Overview
The black scholes formula is a formula that solves the black scholes equation for plain vanilla European calls and puts.The formula for a plain vanilla call is given as
{% C(S_t, t) = N(d_1)S_t - N(d_2)Ke^{-r(T-t)} %}
{% d_1 = \frac{1}{\sigma \sqrt{T-t}} [ln(\frac{S_t}{K}) + (r + \frac{\sigma^2}{2})(T-t)] %}
{% d_2 = d_1 - \sigma \sqrt{T-t} %}
Here, {% C %} is the prices of option, {% S %} is the price of the underlying, {% r %} is the rate of return on cash,
{% T-t %} is the time to expiry, and {% \sigma %} is the underlying asset volatility.
{% K %} is the option strike price and N is the cumulative normal distribution with mean 0 and standard deviation of 1. (standard normal)