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Local Volatility Surface and No-arbitrage

Suppose that the local volatility function σ(K, T) is somehow known. Also, suppose that given σ(K, T), there exist theoretical option prices that solve the Dupire equation derived in the previous chapter. Finally, suppose that for a given set of option strikes K and maturities T these theoretical prices exactly coincide with the corresponding market prices. Then one can build a local volatility surface by using the values of σ(K, T) at the given set of [K, T], and say that this surface represents the given set of the market quotes. Note, that in practice, we usually solve the inverse problem. This is, given the market quotes, find the local volatility function that being used in the Dupire equation produces the theoretical option prices such that the difference between this set of option prices and the corresponding market prices reaches minimum under some suitable norm…