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 $(\sigma^i_{implid},K^i),\ (C^i,K^i).$