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Can some one explain the Local Volatility (Dupire's Formula) for me?

When I was reading the book Martingale method in financial modeling. In the local volatility part(P248-P249 of 2nd version), author gives me two formulas. One take the derivatives of the call with respect to K, another with respect to s. How to interpret this and explain this? Can anyone help?
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Bastian Gross

German Mathquant
Hi,
derivatives of the call are not only taken with respect to strike K but also with respect to maturity T in place of price S and time t. This is made by an adjoint/backward argument. Maybe this paper by Jim Gatheral will help you (especially section 2.2).
 
Hi,
derivatives of the call are not only taken with respect to strike K but also with respect to maturity T in place of price S and time t. This is made by an adjoint/backward argument. Maybe this paper by Jim Gatheral will help you (especially section 2.2).
Thanks Bastian.
I am not sure about adjoint/backward argument. But dose that mean we take the derivatives with respect to K and T, in which the stock price at time T also equals to K. So take both derivatives comes the same result but means two different things.
 

Bastian Gross

German Mathquant
You're welcome to ask!
I wrote a short explanatory approach for you.
 

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