Gram–Charlier methods, regime-switching and stochastic volatility in exponential Lévy models. Issue 4 (3rd April 2022)

• Record Type: Journal Article
• Authors: Asmussen, Søren; Bladt, Mogens
• Is Part Of: Quantitative finance. Volume 22:Issue 4(2022)
• Page Start: 675
• Page End: 689
• Publication Date: 2022-04-03
• Subjects: Bell polynomials -- CGMY process -- Cumulants -- European call option -- Faà di Bruno's formula -- Integrated CIR process -- Markov additive process -- Markov-modulation -- Matrix-exponentials -- Normal inverse Gaussian distribution -- Risk neutrality -- Tempered stable distribution; C02 -- C6 -- C63 -- C65 -- G10 -- G12; Finance -- Periodicals; Business mathematics -- Periodicals; Finance -- Mathematical models -- Periodicals; Investments -- Mathematics -- Periodicals; Economics -- Periodicals; Finances -- Modèles mathématiques -- Périodiques; 332.015118
• DOI: 10.1080/14697688.2021.1998585
• Languages: English
• ISSNs: 1469-7688

Abstract

Abstract : The Gram–Charlier expansion of a target probability density, f ( x ), is an L 2 -convergent series f ( x ) = ∑ 0 ∞ c n p n ( x ) f ∗ ( x ) in terms of a reference density f ∗ ( x ) and its orthonormal polynomials p n ( x ) . 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 f ( x ) 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 f ∗ ( x ) as normal with the same mean and variance as f ( x ) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, f ∗ ( x ) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.