 # Use forward contract to prove put-call parity

Consider two portfolios: (call and put has same exercise price K, forward contract has delivery price K, call and put expire at time T, forward contract delivers at time T) 1. Long one call 2. Long one put and long one forward contract At time 0 forward price F_0 = S_0*exp(r*T) value of one forward contract f_0 = (F_0 - K)*exp(-r*T) = S_0 - K*exp(-r*T) At time T value of one forward contract f_T = (F_T - K)*exp(-r*0) = S_T - K portfolio 1. becomes max(S_T - K, 0) protfolio 2. becomes max(K - S_T, 0) + S_T - K = max(0, S_T - K) so the two portofolios are equivalent Since the call, put and forward can’t be exercised prior to time T, so two portfolios must also have the same value currently. So C = P + f_0 = P + S_0 - K*exp(-r*T) Does that sound right?

Its easier to remember the formula for put-call parity (in the below format to start with!) C + Bond having a Future Value equal to exercise price = P + Spot position in asset Long positions in the Call and bond equal the long positions in put and the asset. both the options have same exercise price. (I’ll come the forward contract in a minute) LHS of the equation : Call, C captures the asset’s upside move and requires a small investment, the call premium. Bond position ensure that u get atleast the exercise price since the call can only expire worthless. RHS of the equation : The long asset position in spot ensures u capture the upside of the move in prices. P ensures that the downside is protected leaving u with atleast the exercise price. please note that each side of the equation requires a smaller investment in a derivative and a larger investment in either a bond or the asset itself and therefore the two sides match. I found this equation very intuitive and helped me answer any related question. Now the forward contract part. one can rewrite the equation as C = P + (spot position in asset - bond having a future value equal to the exercise price). Remember that a forward contract is equal to the spot price adjusted for the time value of money and that is what the above does. I have carefully tried to avoid using notations (Every book has its own). Hope this satisfies your query.

Whoa. A forward contract is a really lousy place to work on put-call parity. First “a forward contract is equal to the spot price adjusted for the time value of money” is not close to true. A forward contract is priced differently because: a) The underlier might produce income b) The underlier might cost money to hold c) The underlier available now might be different than the underler available at expiration (e.g., corn) d) It might be impossible or very costly to hold the underlier (tin) e) The forward contract contains asymmetric credit risk f) There is a convenience yield for having the underlier now. and probably a bunch of other reasons. If you’re working on put-call parity with the forward contract, it needs to be an asset substitute and have the same expiration date as the options (or there is a valuation problem) and needs to have no credit risk (not possible in the real-world, though there are plenty of bonds, calls, and puts that have essentially no counterparty risk). You need to be careful with forward contracts. I actually graded a question a few years back about the difference between an American call and a European call on a forward contract. Of course, early exercise of the American call never makes sense but more than half missed the question completely by not recognizing that (and wrote tons of BS instead of “Early exercise of the American call makes no sense because the forward contract can’t be exercised early. Hence, value same” = 5 pts = 30 seconds.

Thans Joey, I stand corrected. In addition to time value of money, carry costs come into play in case of futures and credit risk in case of forwards, interest rates diff in case of currencies, dividends for stocks, curve shape and coupon for FI securities, convenience yield on account of storage costs for physical assets etc. I understand there are these other issues, but thought the reply didnt need the rigour that you have given in the rejoinder. May be thats why i am an L2 repeater : Understand that analytical rigour is required and yet i somehow am a sucker for punishments.

The textbook proof is based on what jonnash outlined I think it is kinda fun to use another way to prove it. It also helps me understand forward contract better. 