| This dissertation applies techniques of stochastic calculus to solve problems in two areas of research: Schramm-Loewner evolution (SLE) and option pricing.;In relation to SLE, we review one method for estimating the modulus of continuity of an SLE curve in terms of the inverse Loewner map. Then we prove estimates about the distribution of the inverse Loewner map, which underpin the difficulty in bounding the modulus of continuity of SLE for kappa = 8. The main idea in the proof of these estimates is applying the Girsanov theorem to reduce the problem to estimates about one-dimensional Brownian motion.;In the area of option pricing, we consider a family of models written in terms of a parameter epsilon, which result from perturbing a model where option prices can be computed exactly. We compute option prices rigorously as a series in epsilon using two methods: characteristic functions, and martingales. When a series expansion for the option price is available, we derive an expansion for the implied volatility. We compute option prices numerically for a number of models, and analyse the approximation of the resulting expansion. |
