Overview
The Black Sholes equation is
{% \frac{\partial V}{\partial t} + \frac{1}{2} \sigma ^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} -rV = 0 %}
where
- {% V %} is the value of the option
- {% S %} is the price of the underlying asset.
- {% \sigma %} is the volatility of the underlying asset
- {% r %} is the interest rate
Derivation of Black Scholes
Consider a European call option, and let {% C(S,t) %} denote its value as a function of its underlying stock price, {% S %} and time {% t %}.
Now consider a portfolio that is short the call option and long {% \Delta %} shares of the stock. The remainder, {% C -\Delta S %}
is invested in the risk free asset, with rate {% r %}. Then the change of the value of the portfolio is
{% d \Pi = -dC + \Delta dS + (C- \Delta S)r %}
we know from Itos lemma that
{% dC = \frac{\partial{C}}{\partial{S}} dS + \frac{\partial{C}}{\partial{t}} dt + \frac{1}{2} \frac{\partial^2{C}}{\partial{S}^2} dt %}
Substituing this into the portfolio equation, we find that we can eliminate the {% dS %} term is we take
{% \Delta = \frac{\partial{C}}{\partial{S}} %}
If we trade the portfolio such that delta is given as above, then the portfolio has no risk, and as such, should grow at the risk
free rate.