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!