The role of characteristic functions in finance has been strongly amplified by the development of the general option pricing formula by Carr and Madan. As these functions are defined and operating in the complex plane, they potentially encompass a few well known numerical
issues due to ”branching”. A number of elegant publications have emerged tackling these effects specifically for the Heston model. For the latter however we have two specifications for the characteristic function as they are the solutions to a Riccati equation. In this article
we put the i’s and cross the t’s by formally pointing out the properties of and relations between both versions. For the first specification we show that for nearly any parameter choice, instabilities will occur for large enough maturities. We subsequently establish - under an additional parameter restriction - the existence of a “threshold” maturity from which the complex operations become a spoil-sport. For the second specification of the characteristic function it is proved that stability is guaranteed under the full dimensional and unrestricted parameter space. We blend the theoretical results with a few examples.