# Proof of arbitrage-free implied volatility surface in relation to local volatility surfaces

I'm looking for proof of the following statement:

"The existence of an arbitrage-free implied volatility surface is equivalent to the existence of a well-defined local volatility surface."

• I would imagine any derivation of Dupire's formula should suffice - Dupire's formula is nonsense in the face of arbitrage. – FinanceGuyThatCantCode May 3 '17 at 16:00
• Agreed. And this transpires through Dupire's local volatility formula: the local variance cannot be negative (otherwise local vol would be complex) which is true if both numerator and denominator remain positive . Numerator positivity is equivalent to absence of calendar arbitrage (total variance should be constant in forward moneyness axis). Denominator positivity is equivalent to absence of butterfly arbitrage. – Quantuple May 3 '17 at 18:12
• Please make this an answer so I can accept. – pyCthon May 10 '17 at 23:03

In the other direction, there are local volatility functions which give a strict local martingale spot dynamics. For example, let the local volatility be $\sigma (S)=S$, so $dS=S^2 dW$. According to the usual theories of arbitrage pricing, this model has arbitrage. For instance there is a delta-hedging strategy to replicate $S$ at time $T$ for less cost than $S (0)$. Such local vol models are only arbitrage free under a different numeraire. For example taking a basket of one dollar and one stock as numeraire, the local vol model allows the price of the dollar to become worthless with respect to the basket, but the arbitrage is no longer present.