Calibration for Weak Variance-Alpha-Gamma Processes

TL;DR

This paper evaluates calibration methods for the weak variance-alpha-gamma process, a multivariate Lévy process, demonstrating that maximum likelihood estimation and digital moment estimation have different advantages depending on Fourier invertibility conditions, with DME fitting best on real market data.

Contribution

It introduces and compares three calibration methods for the weak variance-alpha-gamma process, highlighting the conditions under which each method performs best and demonstrating its superior fit on financial data.

Findings

MLE outperforms DME when Fourier invertibility condition holds.

DME provides better fit when Fourier invertibility condition is violated.

Weak variance-alpha-gamma process captures a wider dependence range and fits market data better.

Abstract

The weak variance-alpha-gamma process is a multivariate L\'evy process constructed by weakly subordinating Brownian motion, possibly with correlated components with an alpha-gamma subordinator. It generalises the variance-alpha-gamma process of Semeraro constructed by traditional subordination. We compare three calibration methods for the weak variance-alpha-gamma process, method of moments, maximum likelihood estimation (MLE) and digital moment estimation (DME). We derive a condition for Fourier invertibility needed to apply MLE and show in our simulations that MLE produces a better fit when this condition holds, while DME produces a better fit when it is violated. We also find that the weak variance-alpha-gamma process exhibits a wider range of dependence and produces a significantly better fit than the variance-alpha-gamma process on an S&P500-FTSE100 data set, and that DME produces the best fit in this situation.