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This paper gives on equation 22 an estimator for the mean reversion level of the variance process under the real world measure. The context is the Heston model, where the variance is stochastic and the paper is trying to give a proxy for the determination of the Heston model parameters under the real word measure.

My question is: I perfectly understand equation 22, but it is not really clear to me if I should use the historical log-returns to do the computation. Could you please confirm that all I need to compute the estimator in equation 22 are the historical log-returns?

The equation 22, wich I refer to, is the following: $$\hat{\bar{\nu}}^{*} := \frac{1}{T}\sum_{k=1}^{K}\left[\ln\left( \frac{S(t_k)}{S(t_{k-1})}\right)\right]^2$$

Thank you

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  • $\begingroup$ Yes you should use log-returns observed under the $\Bbb{P}$ measure, for this equation amounts to computing the stationary realised variance of a pure diffusion process, which Heston is (oversimplifying, $K$ sufficiently large guarantees that the sum converges to the quadratic variation of log-returns and $T$ sufficiently large means that you focus on stationary variance level). $\endgroup$
    – Quantuple
    Commented May 23, 2017 at 8:17
  • $\begingroup$ As a side note, moving back to the risk-neutral parameters will require an assumption regarding the market price of volatility risk since the Heston model is incomplete. $\endgroup$
    – Quantuple
    Commented May 23, 2017 at 8:17
  • $\begingroup$ @Quantuple but I am solving this equation in order to get the $\Bbb{P}$-parameters. You are saying that I have to do a simulation under that $\Bbb{P}$-measure to solve the equation, but I do ot even have the $\Bbb{P}$-parameters (I am seeking for them). Hence, I am assuming that the unique way of doing this is by using historical log-returns as a proxy. Am I right? Should I use historical data for the calculation (say, T=2 years and K=number of days)? $\endgroup$
    – Joanna
    Commented May 23, 2017 at 23:30
  • $\begingroup$ I think you don't understand the concept of P measure? You need no simulations. P is the real world. You just need historical time series. $\endgroup$
    – Quantuple
    Commented May 24, 2017 at 6:11

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Answering my own question: Yes, on page 27 of the article the author says that equation 22 should indeed be computed from historical data.

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    $\begingroup$ So Quantuple had the right answer in his comment above. $\endgroup$
    – nbbo2
    Commented May 24, 2017 at 13:35

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