Is Dupire's equation defined when the T input (the expiration) is equal to 0 (the current time)?
If it's not, how would you determine an appropriate local volatility to use at that time when modelling stock price movements?
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Well, at time zero, the second derivative is a delta function, so you'll get zero. But then if you use something like a milstein scheme, your vol will be a mix with that at the next time step, so you don't get zero... – will Dec 01 '16 at 22:05