Question

Prove the result in equation c+D+Ke^(-rT)=p+S0

(Hint: For the first part of the relationship, 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

Using numbers to prove the equation

Consider the payoff of a trader who has the following position:

• A call option with a strike price of 5

• A bond or cash with a maturity value of 5

Share price at expiration	Call pay-off	Strike price	Bond value at maturity	Bond + call
0-5	0	5	5	5
6	1	5	5	6
7	2	5	5	7
8	3	5	5	8
9	4	5	5	9
10	5	5	5	10

Now consider the position of a trader who has:

• A put option with a strike price of 5 and

• An equivalent unit of the stock

Share Price at expiration	put pay-off	strike price	Stock pay - off	Stock + put
0	5	5	0	5
1	4	5	1	5
2	3	5	2	5
3	2	5	3	5
4	1	5	4	5
5-10	0	5	5-10	5-10

We can see from both the tables that the payoff of the R.H.S is equal to the Payoff of the L.H.S .

Hence the equation is proved.