Abstract
Having shown in the previous sections how statistical parameters such as the volatility can be obtained implicitly from the prices of derivatives traded in the market (if they are not quoted directly anyway), we now proceed with presenting in the following section, how such statistical figures, one of which is the volatility, could be determined by analyzing the historical time series.
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Notes
- 1.
The factor \(\left ( n-1\right ) /n\) of the variance is necessary if the estimator for the variance is to be unbiased. See any introductory statistics textbook for more on this subject.
- 2.
Here, the δ in δt denotes the length of a time interval between two data points in the time series and not the “error in t”.
- 3.
Autocorrelations do not appear merely in certain measurement methods but are in general inherent to non-Markov processes, i.e. for processes whose current value is influenced by past values. See Sect. 32.1 for more on this topic. It should be noted that correlation measures only linear dependencies, though.
- 4.
This means intuitively that the parameters describing the time series are time independent, see Chap. 32.
- 5.
Of course, this estimator is strongly autocorrelated since one time step later N − m out of the N − m + 1 values in the sum are still the same.
- 6.
On one hand we get with the index transformation i := N − k
$$\displaystyle \begin{aligned} \sum_{k=0}^{N-1}\left( N-k\right) =\sum_{i=N}^{i=1}i=\sum_{i=1}^{N}i;. \end{aligned}$$On the other hand we have
$$\displaystyle \begin{aligned} \sum_{k=0}^{N-1}\left( N-k\right) =N^{2}-\sum_{k=0}^{N-1}k=N^{2}-\sum _{i=1}^{N}(i-1)=N^{2}+N-\sum_{i=1}^{N}i\;. \end{aligned}$$Equating both results yields
$$\displaystyle \begin{aligned} \sum_{i=1}^{N}i=N^{2}+N-\sum_{i=1}^{N}i\Longrightarrow\sum_{i=1} ^{N}i=N(N+1)/2\;. \end{aligned}$$
References
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972)
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Deutsch, HP., Beinker, M.W. (2019). Market Parameter from Historical Time Series. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_31
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DOI: https://doi.org/10.1007/978-3-030-22899-6_31
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