Asymptotics of Wiener Functionals and Applications to Mathematical Finance

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Title: Asymptotics of Wiener Functionals and Applications to Mathematical Finance
Authors: Violante, Sean
Item Type: Thesis or dissertation
Abstract: In this thesis we study asymptotic expansions for option pricing with emphasis on small noise “singular perturbations” which are, as we shall see, better suited than the more popular small time asymptotics to approximate typical stochastic volatility models. In particular, we argue that analytic solutions are unlikely for more advanced models, and therefore numerical methods of calculation are required. The following are the main results of the thesis. We show that zeroth order implied volatility is given by the large deviation rate function under minimal assumptions. We then show a small noise sample path large deviations principle for a class of two dimensional positive diffusions of relevance to finance. We numerically calculate the large deviations rate function for an example process, Gatheral’s Double CEV model, and highlight the speed and accuracy of the approximation. We then investigate Yoshida-Watanabe asymptotic expansions and develop a Mathematica program to derive them automatically. Lastly, we develop a small noise asymptotic expansion for marginal densities of solutions of SDEs (joint work). Using this we determine the large strike implied volatility for the Stein-Stein model and the Schobel and Zhu model by rescaling into a small noise problem.
Issue Date: 2012
Date Awarded: Mar-2013
Supervisor: Crisan, Dan
Department: Mathematics
Publisher: Imperial College London
Qualification Level: Doctoral
Qualification Name: Doctor of Philosophy (PhD)
Appears in Collections:Mathematics PhD theses

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