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.

/pdf/roughing-it-up-including-jump-components-in-the-measurement-35xg2w5blh.pdf

Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility

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 and risk-free discount rates. It has become increasingly evident that such "classical" models fail to account adequately for the underlying dynamic evolution of asset prices and interest rates. Not surprisingly, the inadequacy of these specifications also shows up in bond and derivatives pricing, where the standard representations falter systematically. For example, the BlackScholes pricing formula, although widely used by practitioners, is well known to produce pronounced and persistent biases in the pricing of options. Devi

/pdf/an-empirical-investigation-of-continuous-time-equity-return-3hlodksuv4.pdf

An empirical investigation of continuous-time equity return models

/pdf/roughing-it-up-including-jump-components-in-the-measurement-kou7unini2.pdf

Roughing it Up: Including Jump Components in the Measurement, Modeling and Forecasting of Return Volatility

Publisher Summary This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. It defines different volatility concepts; the notional volatility corresponding to the sample-path return variability over a fixed time interval; the expected volatility over a fixed time interval; and the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently popularized realized volatility measures for fixed-length time intervals.

/pdf/parametric-and-nonparametric-volatility-measurement-17a3b400ep.pdf

Parametric and Nonparametric Volatility Measurement

The Black-Scholes call option pricing model exhibits systematic empirical biases. The Merton call option pricing model, which explicitly admits jumps in the underlying security return process, may potentially eliminate these biases. We provide statistical evidence consistent with the existence of lognormally distributed jumps in a majority of the daily returns of a sample of NYSE listed common stocks. However, we find no operationally significant differences between the Black-Scholes and Merton model prices of the call options written on the sampled common stocks.

On Jumps in Common Stock Prices and Their Impact on Call Option Pricing

The specification of the stochastic process for stock prices is an important assumption underlying most equilibrium and option-pricing models in modern finance. This is particularly true of the continuous-time capital asset pricing models where it is assumed that asset prices follow diffusion processes with continuous sample paths (see Merton 1973a; Breeden 1979; Grossman and Shiller 1982). The extent to which the implications of these models generalize to discontinuous sample paths is an unanswered question in theoretical finance. However, the answer to this question is of more than just theoretical interest since accumulated empirical evidence appears to be inconsistent with the continuous sample path assumption (see Rosenberg 1972; Oldfield, Rogalski, and Jarrow 1977; Rosenfeld 1982). One purpose of this paper is to provide some answers to this question. We do this by extending Merton's (1973a) intertemporal asset pricing model, in the special case of a constant investment opportunity set, to include discontinuous sample paths for asset prices. The constant investment opportunity set assumption is imposed because we are interested in obtaining sufficient conditions under which an instantaneous capital asset pricing model (CAPM) results. The extension of our model to a stochastic opportunity set is discussed briefly in the text. This paper investigates an economy where stock prices follow a jump-diffusion process. The first part of the analysis derives a sufficient condition under which an instantaneous CAPM will characterize equilibrium expected returns. The sufficient condition is that the jump component of a stock's return must be "unsystematic" and diversifiable in the market portfolio. The second part of the paper is an empirical investigation of the satisfaction of this condition. The evidence is seen to be inconsistent with the satisfaction of this hypothesis, that is, the market portfolio appears to contain a jump component.

Jump Risks and the Intertemporal Capital Asset Pricing Model

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Jump Risks and the Intertemporal Capital Asset Pricing Model