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Orthogonal Polynomials and Their Applications in Financial and Actuarial Models Open Access


Other title
Pearson's equation
orthogonal polynomials
Rodrigues formula
actuarial and financial models
Type of item
Degree grantor
University of Alberta
Author or creator
Li, Hao
Supervisor and department
Melnikov, Alexander (Mathematical and Statistical Sciences)
Examining committee member and department
Schmuland, Byron (Mathematical and Statistical Sciences)
Cadenillas, Abel (Mathematical and Statistical Sciences)
Litvak, Alexander (Mathematical and Statistical Sciences)
Rachev, Svetlozar (Applied Mathematics and Statistics)
Melnikov, Alexander (Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Mathematical Finance
Date accepted
Graduation date
Doctor of Philosophy
Degree level
It is well known that the normal return estimation for financial asset prices is defective. In order to find better models to estimate the prices behavior of financial assets, people need probabilistic distribution that can capture fat-tails, non-constant moments, etc. This thesis find some distributions that can be utilized to model the financial asset returns and the actuarial claim sizes, with the help of some orthogonal polynomials. We use the Pearson's differential equation as a way for orthogonal polynomials construction and solution. The generalized Rodrigues formula is used for this goal. Deriving the weight function of the differential equation, we use it as a basic distribution density of variables like financial asset returns or insurance claim sizes. This density function is adjusted using the product with a polynomial, which is expressed as a linear combination of the orthogonal polynomials we find as the solutions of the Pearson's differential equation. Using this method, we create the Polynomial-Normal model, Polynomial-T-Distribution model and some further extensions. We derive explicit formulae for option prices as well as for insurance premiums. The numerical analysis shows that our new models provide a better fit than some previous actuarial and financial models.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
Li, H. and Melnikov, A. (2012) On the Polynomial-Normal Model and Option Pricing. Advances in Statistics, Probability and Actuarial Science, 1, 285-302. World Scientific Published, Singapore.

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