Local volatility implied spot vol correlation

I have a question about local volatility models.

In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility models where you can control this correlation.

Is there a proof demonstrating how this correlation is -1 ?

I tried by computing the covariance and replacing sigma with Dupire formula but cannot find -1.

Thanks

• In a LV model, inst. vol is a deterministic function of $S_t$, $\sigma_t = \sigma(t,S_t)$. Applying Ito you can then write $d\sigma_t = \cdot dt + d\sigma(t,S_t)/dS dS_t$. The target correlation (computed from the indiv quadratic variation and covariation) is then simply $\rho=\text{sign}(d\sigma(t,S_t)/dS)$. For equity markets that are negatively skewed, this usually yields -1 May 7, 2022 at 9:44