May 4, 2009

Black Scholes using Hedging and the Partial Differential Equation

Using the Black Scholes model assumptions, we assume that the underlying stock follows Geometric Brownian Motion.

\mathrm{d}S = \mu S \mathrm{d}t + \sigma S \mathrm{d}X.

Use \Pi to denote the value of a portfolio of one long option position and a short position in some quantity \Delta of the underlying:

\Pi = V(S, t) - \Delta S

As the time t to t + \mathrm{d}t the value of the portfolio will change. This change in value is partly due to the change in the option value and partly to the change in the underlying

\mathrm{d} \Pi = \mathrm{d} V - \Delta \mathrm{d} S

From Ito's Lemma we have

\mathrm{d} V = \frac{\partial V}{\partial t} \mathrm{d} t + \frac{\partial V}{\partial S} \mathrm{d} S + \frac{1}{2} \sigma^2 S^2 \frac{\partial ^2 V}{\partial S^2} \mathrm{d} t

Substituting, the value of \mathrm{d} V in the previous equation and grouping for \mathrm{d} t and \mathrm{d} S terms.

\mathrm{d} \Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial ^2 V}{\partial S^2} \right) \mathrm{d} t + \left( \frac{\partial V}{\partial S} - \Delta \right) \mathrm{d} S

\mathrm{d} t, terms are deterministic terms and \mathrm{d} S terms are random.

We can eliminate the random terms by choosing -

\Delta = \frac{ \partial V}{\partial S},

Thus by choosing a suitable value of \Delta, our portfolio change is deterministic and completely riskless. This riskfree change must be equal to the growth we would get by putting the equivalent amout in a risk-free asset.

\mathrm{d} \Pi = r \Pi \mathrm{d} t

Putting it all together and elimiating \Delta and \Pi, we get

\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 Black-Scholes PDE.