Financial Markets with Delay

Abstract

This thesis focuses on the study of financial markets with delay and in a broader sense, the study of stochastic differential equations with delay. We describe a continuous model, argue the existence, uniqueness and positivity of a solution to the stochastic differential equation chosen. We then prove non-arbitrage property as well as the completeness of the market. Numerical approaches are developed: the classic Euler-Maruyama scheme for delay equations and a so-called logarithmic Euler-Maruyama scheme. A pure jump model is then considered. Existence, uniqueness and positivity of the solution to the stochastic differential equation describing the stock price dynamics are proven. We prove the non-arbitrage property and the incompleteness of the market. An extension of the classic Euler-Maruyama scheme with jumps is developed and an approach to hedging in such cases is then discussed.