Estimation and Simulation for Multivariate Tempered Stable Distributions with Applications To Finance

1 online resource (117 pages) : PDF

2022

University of North Carolina at Charlotte

In this thesis, we introduce a methodology for the simulation and parameter estimation of multivariate tempered stable distributions. Using the fact that tempered stable distributions can be specified indirectly by a Lévy measure, our approach is based on an approximation due to a discretization of the Lévy measure. We derive this discretization in general and give an explicit construction of the discretization in the bivariate case. Also, our approximation results hold for a wide class of multivariate infinitely divisible distributions. Based on our main approximation, we develop a method for simulations, which we call the discretization and simulation (DS) method. To demonstrate how well the method works, we perform a series of simulations in the bivariate cases and compare it with another approximate simulation method developed by Rosiński. Further, we use our discretization for parameter estimation by minimizing the distance between the characteristic function of the multivariate tempered stable distribution and the empirical characteristic function. We then apply our methodology to two bivariate financial datasets related to exchange rates. The first is comprised of exchange rates between standard currencies, while the second is based on exchange rates related to cryptocurrencies. We also perform goodness-of-fit tests to show that the multivariate tempered stable model does a good job fitting the model. Further, we apply our model for the pricing of the bivariate basket option. Toward this end, we provide theoretical results on the existence of equivalent martingale measures. Then combining this with our model for parameter estimation and the DS method for simulation, we develop a Monte Carlo based method for option pricing. We apply it to the pricing of European call options with different strikes and the pricing of the Multi-asset rainbow option.

doctoral dissertations

Mathematics

Ph.D.

Discretization
Infinitely Divisible Distributions
Tempered Stable Distribution

Applied Mathematics

Grabchak, Michael

Molchanov, Stanislav
Sonin, Isaac
Tian, Weidong

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2022.

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Xia_uncc_0694D_13066

http://hdl.handle.net/20.500.13093/etd:2668