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

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 ${Z(t)}_{t\in[0,1]}$.