TY - JOUR

T1 - Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models

AU - Asmussen, Søren

AU - Bladt, Mogens

N1 - Publisher Copyright:
© 2021 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2022

Y1 - 2022

N2 - The Gram–Charlier expansion of a target probability density, (Formula presented.), is an (Formula presented.) -convergent series (Formula presented.) in terms of a reference density (Formula presented.) and its orthonormal polynomials (Formula presented.). We implement this for the density of a regime-switching Lévy process at a given time horizon T. The main step is the evaluation of moments of all orders of (Formula presented.) in terms of model primitives, for which we give a matrix-exponential representation. A number of numerical examples, in part involving pricing of European options, are presented. The traditional choice of (Formula presented.) as normal with the same mean and variance as (Formula presented.) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, (Formula presented.) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.

AB - The Gram–Charlier expansion of a target probability density, (Formula presented.), is an (Formula presented.) -convergent series (Formula presented.) in terms of a reference density (Formula presented.) and its orthonormal polynomials (Formula presented.). We implement this for the density of a regime-switching Lévy process at a given time horizon T. The main step is the evaluation of moments of all orders of (Formula presented.) in terms of model primitives, for which we give a matrix-exponential representation. A number of numerical examples, in part involving pricing of European options, are presented. The traditional choice of (Formula presented.) as normal with the same mean and variance as (Formula presented.) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, (Formula presented.) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.

KW - Bell polynomials

KW - CGMY process

KW - Cumulants

KW - European call option

KW - Faà di Bruno's formula

KW - Integrated CIR process

KW - Markov additive process

KW - Markov-modulation

KW - Matrix-exponentials

KW - Normal inverse Gaussian distribution

KW - Risk neutrality

KW - Tempered stable distribution

U2 - 10.1080/14697688.2021.1998585

DO - 10.1080/14697688.2021.1998585

M3 - Journal article

AN - SCOPUS:85121350652

VL - 22

SP - 675

EP - 689

JO - Quantitative Finance

JF - Quantitative Finance

SN - 1469-7688

IS - 4

ER -