Why the Rotation Count Algorithm Works

Tinbergen Institute Discussion Paper No. 2006-065/2

33 Pages Posted: 3 Aug 2006

See all articles by Roger Lord

Roger Lord

Cardano Risk Management

Christian Kahl

University of Wuppertal; ABN-Amro Bank, United Kingdom

Date Written: July 2006


The characteristic functions of many affine jump-diffusion models, such as Heston's stochastic volatility model and all of its extensions, involve multivalued functions such as 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 under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jäckel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston's formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schöbel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin's SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances.

Keywords: Complex logarithm, affine jump-diffusion, stochastic volatility, Heston, characteristic function, moment stability, option pricing

JEL Classification: C63, G13

Suggested Citation

Lord, Roger and Kahl, Christian, Why the Rotation Count Algorithm Works (July 2006). Tinbergen Institute Discussion Paper No. 2006-065/2, Available at SSRN: https://ssrn.com/abstract=921335 or http://dx.doi.org/10.2139/ssrn.921335

Roger Lord (Contact Author)

Cardano Risk Management ( email )

Rotterdam 3011 AA

Christian Kahl

University of Wuppertal ( email )

Gaußstraße 20
42097 Wuppertal

ABN-Amro Bank, United Kingdom ( email )

London EC2N 4BN
United Kingdom

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