Stochastic Calculus for Quants | Understanding Geometric Brownian Motion using Itô Calculus

In this tutorial we will learn the basics of Itô processes and attempt to understand how the dynamics of Geometric Brownian Motion (GBM) can be derived. First we learn what an Itô integral is and how it differs from a regular integral. This leads us to discussing the dynamics of Itô processes and then a special type of calculus for financial mathematics based on Brownian Motion called Itô Calculus.

We will understand why this is different to ordinary calculus in terms of the accumulation of quadratic variation. Also, we discuss how to use Taylor series expansion and using Itô’s Lemma to understand the dynamics of a particular function (valuation) given a defined Itô process, or stochastic differential equation (SDE) that has been defined for the underlying.

We briefly discuss a generic drift diffusion model and the Itô-Doeblin formula for Itô processes. This then leads to a derivation of the dynamics of Geometric Brownian Motion, and it’s explicit formulation which can be used for simulating GBM paths.

00:00 Intro
01:34 Itô Integrals
06:30 Itô processes
09:10 Contract/Valuation Dynamics based on Underlying SDE
12:24 Itô's Lemma
13:35 Itô-Doeblin Formula for Generic Itô Processes
18:04 Geometric Brownian Motion Dynamics

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