The asset price model

The underlying asset follows Geometric Brownian Motion (GBM):

\[dS_t = \mu S_t\, dt + \sigma S_t\, dW_t\]

where $W_t$ is a Wiener process, $\mu$ is the drift, and $\sigma$ is the volatility.

The Black–Scholes PDE

By forming a delta-hedged portfolio $\Pi = V - \Delta S$ and applying Itô’s lemma, all Brownian terms cancel under the right hedge ratio $\Delta = \partial V/\partial S$. No-arbitrage then gives:

\[\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0\]

The closed-form formula

For a European call with strike $K$ and expiry $T$:

\[C = S_0 \Phi(d_1) - K e^{-rT} \Phi(d_2)\] \[d_1 = \frac{\ln(S_0/K) + (r + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}\]

where $\Phi$ is the standard normal CDF.

The Greeks

Greek	Definition	Meaning
Delta $\Delta$	$\partial C/\partial S$	Sensitivity to spot price
Gamma $\Gamma$	$\partial^2 C/\partial S^2$	Rate of change of delta
Vega $\mathcal{V}$	$\partial C/\partial \sigma$	Sensitivity to volatility
Theta $\Theta$	$\partial C/\partial t$	Time decay