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Suppose a Stock follows an Itô process with instantaneous volatility $\sigma(S(t),t)$. Precisely

$$dS(t)=\mu S(t)dt+\sigma(S(t),t)S(t)dW(t)$$

I have a historical data for the values of $S(t)$.How can I estimate the instantaneous volatilities $\sigma(S(t),t)$ that took place on each day in this historical daily data series?

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  • $\begingroup$ The frequency is daily data $\endgroup$ – John Jul 26 '17 at 18:10
  • $\begingroup$ It sounds like you want to choose a stochastic volatility model and estimate parameters?(eg. GARCH) $\endgroup$ – Matthew Gunn Jul 26 '17 at 18:24
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    $\begingroup$ Note: in the GBM case, the volatility is assumed to be constant, so it's even wrong to write $\sigma(S(t),t)$. Your question makes more sense if considered for a generic Ito process - such as the Heston model. However, the standard procedure for estimating instantaneous volatility (and even constant volatility) is usually carried out with maximum likelihood under the measure P. Look here for more details: princeton.edu/~yacine/stochvol.pdf $\endgroup$ – james42 Jul 26 '17 at 19:00
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This is upto you how you defined $\sigma(S(t), t)$. As @james42 has pointed out, if volatility is constant, ie $\sigma(S(t), t) = \sigma $, then you can compute $\sigma$ by taking standard deviation of daily log return $r_t$ defined $r_t = ln(S_t / S_{t-1})$, where $ln$ is natural logarithm. If there are empirical evidences to suggest that volatility is not constant then you can use plethora of GARCH family models available in the literature.

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