1
$\begingroup$

As mentioned in Carr-Madan's paper, here, the European call option is: $$ C_T(k)=\frac{e^{\alpha k}}{\pi}\int_0^\infty\mathcal{Re}\left(e^{-iuk}\psi(u)\right)du $$ where $$ \psi(u)=e^{-rT}\frac{\phi_T(u-(\alpha+1)i)}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} $$ and $\phi_T(u)$ is the characteristic function for a given process. Please refer to the paper all parameters.

So my question is, this derivation is based on the factor $S_0=1$ and define $k=\ln(K)$.

I am trying to derive the formula for any $S_0$ starting with $k=\ln(K/S_0)$ and $x=\ln(S_T/S_0)$, then end up with something like: $$ \psi(u)=e^{-rT}S_0\frac{\phi_T(u-(\alpha+1)i)}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} $$ Intuitively, it looks OK to me. But is there any other sources with a general $S_0$? Any help will be appreciated! Thank you!

Improve this question
$\endgroup$
5
  • $\begingroup$ I may be wrong but I remember that formula to allow for a general $S_0$? If I remember correctly, Carr and Madan only assume $S_0=1$ for the second (time value) approach? I once wrote down a general formula for the second case. But the first ``standard'' approach from Carr and Madan incorporates a general $S_0$ via the characteristic function of $S_T$, i.e. $\phi_T(u)$? I may be wrong though and will look at the paper again. $\endgroup$
    – Kevin
    Apr 1, 2020 at 8:53
  • $\begingroup$ Thank you @KeSchn, I think you may be correct that Carr and Madan incorporate a general $S_0$ via the characteristic function of $S_T$. I'll take a look at details as well, but feel free to answer this question. $\endgroup$
    – Pandaaaaaaa
    Apr 1, 2020 at 12:40
  • $\begingroup$ Given that you can incorporate a general $S_0$ via $\phi_T(u)$, why would you still need the Fourier transform of the call option with respect to $\ln\left(\frac{K}{S_0}\right)$? Can't you simply take their formula and plug in your general $S_0$? $\endgroup$
    – Kevin
    Apr 1, 2020 at 12:51
  • $\begingroup$ I think I need to make sure the meaning of $k=\ln(K/S_0)$ is correct before I can plug into the formula? Here is the part I felt confused... $\endgroup$
    – Pandaaaaaaa
    Apr 1, 2020 at 13:07
  • $\begingroup$ Just take the formulae from the paper, use $\phi_T(u)$ according to your model which includes $S_0$ and set $k=\ln(K)$. If you follow the formulae from the paper, you’ll get correct option prices $\endgroup$
    – Kevin
    Apr 1, 2020 at 14:44

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.