This paper examines continuous-time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood-based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998. Using formal and informal diagnostics, we ¢nd strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing.

This paper examines the empirical performance of jump diffusion models of stock price dynamics from joint options and stock markets data. The paper introduces a model with discontinuous correlated jumps in stock prices and stock price volatility, and with state-dependent arrival intensity. We discuss how to perform likelihoodbased inference based upon joint options/returns data and present estimates of risk premiums for jump and volatility risks. The paper finds that while complex jump specifications add little explanatory power in fitting options data, these models fare better in fitting options and returns data simultaneously.THE STATISTICAL PROPERTIES of stock returns have long been of interest to financial decision makers and academics alike. In particular, the great stock market crashes of the 20 th century pose particular challenges to economic and statistical models. In the past decades, there have been elaborate efforts by researchers to build models that explicitly allow for large market movements, or "fat tails" in return distributions. The literature has mainly focused on two approaches: (1) time-varying volatility models that allow for market extremes to be the outcome of normally distributed shocks that have a randomly changing variance, and (2) models that incorporate discontinuous jumps in the asset price.Neither stochastic volatility models nor jump models have alone proven entirely empirically successful. For example, in the time-series literature, the models run into problems explaining large price movements such as the October 1987 crash. For stochastic volatility models, one problem is that a daily move of −22% requires an implausibly high-volatility level both prior to, and after the crash. Jump models on the other hand, can easily explain the crash of 1987 by a parameterization that allows for a sufficiently negative jump. However, jump models typically specify jumps to arrive with constant intensity. This assumption poses problems in explaining the tendency of large movements to cluster over time. In the case of the 1987 crash, for example, there were large movements both prior to, and following the crash. With respect to option prices, * Eraker is from Duke University. I am indebted to Rick Green (the editor) and an anonymous referee for their helpful comments. I also thank

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The paper examines equilibrium models based on Epstein-Zin preferences in a framework in which exogenous state variables follow affine jump diffusion processes. A main insight is that the equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by imposing parametric restrictions on market prices of risk, determined inside the model by preference and model parameters. An appealing characteristic of the general equilibrium setup is that the state variables have an intuitive and testable interpretation as driving the consumption and dividend dynamics. We present a detailed example where large shocks (jumps) in consumption volatility translate into negative jumps in equilibrium prices of the assets as agents demand a higher premium to compensate for higher risks. This endogenous "leverage effect," which is purely an equilibrium outcome in the economy, leads to significant premiums for out-of-themoney put options. Our model is thus able to produce an equilibrium "volatility smirk," which realistically mimics that observed for index options. Disciplines Finance | Finance and Financial Management CommentsAt the time of publication, author Ivan Shaliastovich was affiliated with Duke University. Currently, he is a faculty member at the Wharton School at the University of Pennsylvania.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/fnce_papers/307 An Equilibrium Guide to Designing Affine Pricing ModelsBjørn Eraker and Ivan Shaliastovich * Duke University AbstractWe examine equilibrium models based on Epstein-Zin preferences in a framework where exogenous state variables which drive consumption and dividend dynamics follow affine jump diffusion processes. Equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by imposing parametric restrictions on market prices of risk, determined by preference and model parameters. We present a detailed example where large shocks (jumps) in consumption volatility translate into negative jumps in equilibrium prices of the assets. This endogenous "leverage effect" leads to significant premiums for out-of-the-money put options. Our model is thus able to produce an equilibrium "volatility smirk" which realistically mimics that observed for index options.

No-arbitrage models are extremely flexible modelling tools but often lack economic motivation. This paper describes an equilibrium consumption-based CAPM framework based on Epstein-Zin preferences, which produces analytic pricing formulas for stocks and bonds under the assumption that macro growth rates follow affine processes. This allows the construction of equilibrium pricing formulas while maintaining the same flexibility of state dynamics as in no-arbitrage models. In demonstrating the approach, the paper presents a model that incorporates inflation such that asset prices are nominal. The model takes advantage of the possibility of non-Gaussian shocks and model macroeconomic uncertainty as a jump-diffusion process. This leads to endogenous stock market crashes as stock prices drop to reflect a higher expected rate of return in response to sudden increases in risk. The nominal yield curve in this model has a positive slope if expected inflation growth negatively impacts real growth. This model also produces asset prices that are consistent with observed data, including a substantial equity premium at moderate levels of risk aversion.finance, investment, asset pricing, probability, diffusion, stochastic model applications

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Economic data are collected at various frequencies but econometric estimation typically uses the coarsest frequency. This paper develops a Gibbs sampler for estimating VAR models with mixed and irregularly sampled data. The Gibbs sampler allows efficient likelihood inference and uses simple conjugate posteriors even in high dimensional parameter spaces, avoiding a non-Gaussian likelihood surface even when the Kalman filter applies. Two examples studying the relationship between financial data and the real economy illustrate the methodology and demonstrates efficiency gains from the mixed frequency estimator.