Mathematical Finance Seminar

March 22, 2001 , 5:30 PM to 7:00 PM

Vadim Linetsky, University of Michigan

Eigenfunction Expansion Methods in Asset Pricing

We develop an eigenfunction expansion approach to pricing derivative securities written on diffusion processes. Contingent claims are unbundled into portfolios of primitive securities, termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator and diagonalize the pricing semigroup on the Hilbert space of all contingent claims written on the underlying diffusion (with given boundary and/or integrability conditions). The computational work is at the stage of determining eigenvalues and eigenvectors of the pricing operator. This is accomplished by appealing to the spectral theory of second-order differential operators (Sturm-Liouville theory for scalar diffusions in particular). The pricing is then immediate by the linearity property of the pricing operator and the eigenvector property of eigensecurities. To illustrate the power of the method, we develop a series of applications to problems in asset pricing: · Pricing vanilla, single- and double-barrier and lookback options under the constant elasticity of variance (CEV) process exhibiting volatility smiles/skews · Pricing interest rate knock-out options in the Cox-Ingersoll-Ross term-structure model · Deriving a new closed-form pricing formula for the arithmetic Asian option under the lognormal process (effectively inverting the Geman-Yor Laplace transform) · Pricing occupation time derivatives (including step options) under the CEV process The talk is based on joint work with Dmitry Davydov, Global Equity Derivatives Research, UBS Warburg.