Hybrid scheme for Brownian semistationary processes

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 appro...

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Bibliographic Details
Published in:Finance and stochastics 2017-10, Vol.21 (4), p.931-965
Main Authors: Bennedsen, Mikkel, Lunde, Asger, Pakkanen, Mikko S.
Format: Article
Language:English
Subjects:
Computer simulation
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Finance
Insurance
Integrals
Kernel functions
Management
Mathematical models
Mathematics
Mathematics and Statistics
Monte Carlo simulation
Power
Probability Theory and Stochastic Processes
Quantitative Finance
Simulation
Statistics for Business
Step functions
Turbulence models
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Summary: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.
ISSN:0949-2984
1432-1122
DOI:10.1007/s00780-017-0335-5