Path-dependent functionals of constant elasticity of variance and related processes : distributional results and applications in finance
The present thesis provides an analysis of some path-dependent functionals of Constant Elasticity of Variance (CEV) processes. More precisely, we study the continuous arithmetic average of the process over time, plain or sometimes multiplied by a knock-out indicator. We start by describing its mathematical properties and provide new distributional results (moments, densities, moment generating function among others). Some of these results also pertain to the joint distribution of the integral and the process itself. The versatility of the process enables us to consider diverse financial applications: fixed and floating strike Asian options on equities, European vanilla options on equity in the presence of stochastic volatility as well as zero-coupon bonds, guaranteed endowment options and average-rate claims under stochastic interest rates. We devote a great part of the present work to the square-root process and the geometric Brownian motion, two important subcases of the CEV process. For both these nested diffusions, a number of mathematical and financial quantities have been solved for in the literature in closed-form, in terms of Laplace transforms. In this thesis, we derive these quantities in a fully explicit form, which is advantageous both from a theoretical point of view, to gain insight in their mathematical structure and from a practical stand, as the numerical evaluation of our formulae appear more robust and efficient than other numerical methods for some ranges of parameters. In the general CEV case, for which the integrated process has scarcely been considered in the literature, we derive semi-closed form expressions.