put call parity-question

[amir]

I have a Eu put and call written on the same div-free stock, with same expiring (T) and same strike K. Assume C and P are current price of them.
How can I show that if the current price of stock is K, and C - P > Kr, then an arbitrage opportunity exist? (r > 0).

 

[koupparis]

Google is your friend, my friend. You have everything you need in your statement, just look up Put-Call parity and digest. Or think up a way to exhibit arbitrage from what you're given. Just from a quick look is this correct C - P > Kr?

 

[amir]

I know that if I let Kr > K * {1 - exp ( - r ( T - t ) )} which requires
r > 1 - exp ( - r ( T - t ) ), (which requires T - t must be less than 1) then
there exist an arbitrage opportunity; because C + Kexp ( - r ( T - t ) ) > P + K

Is that right? I think that there might be more simple solution, since in the above argument Its not easy to show that r > 1 - exp ( - r ( T - t ) ) without using numerical illustration.

 

[koupparis]

That is correct, that is why I'm wondering if Kr is correct, and if it shouldn't be Kr(T-t) or something to that effect.

 

[bob]

\(C-P=S_0-Ke^{-rT}\)
Since \(S_0 = K\) is given:
\(C-P = K(1-e^{-rT})\)
\(e^x\) is convex, so:
\(C-P <= K[1 - (1-rT)]\)
\(C-P <= K*rT\)

Can't see how you get rid of \(T\), though.

 

[amir]

thanks Koupparis and Bob
I think if we assume current time as t, then following Bob's solution we get
C - P <= K * r * (T-t) < K * r
hmm!

 

[koupparis]

Not unless T-t < 1.

 

[pruse]

if \(T\leq 1\) then \(1-e^{-rT}\leq r\), so PCP is violated and the conclusion follows.

if \(T>1\), however, then \(1-e^{-rT}>r\) is possible, so the conclusion can't be drawn without additional information. you must be missing something.