Changes

at time <math>T</math>. Let <math>\Phi(t)</math> denote the fair value of this contract at time <math>t< T</math>. In deriving a formula for <math>\Phi(t)</math>, Black and Scholes' key insight was that by forming a portfolio with the exact right balance of <math>S</math> and the call option, one can completely eliminate risk associated to movements in the stock price <math>S</math>. Moreover, the resulting portfolio, being risk-free, has to grow at the risk free rate. These observations implied that the fair price of the call option had to satisfy the differential equation:

at time <math>T</math>. Let <math>\Phi(t)</math> denote the fair value of this contract at time <math>t< T</math>. In deriving a formula for <math>\Phi(t)</math>, Black and Scholes' key insight was that by forming a portfolio with the exact right balance of <math>S</math> and the call option, one can completely eliminate risk associated to movements in the stock price <math>S</math>. Moreover, the resulting portfolio, being risk-free, has to grow at the risk free rate. These observations implied that the fair price of the call option had to satisfy the differential equation:

<math>\frac{\partial\Phi}{\partial t}+\mu S\frac{\partial \Phi}{\partial S}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Phi}{\partial S^2} = r\Phi

<math>\frac{\partial\Phi}{\partial t}+r S\frac{\partial \Phi}{\partial S}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Phi}{\partial S^2} = r\Phi

</math>

</math>

where <math>r</math> is the continuously compounded risk-free interest rate. The solution to this differential equation, satisfying the boundary condition

where <math>r</math> is the continuously compounded risk-free interest rate, and <math>\sigma</math> is the volatility of the stock. The solution to this differential equation, satisfying the boundary condition

<math>\Phi(T) = \hbox{max}(S(T)-K,0)</math>

<math>\Phi(T) = \hbox{max}(S(T)-K,0)</math>