Local Volatility Model Limitations and Implied Smile Dynamics

Summary

The document raises questions about local volatility models used for equity derivatives. It asks why forward smiles under local volatility are said to become flatter and higher, whether the model’s calibration to an initial volatility surface makes its dynamics effectively static, and why trading desks continue to use it despite these concerns. It also asks what it means to say that the model depends on the underlying’s terminal state.

The post offers an intuitive but tentative picture of the local volatility surface as fixed from its initial construction, along with a discrete-time sketch of underlying dynamics. It does not provide proofs, answers, numerical tests, or a comparison with alternative models. As a result, it is most useful as a set of prompts about the distinction between matching current vanilla prices and representing future smile behavior. The claims about forward-smile changes and static behavior are posed for investigation, not established by evidence in the document.

Key ideas

• The post questions how local volatility models imply changes in forward implied-volatility smiles.
• It asks whether calibration to an initial surface limits the model’s representation of future smile dynamics.
• The underlying’s modeled evolution depends on the local volatility evaluated at its current state and time.
• The post asks why local volatility remains useful for derivative pricing despite its perceived limitations.
• Its claims and intuitions are questions rather than demonstrated results.

Tags

# Questions on limitations of local volatility model


# Questions on limitations of local volatility model












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.

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