Risk free interest rate=r

volatility of stock A=

continuous dividend rate=q

price of stock A=S

a>0,K>0

There is an option that pays a(S_T) if the stock price is less than K at maturity T and otherwise becomes zero.

Show that the current value of this option v_0 is

v_0 = a(S_0)e^(qT) (d)

where

d = ln (S_0/K) + (r q + (^2)/2 )T/( T)

and

is the cdf of standard normal distribution