Complex Logarithms and the Piecewise Constant Extension of the Heston Model

The (Heston, 1993) model features an analytical characteristic function. It is well known that discontinuities can arise in the original formulation of the function, when the complex logarithm therein is restricted to its principal branch. In recent years, however, an alternative formulation has emerged. For this alternative, it has been established that discontinuities cannot arise, within the strip of regularity, when restricting the corresponding complex logarithm in this manner. For a region of this strip, we extend the analysis by allowing for piecewise constant parameters. Within this semi‐analytical framework, our results cater for both European and Forward Starting options.1