H\"older regularity and series representation of a class of stochastic volatility models

TL;DR

This paper investigates the regularity and series representation of a class of stochastic volatility models where the log-price process is driven by a Gaussian process and Brownian motion, establishing H"older regularity and providing a series expansion for simulation.

Contribution

It establishes the H"older regularity of the process and derives a Haar basis series expansion, enabling efficient simulation of the stochastic volatility models.

Findings

Typical trajectories have H"older regularity 1/2.

Series expansion converges geometrically in H"older spaces.

Provides an efficient iterative simulation method.

Abstract

Let $\Phi:\R\rightarrow\R$ be an arbitrary continuously differentiable deterministic function such that $|\Phi|+|\Phi'|$ is bounded by a polynomial. In this article we consider the class of stochastic volatility models in which ${Z(t)}_{t\in [0,1]}$, the logarithm of the price process, is of the form $Z(t)=\int_{0}^t \Phi(X(s)) dW(s)$, where ${X(s)}_{s\in[0,1]}$ denotes an arbitrary centered Gaussian process whose trajectories are, with probability 1, H\"older continuous functions of an arbitrary order $\alpha\in (1/2,1]$, and where ${W(s)}_{s\in[0,1]}$ is a standard Brownian motion independent on ${X(s)}_{s\in [0,1]}$. First we show that the critical H\"older regularity of a typical trajectory of ${Z(t)}_{t\in[0,1]}$ is equal to 1/2. Next we provide for such a trajectory an expression as a random series which converges at a geometric rate in any H\"older space of an arbitrary order $\gamma<1/2$; this expression is obtained through the expansion of the random function $s\mapsto \Phi(X(s))$ on the Haar basis. Finally, thanks to it, we give an efficient iterative simulation method for ${Z(t)}_{t\in[0,1]}$.