[Paper Review] Hybrid scheme for Brownian semistationary processes

Mikkel Bennedsen, Asger Lunde+1 more|RePEc: Research Papers in Economics|Jul 10, 2015

Stochastic processes and financial applications4 citations

TL;DR

This paper proposes a hybrid discretization scheme for simulating Brownian semistationary (BSS) processes with kernels exhibiting power-law behavior near zero. By combining a power-function approximation near zero and a step-function approximation elsewhere, the scheme achieves superior mean square error performance—significantly outperforming standard Riemann-sum methods—while maintaining identical computational complexity.

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(6), 887-904, 2016], respectively.

Motivation & Objective

• Address the poor convergence of standard Riemann-sum discretization for BSS processes with singular kernels near zero.
• Develop a simulation scheme that captures the steepness of regularly varying kernels near zero more accurately.
• Maintain computational efficiency while improving mean square error (MSE) in the discretization of BSS processes.
• Enable accurate finite-sample inference and Monte Carlo pricing in models with rough volatility, such as the rough Bergomi model.

Proposed method

• Approximate the kernel function $ g $ in the stochastic integral representation of BSS processes using a power function near zero and a step function for larger lags.
• Construct the hybrid scheme as a linear combination of Wiener integrals (with respect to Brownian motion) and Riemann sums.
• Use the theory of regular variation to rigorously characterize the behavior of $ g(x) \propto x^\alpha $ near zero for $ \alpha \in (-\frac{1}{2}, \frac{1}{2}) \setminus \{0\} $.
• Derive asymptotic expressions for the mean square error (MSE) of the hybrid scheme using integral identities involving the Gauss hypergeometric function $ {}_2F_1 $.
• Establish theoretical bounds on the MSE components using Potter’s bound and properties of slowly varying functions.
• Validate the scheme through numerical experiments on rough volatility estimation and option pricing in the rough Bergomi model.

Experimental results

Research questions

• RQ1 How can the discretization error of BSS processes be reduced when the kernel function has a power-law singularity at zero?
• RQ2 What is the asymptotic behavior of the mean square error for a hybrid discretization scheme combining power-law and step-function kernel approximations?
• RQ3 Does the proposed hybrid scheme achieve better accuracy than the standard forward Riemann-sum scheme without increasing computational cost?
• RQ4 Can the hybrid scheme be effectively applied to finite-sample estimation of the roughness parameter $ \alpha $ in rough volatility models?
• RQ5 How does the hybrid scheme perform in Monte Carlo simulations for option pricing under the rough Bergomi model?

Key findings

• The hybrid scheme achieves a mean square error that decays as $ n^{-(2\alpha+1)}L_g(1/n)^2 $, where $ L_g $ is a slowly varying function, matching the theoretical optimal rate.
• The mean square error of the hybrid scheme is asymptotically dominated by the error from the power-function approximation near zero, which is significantly smaller than the error from the Riemann-sum scheme.
• The hybrid scheme improves accuracy substantially over the standard forward Riemann-sum method, even though both have the same computational complexity.
• Numerical experiments confirm that the hybrid scheme yields more accurate finite-sample estimates of the roughness parameter $ \alpha $ in BSS processes.
• In Monte Carlo option pricing under the rough Bergomi model, the hybrid scheme reduces bias and improves convergence compared to standard discretization methods.
• Theoretical analysis confirms that the hybrid scheme’s error is asymptotically equivalent to $ \mathbb{E}[\sigma(0)^2] D_n' $, where $ D_n' \sim n^{-(2\alpha+1)}L_g(1/n)^2 $, establishing its optimality in the asymptotic regime.