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A noise-robust estimator of volatility based on interquantile ranges

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Abstract

This paper proposes a new class of estimators based on the interquantile range of intraday returns, referred to as interquantile range based volatility (IQRBV), to estimate the integrated daily volatility. More importantly and intuitively, it is shown that a properly chosen IQRBV is jump-free for its trimming of the intraday extreme two tails that utilize the range between symmetric quantiles. We exploit its approximation optimality by examining a general class of distributions from the Pearson type IV family and recommend using IQRBV.04 as the integrated variance estimate. Both our simulation and the empirical results highlight interesting features of the easy-to-implement and model-free IQRBV over the other competing estimators that are seen in the literature.

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Notes

  1. On the recent empirical literature, in bond market, Wright and Zhou (2009) find evidence that jumps can help predict future excess returns and Dungey and Hvozdyk (2012) show that the spot and future prices are affected by jumps. While Kim et al. (2007) and Psychoyios et al. (2010) argue the salient role of jumps in characterizing return dynamics and pricing volatility derivatives, Evans (2011) and Rangel (2011) show that intraday jumps are associated with macroeconomic news announcements.

  2. The effects of microstructure noise may be alleviated to some extent by, e.g., finding an optimal sampling frequency, pre-whitening the dependence of data to get efficient price innovations, or irregularly-spaced modelling; see, Zhou (1996), Andersen et al. (2001, 2003), Corsi et al. (2001), Bollen and Inder (2002), Bandi and Russell (2006), Oomen (2006), and Zhang et al. (2005).

  3. For instance, see, Moder and Rodgers (1968), Keefer and Bodily (1983), Keefer and Verdini (1993), Keefer (1994), and recently Taylor (2005), among many others.

  4. One can show that the expected value of the squared range is

    $$ E[{\rm range}_t^2]=4\hbox{ln}(2)\sigma, $$

    assuming constant volatility across the day. A range-based proxy of variance can simply be constructed as a constant, \(\frac{1}{4\hbox{ln}(2)}=0.361\), times the squared price range.

  5. Alizadeh et al. (2002) have referred to the results from Feller (1951) to show the distribution of the log range under driftless jump free diffusion to be

    $$ \hbox{Prob}\left[ \hbox{ln}\left( \sup_{0<t\leq \tau} p_t- \inf_{0<t\leq \tau} p_t \right)\in dy\right] =8\sum_{k=1}^{\infty}(-1)^{k-1}\frac{k^2e^y}{\sigma\sqrt{\tau}}\varphi\left(\frac{k^2e^y}{\sigma\sqrt{\tau}}\right)dy, $$

    where \(\varphi(\cdot)\) denotes a standard normal density. It is not hard to imagine that the log range will include additional terms to reflect the presence of jumps and will thus overestimate the underlying volatility under general jump diffusion processes.

  6. Christensen et al. (2010) also adapt the idea of quantile-based information and provide the realized quantile-based estimation (QRV). Both IQRBV and QRV are applicable with activity jumps and outliers in the price series. In comparison to IQRBV, their estimator is more efficient; however, implementation of the QRV is required to choose some tuning parameters including the number of blocks or block length, the quantiles, and the quantile weights. The IQRBV is much easier to apply in the empirical tests and remains several nice properties.

  7. Witnessed by C. P. Winsor during World War II for χ 2 and log χ 2 distributions.

  8. Including both symmetric bell-shaped (normal, central t, stable) or skewed (noncentral t, stable), J-shaped (log-normal, χ 2, log χ 2) distributions with finite fourth moments. A total of 29 distributions were being investigated by varying these distributions’ coefficients of skewness and kurtosis ranging from [1,12] to [0,4], respectively.

  9. Andersen et al. (2002) show that jump intensity is about 0.014, that is, 14 times every one thousand trading days on average, for the daily S&P 500 cash index. Using different high-frequency data, Andersen et al. (2007a, b) find the jump intensities to be roughly 0.058 and 0.082. Andersen et al. (2002) also estimate the jump size parameter σ j to be around 1.5 %. In the simulation conducted by Huang and Tauchen (2005), the range for σ j varies from 0 to 2.5 %.

  10. All estimates in Fig. 4 are based on the simulated results in Fig. 3.

  11. The empirical results for this specific period are removed from this version but available upon request from the authors.

  12. If we process the IBM case with a 15-s sampling interval in the full-period, 70 intraday returns are ignored (about 0.002 % of all data). Moreover, we inspect these data carefully and believe almost all of them are typos. The other companies have less ignored data than the IBM case.

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Authors and Affiliations

  1. Department of Finance, National Central University, Taoyuan, Taiwan

    Jin-Huei Yeh

  2. Department of Finance, Minghsin University of Science and Technology, Hsinchu, Taiwan

    Jying-Nan Wang

  3. Department of Finance, National Taiwan University, Taipei, Taiwan

    Chung-Ming Kuan

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  1. Jin-Huei Yeh
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  2. Jying-Nan Wang
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Correspondence to Jin-Huei Yeh.

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Yeh, JH., Wang, JN. & Kuan, CM. A noise-robust estimator of volatility based on interquantile ranges. Rev Quant Finan Acc 43, 751–779 (2014). https://doi.org/10.1007/s11156-013-0391-7

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