Call price hedging setup by Merton

Stock price follows a geometric Brownian motion:

\[\begin{equation} dS = \mu S d_t + \sigma S dz \end{equation}\]

Using Ito’s lemma, call price must satisfy the PDE function:

\[dc = \bigg( \frac{\partial c}{\partial t} + \frac{\partial c}{\partial S} \mu S + \frac{1}{2} \frac{\partial^2 c}{\partial S^2} \sigma^2 S^2 \bigg)dt + \frac{\partial c}{\partial S} \sigma S dz\]

Value of the portfolio with delta hedging (short 1 option and long \(\displaystyle\frac{\partial c}{\partial S}\) derivatives or vice versa):

\[V = -c + \displaystyle\frac{\partial c}{\partial S} S\]

The change of the portfolio value:

\[dV = -dc + \displaystyle\frac{\partial c}{\partial S} dS\]

Replace the change in the portfolio value into the PDE we have:

\[dV = - \bigg( \frac{\partial c}{\partial t} + \frac{\partial c}{\partial S} \mu S + \frac{1}{2} \frac{\partial^2 c}{\partial S^2} \sigma^2 S^2 \bigg)dt - \frac{\partial c}{\partial S} \sigma S dz + \frac{\partial c}{\partial S} (\mu S dt + \sigma S dz)\] \[= - \frac{\partial c}{\partial t} dt - \frac{1}{2} \frac{\partial^2 c}{\partial S^2}\sigma^2 S^2 dt\]

This portfolio does not include the stochastic/Brownian component, it should earn the risk free rate:

\[dV = rVdt = r \bigg(-c + \frac{\partial c}{\partial S} S \bigg) dt = - \frac{\partial c}{\partial t} dt - \frac{1}{2} \frac{\partial^2 c}{\partial S^2}\sigma^2 S^2 dt\]

Re-arrange the equation, we have:

\[\frac{\partial c}{\partial t} + \frac{\partial c}{\partial S} S + \frac{1}{2} \frac{\partial^2 c}{\partial S^2}\sigma^2 S^2 = rc\]

Based on this PDE, we transform and solve the heat equation using boundary conditions. The result is Black-Scholes equation for option pricing. Put price is derived using put-call parity