# Prove the following Call and Put relationship: [duplicate]

I need to prove that

$$c(S,X,T)=\frac{X}{F}p(S,\frac{F^2}{X},T)$$

where $$F=Se^{(r-q)(T-t)}$$

I am having trouble proving this relationship. Is this relationship even possible? If so, can someone please assist me?

Thank you.

• Hello Ruan and welcome to SE. Could you clarify a bit the setup and the notations? I assume that S is lognormal, and that $c$ and $p$ are the european call and put prices? Could you also tell us what you attempted and where did you get stuck? Here are some hints: start with the case where r - q = 0 which is easier and write the call and put prices as expectations... you should be able to conclude in this case by remarking that a lognormal process starting at $K$ is the same as a lognormal process starting at $K$ multiplied by $\frac{K}{S}$... – byouness Jul 8 '19 at 10:13
• This is with Black Scholes I assume? I think the question would want you to just plug in the values into the BS put equation and you'd get the BS call equation. If this isn't with BS, maybe you can try setting up a portfolio of a call with strike $X$ and a portfolio of $X/F$ puts with the other strike and then show they have equal values in the future, so by no arbitrage they would be equal. But I am not sure if that's true in the general case (like how put call parity is true even without Black Scholes assumptions) – Slade Jul 8 '19 at 10:14
• This is called put-call symmetry, see the following thread, in particular the answer by Peter Carr: quant.stackexchange.com/questions/40070/put-call-symmetry/40071 – ilovevolatility Jul 8 '19 at 10:48