Editing Black–Scholes model (section)

===Instruments paying continuous yield dividends===
For options on indices, it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

The dividend payment paid over the time period <math>[t, t + dt]</math> is then modelled as:
:<math>qS_t\,dt</math>

for some constant <math>q</math> (the [[dividend yield]]).

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be:
:<math>C(S_t, t) = e^{-r(T - t)}[FN(d_1) - KN(d_2)]\,</math>

and
:<math>P(S_t, t) = e^{-r(T - t)}[KN(-d_2) - FN(-d_1)]\,</math>

where now
:<math>F = S_t e^{(r - q)(T - t)}\,</math>

is the modified forward price that occurs in the terms <math>d_1, d_2</math>:
:<math>d_1 = \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r - q + \frac{1}{2}\sigma^2\right)(T - t)\right]</math>

and
:<math>d_2 = d_1 - \sigma\sqrt{T - t} = \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r - q - \frac{1}{2}\sigma^2\right)(T - t)\right]</math>.<ref name="finance.bi.no, 2017">{{cite web|title=Extending the Black Scholes formula|url=http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html|website=finance.bi.no|access-date=July 21, 2017|date=October 22, 2003}}</ref>