# Can we observe smile arbitrage from the implied and local volatility?

Here are graphs of implied volatility and local volatility. Our prof mentioned that we can observe that the short end low strike region has some smile arbitrage. I would like to know how?

Thanks

Smile arbitrage is the presence of a butterfly spread arbitrage in a given maturity of your surface, i.e. if your call prices are non-convex leading to an arbitrage. An easy way to spot the arbitrage is to build the call prices and check for strictly convex prices in strike.

If you have a parametrisation of the implied volatility $$\sigma(K)$$ then you can derive the probability density function and show that it is negative in some regions to find the arbitrage. You can do this by using the formula $$p(K)=\frac{\partial^2C(\sigma(K))}{\partial K^2}$$ and apply the chain rule.

• Thanks! I understand that we can tell whether there is a smile arbitrage opportunity by checking whether the call prices are strictly convex over in strike. My question is if we can tell the smile arbitrage from implied/local volatility surface? Mar 1 '21 at 16:36
• Yes, the term above is from the denominator of the local vol formula. One way to do it is to write the parametrisation of the volatility surface in terms of strike, in your case you have a cubic spline so you will need the coefficients and write a formula for it, along with first and second derivatives and apply section 2.2 of papers.ssrn.com/sol3/papers.cfm?abstract_id=2033323 Then find if g(k) is negative anywhere. I can't see your plot clearly, but if your local vol is negative anywhere but passes calendar spread check then it's a smile arbitrage. Mar 1 '21 at 18:15