Discrete sine transform for multi-scale realized volatility measures

Abstract

In this study we present a new realized volatility estimator based on a combination of the multi-scale regression and discrete sine transform (DST) approaches. Multi-scale estimators similar to that recently proposed by Zhang (2006) can, in fact, be constructed within a simple regression-based approach by exploiting the linear relation existing between the market microstructure bias and the realized volatilities computed at different frequencies. We show how such a powerful multi-scale regression approach can also be applied in the context of the Zhou [Nonlinear Modelling of High Frequency Financial Time Series, pp. 109–123, 1998] or DST orthogonalization of the observed tick-by-tick returns. Providing a natural orthonormal basis decomposition of observed returns, the DST permits the optimal disentanglement of the volatility signal of the underlying price process from the market microstructure noise. The robustness of the DST approach with respect to the more general dependent structure of the microstructure noise is also shown analytically. The combination of the multi-scale regression approach with DST gives a multi-scale DST realized volatility estimator similar in efficiency to the optimal Cramer–Rao bounds and robust against a wide class of noise contamination and model misspecification. Monte Carlo simulations based on realistic models for price dynamics and market microstructure effects show the superiority of DST estimators over alternative volatility proxies for a wide range of noise-to-signal ratios and different types of noise contamination. Empirical analysis based on six years of tick-by-tick data for the S&P 500 index future, FIB 30, and 30 year U.S. Treasury Bond future confirms the accuracy and robustness of DST estimators for different types of real data.

§Earlier versions of this paper were circulated under the title ‘A discrete sine transform approach for realized volatility measurement’.

Keywords:

• High Frequency data
• Realized volatility
• Market microstructure
• Bias correction

Notes

§Earlier versions of this paper were circulated under the title ‘A discrete sine transform approach for realized volatility measurement’.

†Studies on the bid–ask spread are largely developed within the framework of quote-driven markets. However, the bid–ask spread is not unique to the dealer markets: Cohen et al. (1981) and Glosten (1994) establish the existence of the bid–ask spread also in a limit-order market because of transaction costs and asymmetric information.

‡More recently, this approach has been revived by Oomen (2005) and Hansen and Lunde (2006).

§Barndorff-Nielsen et al. (2004) show that a direct link between the multi-scale and the kernel-based estimators exists.

†Alternatively, a pure jump process such as the compound Poisson process proposed by Oomen (2006) could be employed to model the dynamics of the true price process.

‡We use the notation (t) to indicate an instantaneous variable, while subscript t denotes daily quantities.

§That is, a time scale having the number of trades as its directing process (here we do not make the distinction between tick time and transaction time).

¶In tick time, even a simple constant volatility process can reproduce stylized facts observed in physical time such as heteroskedasticity, volatility clustering, fat tails and others. Hence, the hypothesis of homoskedastic processes in tick time is far less restrictive than the same hypothesis made for processes defined in physical time where this assumption would clearly be violated by the empirical data.

†Also known as a Karhunen–Loéve expansion or a Hotelling transformation.

†See Corsi et al. (2001) for an empirical example and Oomen (2006) for a detailed theoretical analysis.

‡As mentioned, for returns with a length in ticks k _j  > 1, subsampling and averaging (i.e. a full overlapping scheme) is adopted.

§This regression approach has also recently been applied by Nole and Voev (2008).

¶Both in simulation and empirical applications, we found that the performance of the multi-scales estimator was very robust (giving very similar numerical results) to the precise choice of the frequencies, provided that they are reasonably distributed along the frequency spectrum. In our applications we chose the following set of frequencies: {1, 4, 8, 12, 16, 20, 25, 30, 60, 90, 120}.

†Following Oomen (2006) we define the noise-to-signal ratio as the standard deviation of the noise divided by the average standard deviation per tick of the true price process. This standardization has been chosen first because it seems reasonable to normalize both the noise and signal standard deviation with respect to the same time interval and second because doing that at the tick-by-tick level facilitates comparison across different assets and over time, and makes such a ratio unaffected by different market activities.

†Concerning the values chosen for the frequency ratio α, Bandi and Russel (2010) show that, for realistic sample sizes encountered in practical applications, the asymptotic criteria suggested by Zhang et al. (2005) do not give sufficient guidance for practical implementation, as they provide unsatisfactory representations of the finite sample properties of the estimators. We then preferred to determine the optimal scales in the two-scales estimator performing, by mean of simulations, an extensive grid search over the return frequencies of the slower scale.

‡Comparison with the recently proposed high-frequency range, the so-called ‘realized range-based variance’ (Christensen and Podolskij 2007, Martens and van Dijk 2007), will be deferred to future research.

†Hasbrouck and Ho (1987) suggest that positive autocorrelation at lag lengths greater than one may be the result of traders working an order: “a trader may distribute purchases or sales over time”. However, also significant negative autocorrelation at lag two are often observed.

†The analysis of the market microstructure determinants and specific institutional constraints that would lead to such empirical evidence is beyond the scope of the present study.

†This choice of a somewhat lower frequency of 30 minutes instead of a higher frequency is motivated by the need of having an unbiased estimator of the daily volatility. Higher-frequency realized volatility measures (such as with 1 minute or 5 minute returns) would in fact suffer, with this data set, from a large positive bias that, by itself, would heavily distort the criterion considered here.

‡The EMA filter estimator has not been included here because, as shown in the simulations, it is sensitive to the presence of significant higher-order autocorrelation in the tick-by-tick returns which results to be significantly different from zero in all three series considered here. While, with this kind of data, the simple MS estimators give results that, under these weak empirical tests, are almost indistinguishable from those of MS-DST, thus confirming the results of the Monte Carlo simulations where the MS-DST and the simple MS estimators were all very similar when the level of noise is moderate and the number of observations relatively high.

†In fact, 𝔼[CC ^⊤] = 𝔼[Ψ^⊤ RR ^⊤ Ψ] = Ψ^⊤ 𝔼[RR ^⊤]Ψ = Ψ^⊤ ΩΨ = Λ.