Black-Scholes Derivation — Delta Hedging Argument

source: https://thefinancialengineer.org/financial-engineering/volatility-engineering-and-volatility-trading/

In order to derive the Black Scholes PDE from the Brownian Motion using the Delta-Hedging Argument, we have to set up our self-financing portfolio first. This portfolio will be comprised of an option, ∆ units of the underlying stock, and a bank account, B. The goal is to make our portfolio risk-free, i.e. insensitive to changes in the price of the security.

We start by writing down the equations of the individual components of our portfolio.

a. Price of the Stock: Is assumed to follow the Geometric Brownian Motion.

b. Bank Account: the interest is assumed to be constant

c. Option Value: the value of which will change according to changes in time and the underlying stock movements.

For the option value, we will use its differential form derived by applying Ito’s Lemma:

Using Delta-Hedging to Eliminate the Stochastic Components

Delta hedging means that we are reducing the risk of the option by trading ∆ units of the underlying stock. This will make the value of the portfolio remain unchanged when small changes occur to the value of the underlying.

This means that we have to construct a strategy that involves the combination of the option value and…