Complex Logarithms in Heston-Like Models

30 Pages Posted: 17 Mar 2008

See all articles by Roger Lord

Roger Lord

Cardano Risk Management

Christian Kahl

Commerzbank Corporates & Markets; University of Wuppertal; ABN-Amro Bank, United Kingdom

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Date Written: February 21, 2008


The characteristic functions of many affine jump-diffusion models, such as Heston's stochastic volatility model and all of its extensions, involve multivalued functions like the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper we prove without any restrictions that there is a formulation of the characteristic function in which the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than the so-called rotation count algorithm of Kahl and Jäckel [2005], we solely focus on its stability in this article. The remainder of this paper shows how complexd iscontinuities can be avoided in the Variance Gamma and Schöbel-Zhu models, as well as in the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya.

Keywords: Complex logarithm, affine jump-diffusion, stochastic volatility, Heston, characteristic function, option pricing, Fourier inversion, Variance Gamma, Schöbel-Zhu,exact simulation

JEL Classification: C63, G13

Suggested Citation

Lord, Roger and Kahl, Christian, Complex Logarithms in Heston-Like Models (February 21, 2008). Available at SSRN: or

Roger Lord (Contact Author)

Cardano Risk Management ( email )

Rotterdam 3011 AA

Christian Kahl

Commerzbank Corporates & Markets ( email )

60 Gracechurch Street
London, EC3V 0HR
United Kingdom

University of Wuppertal ( email )

Gaußstraße 20
42097 Wuppertal

ABN-Amro Bank, United Kingdom ( email )

London EC2N 4BN
United Kingdom

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