Changes

is given by:

is given by:

<math>\Phi(0) = S(0)N(d_1) - Ke^{-rT}N(d_2)</math>

<math>\Phi(t) = S(t)N(d_1) - Ke^{-r(T-t)}N(d_2)</math>

Here <math>N(x)</math> is the cumulative normal distribution function,   

Here <math>N(x)</math> is the cumulative normal distribution function,   

<math>d_1 = \frac{\log(S(0)/K)+(r+\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}</math>

<math>d_1 = \frac{\log(S(t)/K)+(r+\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}</math>

and <math>d_2 = d_1-\sigma\sqrt{T}</math>

and <math>d_2 = d_1-\sigma\sqrt{T}</math>

This is the famous Black-Scholes formula for the price of a European call. Note that all the  variables except for <math>\sigma</math> can be observed in directly in the market. The volatility, <math>\sigma</math> of the stock must be estimated using either statistical data, or inferred from the price of options being sold in the market.

This is the famous Black-Scholes formula for the price of a European call. Note that all the  variables except for <math>\sigma</math> can be observed in directly in the market. The volatility, <math>\sigma</math> of the stock must be estimated using either statistical data, or inferred from the price of options being sold in the market.