This article presents convenient reduced-form models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a credit-spread option

In the setting of ‘‘affine’ ’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks ’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

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I find that the standard class of a#ne models produces poor forecasts of future changes in Treasury yields. Better forecasts are generated by assuming that yields follow random walks. The failure of these models is driven by one of their key features: The compensation that investors receive for facing risk is a multiple of the variance of the risk. This means that risk compensation cannot vary independently of interest rate volatility. I also describe and empirically estimate a class of models that is broader than the standard a#ne class. These "essentially a#ne" models retain the tractability of the usual models, but allow the compensation for interest rate risk to vary independently of interest rate volatility. This additional flexibility proves useful in forming accurate forecasts of future yields. Address correspondence to the University of California, Haas School of Business, 545 Student Services Building #1900, Berkeley, CA 94720. Phone: 510-642-1435. Email address: du#ee@haas.b...

This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the trade-off between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor ~SDF! that prices all assets in the economy. The behavior of the term structure of real interest rates restricts the conditional mean of the SDF, whereas patterns of risk premia restrict its conditional volatility and factor structure. Stylized facts about interest rates, aggregate stock prices, and cross-sectional patterns in stock returns have stimulated new research on optimal portfolio choice, intertemporal equilibrium models, and behavioral finance. This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work. Theorists develop models with testable predictions; empirical researchers document “puzzles”—stylized facts that fail to fit established theories—and this stimulates the development of new theories. Such a process is part of the normal development of any science. Asset pricing, like the rest of economics, faces the special challenge that data are generated naturally rather than experimentally, and so researchers cannot control the quantity of data or the random shocks that affect the data. A particularly interesting characteristic of the asset pricing field is that these random shocks are also the subject matter of the theory. As Campbell, Lo, and MacKinlay ~1997, Chap. 1, p. 3! put it: What distinguishes financial economics is the central role that uncertainty plays in both financial theory and its empirical implementation. The starting point for every financial model is the uncertainty facing investors, and the substance of every financial model involves the impact of uncertainty on the behavior of investors and, ultimately, on mar-* Department of Economics, Harvard University, Cambridge, Massachusetts

Despite powerful advances in yield curve modeling in the last twenty years, comparatively little attention has been paid to the key practical problem of forecasting the yield curve. In this paper we do so. We use neither the no-arbitrage approach, which focuses on accurately fitting the cross section of interest rates at any given time but neglects time-series dynamics, nor the equilibrium approach, which focuses on time-series dynamics (primarily those of the instantaneous rate) but pays comparatively little attention to fitting the entire cross section at any given time and has been shown to forecast poorly. Instead, we use variations on the Nelson-Siegel exponential components framework to model the entire yield curve, period-by-period, as a three-dimensional parameter evolving dynamically. We show that the three time-varying parameters may be interpreted as factors corresponding to level, slope and curvature, and that they may be estimated with high efficiency. We propose and estimate autoregressive models for the factors, and we show that our models are consistent with a variety of stylized facts regarding the yield curve. We use our models to produce term-structure forecasts at both short and long horizons, with encouraging results. In particular, our forecasts appear much more accurate at long horizons than various standard benchmark forecasts. Finally, we discuss a number of extensions, including generalized duration measures, applications to active bond portfolio management, and arbitrage-free specifications. Acknowledgments: The National Science Foundation and the Wharton Financial Institutions Center provided research support. For helpful comments we are grateful to Dave Backus, Rob Bliss, Michael Brandt, Todd Clark, Qiang Dai, Ron Gallant, Mike Gibbons, Da...

We construct a model for pricing sovereign debt that accounts for the risks of both default and restructuring, and allows for compensation for illiquidity. Using a new and relatively efficient method, we estimate the model using Russian dollar-denominated bonds. We consider the determinants of the Russian yield spread, the yield differential across different Russian bonds, and the implications for market integration, relative liquidity, relative expected recovery rates, and implied expectations of different default scenarios. THIS PAPER DEVELOPS A MODEL of the termstructure of credit spreads on sovereign bonds that accommodates: (i) Default or repudiation: The sovereign announces that it will stop making payments on its debt; (ii) Restructuring or renegotiation: The sovereign and the lenders ‘‘agree’ ’ to reduce (or postpone) the remaining payments; and (iii) A‘‘regime switch,’’such as a change of government or the default of another sovereign bond that changes the perceived risk of future defaults.We build on the framework of Duffie and Singleton (1999), showing that

This paper studies the empirical performance of jump-diffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the in-sample estimation period. This contrasts previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts out-of-sample. This evidence points towards a too simplistic specification of the mean dynamics of volatility.

We identify and estimate the sources of risk that cause corporate bonds to earn an excess return over default-free bonds. In particular, we estimate the risk premium associated with a default event. Default is modelled using a jump process with stochastic intensity. For a large set of firms, we model the default intensity of each firm as a function of common and firm-specific factors. In the model, corporate bond excess returns can be due to risk premia on factors driving the intensities and due to a risk premium on the default jump risk. The model is estimated using data on corporate bond prices for 104 US firms and historical default rate data. We find significant risk premia on the factors that drive intensities. However, these risk premia cannot fully explain the size of corporate bond excess returns. Next, we estimate the size of the default jump risk premium, correcting for possible tax and liquidity effects. The estimates show that this event risk premium is a significant and economically important determinant of excess corporate bond returns.

We develop a term structure model where the short interest rate and the market price of risks are subject to discrete regime shifts. Empirical evidence from efficient method of moments estimation provides considerable support for the regime shifts model. Standard models, which include affine specifications with up to three factors, are sharply rejected in the data. Our diagnostics show that only the regime shifts model can account for the well-documented violations of the expectations hypothesis, the observed conditional volatility, and the conditional correlation across yields. We find that regimes are intimately related to business cycles. MANY PAPERS DOCUMENT THAT THE UNIVARIATE short interest rate process can be reasonably well modeled in the time series as a regime switching process ~see Hamilton ~1988!, Garcia and Perron ~1996!!. In addition to this statistical evidence, there are economic reasons as well to believe that regime shifts are important to understanding the behavior of the entire yield curve. For example, business cycle expansion and contraction “regimes ” potentially