Defaultable Zero-coupon Bond Research Articles

Pricing a Defaultable Zero-Coupon Bond under Imperfect Information and Regime Switching

We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and risky debt under regime switching with a regime switching interest rate. While the value of the firm’s equity is observed continuously, we assume that the total value of the firm is only observed at discrete times, such as the dates of the release of the firm’s annual reports, or quarterly reports. This uncertainty about the true value of the firm results in credit spreads that do not approach zero as the debt approaches maturity, which is a problem with many structural models. The firm’s value is typically decomposed into its equity and debt; however, we consider the asset–to–equity ratio, an accounting ratio used to examine a firm’s financial well-being. The parameters in our model are regime switching, where the regime can be thought of as the state of the economy. A Markov chain with a constant transition rate matrix produces the regime switching.

THE JARROW AND TURNBULL SETTING REVISITED

We consider a financial market with zero-coupon bonds that are exposed to credit and liquidity risk. We revisit the famous Jarrow & Turnbull (1995) setting in order to account for these two intricately intertwined risk types. We utilize the foreign exchange analogy that interprets defaultable zero-coupon bonds as a conversion of nondefaultable foreign counterparts. The relevant exchange rate is only partially observable in the market filtration, which leads us naturally to an application of the concept of platonic financial markets as introduced by Cuchiero et al. (2020). We provide an example of tractable term structure models that are driven by a two-dimensional affine jump diffusion. Furthermore, we derive explicit valuation formulae for marketable products, e.g. for credit default swaps.

A three-factor hazard rate model for single-name credit default swap pricing

We propose a reduced-form model that integrates the modeling of interest rates, equity and default hazard rates. In this model the evolution of the risk-neutral hazard rate is driven by three risk factors: the reference firm’s equity price (in logarithmic form), the risk-free short rate and a latent risk premium factor that is orthogonal to the first two. The equity price and the short rate are considered to be the two primary risk factors, while the risk premium factor is introduced to capture the remaining risks for the hazard rate (such as the liquidity risk). The equity price follows a diffusion process with the possibility of a jump to default, the short rate is governed by a Vasicek model, and the dynamics of the risk premium factor are modeled as a Cox– Ingersoll–Ross process. Under the three-factor model, we derive computable pricing formulas for defaultable zero-coupon bonds, European call options and standard running credit default swaps (CDSs).We then discuss the calibration of the model using CDS and equity option data. Our numerical results reveal that the model is able to interpret a diverse set of CDS spread term structures.

Closed-Form Solution for Defaultable Bond Options under a Two-Factor Gaussian Model for Risky Rates Modeling

In this article, the authors provide a closed-form solution for defaultable bond options under a two-factor Gaussian model for risky rates. The key feature of the proposed stochastic model is the introduction of two stochastic dynamics to address the behavior of both risk-free interest rates and credit spreads where the two sources of risk are correlated. Moreover, the model can match exactly the term structure of defaultable interest rates. In order to model credit-sensitive bonds, the authors assume an intensity-based reduced-form framework where the approach based on the recovery of market value is considered and where the recovery rate is one of the model’s input. Under this framework, the authors derive closed formulas for the price of European options having as their underlying both defaultable zero-coupon bonds and coupon-bearing bonds, which the authors assume are approximately log-normally distributed. Moreover, the options are assumed to be knocked out upon a default event of the bond, with zero value to the option holder. An empirical analysis is performed to illustrate the calibration process of the proposed model using financial market data to price call and put options across different credit ratings and different levels of the loss given default. TOPICS:Factor-based models, derivatives, options Key Findings • An option pricing model that considers the dynamic of risky rates under a two-factor Gaussian model where risky rates are the sum of risk-free rates and credit spreads is proposed. This distributional hypothesis is consistent with negative interest rates and benefits from analytical tractability. • Under this framework, closed formulas for the price of European options having both defaultable zero-coupon bonds and coupon-bearing bonds as their underlying are derived under an approximately log-normal distribution assumption. • A detailed two-step calibration procedure is provided. The first step estimates the parameters of the one-factor Hull-White model for the risk-free rates and the second step estimates the parameters of the model’s credit component under a grid of correlation values.

A GENERALIZED CONTAGION PROCESS WITH AN APPLICATION TO CREDIT RISK

We introduce a class of analytically tractable jump processes with contagion effects by generalizing the classical Hawkes process. This model framework combines the characteristics of three popular point processes in the literature: (1) Cox process with CIR intensity; (2) Cox process with Poisson shot-noise intensity; (3) Hawkes process with exponentially decaying intensity. Hence, it can be considered as a self-exciting and externally-exciting point process with mean-reverting stochastic intensity. Essential probabilistic properties such as moments, the Laplace transform of intensity process, and the probability generating function of point process as well as some important asymptotics have been derived. Some special cases and a method for change of measure are discussed. This point process may be applicable to modeling contagious arrivals of events for various circumstances (such as jumps, transactions, losses, defaults, catastrophes) in finance, insurance and economics with both endogenous and exogenous risk factors within one framework. More specifically, these exogenous factors could contain relatively short-lived shocks and long-lasting risk drivers. We make a simple application to calculate the default probability for credit risk and to price defaultable zero-coupon bonds.

Jump-diffusion CIR model and its applications in credit risk

In this paper, the author discusses the distribution of the jump-diffusion CIR model (JCIR) and its applications in credit risk. Applying the piecewise deterministic Markov process theory and martingale theory, we first obtain the closed forms of the Laplace transforms for the distribution of the jump-diffusion CIR model and its integrated process. Based on the obtained Laplace transforms, we derive the pricing of the defaultable zero-coupon bond and the fair premium of a Credit Default Swap (CDS) in a reduced form model of credit risk. Some numerical calculations are also provided.

The pricing of credit risky securities under stochastic interest rate model with default correlation

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R.A. Jarrow and F.Yu (2001), S.Y.Leung and Y.K.Kwok (2005), A.Wang and Z.Ye (2011)). By using the method of change of measure and the technology (H. S.Park (2008), R.Hao and Z.Ye (2011)) of dealing with jump diffusion processes, we obtain the analytic pricing formulas of defaultable zero-coupon bonds. Moreover, by the “total hazard construction”, we give the analytic pricing formulas of credit default swap (CDS).

On the conditional default probability in a regulated market with jump risk

In this paper, we introduce tractable dynamic models for financial variables (such as interest rates, foreign exchange rates, commodity prices, etc.) with capturing both jump risk and boundedness of the price fluctuation in a regulated market. For the jump risk, we use a compound Poisson process with double exponential jump amplitude to describe the jumps. As for the boundedness of the fluctuation, we introduce two regulators to regulate (or control) the price dynamics. Based on such a dynamics, we then study the default events under the structural framework for credit risk. The explicit expression for the Laplace transform of the default time is derived. For exploring the effect of the jump risk and the regulation, we perform some numerical simulation. As a practical application, an analytic valuation of defaultable zero-coupon bonds is presented. It turns out that our model can produce a variety of strictly positive credit spread term structures, including downward, humped and upward styles.

Asymptotic Analysis for One-Name Credit Derivatives

We propose approximate solutions to price defaultable zero-coupon bonds as well as the corresponding credit default swaps and bond options. We consider the intensity-based approach of a two-correlated-factor Hull-White model with stochastic volatility of interest rate process. Perturbations from the stochastic volatility are computed by using an asymptotic analysis. We also study the sensitive properties of the defaultable bond prices and the yield curves.

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.

Robust Recovery Risk Hedging: Only the First Moment Matters

Credit derivatives are subject to at least two sources of risk: the default time and the recovery payment. This paper examines the impact of modeling the recovery payment on hedging strategies in a reduced-form model as well as a structural model. We show that all hedging approaches based on a quadratic criterion do only depend on the expected recovery payment at default and not the whole shape of the recovery payment distribution if the underlying hedging instrument (say, a defaultable zero coupon bond) jumps to or reaches a pre-specified value when the credit event occurs. This justifies assuming a \emph{certain} recovery rate conditional on default time and interest rate level. Hence, this result allows a simplified modeling of credit risk.

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.

Optimal Investment with Default Risk

In this paper, we investigate how investors who face both equity risk and credit risk would optimally allocate their financial wealth in a dynamic continuous-time setup. We model credit risk through the defaultable zero-coupon bond and solve the dynamics of its price after pricing it. Using stochastic control methods, we obtain a closed-form solution to this investment problem and characterize its variation with respect to different factors in the economy. We find that non-zero recovery rate of the credit-risky bond affects investors' decision in a fundamental way. Because of this, investors try to time the market conditions in their decision making process. It also induces hedging term in this setup of otherwise deterministic investment opportunity set. Through numerical examples, we show that the inclusion of credit market is shown to be able to enhance investors' welfare.

Integrating Market Risk and Credit Risk: A Dynamic Asset Allocation Perspective

This paper investigates how investors who face both market risk (interest rate risk) and credit risk in addition to equity risk would optimally allocate their financial wealth in a dynamic, no arbitrage, continuous-time setup. I model credit risk through a defaultable zero-coupon bond and solve the stochastic differential equation of it under the recovery to market value scheme. I obtain a closed-form solution to this investment problem, which enables me to analyze the impact on investors' decisions of various risk parameters. One interesting insight of this paper is that a non-zero recovery rate of the credit-risky bond affects investors' decision in a fundamental way. This is manifested in a dividend-like adjustment term in the drift of the stochastic differential equation (SDE) of the defaultable zero-coupon bond's return process. The optimal asset allocation involves the separation effect and effect. As a result, the relation between myopic demands for bonds and market prices of risk becomes relatively complicated compared with that in the traditional setup. In addition, I show the cross-markets correlation is an important factor in the asset allocation decision. In particular, it affects investors' ability to hedge against or speculate on the stochastic risk premium of the defaultable bond. Numerical examples show that the inclusion of credit markets significantly enhances investors' welfare. I also find modeling interest rate as a stochastic process greatly reduces model risk, giving investors more realistic prospects in their investment decisions. Key Words: integration of market risk and credit risk, asset allocation, welfare analysis, corporate bond, model risk

An Econometric Model of the Term Structure of Interest-Rate Swap Yields

This paper develops a multi-factor econometric model of the term structure of interest-rate swap yields. The model accommodates the possibility of counterparty default and any differences in the liquidities of the Treasury and Swap markets. By parameterizing a model of swap rates directly, we are able to compute model-based estimates of the defaultable zero coupon bond rates implicit in the swap market without having to specify a priori the dependence of these rates on default hazard or recovery rates. The time series analysis of spreads between zero-coupon swap and treasury yields reveals that both credit and liquidity factors were important sources of variation in swap spreads over the past decade.

Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analyzed in the general equilibrium model of Cox, Ingersoll, and Ross (Cir, 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 derived in a simple discrete-time setting while a more rigorous treatment, in a continuous-time setting, is contained in the Appendix A. Defaultable bonds may be valued by discounting the promised terminal payoff at a default-risk-adjusted interest rate, i.e. the risk-free rate plus a default-risk premium, or by discounting the expected terminal payoff at a risk-free interest rate. 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. Valuation is performed under different contractual provisions dealing with the event of default: their impact on options prices is investigated and several numerical examples are given. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.