This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of time-varying intensity. We find that any reasonably descriptive continuous-time model for equity-index returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equity-index returns and the stylized features of the corresponding options market prices. MUCH ASSET AND DERIVATIVE PRICING THEORY is based on diffusion models for primary securities. However, prescriptions for practical applications derived from these models typically produce disappointing results. A possible explanation could be that analytic formulas for pricing and hedging are available for only a limited set of continuous-time representations for asset returns

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 proposes a new method for estimation of parameters in diffusion processes from

Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, since the transition density is generally unknown, one is forced to approximate it. The simulation-based approach suggested by Pedersen (1995) has great theoretical appeal, but previously available implementations have been computationally costly. We examine a variety of numerical techniques designed to improve the performance of this approach. Synthetic data generated by a CIR model with parameters calibrated to match monthly observations of the U.S. short-term interest rate are used as a test case. Since the likelihood function of this process is known, the quality of the approximations can be easily evaluated. On data sets with 1000 observations, we are able to approximate the maximum likelihood estimator with negligible error in well under one minute. This represents something on the order of a 10,000-fold reduction in computational effort as compared to implementations without these enhancements. With other parameter settings designed to stress the methodology, performance remains strong. These ideas are easily generalized to multivariate settings and (with some additional work) to latent variable models. To illustrate, we estimate a simple stochastic volatility model of the U.S. short-term interest rate.

This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for short-dated options.

This paper develops an arbitrage-free time-series model of yields in continuous time that incorporates central bank policy. Policy-related events, such as FOMC meetings and releases of macroeconomic news the Fed cares about, are modeled as jumps. The model introduces a class of linear-quadratic jump-diffusions as state variables, which allows for a wide variety of jump types but still leads to tractable solutions for bond prices. I estimate a version of this model with U.S. interest rates, the Federal Reserve’s target rate, and key macroeconomic aggregates. The estimated model improves bond pricing, especially at short maturities. The “snake-shape ” of the volatility curve is linked to monetary policy inertia. A new monetary policy shock series is obtained by assuming that the Fed reacts to information available right before the FOMC meeting. According to the estimated policy rule, the Fed is mainly reacting to information contained in the yield-curve. Surprises in analyst forecasts turn out to be merely temporary components of macro variables, so that the “hump-shaped” yield response to these surprises is not consistent with a Taylor-type policy rule.

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This paper is a critical survey of models designed for pricing fixed income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by overviewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have “switching regimes. ” Then the goodness-of-fits of these models are assessed relative to their abilities to: (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative fits to historical yield spreads. 1

A new Bayesian method is proposed for the analysis of discretely sampled diffusion processes. The method, which is termed high frequency augmentation (HFA), is a simple numerical method that is applicable to a wide variety of univariate or multivariate diffusion and jump-diffusion processes. It is furthermore useful when observations are irregularly observed, when one or more elements of the multivariate process are latent, or when microstructure effects add error to the observed data. The Markov chain-Monte Carlo-based procedure can be used to attain the posterior distributions of the parameters of the drift and diffusion functions as well as the posteriors of missing or latent data. Several examples are explored. First, posteriors of the parameters of a geometric Brownian motion are attained using HFA and compared with those obtained using standard analytical methods in a short Monte Carlo study. Second, a stochastic volatility model is estimated on a sample of S&P500 returns, a problem for which posteriors are analytically intractable. Third, it is shown how the method can be used to estimate an interest rate process using data that suffer from severe rounding. Finally, extension of the method to jump-diffusions is described and applied to the analysis of the U.S dollar/German mark exchange rate.

Introduction Estimating functions provide a general framework for finding estimators and studying their properties in many di#erent kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obtained by equating the estimating function to zero and solving the resulting estimating equation with respect to the parameter. The idea of using estimating equations is an old one and goes back at least to Karl Pearson's introduction of the method of moments. The term estimating function may have been coined by Kimball (1946). The estimating function approach has turned out to be very useful in obtaining, improving and studying estimators for discretely sampled parametric di#usion-type models, where the likelihood function is usually not explicitly known. Estimating functions are often constructed by combining relationships (dependent on the unknown parameter) between an observa