Abstract
A cumulative distribution function (CDF) and the price of a European option can be expressed as d-dimensional integrals. The CDF is equal to the integral of the product of a density and an indicator function. Similarly, the price of a European option can be expressed as the integral of the product of a density and a payoff function.
We review the literature on how to compute such integrals when the density is unknown but the characteristic function is given in closed form. We introduce the novel damped COS method to solve these integrals efficiently up to five dimensions. We show that the COS method converges exponentially when the characteristic function decays exponentially.
To obtain the quantile function in one dimension, one typically first calculates the CDF using the Gil-Pelaez formula or the COS method, and then numerically inverts the CDF. We show theoretically and empirically that the numerical error of the quantile function is typically several orders of magnitude larger than the numerical error of the CDF for probabilities close to zero or one.
The talk is based on:
[1] Junike, G. and Stier, H. (2025). From characteristic functions to multivariate distribution functions and European option prices by the damped COS method. Preprint
[2] Junike, G. (2025). Precise quantile function estimation from the characteristic function. Statistics and Probability Letters
Organized by: Umberto Cherubini