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First, to make that clear: The Heston model does not generate negative volatility, but - for example - an Euler discretization of the Heston model may generate negative volatility (or variance). It is not a problem of the model. It is a problem of the numerical scheme. If you use an Euler scheme which generates negative volatility and then use any of the ...

I highly recommend you to stick with the error function (RMSE) value minimization approach. I love MC techniques for this and related problem solving and thus do not recommend you to use anything else because of its simplicity and transparency. It comes down to using the right discretization function and to possibly implement variance reduction approaches. ...

Doesn't the Heston model have some Fourier transform formulae for pricing vanillas? I think one could use those to calibrate to the vanillas. Can't provide references at this moment, on the road. Edit: check out http://www.visixion.com/dok/Visixion_Calibrating_Heston.pdf -- I haven't read this closely but it sounds familiar

Expanding a bit on chrisaycock's answer, and noting in particular from the abstract In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset. we can see that this would be used to price those few rare cases of perpetual options. ...

From this abstract: The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic ...

It is not necessarily something that must be wrong with your model. Inherent in the Heston discretization methods of its continuous time dynamics is the possibility of negative values in the variance process. Here are couple solutions you can look at in order to "fix" your problem: Usage of different Euler schemes, such as the Full Truncation scheme. ...

I would say Start with Black Scholes to look at accuracy. In particular, you have a closed formula and you know what the characteristic function for lognormal is. Running FFT and comparing FFT pricing with the closed formula will give you an idea of what are the convergence issues, what is the behaviour at the boundaries (extreme strikes) etcetera. Then ...

You can find the derivation of the Heston characteristic function (its Fourier Transform) in Gatheral (2006). Using the characteristic function, you can optimize the model on the prices. There are multiple approaches to optimize, among others pattern search (which is very slow) and stochastic optimization (randomly jump around and stop after n iterations), ...

Here's a decent study of calibration performance using fast fourier transforms versus other techniques. It concludes Gaussian quadrature works better than other techniques. http://www.frankfurt-school.de/dms/publications-cqf/CPQF_Arbeits6.pdf Edit: AZhu points out the link above is dead and that a working link is ...