Black Scholes Equation

Statement

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

Assumptions

Stock price dynamics

The underlying stock follows a geometric Brownian motion:

\[dS = \mu S dt + \sigma S dW.\]

Self-financing portfolio

Consider a self-financing portfolio $\Pi$ consisting of

• a long position in one derivative whose value is $V = V(S_t, t)$, and
• a short position in $\Delta = \frac{\partial V}{\partial S}$ shares of the stock.

The portfolio value is

\[\Pi = V - \Delta S.\]

Ito’s lemma

Applying Ito’s lemma to $V(S,t)$, we have

\[\begin{align*} d\Pi &= dV - \frac{\partial V}{\partial S}dS\\ &= \left( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt. \end{align*}\]

Therefore the portfolio is risk-free.

No-arbitrage condition

The portfolio must earn the risk-free rate $r$:

\[d\Pi = r\Pi dt.\]

Substituting $\Pi = V - \frac{\partial V}{\partial S} S$ yields the Black-Scholes equation.