*7.3. Discrete Price Effect*

The instantaneous volatility standard deviation computed for four different directional-change thresholds has an extremely high value (column *<sup>σ</sup>*<sup>−</sup>*DC* in Table 2). This indicates that in contrast to the realised volatility capable of the scaling, together with the time interval Δ*t*, the instantaneous one does not scale together with the threshold size *δ*.

The price discontinuity typical for all real markets is the cause of the high standard deviation of the instantaneous volatility computed by the directional-change approach. Conventional exchange architecture restricts the price quotations to be a multiple of some constant, for example, 0.001 of a USD. This discreteness caused substantial debates in the scientific literature with regards to the accuracy of the "natural" estimators and on the extent to which they overestimate the actual volatility of the studied process (French and Roll 1986; Gottlieb and Kalay 1985). Equation (19) connects the number of directional changes observed per period of time and the instantaneous volatility and is built on the assumption of the continuous Brownian process. It has no adjustment factors to the discreteness of the analysed data. In reality, the directional-change intrinsic time does not precisely tick at the level where the size of the return is equal to the size of the threshold *δ*. Instead, in most cases, a new directional-change event becomes registered when the price has already jumped over the expected level. This discreteness effect becomes more pronounced when the size of the elementary price move (tick) is relatively big. That is, two factors contribute to the size of the instantaneous volatility of the discrete data-sample computed by the novel approach: the scale of the selected threshold *δ* and the tick size in the given sample (discreteness). Further, we provide results of a set of experiments where the impact of the price discreteness and the threshold size *δ* on the computed instantaneous volatility is observed.

Three time series were generated by GBM with the various density of ticks per period of time. Variation of the number of price changes in the simulation is equivalent to changing the simulated tick size having fixed a one-year time interval and volatility of the generated process fixed to be 15%. Forty gradually increasing directional-change thresholds were applied to all three GBMs. The thresholds range from 0.01% to 0.29%. We provide the plot of the computed instantaneous volatility of simulated time series in Figure 5. Two particular properties can be noticed. First, one brings the generated time series closer to the continuous process by making the size of a tick smaller (increasing the number of price changes per period in the sample). In this case, the values of the estimated instantaneous volatility *σDC* become closer to the volatility *σ* embedded in the model. Second, bigger thresholds are less sensitive to the discreetness of the given set of prices. The slippage effect of the price jump over the expected intrinsic time level becomes less pronounced, and the obtained result also approaches the value *σ* when the tick size represents a small fraction of the directional-change thresholds. A more comprehensive analysis should be performed in further research works to bridge the gap between the realised and instantaneous volatilities.

**Figure 5.** Instantaneous volatility of three time series generated by GBM with various tick frequencies and fixed volatility (15%). The volatility is computed by the directional-change approach (Equation (19)). Sizes of the directional change thresholds, used to calculate the volatility values, are put on the X-axis. Red dashed line marks the 15% level.
