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.