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We already know the equivalence between local vol, implied vol and option price and there are one-one maps between pairs: $$(\sigma_{local},K),\ (\sigma_{implid},K),\ (C,K)$$ here $K$ is the strike and $C$ is the call price. And the one-one maps are determined by Black-Scholes formula.

And we know that one of the no-arbitrage condition is that call price $C$ should be a convex function against the strike $K.$

So, given some market equivalent sample points $$(\sigma^i_{local},K^i),\ (\sigma^i_{implid},K^i),\ (C^i,K^i)$$ assume they meet the no arbitrage condition as precondition, then we know that linear interpolation of implied vol $\sigma^i_{implid}$ is not allowed.

My question is whether linear interpolation of local vol $\sigma^i_{local}$ is allowed? Since linear interpolation of market price $C^i$ seems allowed.

If linear interpolation of local vol $\sigma^i_{local}$ is allowed, what's the reason? It seems related to the convexity of the maps between $(\sigma^i_{local},K^i)\ (C^i,K^i)$ and $(\sigma^i_{implid},K^i),\ (C^i,K^i).$

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Any specification of the local volatility surface produces non-arbitrageable prices (as long as it is guaranteed that resulting local volatilities are always positive). The reason is that any (positive) lv surface specifies a unique, consistent, arbitrage-free (Dupire) model, which, by construction, produces mutually consistent, arbitrage-free prices. By contrast, the surface of call prices (or equivalently of implied volatilities) does not constitute the parameter set of a unique, arbitrage-free model, therefore any (positive) surface of prices or ivs is not guaranteed arbitrage-free. It is well known that call prices must be above intrinsic, increasing in maturity, decreasing and convex in strike, which translates in non-trivial ways in terms of iv. May I refer to my volatility lectures for further details? https://www.slideshare.net/AntoineSavine/lecture-notes-from-volatility-modelling-lectures-at-copenhagen-university

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