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I'm looking to understand the practical details of calibrating local volatility to option prices for a range of different expiries using the Dupire local volatility equation. Would appreciate some guidance on the following:

a) Is there a good reference that outlines the key details of local vol calibration? Having never attempted this before, one key confusion I have is whether the calibration is via a global optimisation of $\sigma(t, S(t)$ over a range of expiries and strikes, or is there a sequential bootstrap-type of approach that is possible?

b) When implementing the local vol model via MC simulation, how is the resulting $\sigma(t,S(t))$ surface used? For example, given a series of expiries, $T_1,T_2,...,T_N$, do you simulate from $T_{i-1}$ to $T_i$ using a forward volatility determined from the $\sigma(t,S(t))$ surface, or something else?

Thanks in advance!

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