Pricing multivariate financial derivatives using polynomial approximations

Since the early days of the theory of option pricing, pioneered by Black, Scholes and Merton ([3] and [11]) the use of complex financial derivatives has increased tremendously. Finding prices of financial derivatives accurately and efficiently is an important problem for the financial industry for obvious reasons. It is also a challenging mathematical problem as we do not have explicit expressions for option prices in many situations, and the use of numerical methods is almost unavoidable. In the available literature there are very general methods to price financial derivatives. Deriving and solving the corresponding pricing PDE can be used in many instances (see [6] for instance) but the numerical methods to solve PDEs are not efficient as the dimension of the problem increases. Monte Carlo methods are also popular (see [8]) and do not suffer from "curse of dimensionality" issues, but they can still be slow, and require many simulations to get pricing results within the desired accuracy. Closed-form accurate approximations of option prices are preferable than other computationally expensive numerical methods. In the case of spread options (that depend on two assets) some initial works in this direction are [10], [4] and [5]. This honours project concerns the pricing of multivariate financial instruments. The objective of this work is to develop a pricing methodology that is accurate, computationally efficient, and that admits generalization to price derivatives that depend on an arbitrary number of assets.