Elsevier

Journal of Empirical Finance

Volume 6, Issue 5, December 1999, Pages 457-477
Journal of Empirical Finance

Forecasting financial market volatility: Sample frequency vis-à-vis forecast horizon

https://doi.org/10.1016/S0927-5398(99)00013-4Get rights and content

Abstract

This paper explores the return volatility predictability inherent in high-frequency speculative returns. Our analysis focuses on a refinement of the more traditional volatility measures, the integrated volatility, which links the notion of volatility more directly to the return variance over the relevant horizon. In our empirical analysis of the foreign exchange market the integrated volatility is conveniently approximated by a cumulative sum of the squared intraday returns. Forecast horizons ranging from short intraday to 1-month intervals are investigated. We document that standard volatility models generally provide good forecasts of this economically relevant volatility measure. Moreover, the use of high-frequency returns significantly improves the longer run interdaily volatility forecasts, both in theory and practice. The results are thus directly relevant for general research methodology as well as industry applications.

Introduction

The expected future volatility of financial market returns is the main ingredient in assessing asset or portfolio risk and plays a key role in derivatives pricing models. Thus, not surprisingly, much effort has been devoted to modeling return volatility dynamics. Meanwhile, it is difficult to extract a coherent set of prescriptions concerning the most appropriate empirical procedure for volatility forecasting from the existing literature. In fact, some recent surveys, e.g., Figlewski (1997), have come to the conclusion that simple averages of historical volatility generally predict future volatility better than standard volatility models, such as the popular GARCH(1,1) specification. Moreover, it has often been suggested that there is little, if any, gain from using higher frequency return data when forecasting volatility over longer horizons. On a similar note, Canina and Figlewski (1993)conclude that implied volatility from a liquid equity-index option market contains no useful information about the future volatility. Furthermore, a string of recent studies, e.g., Cumby et al. (1993), Jorion (1995), and West and Cho (1995), all point to a low degree of explanatory power provided by volatility forecasts vis-à-vis the subsequent realized squared or absolute returns. Taken together, these empirical findings call into question the practical usefulness of standard volatility models.

Closer scrutiny reveals that the seemingly poor predictive performance of standard volatility models actually is to be expected, even when the specific volatility model entertained constitutes the true data generating process. In particular, as argued by Andersen and Bollerslev (1998b), henceforth AB, the standard expedient of measuring ex-post volatility by the corresponding squared or absolute realized returns is destined to be very inaccurate, and this affects any associated judgment regarding the quality of competing volatility forecasts. Although the squared returns constitute an unbiased estimator for the latent volatility factor, they also embody a large idiosyncratic component that is unrelated to the actual volatility driving the market over the observation interval.3 As such, gauging the usefulness of volatility forecasts requires a more refined articulation of the relevant volatility concept as well as the construction of a volatility measure that captures this notion in an empirically sensible fashion. AB propose such a measure based on the cumulative squared returns obtained from high-frequency data. This integrated volatility measure is closely related to the notion of the squared variation process of a continuous-time diffusion, and it corresponds directly to the notions of volatility entertained in diffusion models, as developed formally by Barndorff-Nielsen and Shephard (1998). This concept of volatility is also consistent with the type of volatility measures that have been emphasized in the stochastic volatility option pricing literature, e.g., Hull and White (1987). More importantly in the present context, the integrated ex-post volatility measure allows for the construction of more meaningful and accurate volatility forecast evaluation criteria. Building on these ideas, AB show that for one-day-ahead exchange rate volatility forecasts, a daily GARCH(1,1) model vastly outperforms the corresponding historical unconditional volatility forecasts.

The present paper extends the analysis in AB in several important new directions. First, whereas AB are concerned exclusively with the performance of one-day-ahead forecasts based on a model for daily returns, we explore how much modeling of high-frequency intraday returns may improve daily, or longer-run, volatility forecasts. Our theoretical results exploit the GARCH(1,1) diffusion approximations of Nelson (1990)and Drost and Werker (1996). This approach ensures that standard discrete-time inference remains valid, even when the data are generated by an underlying diffusion, thus allowing for a direct analysis of the forecast performance of the corresponding “weak” GARCH(1,1) models across different forecast horizons and sample frequencies. The theoretical results indicate that the improvements obtained by increasing the sampling frequency may be quite dramatic. Second, the analysis in AB is restricted to a 1-day forecast horizon only. However, the majority of practical applications, including most value-at-risk (VaR) type calculations, involve horizons beyond one day. Here, we show that even over a 1-month horizon, moving from, say, a monthly to a daily to an hourly sampling frequency may produce quite significant improvements in terms of the volatility forecast errors.4 Third, the forecast comparisons reported in AB are based exclusively on the R2 from the regression of the model forecast on the ex-post realized volatility measure. Although this is arguably the most commonly employed criteria in the existing literature, it is not necessarily the best criteria to adopt when evaluating nonlinear volatility forecasts. Consequently, we also report results from a number of alternative more robust evaluation criteria. Lastly, the empirical analysis in AB is based on a relatively short 1-year sample of 5-minute exchange rate returns. In contrast, the empirical results reported in the present paper rely on a much longer 10-year sample of more than 700,000 5-minute Deutschemark–US dollar (DM–$) returns. Anticipating the empirical results, we find that the qualitative theoretical predictions from the simple discrete-time GARCH(1,1) models hold up reasonably well for sampling frequencies ranging down to about an hour, thus confirming the potential for important improvements in actual forecast performance from explicitly incorporating the information in high-frequency data when calculating longer-run interdaily forecasts. However, the theoretical model predictions falter dramatically for the very shortest intraday sampling frequencies. We attribute these findings to a host of important market microstructure features, including a pronounced intraday volatility pattern, a more complicated multiple volatility component structure, and the existence of discrete price jumps.

The plan for the rest of the paper is as follows. Section 2establishes the notation and reviews the framework for both continuous-time and discrete-time volatility modeling and forecasting. Section 3reports on the simulated theoretical forecast performance of the discrete-time weak GARCH(1,1) models for the different sampling frequencies and forecasting horizons. The corresponding empirical findings for the 10-year sample of 5-minute DM–$ returns are detailed in Section 4. Section 5concludes.

Section snippets

Notation

To set out the notation, let pt denote the continuous time t≥0 logarithmic price of the asset, where the unit interval corresponds to one day. The discretely observed time series process of continuously compounded returns with m observations per day, or a return horizon of 1/m, is then defined byr(m),t≡pt−pt−1/m,where t=1/m, 2/m, …. The corresponding instantaneous return will be denoted by dpt, or r(∞),t. We shall throughout assume that expected returns are zero.

Theoretical volatility forecast evaluation

In this section we report on the accuracy of the forecasts for the latent integrated volatility factor, ∫0hσt+τ2dτ, obtained from the discrete-time weak GARCH(1,1) approximations in Eq. (9). For all of the forecasts, the length of the forecast horizon exceeds the sampling frequency, so that h≥1/m. Also, the forecasts based on a constant variance will be denoted by m=0.

Deutschemark–US dollar volatility

Section 3confirms and extends the theoretical results in AB: standard volatility models do have the ability to capture a significant portion of the predictability of the volatility process, even out to a 1-month horizon. In addition, the theoretical results in Table 2, Table 3, Table 4, Table 5, Table 6 also illustrate the significant gains afforded by the use of high-frequency intraday returns when forecasting the volatility process. Do these theoretical observations carry any significance for

Concluding remarks

One would generally expect a broad consensus regarding the basic concepts underlying empirical measurement and forecasting of financial market volatility. After all, volatility modeling has been the subject of a voluminous literature for more than a decade now. Nonetheless, upon reflection some potentially important drawbacks associated with the notions implicitly or explicitly adopted by standard models surface. One problem is that the concept of volatility used in empirical work remains

Acknowledgements

This work was supported by a grant from the NSF to the NBER. We are grateful to Olsen and Associates for making the intradaily exchange rate quotations available. We also thank seminar participants at Stanford University, the High Frequency Data in Finance-II conference in Zürich, Switzerland, and the Isaac Newton Institute Workshop on Econometrics and Financial Time Series in Cambridge, UK, for helpful comments.

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