Bar data has an honourable history. For decades, high-low-open-close data has been one of the preferred ways of viewing market moves. A simple bar, which just contains the high and the low of a particular period, is an excellent indicator of market activity and volatility in the period. Highly volatile markets have much larger bars than quieter markets. However, very little work has been done on the analysis of bar data. If bar lengths could be estimated, then they could prove to be a quick and robust indicator of market activity. For this to be possible, however, we should be able to estimate bar lengths as a function of time period, so that it is clear when the market is in a regime with unusually large bars.
Naturally, bar length will increase as the time period value increases, as the market has more opportunity to move further in a longer time. However, this change is unlikely to be linear, given the complex nature of financial data distributions. Also, it seems likely that different currencies will have different bar characteristics.
LOGNORMAL FIT
In Figure 1 we can see the fit of the daily bar lengths to a lognormal distribution. All of the currencies are quite reasonably fitted by the lognormal distribution, though all show signs of leptokurtosis, with a higher probability both of very large and very small moves than would be indicated by the lognormal distribution.
The hourly data has rather different characteristics, as shown in Figure 2. The width of the distributions of hourly data is some 50% smaller than the width of the corresponding daily distributions. If in fact the currencies underwent Brownian motion, which is a popular assumption in many models, then we would expect them to be only 20% of the daily distribution. This number is arrived at by scaling by the square root of 24, as these currencies are traded 24 hours of the day.
The distribution of hourly bars is less well-fitted by the lognormal distribution, with a greater degree of fat tails. The risk of a large move in a very short time period is far greater than can be modeled by the lognormal distribution.
VOLATILITY SMILES
To look in more detail at the nature of the deviations from the lognormal distribution, in Figure 3 we plot the log of the daily bar lengths against their deviation from the fitted normal distribution. We recall that if the bars themselves are lognormally distributed, the logs of the bars should be normally distributed.
The results are startlingly reminiscent of the implied volatility smile which appears when pricing out-of-the-money options. Indeed, they are representing similar distributional features, though the implied volatility smile is a feature of the implied distribution, and Figure 3 represents actual historical data. It can be seen that apart from the 'smile,' indicating that very small and very large moves are found more frequently than the lognormal distribution indicates, there is a sharp dip for the very smallest bar lengths. This is simply indicative of the fact that for active trading days, there will nearly always be some measurable move, and so small moves approaching invisible are unlikely.
The picture is rather different for hourly bars (Figure 4). Rather than a smile there is a volatility skew, with a low probability of small bars and a high probability of large bars relative to the lognormal distribution. This is typical of a distribution which is simply fatter than the lognormal distribution, rather than both fatter and more sharply peaked, which is the leptokurtotic characteristic of the daily data.
QUANTIZATION & OCCUPANCY FEATURES
A fascinating feature of Figures 3 and 4 is the 'sawtooth' pattern seen in all the data apart from the sterling/Deutsche mark cross rate. This is due to two distinct effects. The first is simply the quantization of data into units or basis points, which are generally about 1/10,000 of the absolute value of the currency. Thus for U.S. dollar/Japanese yen, which is of order 100 yen per dollar, a basis point is 0.01 yen per dollar, while for sterling/U.S. dollar, which is of order 1, a basis point is 0.0001 dollar per pound sterling. This quantization is what leads to the regular spikes, as all bars will be a whole number of basis points long. However, this does not explain the fact that approximately every fifth spike is much larger. This is an effect which we call occupancy. Foreign exchange rates tend to 'stick' at values which end in a five or a zero, and so these levels are more highly 'occupied' than their neighbors.
The occupancy effect is absent in sterling/ Deutsche mark because the cross is subservient to the major foreign exchange rates, and they cannot mathematically all exhibit the effect. Thus the cross rate just has to be whatever is dictated by arbitrage considerations between the major currencies. The lack of quantization is caused by the data generation process. The cross rate is generated from the majors, as cross data was unavailable for the same period, and thus there is no quantization, as levels of a fraction of a basis point have been produced. We expect that the quantization approach would be visible in traded data for the cross rate.
This week's Learning Curve was written by Jessica James in the strategic risk management advisory group at Bank Onein London.
