Articlehybrid2017

Hybrid scheme for Brownian semistationary processes

128 citations0/0 files storedFinance and Stochastics

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Stochastic processes and financial applications Numerical methods for differential equations Mathematical Approximation and Integration

Abstract

We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel function by a power function near zero and by a step function elsewhere. The resulting approximation of the process is a combination of Wiener integrals of the power function and a Riemann sum, which is why we call this method a hybrid scheme. Our main theoretical result describes the asymptotics of the mean square error of the hybrid scheme, and we observe that the scheme leads to a substantial improvement of accuracy compared to the ordinary forward Riemann-sum scheme, while having the same computational complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we examine the finite-sample properties of an estimator of the roughness parameter of a Brownian semistationary process and study Monte Carlo option pricing in the rough Bergomi model of Bayer et al. (Quant. Finance 16:887–904, 2016), respectively.

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References

2007Stochastic Simulation 2008Stochastic Calculus for Fractional Brownian Motion and Related Processes 2015Pricing under rough volatility 2004Stochastic Models That Separate Fractal Dimension and the Hurst Effect 2007On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

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