Orthogonal Polynomials and Their Applications in Financial and Actuarial Models

• Author / Creator

  Li, Hao

• 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.

• Subjects / Keywords

  • Orthogonal polynomials
  • Rodrigues formula
  • Actuarial and financial models
  • Pearson's equation

• Graduation date

  Spring 2014

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R37383

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.