Estimating Stochastic Volatility: The Rough Side to Equity Returns

Editor’s note: This post is part of a series showcasing Barcelona GSE master projects by students in the Class of 2017. The project is a required component of every master program.


Lukas Grimm, Jonathan Haynes and Daniel Schmitt

Master’s Program:


Paper Abstract:

This Project evaluates the forecasting performance of a Brownian Semi-Stationary (BSS) process in modelling the volatility of 21 equity indices. We implement a sophisticated Hybrid Scheme to simulate BSS processes with high efficiency and precision. These simulations are useful to price derivatives, accounting for rough volatility. Then we calibrate the BSS parameters for the realised kernel of 21 equity indices, using data from the Oxford-Man Institute. We conduct one- and ten-step ahead forecasts on six indices and find that the BSS outperforms our benchmarks, including a Log-HAR specification, in the majority of cases.


This project confirms the findings of Gatheral et al. (2014) and Bennedsen et al. (2016) that volatility is indeed both rough and persistent across a wide range of equity indices. We have explored the advantage of using a Brownian Semi-Stationary (BSS) process to model volatility enabling the user to calibrate both stylised facts in contrast to previous generations of fractal processes, like Fractional Brownian Motion. We have successfully implemented simulation methods so that a BSS process can be incorporated within a continuous time asset pricing equation to price options and other exotic derivatives. We then calibrated the parameters for the BSS model using the realised kernel of 21 equity indices. Our parameter estimates confirm the expected roughness and persistence in the series. The parameter for roughness, α, was quite stable across the cross-section of indices, but fluctuated over time. α averaged -0.37 and ranged from −0.33 to −0.42, implying much more roughness than the α = 0 implied by Standard Brownian Motion. Estimates of the long memory parameter, λ, were less stable, ranging from 0.0041 to 0.0230. We identify an issue when using MoM estimation that suggests MoM may be sub-optimal for BSS-Gamma forecasting. We forecast with six indices that cover a broad geographical spread and have stable lambda estimates. For the one-step ahead forecast we find that the BSS model outperformed two of our three benchmarks consistently under both MSE and QL loss functions. The BSS beat the Log-HAR benchmark in the case of the index with the longest memory, while it was slightly worse for the other five indices. For the ten-step ahead forecast, under the MSE loss function, the BSS model outperformed all benchmarks consistently for five out of six indices. Under the QL loss function the BSS outperforms all benchmarks, and this outperformance is always statistically significant.
Areas for further research would include investigating the forecasting accuracy of the BSS Power Kernel using a wider range of asset class, such as commodities, real estate funds and foreign exchange rates. Further robustness checks could test the performance of BSS against the family of fractional volatility models. It would also be interesting to further explore the relationship of ξ and its link with the variance swap curve.


The full version of this Master Project can be found here



4 thoughts on “Estimating Stochastic Volatility: The Rough Side to Equity Returns”

    1. Dear Nicolas, A link to the full version of this Master Project has been added to end of the post. Thank you for your interest in the piece.

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