Extending the Carr–Madan Call Pricing Formula to Arbitrary Spot Prices

Summary

The document asks how to adapt the Carr–Madan Fourier transform method for European call options when the initial asset price is not normalized to one. It presents the damped Fourier pricing integral and its characteristic-function term, then describes changing variables to log moneyness, defined relative to the initial spot, and log return from that spot.

The proposed adjustment places the initial spot as a multiplicative factor in the transform expression. The author says this appears intuitive and seeks a reference or confirmation. No derivation, numerical example, or reply is included, so the proposed formula is not verified in the document. The key issue is consistent scaling between log strike, log return, the characteristic function, and the call price when moving from a unit spot normalization to a general initial price.

Key ideas

• The Carr–Madan method prices European calls using a damped Fourier integral.
• The question concerns generalizing a derivation that assumes an initial spot of one.
• The proposed setup expresses strike relative to spot and terminal value as a log return.
• The document proposes a spot scaling factor but provides no proof or validation.

Tags

# Fourier transform Carr-Madan method on an arbitrar initial $S_0$ values


# Fourier transform Carr-Madan method on an arbitrar initial $S_0$ values












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!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.