Defaultable Bond Prices Research Articles

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.

Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility

This paper presents a class of defaultable term structure models within the HJM framework with stochastic volatility. By modelling the connection between defaultfree and defaultable term structures, namely the credit spread, a correlation structure between the credit spread, the default-free interest rate and the stochastic volatility is also accommodated. Under certain volatility specifications, the model admits finite dimensional Markovian structures and consequently provides tractable solutions for default-free and defaultable bond prices. Furthermore, a bond pricing formula is obtained in terms of market observable quantities, specifically in terms of discrete tenor defaultable forward rates. The effect of stochastic volatility and of correlations between the stochastic volatility, defaultable short rate and credit spreads on the defaultable bond prices and returns is also investigated.

Explicit Solutions to Optimal Risk-Averse Trading of Defaultable Bonds Under Heterogeneous Beliefs

This paper studies the problem of pricing and trading of defaultable bonds among investors with heterogeneous risk preferences and beliefs. Based on the utility indifference pricing methodology, we first construct the risk-averse bid-ask spread, which naturally widens as risk aversion or trading volume increases. In addition, we analyze the defaultable bond buyer's optimal static trading position under different market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty - single or multiple sellers - may have different risk preferences and beliefs. In each of these cases, we provide an explicit formula for the optimal trading position and examine the combined effects of risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade.

On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing

This paper considers the estimation of an unknown function h that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator.

An Integral-Equation Approach for Defaultable Bond Prices with Application to Credit Spreads

We study defaultable bond prices in the Black–Cox model with jumps in the asset value. The jump-size distribution is arbitrary, and following Longstaff and Schwartz (1995) and Zhou (2001) we assume that, if default occurs, the recovery at maturity depends on the ‘severity of default’. Under this general setting, the vehicle for our analysis is an integral equation. With the aid of this, we prove some properties of the bond price which are consistent numerically and empirically with earlier works. In particular, the limiting credit spread as time to maturity tends to 0 is nonzero. As a byproduct, we show that the integral equation implies an infinite-series expansion for the bond price.

An Integral-Equation Approach for Defaultable Bond Prices with Application to Credit Spreads

We study defaultable bond prices in the Black–Cox model with jumps in the asset value. The jump-size distribution is arbitrary, and following Longstaff and Schwartz (1995) and Zhou (2001) we assume that, if default occurs, the recovery at maturity depends on the ‘severity of default’. Under this general setting, the vehicle for our analysis is an integral equation. With the aid of this, we prove some properties of the bond price which are consistent numerically and empirically with earlier works. In particular, the limiting credit spread as time to maturity tends to 0 is nonzero. As a byproduct, we show that the integral equation implies an infinite-series expansion for the bond price.

Empirical Evaluation of Hybrid Defaultable Bond Pricing Models

A four‐factor model (the extended model of Schmid and Zagst) is presented for pricing credit risk related instruments such as defaultable bonds or credit derivatives. It is an advancement of an earlier three‐factor model. In addition to a firm‐specific credit risk factor, a new systematic risk factor in the form of GDP growth rate is included. This new model is set in the context of other hybrid defaultable bond pricing models and empirically compared to specific representatives. We find that a model based only on firm‐specific variables is unable to capture changes in credit spreads completely. However, it is shown that in this model, market variables such as GDP growth rates, non‐defaultable interest rates and firm‐specific variables together significantly influence credit spread levels and changes.

The Valuation of Corporate Debt With Default Risk

This article values equity and corporate debt by taking into account the fact that in practice the default point differs from the liquidation point and that it might be in the creditors' interest to delay liquidation. The article develops a continuous time asset pricing model of debt restructuring which explicitly considers the inalienability of human capital. The study finds that even though in general the creditors will not liquidate the firm on the incidence of default, but nevertheless would liquidate the firm prematurely relative to the first best threshold. This agency problem leads to the breakdown of the capital structure irrelevance result. 1. Theoretical Foundation The literature on the pricing of defaultable bonds started with Merton (1974) who applied the contingent claims valuation insight of Black and Scholes (1973) to the pricing of corporate bonds. He obtained closed form valuation expressions for (zero-coupon) risky bonds, by taking the lower reorganization boundary as given. Thus in his model, default occurred at maturity if the value of the firm was less than the payment promised to the bondholders. Furthermore, at maturity the compensation received by the creditors is fixed and they receive the minimum of the value of the firm or the contracted payment. In the event of a default, the control of the firm is transferred to the creditors and this default point can also be interpreted as a liquidation point with zero bankruptcy costs. Thus, in Merton's model, both the lower reorganization boundary and the compensation received by the bondholders is taken to be exogenous. In practice, however, the compensation received by the creditors may vary continuously with the value of the firm if the firm is in financial distress. Furthermore, Merton does not make the distinction between default and liquidation. There have been a number of extensions of the Merton model. Black and Cox (1976) incorporate bond indenture provisions. Jones, Mason and Rosenfeld (1984) consider mul- tiple issue of callable coupon debt. In both of these models however the induced lower boundary at which the firm is liquidated is taken to be exogenous. Further, default is again tantamount to bankruptcy in these models. Leland (1993) and Leland and Toft (1996) extend Merton's analysis by endogenising the lower reorganization boundary. By assuming that the equityholders will always issue equity to prevent a default, they obtain expressions for the values of risky debt via the smooth-pasting condition. Thus in their models, the equityholders keep on issuing equity (when necessary) to avoid a default until the value of equity falls to zero. Thus default occurs when the value of equity falls to zero and this default point is again synonymous

Default-risky bond prices with jumps, liquidity risk and incomplete information

This paper provides a default-risky bond valuation model, which assumes that the issuer’s credit quality, modelled by the intensity of default, is driven by a continuous-time Markov chain. The model accounts for default and liquidity risk as well as incomplete information. A full-information semimartingale representation of a liquid defaultable bond price, which separates three different types of risks—default, interest-rate and credit-quality, is obtained. The illiquidity is modelled as exogenously specified stochastic reduction in the price of the bond, which adds more risks for the investors. A model of a market with partially informed investors, belonging to specific investor classes and having access to discrete information sets about credit quality, was specified. Valuations of defaultable bonds in this market were provided as well as price impacts of the new information releases.

AN EXTENSION OF THE BRODY–HUGHSTON–MACRINA APPROACH TO MODELING OF DEFAULTABLE BONDS

The innovative information-based framework for credit risk modeling, proposed recently by Brody, Hughston, and Macrina, is extended to a more general and practically important setup of random interest rates. We first introduce the market model, and we derive an explicit expression for defaultable bond price. Next, the dynamics of the information process and dynamics of defaultable bond are found for both deterministic and random interest rates. Finally, the valuation and hedging of derivative securities are briefly examined. In particular, the valuation formula for a European option on a defaultable bond is established.

CORRELATED DEFAULTS IN INTENSITY‐BASED MODELS

This paper presents an intensity‐based model of correlated defaults with application to the valuation of defaultable securities. The model assumes that the intensities of the default times are driven by common factors as well as other defaults in the system. A recursive procedure called the “total hazard construction” is used to generate default times with a broad class of correlation structures. This approach is compared to standard reduced‐form models based on conditional independence as well as alternative approaches involving copula functions. Examples are given for the pricing of defaultable bonds and credit default swaps of the regular and basket type.

Quadratic Models for Portfolio Credit Risk with Shot-Noise Effects

We propose a reduced form model for default that allows us to derive closed-form solutions to all the key ingredients in credit risk modeling: risk-free bond prices, defaultable bond prices (with and without stochastic recovery) and probabilities of survival. We show that all these quantities can be represented in general exponential quadratic forms, despite the fact that the intensity is allowed to jump producing shot-noise effects. In addition, we show how to price defaultable digital puts, CDSs and options on defaultable bonds. Further on, we study a model for portfolio credit risk where we consider both firm specific and systematic risks. The model generalizes the attempt from Due and

What drives credit risk in emerging markets? The roles of country fundamentals and market co-movements

This paper uses bond prices to investigate how the creditworthiness of Argentina, Brazil, Mexico and Venezuela is influenced by global, regional and country-specific factors. Each country's distance-to-default is estimated monthly for 1994–2001, by fitting the structural model of Cathcart and El-Jahel [ Cathcart, L., El-Jahel, L., 2003. Semi-analytical pricing of defaultable bonds in a signalling jump-default model. The Journal of Computational Finance 6, 91–108] with a Kalman Filter to Brady bonds. A small set of variables is able to explain up to 80% of the variance of the estimated distance-to-default for each country. Surprisingly, country-specific variables account for only about 8% of the explained variance; the largest part of the variance (45%) is explained by regional factors, which relate to joint stock-market returns, volatility and market sentiment; global conditions, related mainly to US stock-market returns, explain another 25% of the variance. Of the 20% variance which remains unexplained, more than half is due to another common (but unidentified) factor. The conclusion is that the creditworthiness of these four emerging markets is driven mainly by a common set of factors, which are related closely to stock markets in the region and the United States.

First Passage and Excursion Time Models for Valuing Defaultable Bonds

In this contribution, we study structural models of defaultable bond pricing in which default occurs at the first time a relevant process either reaches the default boundary or has spent continuously (or cumulatively) a fixed time period below that threshold. Unlike first-passage time approaches, excursion time models allow for a non-absorbing state of default. Both the first-passage time and the excursion time approaches can be generalized by defining the default time as the first instant at which the firm value process (or another signaling process) either remains a certain time below the default threshold or hits a lower barrier. This corresponds, for instance, to a situation in which a firm is allowed temporarily to be short of funds, but enters default immediately when the financial distress becomes severe. Moreover, we examine the effects of different default time specifications on bond prices and credit spreads.

Pricing of Defaultable Bonds with Log-Normal Spread: Development of the Model and an Application to Argentinean and Brazilian Bonds During the Argentine Crisis

In this paper we describe a two-factor model for a defaultable discount bond, assuming log-normal dynamics with bounded volatility for the instantaneous short rate spread. Under some simplified hypothesis, we obtain an explicit barrier-type solution for zero recovery and constant recovery. We also present a numerical application for Argentinean and Brazilian Sovereign Bonds during the default crisis of Argentina.

Correlated Defaults in Intensity-Based Models

This paper presents an intensity-based model of correlated defaults with application to the valuation of defaultable securities. The model assumes that the intensities of the default times are driven by common factors as well as other defaults in the system. A recursive procedure called the total hazard construction is used to generate default times with a broad class of correlation structures. This approach is compared to standard reduced-form models based on conditional independence as well as alternative approaches involving copula functions. Examples are given for the pricing of defaultable bonds and credit default swaps of the regular and basket type.

A Markovian Defaultable Term Structure Model with State Dependent Volatilities

The defaultable forward rate is modeled as a jump diffusion process within the Schonbucher (2000, 2003) general Heath, Jarrow and Morton (1992) framework where jumps in the defaultable term structure cause jumps and defaults to the defaultable bond prices. Within this framework, we investigate an appropriate forward rate volatility structure that results in Markovian defaultable spot rate dynamics. In particular, we consider state dependent Wiener volatility functions and time dependent Poisson volatility functions. The corresponding term structures of interest rates are expressed as finite dimensional affine realisations in terms of benchmark defaultable forward rates. In addition, we extend this model to incorporate stochastic spreads by allowing jump intensities to follow a square-root diffusion process. In that case the dynamics become non-Markovian and to restore path independence we propose either an approximate Markovian scheme or, alternatively, constant Poisson volatility functions. We also conduct some numerical simulations to gauge the effect of the stochastic intensity and the distributional implications of various volatility specifications.

The effect of taxes on the pricing of defaultable debt

Empirical studies have documented the dependence of corporate credit spreads on default risk, equity premiums, and taxes. However, taxes have previously not been incorporated into reduced-form credit risk models. Therefore, we first extend the existing literature by considering a default intensity that depends on taxes as well as the default-free short rate and a market index. Consequently, we establish a theoretical basis to explain previous empirical findings regarding the significant impact of taxation on defaultable bond prices. Unlike previous models, tax implications for defaultable debt cannot be constructed from a sum of tax effects on zero coupon bonds. Our empirical tests then illustrate the importance of taxation. In particular, the impact of taxation increases as a function of the debt’s maturity and coupon rate.

Corporate Walkout Decisions and the Value of Default

We present a continuous-time asset pricing model of the levered firm where shareholders select not only the timing but also the form of abandonment. Shareholders can walk out of the firm either by (i) defaulting on their debt obligations or (ii) selling their shares to alternative operators of the technologies, as in a corporation sale. The structural model relates shareholders' ex-post choice to both technological and financial factors. Considering that operators' technological supremacy is not universal, we obtain that whereas default necessarily involves an inefficient timing of ownership transfer, corporation sales do not. Then, the likelihood of default being chosen instead of a corporation sale increases with (i) the degree of leverage displayed by the firm and (ii) its technological supremacy. By ignoring corporation sales, existing defaultable bond pricing models have thus a tendency to exaggerate risk premia and underestimate the borrowing ability (debt capacity) of firms.

Portfolio Optimization Under Credit Risk

Based on the models of Hull & White (1990) for the pricing of non-defaultable bonds and Schmid & Zagst (2000) for the pricing of defaultable bonds we develop a framework for the optimal allocation of assets out of a universe of sovereign bonds with different time to maturity and quality of the issuer. We estimate the model parameters by applying Kaiman filtering methods as described in (Schmid & Kalemanova 2002). Based on these estimates we simulate the prices for a given set of bonds for a future time horizon. For each future time step and for each given portfolio composition these scenarios yield distributions of future cash flows and portfolio values. We show how the portfolio composition can be optimized by maximizing the expected final value or return of the portfolio under given constraints.