I am currently studying local volatility for equity models and I am trying to understand some limitations of the model:
1.
under local volatility, the forward smile gets flatter and higher.
Lorenzo Bergomi uses an approximation in this book to find out that:
But I am wondering if there is any mathematical proof about this fact. When thinking about it the physical way, I can see the local volatility function as a sort of "wavelet" function and thus: $$ \sigma^{\text{local}}_t(K,T;S_t)= \alpha_t * \sigma^{\text{local}}_0(K,T+t;S_0)$$ and thus our local volatility map is kind-of "frozen" and dependant on our initial contruction thus, It wont be able to estimate the forward smile. I don't know if my reasoning is right but all remarks/alternative proofs are welcome.
The LV model is a static model.
Again, I try to convice myself about this fact by writing the dynamic of the underlying: $$ S_{t_{k+1}} = S_{t_k} ( r^{*} \delta + \sigma^{\text{local}}_0(S_{t_k}, t_k) \delta^{\frac{1}{2}} g)$$ where $g \sim \mathcal{N}(0,1)$ So the dynamic over time depends on the initial construction of the local volatility surface, so is it the reason why LV model is said to be static? And why do we say that LV model depends on "terminal state" of the underlying?
- My last question would be: if local volatility suffers from all these issues, why is it still used in trading desk to price derivatives ?
Thank you in advance for your answers, all remarks/resources are welcome.
