scholarly journals On the minimal number of driving Lévy motions in a multivariate price model

2018 ◽  
Vol 55 (3) ◽  
pp. 823-833
Author(s):  
Jean Jacod ◽  
Mark Podolskij
Keyword(s):  
Factor Analysis ◽  
Stochastic Volatility ◽  
Lévy Process ◽  
Levy Process ◽  
Price Model ◽  
Lévy Motion

Abstract In this paper we consider the factor analysis for Lévy-driven multivariate price models with stochastic volatility. Our main aim is to provide conditions on the volatility process under which we can possibly reduce the dimension of the driving Lévy motion. We find that these conditions depend on a particular form of the multivariate Lévy process. In some settings we concentrate on nondegenerate symmetric α-stable Lévy motions.

scholarly journals LÉVY FLIGHT SUPERDIFFUSION: AN INTRODUCTION

2008 ◽  
Vol 18 (09) ◽  
pp. 2649-2672 ◽  
Author(s):  
A. A. DUBKOV ◽  
B. SPAGNOLO ◽  
V. V. UCHAIKIN
Keyword(s):  
Brownian Motion ◽  
Characteristic Function ◽  
Lévy Process ◽  
Levy Process ◽  
Lévy Flight ◽  
Lévy Flights ◽  
Levy Flight ◽  
Barrier Crossing ◽  
Lévy Motion ◽  
Levy Flights

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

Option pricing and hedging under a stochastic volatility Lévy process model

2011 ◽  
Vol 15 (1) ◽  
pp. 81-97 ◽  
Author(s):  
Young Shin Kim ◽  
Frank J. Fabozzi ◽  
Zuodong Lin ◽  
Svetlozar T. Rachev
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility ◽  
Lévy Process ◽  
Process Model ◽  
Levy Process

scholarly journals Markov Regime Switching of Stochastic Volatility Lévy Model on Approximation Mode

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Arthit Intarasit
Keyword(s):  
Stochastic Volatility ◽  
Regime Switching ◽  
Lévy Process ◽  
Stock Exchange ◽  
Asset Returns ◽  
Levy Process ◽  
Martingale Measure ◽  
Large Set ◽  
Proposed Model ◽  
Risk Neutral

This paper deals with financial modeling to describe the behavior of asset returns, through consideration of economic cycles together with the stylized empirical features of asset returns such as fat tails. We propose that asset returns are modeled by a stochastic volatility Lévy process incorporating a regime switching model. Based on the risk-neutral approach, there exists a large set of candidates of martingale measures due to the driving of a stochastic volatility Lévy process in the proposed model which renders the market incomplete in general. We first establish an equivalent martingale measure for the proposed model introduced in risk-neutral version. Regime switching of stochastic volatility Lévy process is employed in an approximation mode for model calibration and the calibration of parameters model done based on EM algorithm. Finally, some empirical results are illustrated via applications to the Bangkok Stock Exchange of Thailand index.

SMALL-TIME ASYMPTOTICS IN GEOMETRIC ASIAN OPTIONS FOR A STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL

2019 ◽  
Vol 22 (02) ◽  
pp. 1950005
Author(s):  
HOSSEIN JAFARI ◽  
GHAZALEH RAHIMI
Keyword(s):  
Stochastic Volatility ◽  
Lévy Process ◽  
Implied Volatility ◽  
Option Price ◽  
Small Time ◽  
Stochastic Volatility Model ◽  
Levy Process ◽  
Asian Option ◽  
Volatility Model ◽  
Jump Diffusion Model

The aim of this paper is to study the small time to maturity of the behavior of the geometric Asian option price and implied volatility under a general stochastic volatility model with Lévy process. The volatility process does not need to be a diffusion or a Markov process, but the future average volatility in the model is a nonadapted process. An anticipating Itô formula for Lévy process and the decomposition of the price (Hull–White formula) are obtained using the Malliavin calculus techniques. The decomposition formula is applied to find the small-time limit of the geometric Asian option price and the implied volatility for the model in at-the-money and out-of-the-money cases.

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