Entropy-Regularised Calibration of the Lévy Measure

Given a cross-section of European call prices $C_{\text{mkt}}(K_i, T_j)$, the calibration problem for a Lévy model is: find the Lévy measure $\nu$ such that the model prices $C^{\text{model}}(K_i, T_j; \nu)$ match the market. Unlike parametric calibration (fitting the Heston or VG parameters), this is a nonparametric inverse problem: $\nu$ is a measure on $\mathbb{R}$, not a finite vector.

Nonparametric inverse problems are ill-posed — without regularisation, infinitely many $\nu$ achieve the same market prices, and small noise in input prices can cause wildly oscillating solutions. The standard regulariser for this problem is relative entropy (Kullback-Leibler divergence): among all Lévy measures that fit the market prices, choose the one closest (in relative entropy) to a prior $\nu_0$.

Learning objectives. By the end of this note you should be able to: