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Hybrid scheme for Brownian semistationary processes

Mikkel Bennedsen, Asger Lunde, Mikko S. Pakkanen

arXiv:1507.03004v4math.PRq-fin.PR

TL;DR

When stochastic volatility is present, Brownian semistationary processes are neither Markovian nor Gaussian, making exact simulation difficult. The paper introduces a time-domain hybrid discretization of the kernel, with improved asymptotic RMSE over forward Riemann sums at the same computational complexity.

  • Problem

    With stochastic volatility, Brownian semistationary processes are neither Markovian nor Gaussian, making exact simulation difficult and requiring approximate methods in the general case.

    Source: §1 · p. 2

  • Method

    The hybrid scheme approximates the kernel by a power function near zero and a step function elsewhere, combining Wiener integrals with a Riemann sum.

    Source: §1 · p. 3

  • Results

    The hybrid scheme reduces asymptotic RMSE by at least 50% for α∈(0,1/2) and at least 80% for α∈(-1/2,0), with reduction close to 100% as α approaches -1/2.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The scheme has the same computational complexity as the Riemann-sum scheme, and rough Bergomi implied-volatility smiles are indistinguishable from those using exact volatility simulation.

    Source: §1 · p. 3

    The paper does not pursue truncation-dominated MSE asymptotics when condition (2.9) fails, which would require additional assumptions.

    Source: §2.4 · p. 10

Abstract

from arXiv · show

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.

1 Introduction

Brownian semistationary processes combine volatility-modulated Brownian noise with a kernel whose behavior can represent roughness and whose local and global properties are decoupled. The paper introduces a time-domain hybrid discretization that improves asymptotic accuracy over the forward Riemann-sum scheme while retaining the same computational complexity.

1 Introduction

  • BSS processes combine Brownian noise modulated by stochastic volatility with a kernel, allowing local roughness and global dependence behavior to be specified independently.The process can also be viewed as a moving average of volatility-modulated Brownian noise.Source: §1 · p. 2
  • With stochastic volatility, exact simulation is difficult because the process is neither Markovian nor Gaussian, while a step-function kernel approximation fails to capture steepness near zero.The paper notes that approximation is therefore needed in the general case.Source: §1 · pp. 2–3
  • The hybrid scheme approximates the kernel by a power function near zero and a step function elsewhere, combining Wiener integrals with a Riemann sum at the same computational complexity as the Riemann-sum scheme.This time-domain approximation targets the step-function scheme’s failure to capture the kernel’s steepness near zero.Source: §1 · p. 3
  • The hybrid scheme reduces asymptotic RMSE by at least 50% for α∈(0,1/2) and at least 80% for α∈(-1/2,0), with reduction close to 100% as α approaches -1/2.The authors say this addresses the poor precision of the Riemann-sum scheme.Source: §1 · p. 3
  • Numerical experiments examine finite-sample estimation of the roughness index and rough Bergomi option pricing, where hybrid-simulated implied-volatility smiles are indistinguishable from those using exact volatility simulation.The rough Bergomi model is identified as a stochastic volatility model.Source: §1 · p. 3

2 The model and theoretical results

This section defines Brownian semistationary processes and develops a hybrid discretization, characterizing its mean square error and relating kernel behavior near zero to process roughness. It also examines truncation error and extends the error analysis to truncated processes, including their local increments.

2.1 Brownian semistationary process

  • The assumptions imply that X is covariance stationary, but it need not be strictly stationary because dependence between σ and W may vary over time.Source: §2.1 · p. 4

2.2 Kernel function

  • Under the kernel assumptions, X’s variogram is regularly varying at zero with index 2α + 1, and X has a locally Hölder-continuous modification.The paper calls α the process’s roughness index.Source: §2.2 · p. 6

2.3 Hybrid scheme

  • The hybrid scheme uses Wiener integrals of a power-function approximation near zero and step-function approximations elsewhere, improving accuracy over forward Riemann sums.The scheme also permits evaluation points within specified grid intervals.Source: §2.3 · pp. 7–8

2.4 Asymptotic behavior of mean square error

  • Under (A1)–(A4), the hybrid scheme’s MSE is asymptotic to J(α, κ, b)E[σ(0)^2]n^−(2α+1)Lg(1/n)^2.The asymptotic error is determined by the kernel’s behavior near zero; the stated condition ensures truncation error is asymptotically smaller.Source: §2.4 · pp. 9–10
  • Increasing κ decreases J, and the sequence b* minimizes it; numerically, increasing κ beyond 2 appears to yield no significant further RMSE reduction.Using b* rather than forward evaluation points is especially important when α ∈ (0, 1/2).Source: §2.4 · pp. 10–11
  • For α ∈ (−1/2, 0), using κ ≥ 1 substantially reduces asymptotic RMSE versus the forward Riemann-sum scheme and avoids its blow-up as α approaches −1/2.For α > 0, the improvement is more modest but still considerable.Source: §2.4 · pp. 10–11, Figure 1 · p. 11

2.5 Extension to truncated Brownian semistationary processes

  • For truncated BSS processes, the time-dependent variogram is regularly varying at zero with index 2α + 1, and increments starting at t > 0 are locally like those of BSS processes.The process lacks covariance-stationary increments, but has a modification with locally Hölder-continuous trajectories.Source: §2.5 · p. 13
  • Under (A1) and (A2), the hybrid scheme’s MSE for truncated BSS processes has the same asymptotic behavior as for BSS processes.The discretization drops the summands corresponding to integrals and increments on the negative real line; the discretized process’s covariance structure approaches that of Y as n → ∞.Source: §2.5 · p. 14

3 Implementation and numerical experiments

The section describes practical implementation of the hybrid scheme and evaluates it in roughness estimation and rough-volatility option pricing. Across these experiments, the hybrid scheme reproduces benchmark behavior more closely than the Riemann-sum scheme in the reported settings.

3.1 Practical implementation

  • The hybrid scheme combines power-function Wiener-integral terms near zero with a Riemann sum elsewhere, and FFT convolution gives O(n^γ+1 log n) complexity under (A4).For T BSS processes, the implementation via (3.7) has O(n log n) complexity.Source: §3.1 · p. 16
  • Using the same Brownian innovations, the κ = 1 and κ = 2 hybrid trajectories nearly coincide, while Riemann-sum trajectories are smoother and can collapse near α = −0.499.Both Riemann-sum variants produce anomalous trajectories with very little variation in that extreme case.Source: §3.1 · p. 16, Figure 2 · p. 17

3.2 Estimation of the roughness parameter

  • With s = 1, the hybrid scheme produces essentially unbiased roughness estimates for α ∈ (−1/2, 0), while increasing κ to 3 removes bias even near α = 1/2.The largest simulated α was 0.49, and the authors expect performance to weaken as α approaches 1/2.Source: §3.2 · p. 18, Figure 3 · p. 19
  • With s = 5, the κ = 1 hybrid scheme performs on par with exact simulation, whereas the Riemann-sum scheme retains considerable bias near α = −1/2.Increasing s improves results with both schemes, but the Riemann-sum improvements are more modest.Source: §3.2 · pp. 18, 20

3.3 Option pricing under rough volatility

  • For rBergomi Monte Carlo pricing, the authors replace exact simulation of Y with the hybrid scheme and use its Brownian increments to simulate Z for Euler discretization of S.The implied-volatility comparison considers short maturity T = 0.041 and long maturity T = 1.Source: §3.3 · pp. 20–21, Figure 4 · p. 21
  • Hybrid-scheme implied volatility smiles are indistinguishable from exact-simulation benchmarks, while Riemann-sum smiles capture shape but miss level and can fail at extreme log-strikes.The comparison uses κ = 1, 2 with b = b∗ for the hybrid scheme and κ = 0 for the Riemann-sum scheme.Source: §3.3 · p. 21, Figure 4 · p. 21

4 Proofs

The proofs establish regular-variation and path-regularity results used in the main analysis, derive the hybrid scheme’s error rate, and justify the identity in Equation (3.5). The latter proof applies a substitution and Lemma 4.3 to both integrals under the stated index inequalities.

4 Proofs

  • The proof framework uses the Potter bound for slow variation at zero, an elementary power-difference inequality, and a Karamata theorem variant for integrals at zero.The Karamata result applies for α > −1 and gives an asymptotic proportional to y^(α+1)L(y).Source: §4 · p. 22

4.2 Proof of Theorem 2.5

  • Theorem 2.5’s proof shows that the D′_n term determines the convergence rate, while the other error terms are negligible under condition (2.10).The argument decomposes the mean-square error and compares the asymptotic orders of its components.Source: §4.2 · pp. 28, 30

4.3 Proof of Equation (3.5)

  • An integral identity for the Gauss hypergeometric function is used to prove equation (3.5), for α > −1 and 0 ≤ a < b.Source: §4.3 · p. 31
  • In the proof of Equation (3.5), the substitution x = ns precedes an equality justified by Lemma 4.3 for both integrals, since 0 < j − 1 < k − 1 and 0 ≤ j − 2 < k − 2.Source: §4.3 · p. 31
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