Abstract: This paper examines the joint time series of the S&P 500 index and near-the-money short-dated option prices with an arbitrage-free model, capturing both stochastic volatility and jumps. Jump-risk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jump-risk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility “smirks” of cross-sectional options data.

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

As is well known, the classic Black-Scholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that time-changed Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

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A rapidly growing literature has documented important improvements in financial return volatility measurement and forecasting via use of realized variation measures constructed from high-frequency returns coupled with simple modeling procedures. Building on recent theoretical results in Barndorff-Nielsen and Shephard (2004a, 2005) for related bi-power variation measures, the present paper provides a practical and robust framework for non-parametrically measuring the jump component in asset return volatility. In an application to the DM/ $ exchange rate, the S&P500 market index, and the 30-year U.S. Treasury bond yield, we find that jumps are both highly prevalent and distinctly less persistent than the continuous sample path variation process. Moreover, many jumps appear directly associated with specific macroeconomic news announcements. Separating jump from non-jump movements in a simple but sophisticated volatility forecasting model, we find that almost all of the predictability in daily, weekly, and monthly return volatilities comes from the non-jump component. Our results thus set the stage for a number of interesting future econometric developments and important financial applications by separately modeling, forecasting, and pricing the continuous and jump components of the total return variation process.

We analyze the specifications of option pricing models based on time-changed Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we must incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.

Prices of equity index put options contain information on the price of systematic downward jump risk. We use a structural jump-diffusion firm value model to assess the level of credit spreads that is generated by option-implied jump risk premia. In our compound option pricing model, an equity index option is an option on a portfolio of call options on the underlying firm values. We calibrate the model parameters to historical information on default risk, the equity premium and equity return distribution, and S&P 500 index option prices. Our results show that a model without jumps fails to fit the equity return distribution and option prices, and generates a low out-of-sample prediction for credit spreads. Adding jumps and jump risk premia improves the fitofthe model in terms of equity and option characteristics considerably and brings predicted credit spread levels much closer to observed levels.

The growth of the option pricing literature parallels the spectacular developments of deriva-tive securities and the rapid expansion of markets for derivatives in the last three decades. Writing a survey of option pricing models appears therefore like a formidable task. To delimit our focus we will put emphasis on the more recent contributions since there are

Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a time-series of asset returns. This paper compares a range of GARCH models along a different dimension, using option prices and returns under the risk-neutral as well as the physical probability measure. We judge the relative performance of various models by evaluating an objective function based on option prices. In contrast with returns-based inference, we find that our option-based objective function favors a relatively parsimonious model. Specifically, when evaluated out-of-sample, our analysis favors a model that besides volatility clustering only allows for a standard leverage effect. JEL Classification: G12

This paper provides a simple unified framework for assessing the empirical linkages between returns and realized and implied volatilities. First, we show that whereas the volatility feedback effect as measured by the sign of the correlation between contemporaneous return and realized volatility depends importantly on the underlying structural model parameters, the correlation between return and implied volatility is unambiguously positive for all reasonable parameter configurations. Second, the lagged return-volatility asymmetry, or the leverage effect, is always stronger for implied than realized volatility. Third, implied volatilities generally provide downward biased forecasts of subsequent realized volatilities. Our results help explain previous findings reported in the extant empirical literature, and is further corroborated by new estimation results for a sample of monthly returns and implied and realized volatilities for the aggregate S&P market index. JEL Classification: G12, C51, C22.