## Partial Differential Equations in Mathematical Finance

These lectures highlight the origins of partial differential equations (PDEs) in math finance and the way we use them.
The content is designed to complement traditional math finance curriculum.

## Lecture 1: PDEs as a Computational Tool

### 8:30-10:30 AM

A financial derivative is a financial product that derives its value from an underlying traded asset. We will use PDEs as a method for pricing financial derivatives and will focus on how to set up the relevant PDE problem sensibly from a mathematical and financial perspective. We will discuss how to solve these pricing PDEs numerically with the explicit finite difference method.

Suggested background knowledge: Calculus (21-120, 21-122) and some familiarity with differential equations at the level of 21-122.
Course descriptions can be found here.

## Lecture 2: Foundations of PDEs from Probabilistic Models

### 1:30-3:30 PM

In Lecture 1, we viewed a PDE as a pricing rule in a "black box" sense: we cared how to use and implement it but didn't care where it came from. Now, we will focus on the underlying probabilistic models for asset prices and derive the PDEs used in Lecture 1. We will discuss some of the main components of stochastic calculus: Brownian motion, stochastic integration, Ito's Lemma (the chain rule), and stochastic differential equations. Using the tools from stochastic calculus, we will connect probabilistic models for stock prices to PDEs using the Feynman-Kac formula. Time-permitting, we will discuss some of the uses of PDEs in modern research.

Suggested background knowledge: Calculus (21-120, 21-122); calculus-based probability would be useful (36-225, 36-217, or 21-325), as would real analysis (21-355).
Course descriptions can be found here.