Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility

this paper we present a tree model for defaultable bond prices which can be used for the pricing of credit derivatives. The model is based upon the two-factor Hull-White (1994) model for default-free interest rates, where one of the factors is taken to be the credit spread of the defaultable bond prices. As opposed to the tree model of Jarrow and Turnbull (1992), the dynamics of default-free interest rates and credit spreads in this model can have any desired degree of correlation, and the model can be fitted to any given term structures of default-free and defaultable bond prices, and to the term structures of the respective volatilities. Furthermore the model can accommodate several alternative models of default recovery, including the fractional recovery model of Duffie and Singleton (1994) and recovery in terms of equivalent default-free bonds (see e.g. Lando (1998)). Although based on a Gaussian setup, the approach can easily be extended to non-Gaussian processes that avoid negative interest-rates or credit spreads

This empirical paper investigates the effect of different distributional assumptions governing defaultable bond price uncertainty on the price of a credit default swap. We value a credit default swap using the two-factor Hull-White (1994) model for the term structure of default-free Spot interest rates and the credit spread process of a Baa-rated bond index and use the fractional recovery model of Duffie-Singleton (1999) and its multiple default extension as given in Schonbucher (1996,1998). The model is implemented using a tree algorithm outlined in Schonbucher (1999) where one factor is used for the spot interest rate and the other factor is for the credit spread or the intensity rate of a Cox process. This tree representation permits correlation between the spot rate and the credit spread dynamics of the Baa bond index, enabling us to provide a model that better fits the term structure of default-free and defaultable bond prices. We compare the values of a credit default swap obtained when the underlying risk factors are modeled with Gaussian and stable non-Gaussian distributions.

This article presents a structural model of default risk under macroeconomic conditions. The macroeconomic conditions are assumed to be a finite state of a Markov chain. The innovation of the author’s model is to characterize the firm default, the default-free pure discount bond price, the defaultable bond price, and the credit spread associated with macroeconomic conditions. By using the Wiener–Hopf factorization, the author shows that there is a risk-neutral measure under regime switching. Under macroeconomic conditions, he derives both the default-free bond price and the price of the defaultable bond governed by fundamental systems of partial differential equations. The defaultable yield to maturity, credit spread, and duration are related to the finite state of the Markov chain.

This paper investigates a two-factor affine model for the credit spreads on corporate bonds. The first factor can be interpreted as the level of the spread and the second factor is the volatility of the spread. The riskless interest rate is modeled using a standard two-factor affine model, thus leading to a four-factor model for corporate yields. This approach allows us to model the volatility of corporate credit spreads as stochastic, and also allows us to capture higher moments of credit spreads. We use an extended Kalman filter approach to estimate our model on corporate bond prices for 108 firms. The model is found to be successful at fitting actual corporate bond credit spreads, resulting in a significantly lower root mean square error than a standard alternative model both in sample and out of sample. In addition, key properties of actual credit spreads such as the stochastic volatility of the credit spreads and the positive skewness of the credit spread distribution are better captured by the model.

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

Explaining Credit Default Swap Spreads With the Equity Volatility and Jump Risks of Individual Firms

Modelling the Evolution of Credit Spreads using the Cox Process within the HJM Framework: A CDS Option Pricing Model

Economic determinants of oil futures volatility: A term structure perspective

This article develops a three-factor defaultable term structure model for the pricing of a wide range of risky debt contracts and derivatives. One of the factors that determine the credit spread is the so-called uncertainty process, which can be understood as an aggregation of all information currently available on the quality of the firm. The authors assume that the spread between a defaultable and a non-defaultable bond is driven to a considerable extent by the uncertainty index, but also that contractual provisions, liquidity, and the premium demanded in the market for similar instruments have a great impact on credit spreads. Credit spreads can be easily related to business cycles by replacing the uncertainty index by some index of macroeconomic variables without any change of the theoretical framework. The analytical solution obtained for defaultable bonds can be implemented easily in practice, as all the variables and parameters can be inferred from market data. The authors demonstrate application of the model to the pricing of a variety of default-related instruments such as credit derivatives.

In the following, we develop a model for pricing defaultable debt that is capable of reflecting real world phenomena realistically. Before we actually introduce the model, we make some heuristic observations to motivate the theory. Throughout the following sections we examine and extend these considerations further: Since defaults actually happen, a bond pricing model must be able to deal with default events: For example, in August 1999 Ecuador defaulted on interest rate payments of Brady bonds with a total amount of about US$ 6 billions. In October 1999 there was a second default of US$ 500 millions. Figure 6.1 shows the yields to maturity and S&P ratings of an Ecuadorian sovereign bond that was downgraded to the rating D in September 1999. Rating agencies can’t check the ratings on a daily basis or even continuously. Ratings of rating agencies can only be described by discrete time processes. But the quality of an obligor usually changes according to a continuous time process (see, e.g., figure 6.1). Thus, the quality or the distance to default of an obligor should be observed continuously. Very often the quality of obligors shows mean reversion properties. In case of a downgrade the management of a firm usually takes actions to improve the quality again. Figure 6.2 shows this phenomenon for the firm Hertz. The mean reversion level of the rating of Hertz is A- and A3 for S&P’s and Moody’s rating, respectively. Many models for pricing defaultable zero coupon bonds don’t allow for positive short term credit spreads: As time to maturity tends to zero the model credit spreads tend to zero, too. Obviously, these models are far away from describing reality. In figure 6.3 we show observed corporate credit spreads for different maturities. Even bonds with a rating of A and time to maturity of seven days can have credit spreads which are higher than 500 basis points. Hence, a model must be capable of generating high credit spreads even for short term debt. Yields and credit spreads usually depend on the quality of the obligor. The better the quality the smaller the yields (compare to figure 6.4). The time series of the yields show high correlations between the rating classes A and AAA. Gains and losses are — at least to some extend — caused by changes in the quality of the obligor. Credit spread processes often show mean reversion properties. See, e.g., figure 6.5. Non-defaultable interest rates are usually mean reverting. See, e.g., figure 6.6. KeywordsDefault RiskCredit SpreadBond PriceShort RateCredit DerivativeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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.

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of orderν> −1 with constant driftμ≠ 0. Whenν> -½ andμ< 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalueλ0= 0, followed by an infinite series of terms corresponding to the higher eigenvaluesλn,n= 1,2,…, as well as an integral over the continuous spectrum aboveμ2/2. When −1 <ν< -½ andμ< 0, there is only one eigenvalueλ0= 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

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