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Exercise 2.11 (Put–call parity). Consider a stock that pays no dividend in an N-period binomial model. A European call has payoff CN = (SN - K)+ at time N. The price Cn of this call at earlier times is given by the risk-neutral pricing formula (2.4.11): Cn = E?n [ CN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1. Consider also a put with payoff PN = (K - SN)+ at time N, whose price at earlier times is Pn = E?n [ PN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1. Finally, consider a forward contract to buy one share of stock at time N for K dollars. The price of this contract at time N is FN = SN - K, and its price at earlier times is Fn = E?n [ FN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1. (Note that, unlike the call, the forward contract requires that the stock be purchased at time N for K dollars and has a negative payoff if SN < K.) (i) If at time zero you buy a forward contract and a put, and hold them until expiration, explain why the payoff you receive is the same as the payoff of a call; i.e., explain why CN = FN + PN. (ii) Using the risk-neutral pricing formulas given above for Cn, Pn, and Fn and the linearity of conditional expectations, show that Cn = Fn + Pn for every n. (iii) Using the fact that the discounted stock price is a martingale under the risk-neutral measure, show that F0 = S0 - K / (1 + r)^N. (iv) Suppose you begin at time zero with F0, buy one share of stock, borrowing money as necessary to do that, and make no further trades. Show that at time N you have a portfolio valued at FN. (This is called a static replication of the forward contract. If you sell the forward contract for F0 at time zero, you can use this static replication to hedge your short position in the forward contract.) (v) The forward price of the stock at time zero is defined to be that value of K that causes the forward contract to have price zero at time zero. The forward price in this model is (1 + r)^N S0. Show that, at time zero, the price of a call struck at the forward price is the same as the price of a put struck at the forward price. This fact is called put-call parity. (vi) If we choose K = (1 + r)^N S0, we just saw in (v) that C0 = P0. Do we have Cn = Pn for every n?

          Exercise 2.11 (Put–call parity). Consider a stock that pays no dividend in an N-period binomial model. A European call has payoff CN = (SN - K)+ at time N. The price Cn of this call at earlier times is given by the risk-neutral pricing formula (2.4.11):

Cn = E?n [ CN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1.

Consider also a put with payoff PN = (K - SN)+ at time N, whose price at earlier times is

Pn = E?n [ PN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1.

Finally, consider a forward contract to buy one share of stock at time N for K dollars. The price of this contract at time N is FN = SN - K, and its price at earlier times is

Fn = E?n [ FN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1.

(Note that, unlike the call, the forward contract requires that the stock be purchased at time N for K dollars and has a negative payoff if SN < K.)

(i) If at time zero you buy a forward contract and a put, and hold them until expiration, explain why the payoff you receive is the same as the payoff of a call; i.e., explain why CN = FN + PN.
(ii) Using the risk-neutral pricing formulas given above for Cn, Pn, and Fn and the linearity of conditional expectations, show that Cn = Fn + Pn for every n.
(iii) Using the fact that the discounted stock price is a martingale under the risk-neutral measure, show that F0 = S0 - K / (1 + r)^N.
(iv) Suppose you begin at time zero with F0, buy one share of stock, borrowing money as necessary to do that, and make no further trades. Show that at time N you have a portfolio valued at FN. (This is called a static replication of the forward contract. If you sell the forward contract for F0 at time zero, you can use this static replication to hedge your short position in the forward contract.)
(v) The forward price of the stock at time zero is defined to be that value of K that causes the forward contract to have price zero at time zero. The forward price in this model is (1 + r)^N S0. Show that, at time zero, the price of a call struck at the forward price is the same as the price of a put struck at the forward price. This fact is called put-call parity.
(vi) If we choose K = (1 + r)^N S0, we just saw in (v) that C0 = P0. Do we have Cn = Pn for every n?

Exercise 2.11 (Put–call parity). Consider a stock that pays no dividend in an N-period binomial model. A European call has payoff CN = (SN - K)+ at time N. The price Cn of this call at earlier times is given by the risk-neutral pricing formula (2.4.11):

Cn = E?n [ CN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1.

Consider also a put with payoff PN = (K - SN)+ at time N, whose price at earlier times is

Pn = E?n [ PN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1.

Finally, consider a forward contract to buy one share of stock at time N for K dollars. The price of this contract at time N is FN = SN - K, and its price at earlier times is

Fn = E?n [ FN / (1 + r)^(N-n) ], n = 0, 1, ..., N - 1.

(Note that, unlike the call, the forward contract requires that the stock be purchased at time N for K dollars and has a negative payoff if SN < K.)

(i) If at time zero you buy a forward contract and a put, and hold them until expiration, explain why the payoff you receive is the same as the payoff of a call; i.e., explain why CN = FN + PN.
(ii) Using the risk-neutral pricing formulas given above for Cn, Pn, and Fn and the linearity of conditional expectations, show that Cn = Fn + Pn for every n.
(iii) Using the fact that the discounted stock price is a martingale under the risk-neutral measure, show that F0 = S0 - K / (1 + r)^N.
(iv) Suppose you begin at time zero with F0, buy one share of stock, borrowing money as necessary to do that, and make no further trades. Show that at time N you have a portfolio valued at FN. (This is called a static replication of the forward contract. If you sell the forward contract for F0 at time zero, you can use this static replication to hedge your short position in the forward contract.)
(v) The forward price of the stock at time zero is defined to be that value of K that causes the forward contract to have price zero at time zero. The forward price in this model is (1 + r)^N S0. Show that, at time zero, the price of a call struck at the forward price is the same as the price of a put struck at the forward price. This fact is called put-call parity.
(vi) If we choose K = (1 + r)^N S0, we just saw in (v) that C0 = P0. Do we have Cn = Pn for every n?

Added by Samuel P.

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