Characteristic function

The characteristic function of a random variable $X$ is $\varphi_X(u) = E[e^{iuX}]$. Since $e^{iu} = \cos u + i \sin u$, it takes a complex value for each real $u$. The MIT lecture notes list as an advantage that it is defined for every random variable and every $u$. The characteristic function of the sum of independent $X$ and $Y$ is the product of their characteristic functions¹.

For example, if $X$ takes the values $+1$ and $-1$ with equal probability 0.5, then $\varphi_X(u) = (e^{iu} + e^{-iu})/2 = \cos u$.

In finance, the characteristic function of the log stock price at maturity has a simple form in many models. The probability density, by contrast, often cannot be written in a formula or is awkward to compute, as Janek et al. explain. This is why option prices are computed from the characteristic function by numerical integration (the Fourier-inversion method of Carr and Madan)². In the Heston model, the characteristic function of the log stock price has a closed form, so the price of a European option can be computed quickly. The formula and its numerical instability are on the related article page.

Notes

1. Scott Sheffield, "Lecture 12: DeMoivre-Laplace and weak convergence" (Massachusetts Institute of Technology, 18.175 Theory of Probability, Spring 2014), page 14, "Characteristic functions". The text replaces the variable $t$ with $u$. ↩

2. Agnieszka Janek et al., "FX smile in the Heston model" (arXiv), section 1.3.4 (page 9). The Carr-Madan method uses the characteristic function and Fourier inversion to find prices; the characteristic function of the log price has a simple form in many models while the density often cannot be written in closed form or is complicated numerically; in the Heston model $E\{\exp(i\varphi \log S_T)\}$ is given by $f_2$ in equation (1.17). ↩