How:
Find a function sigma(S,t) such that the existing market vol. surface is reproduced (fitting to option prices). At each point on the (S,t) plane, the B-S pricing equation is valid.
The subtlety is that although we set out to find sigma(S,t), the observable quantity is sigma(K,T-t).
Pro:
We don't have to introduce new stochastic processes. Market is still complete, and hence preference-free pricing still works (i.e. no need to guesstimate market price of risk of any kind).
Reference:
Derman's GS Research paper
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