# Put-call parity: assorted observations

Posted 12-29-2008 at 12:44 AM by Marid Audran

Updated 12-29-2008 at 01:11 AM by Marid Audran (adding category)

Updated 12-29-2008 at 01:11 AM by Marid Audran (adding category)

Hi there---first time blog poster. I hope I get this right. FYI, I'm studying at present for Exam MFE/3F. My purpose is to think out loud about some of the facts and concepts covered in the exam. When studying, I'll occasionally set aside some notes to post in the blog later on.

- One way of writing put-call parity is

C - P = PV(F - K),

where C is call price, P is put price (and put and call have identical time to expiration and strike price K), and F is the forward price of the underlying asset. An implication is that

C > P*if and only if*F > K.

If F = K, then C = P. Put-call parity says nothing about what C and P are equal to in this case, however; that depends on how volatile the stock is. If F = K and there's zero volatility, then you're risk-free and C = P = 0.

- Another way of writing the parity relationship is

C + PV(K) = P + PV(F).

I had read in a forum post that one of the seminars (TIA? I lost the reference--sorry) expressed it this way and that it made more sense to the poster (but they didn't elaborate). I thought about it, and here's one way to make sense of it: If X is the spot price at expiration, then

*both sides of that equation have a payoff of*max(X,K).

On the one hand, if you buy a call option and lend out the present value of K, then upon expiration you get back K and (if X>K) an additional X-K.

On the other hand, if you buy a put and enter into a prepaid forward contract (the cost of the latter is PV(F)), then upon expiration you can sell the asset with a guaranteed price of K (thanks to the put).

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