This paper presents a consistent and arbitrage-free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric multivariate Markov diffusion process with "stochastic volatility." The yield of any zero-coupon bond is taken to be a maturitydependent affine combination of the selected "basis" set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as wcll as numerical techniques for calculating the prices of term-structure derivative prices. The case of jump diffusions i\ also considered. I . INTRODUCTIONThis paper defines and analyzes a simple multifactor model of the term structure of interest rates. The factors of the model are the yields X = ( X I , Xz, . . . , X , l ) of zero-coupon bonds of n various fixed maturities, I t l , r2. . , . , t,,}. For example, one could think of the current five-year (zero-coupon) yield as a factor. The yield factors form a Markov process, to be described below, that can be thought of as a multivariate version of the single-factor model of Cox, Ingersoll, and Ross ( 1 98%). As opposed to most multifactor term structure models, the factors (Markov state variables) are observable from the current yield curve and their increments can have an arbitrarily specified correlation matrix. The model includes stochastic volatility factors that are specified linear combinations of yield factors. Discount bond prices at any maturity are given as solutions to Ricatti (ordinary differential) equations, and path-independent derivative prices can be solved by, among other methods, an alternating-direction implicit finite-difference solution of the "usual" partial differential equation (PDE). Fully workcd examples of solutions to these Ricatti equations and PDEs are included. Our yield model is "affine" in the sense that there is, for each maturity t, an affine function Y,: R" + R such that, at any time t , the yield of any zero-coupon bond of maturity t is Y , ( X , ) . Indeed, ruling out singularities, cssentially any n yields would serve as the factors, and given the imperfections of any model, it is an empirical issue as to which IZ yields will serve best as such. Likewise, because of linearity, the Markov state variables can be taken to be forward rates at given maturities, so that the model can be viewed as a multifactor Markov parameterization of the Heath, Jarrow, and Morton (HJM) (1992)

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,

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. This article presents a new approach to modeling term structures of bonds and other contingent claims that are subject to default risk. As in previous "reduced-form" models, we treat default as an unpredictable event governed by a hazard-rate process. 1 Our approach is distinguished by the parameterization of losses at default in terms of the fractional reduction in market value that occurs at default. Specifically, we fix some contingent claim that, in the event of no default, pays X at time T. We take as given an arbitrage-free setting in which all securities are priced in terms of some short-rate process r and equivalent martingale measure Q [see Harrison and Kreps (1979) and Harrison and Pliska (1981)]. Under this "risk-neutral" probability measure, we let h t denote the hazard rate for default at time t and let L t denote the expected fractional loss in market value if default were to occur at time t, conditional This article is a revised and extended version of the theoretical results from our earlier article "Econometric Modeling of Term Structures of Defaultable Bonds" (June 1994). The empirical results from that article, also revised and extended, are now found in "An Econometric Model of the Term Structure of Interest Rate Swap Yields" (Journal of Finance, October 1997). We are grateful for comments from many, including the anonymous referee, Ravi Jagannathan (the editor),

Empirical difficulties encountered by representative-consumer models are resolved in an economy with heterogeneity in the form of uninsurable, persistent, and heteroscedastic labor income shocks. Given the joint process of arbitrage-free asset prices, dividends, and aggregate income, satisfying a certain joint restriction, it is shown that this process is supported in the equilibrium of an economy with judiciously modeled income heterogeneity. The Euler equations of consumption in a representative-agent economy are replaced by a set of Euler equations that depend not only on the per capita consumption growth but also on the cross-sectional variance of the individual consumers' consumption growth. Constantinides acknowledges support from a gift to the Graduate School of Business, University of Chicago, by Dimensional Fund Advisors. Duffie acknowledges support from the National Science Foundation under grant SES 90-10062 and from Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 as well as the Gottfried-Wilhelm-Leibniz-Forderpreis. We are grateful for comments from John Cochrane,

We study how intermediation and asset prices in over-the-counter markets are affected by illiquidity associated with search and bargaining. We compute explicitly the prices at which investors trade with each other, as well as marketmakers' bid and ask prices, in a dynamic model with strategic agents. Bid-ask spreads are lower if investors can more easily find other investors or have easier access to multiple marketmakers. With a monopolistic marketmaker, bid-ask spreads are higher if investors have easier access to the marketmaker. We characterize endogenous search and welfare, and discuss empirical implications. Copyright The Econometric Society 2005.