Eigenfunction Expansion Method in Multi-Factor Models

52 Pages Posted: 1 Dec 2004

See all articles by Nina Boyarchenko

Nina Boyarchenko

Federal Reserve Bank of New York

Sergei Levendorskii

Calico Science Consulting

Date Written: November 30, 2004

Abstract

We show that three classes of multi-factor gaussian mean-reverting models: for the dynamics of the (log-)price of a stock, ATSM of the Ornstein-Uhlenbeck type, and QTSM are equivalent, when contingent claims with deterministic life-spans are considered. We provide the reduction of these models to two basic types of QTSMs, which are explicitly solved by the eigenfunction expansion technique. The reduction uses solutions to continuous algebraic Riccati equations and Lyapunov equations. The eigenvalues, eigenfunctions and adjoint functions are calculated by using elements of the representation theory of Lie algebras; the eigenfunctions and adjoint functions are expressed in terms of the Hermite polynomials. We also consider the same classes of models in random time, and show that in the framework of the eigenfunction expansion approach, the models in random time are (almost) as simple as pure gaussian models. We suggest parameters' fitting procedures based on the properties of the asymptotic expansions.

Keywords: Derivative pricing, multi-factor exactly solvable models, eigenfunction expansion, continuous algebraic Riccati equations, Lyapunov equations, representation theory of Lie algebras, Hermite polynomials

JEL Classification: E43

Suggested Citation

Boyarchenko, Nina and Levendorskii, Sergei Z., Eigenfunction Expansion Method in Multi-Factor Models (November 30, 2004). Available at SSRN: https://ssrn.com/abstract=627642 or http://dx.doi.org/10.2139/ssrn.627642

Nina Boyarchenko

Federal Reserve Bank of New York ( email )

33 Liberty Street
New York, NY 10045
United States
212-720-7339 (Phone)
212-720-1582 (Fax)

Sergei Z. Levendorskii (Contact Author)

Calico Science Consulting ( email )

Austin, TX
United States