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Finally I have found the answer on my own. The problem was related to the trasformation of the dataset. The original code used: ${{y}_{t}}=400*(\log ({{P}_{t}})-\log ({{P}_{t-1}}))$ as dataset. Initially I did not care about multipling by 400 because I thought it was usless. Instead it makes a big difference. Now the two series are completely overlapped, I ...

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Great reads to further explore and better understand stochastic volatility models are the series of articles "Smile Dynamics" by Lorenzo Bergomi. As the name indicates the idea is to study stochastic volatility models not only as "smile models" (in the sense that SV models can be used to capture the state of the vanilla market by correctly accounting for ...

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The derivation is in "Managing Smile Risk" by Pat Hagan et al. A copy is here: http://www.math.ku.dk/~rolf/SABR.pdf It is not closed form, but rather an approximation based on expansions.

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It comes from Heat Kernel expansion and differential geometry. See Theorem 6 and Section 8 of http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1717676&download=yes

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