Not-so-complex Logarithms in the Heston Model

Christian Kahl and Peter Jäckel propose a new approach to solve an inherent numerical instability which enables the use of Heston's analytics for practically all levels of parameters and even maturities of many decades

In Heston’s stochastic volatility framework [Heston 1993], semi-analytical formulæ for plain vanilla option prices can be derived. Unfortunately, these formulæ require the evaluation of logarithms with complex arguments during the involved inverse Fourier integration step. This gives rise to an inherent numerical instability as a consequence of which most implementations of Heston’s formulæ are not robust for moderate to long dated maturities or strong mean reversion. In this article, we propose a new approach to solve this problem which enables the use of Heston’s analytics for practically all levels of parameters and even maturities of many decades.

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