Overview
Blacks model gives the prices of options on a forward contract (as opposed to the underlying asset). For this model, we assume that the forward expires at time {% T' %} and the option expires at time {% T \leq T' %}.Formulas
The following are the Black model's formulas
{% Call = P(0,T')F(0)N(d_1) - P(0,T')KN(d_2) %}
{% Put = P(0,T')F(0)N(-d_2) - P(0,T')KN(-d_1) %}
where
{% d_1 = \frac{log(F(0)/K) + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}} %}
{% d_2 = d_1 - \sigma\sqrt{T} %}
and {% P(t,T') %} is the time {% t %} price of one dollar that is paid at time {% T' %}
(see Back pg 133)