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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."

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    $\begingroup$ I would imagine any derivation of Dupire's formula should suffice - Dupire's formula is nonsense in the face of arbitrage. $\endgroup$ – FinanceGuyThatCantCode May 3 '17 at 16:00
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    $\begingroup$ 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. $\endgroup$ – Quantuple May 3 '17 at 18:12
  • $\begingroup$ Please make this an answer so I can accept. $\endgroup$ – pyCthon May 10 '17 at 23:03
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This is not quite true, in either direction.

If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider a spot dynamics where the spot is a martingale that jumps up or down by integer amounts. The spot distribution is discrete, with zero density in between integer levels. The Dupire equation will result in division by zero trying to calculate the corresponding local vol. That is, there is no local vol model that gives exactly the same prices. Local vol models get arbitrarily close, but the limit diffusion is a "gap diffusion" -- what you get when you let the local vol be infinite.

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.

Unfortunately I don't know of a neat way to fix the statement to nicely clean up the edge cases. As written it is "mostly true" or "morally true" but not mathematically true.

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