A question about put-call parity

huayen · January 13th, 2010, 1:33 am

Can someone explain why the put-call parity can be interpreted as following: European call and put with same strike and expiration have the same implied volatilityThanks in advance.

islington · January 13th, 2010, 7:23 am

Because put-call parity tels you that puts and calls are essentially the same thing (gamma-wise, not talking about divs, repo, ...), you can transform one into the other with a static transformation. If they are the same thing, they have to be priced with the same vol.

huayen · January 14th, 2010, 3:59 am

Put-call parity: Taking second order derivative w.r.t S: ===> gamma-wise Therefore, C and P have same dependency on volatility, so they must have same volatility.Is this the logic? Thanks.

Vegawizard · January 14th, 2010, 8:56 am

It all rests on the concept of synthetics: If you buy a call and sell a put of the same strike = future. Buy 100 Calls and sell 100 futures = synthetic Put. As you can (synthetically) convert a call into a put and vice versa, they are essentially equivalent, have the same "optionality", and therefore should be priced with identical volatility. If not, then easy to arb out the difference.