# CHAPTER 9

# Numerical Solution of Multifactor Pricing Problems Using Lagrange-Galerkin with Duality Methods

## 9.1 INTRODUCTION

Many financial derivative products are conveniently modeled in terms of one or more factors, or stochastic spatial variables, and time. Based on the contingent claims analysis developed by Black and Scholes [72] and Merton [73], a partial differential equation (PDE) for the fair price of these derivatives can be obtained. Valuation PDEs for financial derivatives are usually parabolic and of second order. In the more general case, partial differential inequalities (PDIs) arise. The inequality comes when the option has some embedded early-exercise features and the price of the contingent claim must satisfy some inequality constraints in order to avoid arbitrage opportunities. In other words, if the price were to violate those constraints, the option would be exercised, since both the buyer and the seller of an option will try to maximize the value of their rights under the contract. Early-exercise features appear, for example, in American options and in the conversion, call, and put provisions of convertible bonds. These are so-called free boundary problems because there are (a priori) unknown boundaries separating the regions where inequalities are strict from those where they are saturated.

It is almost always impossible to find an explicit solution to a free boundary problem: we need numerical techniques. The extra complication in those problems comes from ...