The Black-Scholes equation is a cornerstone of financial mathematics, providing a theoretical framework for pricing European-style options. This partial differential equation describes how an option's price changes over time and with fluctuations in the underlying asset's price. It considers factors like the current stock price, time to expiration, strike price, risk-free interest rate, and volatility. Crucially, it assumes a no-arbitrage market and that the underlying asset's price follows a geometric Brownian motion. Solving the equation yields the option's fair price, crucial for trading and risk management.

To derive the Black-Scholes equation for the portfolio:

$$ \hat{\Pi}=-p+S \frac{\partial p}{\partial S} $$

🧠We proceed as follows:

Step 1: Define the Stock Price Dynamics (Geometric Brownian Motion) We assume that the stock price $S$ follows the stochastic differential equation (SDE):

$$ d S=\mu S d t+\sigma S d W $$

where:

• $\mu$ is the drift (expected return),
• $\sigma$ is the volatility,
• $d W$ is a Wiener process.

We also assume that the option price $p(S, t)$ is a function of $S$ and time $t$, and it evolves according to Itô's Lemma.

Step 2: Apply Itô's Lemma to the Option Price $p(S, t)$

Using Itô's Lemma for a function $p(S, t)$ :

$$ d p=\left(\frac{\partial p}{\partial t}+\mu S \frac{\partial p}{\partial S}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 p}{\partial S^2}\right) d t+\sigma S \frac{\partial p}{\partial S} d W $$

Step 3: Define the Portfolio $\hat{\Pi}$

The portfolio consists of:

• Short position in the option: $-p$,
• Long position in $S$ : Holding $\frac{\partial p}{\partial S}$ units of stock.

Thus, the portfolio value is:

$$ \hat{\Pi}=-p+S \frac{\partial p}{\partial S} $$