The Delta Term in Fourier Inversion of Option Probabilities
Summary
The document asks how to derive an option-pricing probability expression in Gatheral’s framework by directly performing a complex inverse Fourier integral. It compares that route with the Gil-Pelaez inversion theorem and focuses on the boundary condition at zero time, where the transform is presented as a reciprocal imaginary-frequency term. The author questions whether the inverse transform also needs a contribution at frequency zero.
The edit supplies the key distributional correction: the boundary condition includes both a principal-value reciprocal term and a delta contribution. That delta term accounts for the one-half constant in the real-valued integral expression for the pseudo-probability. This resolves the stated issue at a conceptual level, but the document does not provide a full derivation of the complex integration or discuss convergence and regularity conditions. Its value is the reminder that generalized Fourier transforms may require distributional terms that are not visible in an ordinary integral expression.
Key ideas
- The pseudo-probability formula can be obtained through inverse Fourier transformation or the Gil-Pelaez theorem.
- The zero-time transform requires a principal-value term as well as a delta contribution.
- The delta contribution explains the one-half constant in the real-valued integral form.
- The document gives the correction but not a complete derivation or conditions for the inversion.
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Full text
# Gatheral pseudo-probabilities inverse Fourier transform
# Gatheral pseudo-probabilities inverse Fourier transform
Im having some trouble with the discussion about the pseudo-probabilities in Gatheral's book. In chapter 2, it reads
> Taking the inverse transform using equation (2.8) and performing the complex integration carefully gives the final form of the pseudo-probabilities $P_j$ in the form of an integral of a real-valued function. $$P_j (x, v, \tau ) = \dfrac{1}{2} + \dfrac{1}{\pi} \int^\infty_0 du \, \text{Re} \left\lbrace \dfrac{ \exp \left(C_j (u, \tau ) \bar{v} + D_j (u, \tau ) v + i u x\right)}{ iu} \right\rbrace $$
I have seen the derivation using the Gil Pelaez theorem and get some of the logic. However, if I try to solve this following Gatheral's idea (performing the complex integration directly), I don't see how to proceed. Note that the starting point here is
$$P_j (x, v, \tau ) = \dfrac{1}{2\pi} \int^\infty_0 du \, e^{iux}\dfrac{ \exp \left(C_j (u, \tau ) \bar{v} + D_j (u, \tau ) v\right)}{ iu} $$ since the value of the $\tilde{P}_j$ function at $\tau = 0$ has been set to $\tilde{P}(u, v, 0) = \frac{1}{iu}$. This already seems "funny" to me: doesn't the inverse of the Heaviside function include a delta term to take care of its value at $u=0$?.
Does anyone have a rough idea? Do you know where I could find a proof for this? Thanks in advance!
Edit: A good answer to this has been given previously in https://math.stackexchange.com/a/960560/1222817
The boundary condition should have been $$\tilde{P}_j (u, v, 0) = \text{p.v.}\dfrac{1}{iu} + \pi \delta(u),$$ from where the $1/2$ factor in the expression for $P_j(x, v, \tau)$ follows straight forward.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.