed Income Securities

The purpose of this article is to illustrate the use of reduced-form credit risk models for valuing defaultable coupon-bearing securities, such as risky sovereign bonds, corporate bonds, and retail loans like auto loans and mortgage loans. We focus on issues in the application of these models and not the mathematical derivations of the formulas employed.

A Bottom-up, Reduced Form Credit Risk Model Approach for the Determination of Collateralized Loan Obligation Capital

A Bottom-up, Reduced Form Credit Risk Model Approach for the Determination of Collateralized Loan Obligation Capital

An Exact Structural Model for Evaluating Credit Default Swaps: Theory and Empirical Evidence

Using structural models for credit default swaps has been difficult. Existing models all adopt shortcuts as approximations. In this article, we provide an accurate and efficient solution to the price of the credit default swap. The main result is a Theorem in section 2. In an empirical study, we show how our model can properly capture credit default swap exposure to interest rate volatility and asset volatility. Furthermore, we apply the new model to study (1) the interactions among market, credit, and interest risks; (2) the consistency with the reduced-form credit risk models; and (3) implications to capital structure arbitrage.

Fair Microfinance Loan Rates

AbstractThis paper uses a reduced form credit risk model to determine fair lending rates for microfinance loans. A fair lending rate is defined to be that rate which makes the loan have a net present value of zero to the lender, after the inclusion of the lender's costs of issuing and monitoring the loan.

Fair Microfinance Loan Rates

This paper uses a reduced form credit risk model to determine fair lending rates for micro-finance loans. A fair lending rate is defined to be that rate which makes the loan have a net present value of zero to the lender, after the inclusion of the lender’s costs of issuing and monitoring the loan.

CMBS market efficiency: The crisis and the recovery

This paper presents a reduced form credit risk model to study CMBS pricing and CMBS market efficiency during and after the credit crisis with a comprehensive loan, bond and deal level data set. Using a model determined fair value, an automated trading strategy based on a newly determined risk ratio buys undervalued and sells overvalued CMBS. These strategies result in substantial trading profits between November 2007 and June 2015. Controlling for CMBS sector risk factors, we reject CMBS market efficiency over the entire sample period. When we split the sample into the Crisis and Recovery periods, we observe persistent abnormal returns over both subperiods, which is consistent with an inefficient CMBS market. Because the CMBS market appears to be inefficient, our results suggest that the approach presented in this paper may facilitate the increased financial stability of the CRE sector through the better pricing and risk management of CMBS.

Credit Risk and LGD Modelling

Abstract This paper deals with the methods for estimating credit risk parameters from market prices, e.g. Probability of Default (PD) and Loss Given Default (LGD). Precise evaluation of these parameters is important not only for bank to calculate their regulatory capital but also for investors to price risky bonds and credit derivatives. In this paper, we introduced reduced-form analytical methods for the calculation of LGD to pricing Credit Default Swaps. Reduced-form credit risk models were introduced as a reaction to structural approach, especially trying to decrease informational difficulty when modelling credit risk. In the reduced-form approach, the market value of defaulted bonds is the same as in the fraction recovered from the exposure at default. We use the face value convention, which Hull & White (2000) presented in their model which extended recovery of face value convention for coupon bonds.

A Markov Chain Copula Model for Credit Default Swaps with Bilateral Counterparty Risk

Reduced-form credit risk models are widely used in pricing and hedging credit derivatives. Generating default dependency is the key element in any such model. In this article, we use Markov copulae approach to model the dependence structure of defaults between the three obligors, one is the reference entity, another is the protection seller, the other is the protection buyer(the investor), so we can consider the bilateral counterparty risk of credit default swaps(CDS). In this Markov chain copula model, we obtain the explicit formulas of the CDS premium rates C 1(T) (with unilateral counterparty risk) and C 2(T) (with bilateral counterparty risk). And then we perform some numerical experiments to analyze the difference of the fair spreads between the unilateral case and the bilateral case.

A Model for Dependent Defaults and Pricing Contingent Claims with Counterparty Risk

This paper presents a new, intuitive but mathematically powerful model of dependent defaults and derives a general framework for pricing products whose values depend on credit correlation between the counterparty and the reference entity. The dependence framework is a natural extension of the Gaussian factor approach, which can be applied in the context of reduced form credit risk models, allowing i.a. for stochastic hazard and recovery rates. The prices of plain vanilla credit default swaps, first-to-default swaps and default swaptions are derived as particular examples.

Modeling Credit Risk with Partial Information

This paper provides an alternative approach to Duffie and Lando (2001) for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager's information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the manager's information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azema martingale, and we use excursion theory of Brownian motions to price risky debt.

On a reduced form credit risk model with common shock and regime switching

Reduced form credit risk models are important ones in credit risk theory. In such a model, certain correlated relations are constructed to represent the default dependence structure among the default intensity processes. In this paper, we introduced a reduced form credit risk model in which the default dependence structures among default intensity processes are described by the so-called common shocks with regime-switching. We derive some closed-form expressions for the joint distribution of the default times and for the pricing formulas of the basket default swaps. We also give numerical results to show the applicable aspects of the proposed model.

Subprime mortgage default

This paper constructs a reduced-form credit risk model of mortgage default. The data used is of privately-securitized subprime ARMs (adjustable rate mortgages), originated between 1997 and 2008, and observed between 2000 and 2009. The period studied thus encompasses the beginning of the subprime crisis. Given the estimated model, contractual properties of the loans are then used to infer the market price of default risk for the various quarters of origination. It is empirically determined that a change in the inherent nature of borrowers led to a deterioration in their default performance, a change which can be first detected in late 2004. On the other hand, the evidence also indicates that the secondary mortgage market became aware of this change at about this same time. The large rise in defaults in 2007 cannot, therefore, be attributed to any surprise other than the unexpectedly large fall in housing prices.

Credit market equilibrium theory and evidence: Revisiting the structural versus reduced form credit risk model debate

There are two competing paradigms for modeling credit risk: the structural and reduced form models. This paper applies our knowledge of credit market equilibrium to this debate. We show that credit markets have asymmetric information in the borrowing and lending relationship which influence equilibrium prices. Reduced form models are consistent with asymmetric equilibrium models, but structural models are not. This implies that structural models should not be used for pricing, hedging, or risk management.

CREDIT RISK AND INCOMPLETE INFORMATION: FILTERING AND EM PARAMETER ESTIMATION

We consider a reduced-form credit risk model where default intensities and interest rate are functions of a not fully observable Markovian factor process, thereby introducing an information-driven default contagion effect among defaults of different issuers. We determine arbitrage-free prices of OTC products coherently with information from the financial market, in particular yields and credit spreads and this can be accomplished via a filtering approach coupled with an EM-algorithm for parameter estimation.

Subprime Mortgage Default

Using data on privately-securitized subprime ARMs (adjustable rate mortgages) originated between 1997 and 2008 and observed between 2000 and 2008, and so covering the start of the subprime crisis, this paper constructs a reduced-form credit risk model of default, and then uses contractual properties of the loans to infer the market’s price of default risk at various times of origination. Treating the hazard of default as a process rather than a single realization for each period of origination permits the probability of default to be calculated without knowledge of the still unfolding hazard realization, allowing one to adopt the same position as lenders, who also cannot foretell the future. It is empirically determined that a change in the inherent nature of borrowers caused some deterioration in their default behavior, a change which we can first detect by late 2004, but of which, evidence indicates, the secondary mortgage market became aware at about the same time. The large rise in defaults in 2007 cannot, therefore, be attributed to any surprise other than the unexpectedly large fall in housing prices.

Credit Risk and the Term Structure of Lease Rates: A Reduced Form Approach

Previous research either assumes default free leases or leases subject to default risk using a structural approach. However, structural credit risk models suffer from a common criticism that the firm’s asset value process is unobservable. We develop a reduced form credit risk model for leases that avoids making assumptions regarding unobservable asset valuation processes. Furthermore, we assume a correlated market and credit risk that provides us with a simple analytic formula for valuing defaultable lease contracts. Numerical analysis reveals that tenant credit risk can have a substantial impact on the term structure of leases. Finally, we use the model to demonstrate the implied lease term structure for a set of retail and financial firms in the Fall of 2000.

Time-Varying Credit Risk and Liquidity Premia in Bond and CDS Markets

In this paper, we develop a reduced-form credit risk model that incorporates illiquidity in the bond and the credit default swap (CDS) market. As CDS are derivative contracts, liquidity effects have to be modeled differently than for bonds. In the bond market, illiquidity results in yield premia. In the CDS market, the bid-ask spread constitutes a liquidity signal. This separation has important consequences on the liquidity spill-over between the bond and the CDS market as well as the co-movement of credit risk and liquidity premia. Our most important findings are threefold. First, we find that adding a CDS-specific liquidity component to the model has the important consequence of consistently positive credit risk and liquidity premia in bond markets. Moreover, the size of these premia relative to the yield spread is intuitively plausible. Second, our analysis of the time series of the liquidity premia shows that the bond market's liquidity dries up as default risk increases. For the CDS market, the reaction of the liquidity premia depends on the rating class. The investment grade sector becomes more dominated by protection sellers during times of high default risk while a lower demand pressure decreases liquidity premia in the subinvestment grade sector. Third, the liquidity of the bond market has a direct impact on the liquidity of the CDS market but not vice versa

The delivery option in credit default swaps

Under standard assumptions the reduced-form credit risk model is not capable of accurately pricing the two fundamental credit risk instruments – bonds and credit default swaps (CDS) – simultaneously. Using a data set of euro-denominated corporate bonds and CDS our paper quantifies this mispricing by calibrating such a model to bond data, and subsequently using it to price CDS, resulting in model CDS spreads up to 50% lower on average than observed in the market. An extended model is presented which includes the delivery option implicit in CDS contracts emerging since a basket of bonds is deliverable in default. By using a constant recovery rate standard models assume equal recoveries for all bonds and hence zero value for the delivery option. Contradicting this common assumption, case studies of Chapter 11 filings presented in the paper show that corporate bonds do not necessarily trade at equal levels following default. Our extension models the implied expected recovery rate of the cheapest-to-deliver bond and, applied to data, largely eliminates the mispricing. Calibrated recovery values lie between 8% and 47% for different obligors, exhibiting strong variation among rating classes and industries. A cross-sectional analysis reveals that the implied recovery parameter depends on proxies for the delivery option, primarily the number of available bonds and bond pricing errors. No evidence is found for a direct influence of the bid-ask spread, notional amount, coupon, or rating used as proxies for bond market liquidity.

Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling

We study, in the framework of Back [Rev. Financial Stud. 5(3), 387–409 (1992)], an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm. The market consists of a risk-neutral informed agent, noise traders, and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in the market’s view as in reduced-form credit risk models. However, we show that, in equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider’s trades are inconspicuous.