Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analysed in the general equilibrium model of Cox, Ingersoll, and Ross (1985). This model is extended by means of an additional parameter in order to deal with financial and credit risk simultaneously. The estimation of such a parameter, which can be considered as the market equivalent of an agencies' bond rating, allows to extract from current quotes the market perceptions of firm's credit risk. The general pricing model for defaultable zero-coupon bond is first derived in a simple discrete-time setting and then in continuous-time. The availability of an integrated model allows for the pricing of default-free options written on defaultable bonds and of vulnerable options written either on default-free bonds or defaultable bonds. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.     

Pricing American Put Options on Defaultable Bonds

In the last two decades, the market of credit derivativeshas expanded rapidly, and the importance of pricing problemsfor credit derivatives has been recognized especially in the last decade.Among these securities, the pricing problems of credit derivativeswith an early exercise, such as American put options,have not received enough attention. In view of this need, this paper develops a continuous stochastic modelof American put options on defaultable bonds.The method of obtaining a solution is based on a new result of the optimalstopping problem for a diffusion process with a jump.Some characterizations of American put options are providedusing partial differential equations.     

Valuation Model of Defaultable Bond Values in Emerging Markets

This paper develops a model to value defaultable bonds in emerging markets. Default occurs when some signaling process hits a pre-defined default barrier. The signaling variable is considered to be some macro-economic variables such as foreign exchange rates. The dynamics of the default barrier depend on the volatility and the drift of the signaling variable. We derive a closed-form solution of the defaultable bond price from the model as a function of a signaling variable and a short-term interest rate. The numerical results show that the model values generated by using foreign exchange rates as the signaling variables can broadly track the market credit spreads of defaultable bonds in South Korea and Brazil. Given an expected level of the foreign exchange rate, defaultable bond values under a stressed market situation can be obtained.     

Optimal Portfolios with Credit Default Swaps

Using a continuous-time, stochastic, and dynamic framework, this study derives a closed-form solution for the optimal investment problem for an agent with hyperbolic absolute risk aversion preferences for maximising the expected utility of his or her final wealth. The agent invests in a frictionless, complete market in which a riskless asset, a (defaultable) bond, and a credit default swap written on the bond are listed. The model is calibrated to market data of six European countries and assesses the behaviour of an investor exposed to different levels of sovereign risk. A numerical analysis shows that it is optimal to issue credit default swaps in a larger quantity than that of bonds, which are optimally purchased. This speculative strategy is more aggressive in countries characterised by higher sovereign risk. This result is confirmed when the investor is endowed with a different level of risk aversion. Finally, we solve a static version of the optimisation problem and show that the speculative/hedging strategy is definitely different with respect to the dynamic one.     

A comprehensive structural model for defaultable fixed-income bonds

An exact valuation formula for defaultable corporate coupon bonds is proved. The model incorporates discrete coupons, bankruptcy costs, taxes and the market risk generated by a stochastic risk-free structure. The aim of this paper is twofold: first, we generalise previous pricing models for corporate bonds; second, we provide a comprehensive formula in order to properly disentangle the contribution of several risk factors to credit spreads.     

Arbitrage pricing of defaultable game options with applications to convertible bonds

This paper is the first in a series that we devote to studying the problems of valuation and hedging of defaultable game options in general, and convertible corporate bonds in particular. Here, we present mathematical foundations for our overall study. Specifically, we provide several results characterizing the arbitrage price of a defaultable game option in terms of relevant Dynkin games. In addition, we provide important results regarding price decomposition of defaultable options. These general results are then specified to the case of convertible bonds, yielding in particular a decomposition of convertible bonds in an optional and a bond component.     

Counterparty Risk and the Pricing of Defaultable Securities

Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economy-wide impact, this paper generalizes existing reduced-form models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firm-specific risks that are termed "counterparty risks." Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps.     

Implied Default Probability and Credit Derivatives

Recently many kinds of credit derivatives are traded in the market. The default probability implied in the market becomes important to price some credit derivatives. Also it is useful for managing the credit risk because it includes the market information. In this paper we show how to calculate the implied default probability in the default swap market or the defaultable bond market.This paper is developed from author’s master thesis (Matsumoto, 2000), Graduate School of Systems Management, the University of Tsukuba.     

Asymptotic Expansion for Term Structures of Defaultable Bonds with Non-Gaussian Dependent Innovations

In the context of credit risk, the term structure models that have been studied in the literature are typically models driven by Brownian motion or standard jump diffusions. These models provide coherent modeling that is straightforward to implement. To make these models more flexible, we develop a discrete-time approximation of a continuous-time Vasicek term structure analysis with non-Gaussian and dependent innovations. Higher-order asymptotic theory enables us to evaluate the term structures of defaultable bonds. Numerical examples show that the effects of non-Gaussianity and the dependency of both risk-free rate and default process strongly influence the evaluation of defaultable bonds. As an application, we estimate the parameters of our proposed models for the Japanese corporate credit default swap market.     

Credit rating dynamics in the presence of unknown structural breaks

In many credit risk and pricing applications, credit transition matrix is modeled by a constant transition probability or generator matrix for Markov processes. Based on empirical evidence, we model rating transition processes as piecewise homogeneous Markov chains with unobserved structural breaks. The proposed model provides explicit formulas for the posterior distribution of the time-varying rating transition generator matrices, the probability of structural break at each period and prediction of transition matrices in the presence of possible structural breaks. Estimating the model by credit rating history, we show that the structural break in rating transitions can be captured by the proposed model. We also show that structural breaks in rating dynamics are different for different industries. We then compare the prediction performance of the proposed and time-homogeneous Markov chain models.     

Dynamical analysis of corporate bonds based on the yield spread term-quality surface

Our aim of this research is to propose a model which estimates implied relative credit reliability from the yield spread of defaultable bonds and evaluates their spread risk. We introduce “yield spread term-quality surface” (YSTQS) which is defined on the space of duration and credit reliability of the issuers, and express their yield spread. First, we review the general pricing theorem of defaultable bonds with unpredictable recovery in the no-arbitrage context based on the external hazard rates. Second, we show that the dynamics of state variables determine the shape of the YSTQS, and they drive the YSTQS if the loss-adjusted hazard rates are described by a function of them. Finally, we show an empirical analysis of our model with daily yield spread, duration, and the credit ratings of corporate bonds.

The Credit Risk and Pricing of OTC Options

In the over-the-counter (OTC) markets, the options traded are always subject to credit risk. Therefore the counterparty’s credit risk is a striking factor when pricing options, whereas it is not considered in the classic Black-Scholes models. Based on the first passage time models, this paper develops the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism. The default probability and the pricing formulae of the OTC options are obtained by using partial differential equation (PDE) techniques, especially Green’s function.     

Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives. Nicola Bruti-Liberati: In memory of our beloved friend and colleague.     

On a variational formulation used in credit risk modeling

We consider the credit risk model of Collin-Dufresne and Goldstein (2001). According to this model, the price of a defaultable bond can be efficiently computed using a variational formulation that consists of an integral relation and a Volterra integral equation. In Collin-Dufresne and Goldstein (2001) this integral equation is justified by a probabilistic intuition, but is not proven formally. In this paper we analytically derive the variational formulation used in Collin-Dufresne and Goldstein (2001). This analysis allows to give a correct characterization of the solution of the integral equation. Furthermore the approach proposed in this paper could also be employed for other models of credit risk.     

基于KMV模型的地方政府债券信用风险问题研究

借助于KMV模型的思想建立了地方政府债券信用风险模型,进一步探讨了模型的概率分布形态以提高模型的预测精度;并在确定预测标准后,针对2009年已发行地方政府债券的部分省市计算了各地安全发债规模。研究表明,基于KMV模型的地方政府债券信用风险模型具有很强的应用性和很好的推广前景;实践中,在对发债主体进行科学选择的基础上,通过该模型能实现对发债规模的控制,达到防止其发生信用风险的可能;同时,所选样本省市(除新疆外)的预测安全发债规模和实际发债额是合理的和安全的。因此,应建立一套科学规范的地方政府债券风险防控机制,以保证地方政府债券的健康发展。     

Credit Spreads in Illiquid Markets: Model and Implementation

This paper presents a methodology for estimating a family of credit spread term structures in a market with few transactions. The authors propose partitioning the market into risk classes and modeling credit spread term structures for each risk class using a multifactor Vasicek model with some common and some risk class-specific factors. The approach uses information on the cross section and time series of corporate bonds in all the risk classes to estimate the term structure of credit spreads in each risk class. The model is jointly estimated using an extended Kalman filter and implemented using Chilean corporate and government bonds.     

中国信用风险缓释工具创新试点最新进展

信用风险缓释工具（CRM）是2010年我国债券市场最重要的信用风险管理创新工具,它将短期融资券、中期票据等信用类债券的信用风险剥离定价,并转移给愿意承担风险的投资者。本文通过对CRM试点过程中存在的投资主体单一、市场机制不完善及外部环境建设滞后等问题进行了深度剖析,并有针对性地从投资者培育、信用评级与定价、做市商机制、交易信息披露、信用事件处理、以及监管会计税收法律制度建设等方面提出政策建议,以利于CRM市场的可持续发展。     

Ratings-based credit risk modelling: An empirical analysis

Banks have recently developed new techniques for gauging the credit risk associated with portfolios of illiquid and defaultable instruments. These techniques could revolutionise banks' management of credit risk and could in the longer term serve as a more risk-sensitive basis for calculating regulatory capital on banks' loan books than in Basel 2, the new regulatory capital framework. In this paper we implement a popular credit risk model that exploits the information in credit ratings to determine a portfolio's value-at-risk. Using price data on large eurobond portfolios, we assess, on an out-of-sample basis, how well the model tracks the risks it is supposed to measure.     

Term structure modelling of defaultable bonds

In this paper we present a model of the development of the term structure of defaultable interest rates that is based on a multiple-defaults model. Instead of modelling a cash payoff in default we assume that defaulted debt is restructured and continues to be traded.The term structure of defaultable bond prices is represented in terms of defaultable forward rates similar to the Heath-Jarrow-Morton (HJM) (Heath et al., 1992) approach, and conditions are given under which the dynamics of these rates are arbitrage-free. These conditions are a drift restriction that is closely related to the HJM drift restriction for risk-free bonds, and the restriction that the defaultable short rate must always be not below the risk-free short rate. In its most general version the model is set in a marked point process framework, to allow for jumps in the defaultable rates at times of default.Financial Assistance by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303, at the University of Bonn and the DAAD is gratefully acknowledged.I thank Pierre Mella-Barral, David Lando and David Webb for helpful conversations, and the participants of the FMG Conference on Defaultable Bonds (March 1997) in London and the QMF 97 conference in Cairns for helpful comments. All errors are of course my own.     

Credit rating dynamics and Markov mixture models

Despite mounting evidence to the contrary, credit migration matrices, used in many credit risk and pricing applications, are typically assumed to be generated by a simple Markov process. Based on empirical evidence, we propose a parsimonious model that is a mixture of (two) Markov chains, where the mixing is on the speed of movement among credit ratings. We estimate this model using credit rating histories and show that the mixture model statistically dominates the simple Markov model and that the differences between two models can be economically meaningful. The non-Markov property of our model implies that the future distribution of a firm’s ratings depends not only on its current rating but also on its past rating history. Indeed we find that two firms with identical current credit ratings can have substantially different transition probability vectors. We also find that conditioning on the state of the business cycle or industry group does not remove the heterogeneity with respect to the rate of movement. We go on to compare the performance of mixture and Markov chain using out-of-sample predictions.