Analytical and numerical solution to the partial differential equation arising in financial modeling

In this work we will present a self-contained introduction to the option pricing problem. We will introduce some basic ideas from the probability theory and stochastic differential equations. Later we will move to the partial differential equations since the option pricing problem arising in financial mathematics when asset is driven by a stochastic volatility process and assumed presence of transaction cost leads to solving non-linear partial differential equation. We will also present the complete process from deriving the desired partial differential equation to the proof of existence of a solution and also the numerical simulations. Using techniques form stochastic calculus we will derive the main equation which we are going to analyze for the rest of this work. Later we will show the existence of a solution and at last we will provide numerical results for a set of market parameters