The Math of "The Trillion Dollar Equation"

A deep dive into Veritasium's    • The Trillion Dollar Equation   . Detailed explanation of the classic "Delta Hedging" derivation of the Black-Scholes Partial Differential Equation for European Call Options from scratch.

Here are my notes from I was a PhD student on this stuff (we were allowed to bring in short notes to the exam)

www.math.toronto.edu/mnica/PDE_Finance_Sheet.pdf

www.math.toronto.edu/mnica/Stoch_Calc_Sheet.pdf

0:00 The Trillion Dollar Equation
0:18 Veritasium's Example of the Call Option
2:13 The function v(s t) for the value of the option
3:29 Veritasium's Delta hedged portfolio
4:19 Veritasium's slide by derivation
5:24 Derivation of the Black-Scholes equation from scratch
8:26 Change in portfolio value d Pi t
9:47 Approximate changes by derivatives of v a la Taylor series
15:08 Model for the stock price S t Geometric Brownian Motion
19:42 d S t squared is just a dt a la Itos Lemma
25:44 Set delta to di v di s to perfectly hedge
27:23 Equate to the risk free interest rate equation
29:10 But isnt delta a function of time? Why does this work?