Stochastic Differential Equations for Quant Finance

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*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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