Based on the concept of self-decomposability we extend some recent multidimensional Lévy models by using multivariate subordination. Our aim is to construct multi-asset market models in which the systemic risk instead of affecting all markets at the same time presents some stochastic delay. In particular we derive new multidimensional versions of the well known Variance Gamma and inverse Gaussian processes. To this end, we extend some known approaches keeping their mathematical tractability, we study the properties of the new processes, we derive closed form expressions for their characteristic functions and, finally, we detail how new and efficient Monte Carlo schemes can be implemented. As second contribution of the work, we construct a new Lévy process, termed the Variance Gamma++ process, to model the dynamic of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained relying upon the self-decomposability of the gamma law. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function and transition probability density. These results are instrumental to apply Fourier-based option pricing and maximum likelihood techniques for the parameter estimation. Furthermore, we provide efficient path simulation algorithms, both forward and backward in time. We also obtain an efficient “integral-free” explicit pricing formula for European options. Finally, we illustrate the applicability of our models in the context of gas, power and emission markets focusing on their calibration, on the pricing of spread options written on different underlying commodities and on the evaluation of exotic American derivatives, giving an economical interpretation to the obtained results.

Financial models in continuous time with self-decomposability: application to the pricing of energy derivatives

GARDINI, MATTEO
2022

Abstract

Based on the concept of self-decomposability we extend some recent multidimensional Lévy models by using multivariate subordination. Our aim is to construct multi-asset market models in which the systemic risk instead of affecting all markets at the same time presents some stochastic delay. In particular we derive new multidimensional versions of the well known Variance Gamma and inverse Gaussian processes. To this end, we extend some known approaches keeping their mathematical tractability, we study the properties of the new processes, we derive closed form expressions for their characteristic functions and, finally, we detail how new and efficient Monte Carlo schemes can be implemented. As second contribution of the work, we construct a new Lévy process, termed the Variance Gamma++ process, to model the dynamic of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained relying upon the self-decomposability of the gamma law. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function and transition probability density. These results are instrumental to apply Fourier-based option pricing and maximum likelihood techniques for the parameter estimation. Furthermore, we provide efficient path simulation algorithms, both forward and backward in time. We also obtain an efficient “integral-free” explicit pricing formula for European options. Finally, we illustrate the applicability of our models in the context of gas, power and emission markets focusing on their calibration, on the pricing of spread options written on different underlying commodities and on the evaluation of exotic American derivatives, giving an economical interpretation to the obtained results.
Lévy Processes, Monte Carlo simulations, Energy Derivative Pricing, Fast Fourier Transform, Option Pricing, Calibration, Energy Markets.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/1070581
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