# Dupire Vomma and Stochastic volatility

Suppose that you are short an option on asset $$X_t$$ following a pure diffusion. Suppose you are hedging your position using (Dupire) Local volatility model. Suppose that the option is concave with respect to the volatility, thus the second derivative with respect to the volatility is negative (ie having negative vomma).

Question : at what level of certainty we can say that the local volatility model will produce conservative price ? Is accounting for a stochastic behavior for the volatility will always result in smaller price in this case ?