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There has been a huge amount of work on this. Generally a Fourier transform approach is used. First, be careful to use the form of the characteristic function that does not wind about zero in order to avoid having to count the normal of windings. Second, using contour shifts can make the integral much better behaved. eg integrate along the line with $0.5$...


I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example: Option pricing using fractional FFT


In SV model, it is well-known that the integrand for the call price can sometimes show high oscillation, can dampen very slowly along the integration axis, and can show discontinuities. Remedy The ‘‘Little Trap’’ formulation of Albrecher et al. Also , you can use Fourier transforms Bakshi and Madan (2000) Lewis,(2001). Gatheral (2006) Carr and ...


Sorry if this is late, but this is the bible of Heston (and it has code)


"Monte Carlo convergence" means that you've sampled enough individuals to represent (and understand) a general population. If the probability models behind your Monte Carlo simulation are accurate, then your results will match reality as you increase your sampling size. Monte Carlo convergence becomes difficult when you try to study a low-probability sub-...

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