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 -