On the optimal forecast with the fractional Brownian motion

Abstract

This paper investigates the performance of different forecasting formulas with fractional Brownian motion based on discrete and finite samples. Existing literature presents two formulas for generating optimal forecasts when continuous records are available. One formula relies on a history over an infinite past, while the other is designed for a record limited to a finite past. In reality, only observations at discrete time points over a finite past are available. In this case, the forecasting formula, which has been widely used in the literature, is the one obtained by Gatheral et al. (Volatility is rough. Quant. Finance, 2018, 18(6), 933–949) that truncates and discretizes the formula based on continuous records over an infinite past. The present paper advocates an alternative forecasting formula, which is the conditional expectation based on finite past discrete-time observations. The findings suggest that the conditional expectation approach produces more accurate forecasts than the existing method, as demonstrated by both simulated data and actual daily realized volatility (RV) observations. Moreover, we also provide empirical evidence showing that the conditional expectation approach can lead to larger economic values than the existing method.

Keywords:

• Fractional Brownian motion
• Conditional expectation
• Optimal forecast

Open Scholarship

This article has earned the Center for Open Science badges for Open Data and Open Materials through Open Practices Disclosure. The data and materials are openly accessible at https://fba.um.edu.mo/wp-content/uploads/2023/12/code-and-data-1.zip.

Acknowledgments

We wish to thank Mathieu Rosenbaum, Shuping Shi, Weilin Xiao and Masaaki Fukasawa for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Code and data availability statement

Code can be found online at https://fba.um.edu.mo/wp-content/uploads/2023/12/code-and-data-1.zip. The data that support the findings of this study are available at https://dachxiu.chicagobooth.edu/\#risklab.

Supplemental data

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/14697688.2023.2297730.

Notes

1 https://sites.google.com/site/roughvol/home/risks-1.

2 We are grateful to Mathieu Rosenbaum to confirm that this formula.

3 Although ${\int }_0^T{w}_2(\tau )\mathrm{d}\tau =1$, normalizing $w_2(\tau )$ is necessary as the numerical approximation to ${\int }_0^T{w}_2(\tau )\mathrm{d}\tau$ can be different from one.

4 The performance of alternative discretization schemes will be reported in the Online Supplement simply because these schemes have not been used in the literature. In the Online Supplement, we will also perform k-period-ahead forecasts with $k=2,\dots ,10$.

5 We will also perform k-period-ahead forecasts with $k=2,\dots ,10$ using alternative discretization schemes. The results are reported in the Online Supplement and qualitatively unchanged.

6 Since the daily RV is based on returns from open to close and the conditional variance is based on open to open, following Fleming et al. (2003), we use a bias-correction factor $\delta =\frac{E(r_{T\mathrm{\Delta }}^2)}{E(R{V}_{T\mathrm{\Delta }})}$ to adjust RV, that is, $\mathrm{Va}{r}_{T\mathrm{\Delta }}(r_{m,(T+k)\mathrm{\Delta }})=\delta {\widehat{\mathrm{RV}}}_{(T+k)\mathrm{\Delta }}$.

Additional information

Funding

Wang acknowledges the support from Innovative Research Groups Project of the National Natural Science Foundation of China under No. 72121002, and the support from Shanghai Pujiang Program under No. 22PJC022. Yu and Zhang would like to acknowledge that this research/project is supported by the Ministry of Education, Singapore, under its Academic Research Fund (AcRF) Tier 2 (Award Number MOE-T2EP402A20-0002).