The non-parametric calibration of jump-diffusion lévy models

Van Zyl, jaco

The non-parametric calibration of jump-diffusion lévy models

Thesis / Dissertation

2022

Publisher

University of Cape Town

Department

Accounting and Accountability in Africa

Faculty

Faculty of Commerce

Abstract

This study investigates the effectiveness of relative-entropy-based regularised calibration procedures at addressing the ill-posedness encountered in the calibration of nonparametric jump-diffusion Lévy models. Calibrated models, which have been selected for their capability of simulating realistic price path evolutions, are typically employed to price path-dependent instruments or to perform dynamic hedging. The financial risks associated with using pricing models in real-world transactions can be reduced by selecting an appropriate model together with a suitable calibration procedure. It has been well established that mispricing resulting from improper modelbased pricing has resulted in significant financial losses. Lévy processes aim at improving on the ability of continuous models to represent market-observed price evolutions by allowing highly flexible jump characteristics to be specified, enabling the inclusion of jump-discontinuities in pricing models. The capacity of parametric Lévy models to provide a general class of feature-rich models with jump-discontinuities is further enhanced by the extension to non-parametric Lévy models, where the jump characteristics can more freely be defined. Model calibrations, like most other inverse problems, suffer from ill-posedness. Consequently, optimisation results from a calibration exercise generally do not converge to a unique solution and typically do not vary continuously with input data. The increase in dimension of the solution space associated with the non-parametric approach necessitates further measures to address the ill-posedness. This study evaluates the performance of the relative-entropy-based regularised calibration procedures proposed by Cont and Tankov [CT04, CT06] at addressing the primary concern of ill-posedness encountered in the calibration of non-parametric jump-diffusion Lévy models. We will show that although the procedures provide some stability with respect to the input prices between subsequent calibrations, the procedures are of limited value at addressing the ill-posedness relating to the convergence to a unique solution. In our experiments, we expose the sensitivity of results to both the initial points as well as to the prior measure presented to the optimisation procedures. Our study highlights some deficiencies in the process, showing that the regularised calibration procedure is unreliable, necessitating active user intervention to manage outcomes. We conclude that the observed difficulties are primarily the result of a persistent non-convexity of the regularised objective function at realistic levels of regularisation. Therefore ill-posedness continues to present a risk that needs to be managed by practitioners when applying these procedures to the recovery of non-parametric Lévy models.

Keywords

lévy models

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