Question

3. (10 pts) Suppose the stock S pays dividend ö at time to with to E (0, T). Let CA and PA be the prices of American call and put options, each with the same exercise price X and exercise time T. Use No-arbitrage arguments to prove that: S(0) - der to - X, CA PA where we assume the continuous interest rate is the constant r

Answer 1

Price of American Call = Ca

Price of Put = Pa

Time horizon = T

Dividend = d

Exercise Price = X

Continuous rate = r

Stock = S

The Put Call Parity is stated as

C + Xe-rt = S+P

Now re arranging the equation

C-P = S - Xe-rt

Now, by theory the stock price have to be adjusted for the dividend factor and hence we assume that the d is adjusted while computing the risk free rate and time factor in the X part of the equation. Also the difference between the call and put is equal to stock price and strike price adjusted as per the put call parity theory. But the dividend factor sometimes reduces the value of stock and hence it is a little less than the difference between call and price. Hence, the equation

Ca-Pa>= S - Xe-rt ( hence proved)

The above is just a rearrangement of the asked equation and can be written in both ways i.e. one stated in the question and also the proved.