Question

Prove the following for American option prices: S0 − D − K ≤ C − P ≤ S0 − Ke−rT . (Hint: For the first inequality, consider (a) a portfolio consisting of a European call plus an amount of cash equal to D + K, and (b) a portfolio consisting of an American put option plus one share.)

Answer 1

Let there be two portfolios

(a) a portfolio consisting of a European call plus an amount of cash equal to D + K (D is the present values of dividends), (value today = C+D+K)

(b) a portfolio consisting of an American put option plus one share. (Value today = S0+P)

Now, if one buys portfolio a) and sells portfolio b) and invests the cash

Let the put option be exercised at any time t between 0 and expiry (T)

Value of portfolio b) = K +De^(rt) at time t (K is realised by selling the share using put option and De^rt is dividend till selling the share)   

So Value of portfolio b) at expiry = Ke^(rT-rt) +De^(rt)*e^(rT-rt) =   Ke^(rT-rt) + De^(rT)

Now, at time t the call option is worthless, then

Value of portfolio a) at expiry = (D+K)e^(rT)

Now since  Ke^(rT-rt) + De^(rT) < (D+K)e^(rT)

Value of portfolio a) > value of portolio b)

So, C+D+K> P+S0

=> S0-D-K<C-P

From normal put call parity , we know that

C+Ke^(-rt) = P+S0

If dividends are present, the value of call option is reduced and put option is increased

Hence, C+Ke^(-rt) < P+ S0

So, C-P <  S0- Ke^(-rt)

Combining both we get

S0-D-K<C-P<  S0- Ke^(-rt)