## Fourier Transforms, Option Pricing and Controls

Discuss anything related to quantitative finance which is not directly connected to a particular book here.

### Re: Fourier Transforms, Option Pricing and Controls

the header file complex.h does complex numbers and you can just take log
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Fourier Transforms, Option Pricing and Controls

You say in your introduction that the idea to use control variate for pricing Heston goes back to Andersen and Piterbarg. So, what is new in your article? How did people do before? Compare to a FFT, are you faster?
stronzo

Posts: 11
Joined: Tue Feb 07, 2012 11:28 am

### Re: Fourier Transforms, Option Pricing and Controls

we have a way of picking sigma people tended to just use \sigma_0 before.

We also analyze contour dependence and prove the decay rate.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Fourier Transforms, Option Pricing and Controls

Yes but if I understand well your work, it's not clear that your choice of \sigma is the good one (see 4.4.2. Pricing performance, Test Case 1 page 9 for instance). And concerning the decay rate, it's just due to the use of control-variate as you remarked. But may be, you are the first one to analyze quite deeply (theoretically and numerically) the use of control-variate for pricing (at least for SVI and Heston). For that reason, your paper is very good and important.

Last questions. Did you use a Gauss–Laguerre quadrature? If I understand well, the true price were calculated using 150 points and you say that you just need 10 points of integration to get a result with an error less than 1bp. Is it true?
stronzo

Posts: 11
Joined: Tue Feb 07, 2012 11:28 am

### Re: Fourier Transforms, Option Pricing and Controls

well before our work the recommended thing was to use contour im =1/2 and no one had done any analysis of what was best.

Ultimately we show that with our set-up 10 integration points is sufficient in a wide range of cases.
That there is the odd case where something else works better is not so surprising but very important either.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Fourier Transforms, Option Pricing and Controls

It's amazing. You have implemented Heston using only 10 points of integration! Thank you for all your answers.

PS. However, in Section 3.3 page 6, you should say there that you are assuming that F_T(0)=1.
stronzo

Posts: 11
Joined: Tue Feb 07, 2012 11:28 am

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