16 research outputs found
Multifractal random walks with fractional Brownian motion via Malliavin calculus
We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence of this object and its properties. We then propose a continuous time model in finance that captures the main properties observed in the empirical data, including the leverage effect. We illustrate our result by numerical simulations
Quantitative Breuer-Major Theorems
We consider sequences of random variables of the type S_n= n^{-1/2}
\sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}, , where is a
-dimensional Gaussian process and is a measurable
function. It is known that, under certain conditions on and the covariance
function of , converges in distribution to a normal variable .
In the present paper we derive several explicit upper bounds for quantities of
the type |\E[h(S_n)] -\E[h(S)]|, where is a sufficiently smooth test
function. Our methods are based on Malliavin calculus, on interpolation
techniques and on the Stein's method for normal approximation. The bounds
deduced in our paper depend only on and on simple infinite
series involving the components of . In particular, our results generalize
and refine some classic CLTs by Breuer-Major, Giraitis-Surgailis and Arcones,
concerning the normal approximation of partial sums associated with
Gaussian-subordinated time-series.Comment: 24 page
The Hurst parameter and option pricing with fractional Brownian motion
In the mathematical modeling of the classical option pricing models it is assumed that the underlying stock price process follows a geometric Brownian motion, but through statistical analysis persistency was found in the log-returns of some South African stocks and Brownian motion does not have persistency. We suggest the replacement of Brownian motion with fractional Brownian motion which is a Gaussian process that depends on the Hurst parameter that allows for the modeling of autocorrelation in price returns. Three fractional Black-Scholes (Black) models were investigated where the underlying is assumed to follow a fractional Brownian motion. Using South African options on futures and warrant prices these models were compared to the classical models.Dissertation (MSc)--University of Pretoria, 2012.Mathematics and Applied Mathematicsunrestricte