Multi-asset option pricing problems: a variational approach

Chuang, Chienmin (2012). Multi-asset option pricing problems: a variational approach. University of Birmingham. Ph.D.

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Abstract

Options are important and frequently traded products in the modern financial market. How to price them fairly and reasonably is always an interesting issue for academia and industry. This research is performed under the classical multi-asset Black-Scholes-Merton (BSM) model and can be extended to other exotic models.

We show how to reformulate the multi-asset Black-Scholes-Merton partial differential equation/inequality (BSM PDE/PDI) and provide theorems to justify the unique solution of reformulations. In terms of discretization, we adopt the finite element method (FEM) in space and finite difference method (FDM) in time. Moreover, we develop the closed-form formulas for the elemental matrices used in the finite element assembly process in a general high-dimensional framework.

The discrete systems of option pricing problems are presented in the form of linear system of equations (LSE) and linear complementary problems (LCP) for European and American/perpetual options respectively. Up to six different algorithms for the LCP are introduced and compared on the basis of computational efficiency and errors.

The option values of European, American and perpetual types are calibrated when given various payoffs and up to three assets. Particularly, their numerical free boundaries are identified and presented in the form of (d - 1)-dimensional manifold in a d-assetframework. In the last chapter, we conclude our research with our contributions and potential extension.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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