Suppose over the period 0T a certain stock pays a divided wh

Suppose over the period [0,T] a certain stock pays a divided whose present value at interest rate r is D. Show that the put-call parity relation for European options at t=0, expiring at T, is

C + D + Kd = P + S

where d is the discount factor from 0 to T.

Solution

Put call parity says that

S+P=C+K*e^(-rt)

We know discount factor=e^(-rt)

So, the above can be written as

S+P=C+K*d

One would need to invest present value of zero coupon bond (which is equivalent to strike price) and present value of dividend

hence, the RHS would become

S+P=D+C+K*d