Assume that we have a known dividend d distributed at time T₁, T₁ < T, where T is our maturity date. We can modify the original Black-Scholes-Merton option model by replacing S₀ with S, where:

In the previously discussed example, if we have a known dividend of $1.5 delivered in one month, what is the price of the call?. The price is calculated as follows:

>>>import p4f
>>>s0=40
>>>d=1.5
>>>r=0.015
>>>T=6/12
>>>s=s0-exp(-r*T*d)
>>>x=42
>>>sigma=0.2 
>>>round(p4f.bs_call(s,x,T,r,sigma),2)
1.18

The first line of the program imports the p4f module, which contains the call option model. The result shows that the price of the call is $1.18, which is lower than the previous value ($1.56). It is understandable since the price of the underlying stock would drop roughly by $1.5 in one month. Because of this, the chance that we could exercise our call option will be less, that is...