## Fourier Transforms, Option Pricing and Controls

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### Fourier Transforms, Option Pricing and Controls

Dear Mark Joshi. I have a few questions concerning your recent article "Fourier Transforms, Option Pricing and Controls". Can I ask you them in this thread?
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1941464
stronzo

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

this is the place to ask.
mj
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### Re: Fourier Transforms, Option Pricing and Controls

Thank you very much. First, concerning (3.4), are you assuming that F_T(0) = 0 ?
stronzo

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

E(S_T)= F_T (0)

at some points in the paper, we assume this is 1; see after (4.1)
mj
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### Re: Fourier Transforms, Option Pricing and Controls

Sorry, I wanted to write "=1" instead of "=0".
Yes, but it seems to me that you are assuming here also ((3.4) comes before (4.1)) that F_T(0)=1. Aren't you?
stronzo

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

why do you think that?
mj
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### Re: Fourier Transforms, Option Pricing and Controls

In fact, in all your section 3.3 page 6, you are assuming that F_T(0)=1. Otherwise, there is necessary a term missing in F_T(0) coming from (3.1).
But it's ok, you can do it by (4.1).

In all your section 3.3, you are looking for \sigma such that \hat{\phi}(z)=0 for z\in\Gamma_{\eta} such that Re(z)=0. However, why is it important to have \hat{\phi}(z)=0 at this point ? This property does not imply that e^{-izl}\hat{\phi} is less oscillating on \Gamma_{\eta}. I agree, that value is the more natural value. But, have you tested numerically other values of \sigma?
stronzo

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Joined: Tue Feb 07, 2012 11:28 am

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

well the objective is to make the thing as small as possible and making it vanish there is one way to do so. Making it zero there makes the real part vanish to second order.

All the tests we did indicated that the total modulus of the integrand was minimized this way.
mj
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### Re: Fourier Transforms, Option Pricing and Controls

Page 8, I read : "In some cases, if we restrict the logarithm naively to its principal branch then the characteristic function will become discontinuous, leading to erroneous option prices" and then: "even though we compute the complex logarithm and power using their principal branch, there will not be discontinuity problems in evaluating the characteristic functions." What should I understand ? Do you use or not the principal branch to evaluate complex logarithm?

In Formula (4.5) for d(\xi), do you evaluate \sqrt(a+i b) as \exp((1/2)\ln(a+i b)) ? What is for you the principal branch of the complex logarithm ? For z=a+ib being a complex number which is not a negative real, is it the unique complex number of the forum x+iy with x:=\ln(|z|), y\in ]-\pi,\pi[ and \exp(iy)=z/|z|?

Given (4.4) are we sure that when integrating (3.2) along \Gamma_{\eta} we will never have to consider complex logarithm of negative reals?
stronzo

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

you need to read the Lord--Kahl paper or Gatheral's book where these issues are discussed in depth.
mj
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### Re: Fourier Transforms, Option Pricing and Controls

All right. I'm going to read "Complex logarithms in Heston-like models" (the reference you gave in your bibliography). So does it mean that the answer is "Yes" to all of my questions above i.e., you use the principal branch of the complex logarithm (the one with the argument in ]-\pi,\pi[) and there is no problem?
stronzo

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

basically yes as long as you define the thing you are taking the log of in the right way.
mj
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### Re: Fourier Transforms, Option Pricing and Controls

OK. Is your code available freely somewhere?
stronzo

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

no...

There is some Fourier transform Heston code in QuantLib however.
mj
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### Re: Fourier Transforms, Option Pricing and Controls

But is it easy to use complex numbers in C++? Is there a function which returns the principal branch of the logarithm or at least the argument of a given complex number?
stronzo

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Joined: Tue Feb 07, 2012 11:28 am

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