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I am trying to better understand the Heston model and its implementation. It seems like a lot of people use the FFT method for calculating the call prices during the Heston calibration, but the Monte Carlo method is used to calculate the prices with the calibrated parameters. What is the point in this? Why not just use the FFT method for calculating both prices?

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    $\begingroup$ FFT can be used to compute both prices. You would use MC simulations if you want to price exotic payoffs. Recall that FFT is limited to European-style options. Since these are often used for calibration, FFT is a fast way of finding your model parameters. There are, of course, further alternatives to pricing options under the Heston model (e.g. finite differences, trees and other Fourier methods). It all depends on your application $\endgroup$ – KeSchn Apr 30 at 15:12
  • $\begingroup$ @KeSchn Ok, I think I understand what you are saying. So, the FFT method would work for calibration of parameters since the optimization can just use European call prices because there is no path dependency. And the Monte Carlo method would be needed to find values at specific points for exotic payoffs? $\endgroup$ – Kevin K. Apr 30 at 16:01
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    $\begingroup$ That's right Kevin. Heston (and many other models) give you a closed-form solution for European-style options. Implementing those with FFT gives you a very fast way of computing these option prices. Using liquid observed European option prices, you calibrate your model. Now you got $\kappa$, $\theta$ etc. Then, you can price (almost) every other derivative using either FFT, MC, FD or whatever. By the way, the COS method from Fang and Oosterlee is an even faster Fourier method and even simpler to implement, you may want to check it out :) $\endgroup$ – KeSchn Apr 30 at 16:33
  • $\begingroup$ @KeSchn Great. That helps a lot. Thank you $\endgroup$ – Kevin K. Apr 30 at 16:39
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The big advantage of the Heston (1993) model is that it admits a quasi-analytical formula for the pricing of European options. So, obviously, if you're going to price European options, you should be using some kind of numerical method to approximation the integrals in the quasi-analytical formula. The exact method you use will depend on the case at hand and what matters most -- usually, speed is of the essence.

However, that only works for European options. Outside of that small pond, it might be possible in some cases to think that your European option is a good starting point for calibration. I have never done this personally, but I have done similar things (use simpler methods to reduce my searching time before doing things in a more Kosher manner).

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