Gil-Pelaez Fourier Inversion for Option Exercise Probabilities

Summary

The document explains the source of a one-half term in a Fourier-based option-pricing derivation. It gives the Gil-Pelaez inversion formula, which relates an integral involving the real part of a characteristic function to the probability that the terminal asset price is at or above the strike, adjusted by one-half. The characteristic function is that of the log terminal price under a chosen probability measure.

This result helps connect Fourier transforms of probability distributions to exercise probabilities and clarifies a step that can arise when changing integration order in a derivation. The answer presents the formula but does not show a full derivation or carry it through to an option price. The integral is improper, and the exact probability measure and distributional assumptions matter for applying the identity. It is a focused mathematical explanation rather than a complete pricing recipe.

Key ideas

• Gil-Pelaez inversion links an integral of a characteristic function to a cumulative probability.
• The one-half adjustment appears in the formula for the probability that terminal price meets or exceeds the strike.
• The relevant characteristic function describes log terminal price under a specified probability measure.
• Fourier inversion methods can convert distribution information into option exercise probabilities.
• The document states the identity but does not provide its derivation or a complete pricing example.

Tags

# Option pricing via fourier transform


# Option pricing via fourier transform












can someone help me understand the steps from (1) to the following row? where does the 1/2 comes from, after I apply the fourier transform to the probability density and swithch the integral order I cannot go on.

## Answer by Kevin (score 1)

https://quant.stackexchange.com/a/81517

Gil-Pelaez' (1951) Fourier inversion formula states that for any probability measure $\mathbb{Q}$, \begin{align} \int_0^\infty \mathrm{Re}\left(\frac{e^{-i\ln(K)u}\phi_{\ln(S_T)}^\mathbb{Q}(u)}{iu}\right)\mathrm{d}u = \pi\left(\mathbb{Q}\big[\{S_T\geq K\}\big] - \frac{1}{2}\right). \end{align} Here, $\phi_{\ln(S_T)}^\mathbb{Q}(u)=\mathbb{E}^\mathbb{Q}[e^{iu\ln(S_T)}]$ is the characteristic function of the log process $\ln(S_T)$. $\text{Re}$ is the real part of a complex number.

Indeed, many Fourier methods in finance boil down to this (or a similar) inversion formula. It directly links exercise probabilities to improper integrals of characteristic functions (Fourier transformations of probability density functions).

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.