Modeling and Forecasting Realized Volatility

Abstract

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark / Dollar and Yen / Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quantile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.KEYWORDS: Continuous-time methods, quadratic variation, realized volatility, realized correlation, highfrequency data, exchange rates, vector autoregression, long memory, volatility forecasting, correlation forecasting, density forecasting, risk management, value at risk. _________________ * This paper supercedes the earlier manuscript "Forecasting Volatility: A VAR for VaR." The work reported in the paper was supported by the National Science Foundation. We are grateful to Olsen and Associates, who generously made available their intraday exchange rate quotation data. For insightful suggestions and comments we thank Rob Engle, Atsushi Inoue, Neil Shephard, Clara Vega, Sean Campbell, and seminar participants at Chicago, Michigan, Montreal/CIRANO, NYU, Rice, and the June 2000 Meeting of the Western Finance Association. and covariances (or correlations). When important, the precise meaning will be clear from context.

Computationally efficient bootstrap prediction intervals for returns and volatilities in ARCH and GARCH processes

“…dollar exchange rate, i.e., the number of Yen per U.S. dollar. In order to avoid modeling particular weekend effects, we exclude all of the observations on Saturdays and Sundays (Andersen et al, 2003). Consequently, our full sample includes the daily average Yen/U.S.…”

Spillover effects from London and Frankfurt to Central and Eastern European stock markets

“…The paper uses three approaches to evaluate stock market comovement in the daily returns of European stock exchanges: (1) Following Andersen, et al (2003), the authors define daily returns as …”

Nonlinear models for strongly dependent processes with financial applications

“…Of particular relevance is Diebold and Inoue (2001) who showed that a process with Markov switching regimes can be mistaken for a long memory process; so that long memory can arise from some forms of nonlinearity. A priori it would seem reasonable for the long memory property found in macroeconomic series such as in ‡ation to be due to regime shifts in monetary policy; while realized volatility in frictionless …nancial markets seem to be pure long memory; Andersen, Bollerslev, Diebold and Labys (2003). However, the application of the tests developed by Baillie and Kapetanios (2007) suggested the existence of both nonlinear and long memory components in many economic and …nancial time series.…”