Applications of New Diffusion Models to Barrier Option Pricing and First Hitting Time in Finance

The main focus of this thesis is in the application of a new family of analytical solvable diffusion models to arbitrage-free pricing exotic financial derivatives, such as barrier options. The family of diffusions is the so-called “Drifted Bessel family” having nonlinear (smile-like) local volatility with multiple adjustable parameters. In particular, the drifted Bessel-K diffusion is used to model asset (stock) price processes under a risk-neutral measure whereby discounted asset price are martingales. Closed-form spectral expansions for barrier option values are derived within the Bessel-K family of models. This follow from the closed-form spectral expansions for the transition probability densities which are obtained for the Bessel family of processes with imposed killing boundaries. We also show that the commonly adopted CEV model is recovered as a special parametric limit of our Bessel family of models for the case of zero drift. The rapid convergence of the spectral expansions leads to very efficient numerical implementations of barrier option pricing and sensitivity analysis. We hence carry out various numerical computations in order to study the relative effects of the parameters (state dependencies) of the Bessel family of models with respect to barrier option pricing and hedging. We compare our results with the standard Black-Scholes (GBM) and CEV models, demonstrating that model specification leads to important differences when pricing non-vanilla options, such as barrier options