# Extending the Levy Processes to Multiasset Products Pricing

Levy processes have attained good results in pricing single asset options. Within this dissertation, we present a technique allowing us to extend the single asset pricing approach based upon Levy processes to multiasset cases. In our approach, we suppose the log-return of each asset as a linear sum of independent factors. These factors are determined by the Levy processes, and the specific Levy process we’re researching in this dissertation is the Variance Gamma (VG) process. We recover these factors by a signal processing method known as independent component analysis (ICA), from which we get the physical measure (P measure). To price the contingent claims, we still require the risk-neutral measure (Q measure). We bridge the gap between physical measure and risk-neutral measure by presenting the transformation of measure between the P measure and the Q measure…

## Contents: Levy Processes to Multiasset Products Pricing

1 Introduction
2 Preliminaries
2.1 L´evy Processes: Deﬁnitions and Properties
2.1.1 L´evy Processes and Inﬁnitely Divisible Distributions
2.1.2 L´evy-Khintchine Representation
2.1.3 L´evy-Itˆo Decomposition
2.1.4 Measure Change for L´evy Processes
2.1.5 Subordination of L´evy Processes
2.2 The Variance Gamma Process
2.2.1 Deﬁne the VG Process as Subordinated Brownian Motion
2.2.2 Properties of the VG Process
2.2.3 Simulating the VG Process
2.2.4 The VG Stock Price Process
2.3 The VG Option Prices and FFT method
2.3.1 The Carr-Madan FFT Method
2.3.2 The Fractional FFT Method
3 Independent Component Analysis
3.1 Linear Transformation Methods
3.1.1 The Linear Representations
3.1.2 Principal Component Analysis
3.1.3 Factor Analysis
3.2 Deﬁnition of Independent Component Analysis
3.3 Principles in Estimating ICA
3.3.1 Statistically Independent
3.3.2 Nongaussian Distribution
3.3.3 Unit Variance
3.4 Estimating the ICA Model
3.5 Measurement of Nongaussianity
3.6 Preprocessing the Data
3.6.1 Centering
3.6.2 Whitening
3.7 The FastICA Algorithm
4 The VG Multiasset Option Pricing by ICA
4.1 Linear Representation of the Log-return
4.2 Maximum Likelihood Estimation of Parameters
4.2.1 Method of Maximum Likelihood
4.2.2 MLE for the VG Processes
4.3 The Measure Change for Pure Jump Processes
4.4 Tilting the L´evy Measure
4.5 Risk-neutral Stock Price Dynamics
4.6 Numerical Implementation
5 The VG Multiasset Option Pricing by Copula Method
5.1 Deﬁnitions of Copulas
5.2 Copulas and Multivariate Distribution
5.3 Measures of Dependence
5.3.1 Linear Correlation…

Source: University of Maryland