Introduction
In Financial Mathematics we are interested in pricing financial derivatives (most commonly options) whose payoffs are complicated functions of stochastic pricess processes that are often not solveable analytically. In this project we explore the three main approaches for numerically solving these pricing problems:
- Monte-Carlo (MC) Methods,
- Partial Differential Equations (PDEs), and
- Surrogate Methods.
Throughout the project we will provide example code applying the described method using python. A detailed discussion of the mathematical theory of financial models and option pricing is reserved for a seperate note. This project was informed by materials provided by Prof. Michael Ludkovski through his PSTAT222: Computational Finance course which I took in Spring 2024. The source material for this course was the textbook An Introduction to the Numerical Simulation of Stochastic Differential Equations by Higham & Kloeden. All material produced in this project can be found in the following GitHub repository.
SDE Monte-Carlo Methods
Motivating Example
Assume that a stock price is described by a continuous-time stochastic process with dynamics given by the stochastic differential equation (SDE) \[ dX_t = \]
Consider a call option on the general SD
Suppose that we wish to numerically solve the general SDE \[ dX(t)=f(t,X(t))dt+g(t,X(t))dW(t), \]
on \(0 \leq t \leq T\) with given initial \(X(0)=x\). By definition, the exact solution is given by \[ X(t)-X(t_{0})=\int_{t_{0}}^t f(s,X_{s})ds+\int_{t_{0}}^t g(s,X(s))dW(s). \]