Arbitrage free conditions and extrapolation for option price

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Arbitrage free conditions and extrapolation for option price

Postby niski » Wed Apr 06, 2011 8:43 pm

Hello everyone!

Following is my question

let f(x) be a density function with mean m and let phi(x) a risk neutral transition density (prices may be negative) defined by
phi(x,t) = [1/sqrt(t)]*f(m + (x-m)/sqrt(t))

I want to show that the corresponding call prices given by
C_{K,T} = \int_{-\infty}^{\infty} (x-K)^{+} phi(x,T)dx
are aribtrage free.

Does someone have any idea on how to do it taking into consideration the calendar spread arbitrage (requiring that C_{K,T1} <= C_{K,T2} for T1<T2). I tried to mimic some ideas from Joshi's book but I not sure how to be sure that this prevents arbitrages at all.

Also, I would like to know how to deduce an arbitrage free interpolation for maturities T1 > T, given that I know all the call prices for all the strikes up to a maturity time T.

Thank you in advance!
niski
 
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Re: Arbitrage free conditions and extrapolation for option price

Postby mj » Thu Apr 07, 2011 12:57 am

The easiest way to do this kind of thing is to find a risk-neutral process with these marginals.
mj
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Re: Arbitrage free conditions and extrapolation for option price

Postby niski » Thu Apr 07, 2011 1:53 pm

Hello professor and thanks for your reply!

Do you know where I can read and learn more about that?

Many thanks,

Fabio
niski
 
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