Stochastic Differential Equations for Quant Finance

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*🪐 Jupyter Notebook*
https://github.com/romanmichaelpaolucci/Quant-Guild-Library/blob/main/2025%20Video%20Lectures/37.%20Stochastic%20Differential%20Equations%20for%20Quant%20Finance/solving_stochastic_differential_equations.ipynb

**Roman's Overview of ODE/PDE/SDEs**

*ODEs*: representing a function as its derivative which can be solved via analytical or numerical techniques to recover said function

*PDEs*: can be constructed using a variety of arguments and used to solve for option prices analytically or numerically (finite-differences)

*SDEs*: solutions can be constructed analytically or numerically to produce option prices via the Law of Large Numbers (LLN, Monte Carlo Simulation)

*Black-Scholes Model*: The analytical price is given by the solution to the Black-Scholes equation which can be solved analytically or numerically. The argument assumes a geometric Brownian motion, which can ALSO produce prices via Monte Carlo simulation - I have many videos discussing this idea!

I hope you enjoyed, this was a long one!

- Roman
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*📖 Chapters:*
00:00 - Introduction
02:57 - Understanding Differential Equations (ODEs)
07:15 - How to Think About Differential Equations
09:59 - Understanding Partial Differential Equations (PDEs)
11:31 - Black-Scholes Equation as a PDE
16:49 - ODEs, PDEs, SDEs in Quant Finance
18:17 - Understanding Stochastic Differential Equations (SDEs)
22:30 - Linear and Multiplicative SDEs
23:34 - Solving Geometric Brownian Motion
37:43 - Analytical Solution to Geometric Brownian Motion
40:22 - Analytical Solutions to SDEs and Statistics
43:21 - Numerical Solutions to SDEs and Statistics
47:29 - Tactics for Finding Option Prices
49:46 - Closing Thoughts and Future Topics
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*▶️ Related Videos*

*Ito's Lemma Clearly and Visually Explained*
https://www.youtube.com/watch?v=TgBzqdN24fo

*Ito Integration Clearly and Visually Explained*
https://www.youtube.com/watch?v=dUvZ8m3QpeI&t=6s

*Monte Carlo Simulation and Black-Scholes for Pricing Options*
https://www.youtube.com/watch?v=-1RYvajksjQ

*Why Monte Carlo Simulation Works*
https://youtu.be/-4sf43SLL3A

*Expected Stock Returns Don't Exist*
https://youtu.be/iXNSBn5xqrA

*How to Trade*
https://youtu.be/NqOj__PaMec

*How to Trade with an Edge*
https://youtu.be/NlqpDB2BhxE
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*🗂️ Resources*

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https://quantguild.medium.com/
https://medium.com/quant-guild
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*🛠️ Projects*

*The Gaussian Cookbook:*
https://gaussiancookbook.com

*Recipes for simulating stochastic processes:*
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5332011
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Видео Stochastic Differential Equations for Quant Finance канала Roman Paolucci