The normalized characteristic function of a piecewise constant time dependent Heston model

with time intervals and constant parameters within these intervals

is given by the recurrence relation

and the initial condition

The should be noted that the complex logarithm in this formulation can be restricted to the principal branch without introducing any discontinuities [1]. It is important to know the asymptotic behaviour of in order to calculate the truncation point for the integral over the characteristic function when using the Andersen-Piterbarg approach with control variate [2]. A Mathematica script gives

The implementation of the Andersen-Piterbarg method for the piecewise constant time dependent Heston model is part of the pull request #251.

[1] Afshani, S. (2010) Complex logarithms and the piecewise constant extension of the Heston model.

[2] Andersen, L. and Piterbarg, V. (2010) Interest Rate Modeling, Volume I: Foundations and Vanilla Models, (Atlantic Financial Press London).

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