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Answer to the specific referee's comments:
1) Extensive reviews of the phenomenological approaches to volatility modelling already exist in the literature.
A useful reference (that might perhaps be included for the reader's benefit in a revised version) is the paper of S._H. Poon and C. W. J. Granger (Forecasting volatility in financial markets: A review) in J. of Economic Literature vol. XLI (2003) 478-539.
However, what distinguishes our approach from previously proposed volatility models is that instead of proposing an ad-hoc model with parameters that are then fitted to the data, in our case the model itself is, step by step, constructed from the data and from essentially unique mathematical constraints.
It is also interesting to notice that the data-reconstructed volatility turns out to be driven by fractional noise, not by fractional Brownian motion, as in a few of the existing proposals (ref. 12 for example).
2) Eq. (4) and therefore also (5) are mathematical rigorous relations for F_t adapted processes. However, in practice it is clear that when analysing the data a moving window sample estimate as in (6) must be used.
3) All the analysis we have made of the data of indexes and individual companies in the NYSE and SP500 do not suggest infinite variance. In any case that is not suggested at all by Eq. (9). The linear in t term in right- hand-side is simply a result of the time sum (discrete integration) on the left-hand side and beta is simply related to the average volatility
4) The construction of a stationary process from the market data and the stationarity tests that were performed in the detrended and rescaled data are explained in detail in ref. [8]. The detrended stationary data was used for mathematical consistency, because otherwise some of the mathematical statements in the paper would not be rigorous. However, in practice, because we are dealing with the volatility of a geometric process (the return), the results would be essentially the same with the raw data.
5) When analysing each index or individual company the Hurst exponent of the
(remainder) R_sigma process is obtained with high precision (better than 0.01). However it differs somewhat for different companies in the range 0.8- 0.9.
6) The parameters H, k and beta used in Fig.2 are fitted to the daily NYSE data. Therefore the only thing that is checked is that the shape of the curve is very well described. Then exactly the same parameter values are used for all the other figures.
7) The empirical justification of our model is that with the same parameters many different things are described, which is not always the case with most phenomenological models. In addition the model itself is both mathematical minimal and directly suggested by the data (see remark 1)).
8) Agree. Better captions and description of the figures might be helpful.
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