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Is it possible to estimate the local volatility using the spot price S at time t instead of the strike price K and the expiry date T ?

Any help would be appreciated.

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One does not estimate the local volatility at a given $T$ and $K$. Instead, Dupire's formula actually gives $\sigma(T,K)$ for all $T$ and $K$. $$ \sigma^2(t_0,S_0;T,K)= \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{\frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$ where $C(t_0,S_0;T,K)$ are the call prices for maturity $T$ and strike $K$. You can also express this directly in terms of the whole implied volatility surface $\Sigma : (T,K) \mapsto \Sigma(T,K)$.

Once you computed the function, you can evaluate at $T = t$ and $K = S$ or any other value you want.

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