On the Pricing of Path-Dependent Options and Related Problems

Doctoral thesis, 2003

The thesis considers the pricing of European path-dependent options in a multi-dimensional Black-Scholes model. The thesis focuses mainly on the three different classes of path-dependent options: barrier, Asian, and lookback options. The thesis consists of eight chapters. Chapter 1 gives a brief introduction to the theory of option pricing and describes some path-dependent options. Chapter 2 derives pricing formulas for continuous double barrier options and studies the numerical properties of the formulas obtained. Chapter 3 extends a work by Broadie, Glasserman, and Kou and determines approximation formulas for the price of some discrete barrier options. Chapter 4 estimates the price of discrete barrier options using lattice random walks. Chapter 4 will also discuss the rate of convergence of lattice methods and Besov spaces. Chapter 5 gives a probabilistic interpretation of the theta-method. The theta-method is a class of finite difference methods for the heat equation. Chapter 6 shows, using the isoperimetric inequality for Wiener measure, that the relative error in the Monte Carlo pricing of some path-dependent options is independent of the dimension. Chapter 7 studies a certain class of sublinear functionals of geometric Brownian motion. The chapter discusses convexity properties for the distribution function, tail probabilities, stochastic ordering, moment inequalities, and Stieltjes moment problem. Chapter 8, which is a joint work together with Jenny Dennemark and Håkan Norekrans, describes the Heath-Jarrow model for dividend paying assets and studies how discrete dividends influence the price of some path-dependent options.