On stochastic integration for volatility modulated Lévy-driven Volterra processes (original) (raw)

A Malliavin–Skorohod calculus inL0andL1for additive and Volterra-type processes

Stochastics, 2016

In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the L 0 and L 1 settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in L 1. Our theory is then applied to extend the stochastic integration with respect to volatility modulated Lévy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect to e.g. Volterra processes driven by α-stable processes with α < 2. The presentation focuses on jump type processes.

Stochastic Volatility for Levy Processes

Mathematical Finance, 2003

Three processes re°ecting persistence of volatility are formulated by evaluating three L ¶ evy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating the processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. Our empirical results on index options and single name options suggest advantages to employing higher dimensional L ¶ evy systems for index options and lower dimensional structures for single names. In general, mean corrected exponentiation performs better than employing the stochastic exponential. Martingale laws for the mean corrected exponential are also studied and two new concepts termed L ¶ evy and martingale marginals are introduced. ¤ We would like to thank George Panayotov for assistance with the computations reported in this paper. Dilip Madan would like to thank Ajay Khanna for important discussions and perspectives on the problems studied here. Errors are our own responsibility.

On Lévy processes, Malliavin calculus and market models with jumps

Finance and Stochastics, 2002

Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump-diffusion model are given.

Malliavin Calculus and Anticipative Itô Formulae for Lévy Processes

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2005

We introduce the forward integral with respect to a pure jump Lévy process and prove an Itô formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Itô formula for the Skorohod integral.

Malliavin Calculus for Lévy Processes with Applications to Finance

Malliavin Calculus for Lévy Processes with Applications to Finance, 2009

Extending Gaussian Malliavin derivatives, to a finite moments Levy process framework, using chaos expansions has proven to be a successful approach. In this work the theory is extended by the introduction of the Skorohod integral, and its properties are investigated. From this rather general case the scope is narrowed downstep by step-to a class of jump diffusion processes for which we have a Malliavin derivative chain rule, which is important for the application of Malliavin calculus to variance reduction of Monte Carlo simulation of contingent claim sensitivities. Stochastic weights for these simulations are derived and examples with numerical experiments are presented. These stochastic weights are well known in the continuous case, but the introduction of discontinuous jumps allows for extensions to other asset classes such as credit derivatives. Monte Carlo methods are widely used in particular for credit derivatives, and therefore alternative methods for faster convergence of sensitivities have previously been developed. One such technique is the likelihood ratio method, which is closely linked with the Malliavin weighted method. This discussion is also formalised. CONTENTS 1. Introduction 1.1 Background 8 1.2 Outline of thesis 10 Part I Malliavin calculus for Levy processes 2. Malliavin calculus for Levy processes satisfying a moment condition 2.1 The chaotic representation property 2.1.1 Setup and strongly orthogonal martingales 2.

Option pricing and hedging under a stochastic volatility Lévy process model

Review of Derivatives Research, 2012

In this paper, we discuss a stochastic volatility model with a Lévy driving process and then apply the model to option pricing and hedging. The stochastic volatility in our model is defined by the continuous Markov chain. The risk-neutral measure is obtained by applying the Esscher transform. The option price using this model is computed by the Fourier transform method. We obtain the closed-form solution for the hedge ratio by applying locally risk-minimizing hedging.

A Jump-Diffusion with Stochastic Volatility and Interest Rate

Journal of Mathematics and Statistics, 2013

In this study, we present the application of Time Changed Levy method to model a jump-diffusion process with stochastic volatility and stochastic interest rate. We apply the Lewis Fourier transform method as well as the risk neutral expectation pricing method to derive a formula for a European option pricing. These combining methods give quite a short route to derive the formula and make it efficient to compute option prices. We also show the calibration of our model to the real market with global and local optimization algorithms.