Given your (noisy) data, the fit looks ok to me and you haven't demonstrated any butterfly arb violations in the fit.

Many thanks.In a Jim's previous PPT, he posted b*(1+abs(pho)) <= 4/T to prevent arbitrage. What is the relationship between this condition and g(k) > 0?Should this one be also included in my constraints?

They are distinct conditions; my recollection is that the fomer corresponds to Roger's Lee's asymptotic ([$]|k| \rightarrow \infty[$]), bound on the IV, while the latter is a global condition (all [$]k[$]).My approach in my codes is to simply run the fit without the first condition and check the answer; if satisfied, fine;otherwise re-run with the constraint.

QuoteOriginally posted by: AlanThey are distinct conditions; my recollection is that the fomer corresponds to Roger's Lee's asymptotic ([$]|k| \rightarrow \infty[$]), bound on the IV, while the latter is a global condition (all [$]k[$]).My approach in my codes is to simply run the fit without the first condition and check the answer; if satisfied, fine;otherwise re-run with the constraint.what you mean by 'the first condition' is b*(1+abs(pho)) <= 4/T, right?still a bit confused, is this right that both conditions should be satisfied in order to eliminate butterfly arbitrage?

My recollection was wrong. See http://mfe.baruch.cuny.edu/wp-content/u ... Separately from no butterfly arb, you want all IV's > 0. So, it is an easy exercise to work out the minimum IV and thenrequire that this be > 0.

QuoteOriginally posted by: AlanMy recollection was wrong. See http://mfe.baruch.cuny.edu/wp-content/u ... Separately from no butterfly arb, you want all IV's > 0. So, it is an easy exercise to work out the minimum IV and thenrequire that this be > 0.In Page 9, it said that a+b*sigma*sqrt(1-pho^2) >= 0(not b*(1+abs(pho)) <= 4/T) is the condition that ensures w > 0 for all k

Yes, you're right. I agree it is confusing and I don't have a list of all the correct relationships and their rationale at my fingertips. So, it looks like your task is to spend some time, go through all the literature and/or re-develop carefully all the constraints.Search the forum, too -- there were some older threads on all this

Thank you very much!