A Note on the Discontinuity Problem in Heston's Stochastic Volatility Model

Abstract

Although quasi-analytic formulas can be derived for European-style financial claims in Heston's stochastic volatility model, the inverse Fourier integration involved makes the calculation somewhat complicated. This challenge has puzzled practitioners for many years because most implementations of Heston's formula are not robust, even for customarily-used Heston parameters, as time to maturity is increased. In this article, a simplified approach is proposed to solve the numerical instability problem inherent to the fundamental solution of the Heston model. Specifically, the solution does not require any additional function or a particular mechanism for most software packages or programming library routines to correctly evaluate Heston's analytics.Stochastic volatility model, Heston, discountinuity, options,

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“…7 Guo and Hung (2007) provide the method to avoid the branch cut difficulties arising from the choice of the branch of the complex logarithm in the implementation of the European option pricing formula. 8 The proof has been provided in Appendix B.…”

A generalization of Rubinstein's “Pay now, choose later”

“…We have assured the robustness of the evaluation of formulae by the simple adjustedformula method (see Guo & Hung, 2007).…”

Comment on: A Note on the Discontinuity Problem in Heston's Stochastic Volatility Model

“…Returning to (5), Guo and Hung [2007] recognised the problems associated with this solution, and argued that in the following formulation the logarithm can be restricted to its principal branch:…”