# 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

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.