Financial markets driven by skew diffusions

Abstract

This thesis considers topics in stochastic control theory and in interest rate derivatives theory. These topics are motivated by financial applications. We formulate the stochastic control problem and use the principle of dynamic programming for the optimal strategy. The problem is Merton’s type of portfolio optimisation problem, however, there is an added feature in the price of the risky asset. The risky asset price is modelled such that it exhibits support and resistance levels, particularly we consider a skew geometric Brownian motion. For three special cases we derive closed form solutions. The second topic is the interest rate derivatives theory. We model the short rate by a skew diffusion, with the local time term accounting for the presence of several meanreversion levels. We formulate the model and relying on the Feynman-Kac Theorem and Laplace transformation we derive the price of the zero-coupon bonds under the risk-neutral measure. We show that the bond market does not admit arbitrage.