Calibrating Multivariate Lévy Processes with Neural Networks
This work addresses a domain-specific problem in financial mathematics and statistics for researchers and practitioners dealing with multivariate stochastic processes, offering an incremental improvement over existing methods.
The paper tackles the challenge of calibrating multivariate Lévy processes, which is difficult due to slow decay and high uncertainty in empirical characteristic functions, by approximating the Lévy density with a parametrized functional form using deep neural networks, resulting in robust performance that captures sharp transitions and outperforms piecewise linear and radial basis functions in benchmarks.
Calibrating a Lévy process usually requires characterizing its jump distribution. Traditionally this problem can be solved with nonparametric estimation using the empirical characteristic functions (ECF), assuming certain regularity, and results to date are mostly in 1D. For multivariate Lévy processes and less smooth Lévy densities, the problem becomes challenging as ECFs decay slowly and have large uncertainty because of limited observations. We solve this problem by approximating the Lévy density with a parametrized functional form; the characteristic function is then estimated using numerical integration. In our benchmarks, we used deep neural networks and found that they are robust and can capture sharp transitions in the Lévy density. They perform favorably compared to piecewise linear functions and radial basis functions. The methods and techniques developed here apply to many other problems that involve nonparametric estimation of functions embedded in a system model.