Asymptotics of Wiener Functionals and Applications to Mathematical Finance

摘要

In this thesis we study asymptotic expansions for option pricing with emphasisudon small noise “singular perturbations” which are, as we shall see, better suited thanudthe more popular small time asymptotics to approximate typical stochastic volatilityudmodels. In particular, we argue that analytic solutions are unlikely for more advancedudmodels, and therefore numerical methods of calculation are required. Theudfollowing are the main results of the thesis. We show that zeroth order implied volatilityudis given by the large deviation rate function under minimal assumptions. Weudthen show a small noise sample path large deviations principle for a class of two dimensionaludpositive diffusions of relevance to finance. We numerically calculate theudlarge deviations rate function for an example process, Gatheral’s Double CEV model,udand highlight the speed and accuracy of the approximation. We then investigateudYoshida-Watanabe asymptotic expansions and develop a Mathematica program toudderive them automatically. Lastly, we develop a small noise asymptotic expansionudfor marginal densities of solutions of SDEs (joint work). Using this we determineudthe large strike implied volatility for the Stein-Stein model and the Schobel and Zhuudmodel by rescaling into a small noise problem.