Market heterogeneities and the causal structure of volatility - Part 1

Using over fifty million financial quotes from an eleven year period, we present direct evidence of the different time horizons present in a financial markets. This is achieved by computing time series of historical volatilities (the volatility in the past of t), realized volatilities (the volatility in the future of t), and volatility increments (the change of volatility measured at t, computed with data in the past of t). These three time series depend on one parameter fixing the time horizon over which the volatilities and volatility increments are evaluated. The three time series are evaluated over a range of time horizons, from a few hours to several months. Then, we compute the correlations between historical and realized volatilities, and between the volatility increments and the realized volatilities, for all combinations of time horizons. These correlations measure the dependency between a quantity in the past of t and a quantity in the future of t, and therefore are measuring the susceptibility of the market due to the volatility, or volatility change, at the given time horizons. In this way, we construct two dimensional plots for the correlations, with the respective time horizons on the axises. We call these plots a "mug shot", as they provide for a pictorial signature of a market. These correlation plots show very clearly the existence of market agents operating at certain particular time horizons, essentially of hour, day, week and month. The "mug shot" plots also show the mutual interaction between the trader's groups, in particular the existence of a systematic dependency of a given group on the volatilities at longer time horizons.

In order to better understand the content of these empirical mug shots, we compute them by Monte Carlo simulations for a number of theoretical processes, including GARCH, FIGARCH, another long memory process (Zumbach, 2002), as well as a newly developed process designed to incorporate the present stylized evidences on the market components. This investigation shows very clearly, for example, that the usual GARCH( 1,1) process includes only one time horizon, and this is not enough to replicate the multi-horizons complexity of the real data. In order to compare empirical data and processes, the approach using the "mug shots" is particularly selective, in contrast to the standard econometric approach using a log-likelihood comparison, for example. Indeed, for all the above processes, the log-likelihood function is dominated by sudden (unpredictable) volatility increases. Then, the processes essentially model more or less well the "return to the normal", namely the decrease of the volatility after a burst. Yet, this part account only for a small fraction of the log-likelihood value, and the improvement obtained when increasing the complexity of the processes is disappointingly small (Zumbach, 2002). For example, a long memory process that incorporates the power law decay of the lagged volatility correlation improves the log-likelihood only by 0.07% compare to GARCH(1,1), which has an exponential decaying memory. On the other hand, for empirical data, the "mug shots" provide for an powerful direct diagnostic of the time horizons corresponding to the main market participants. Then, these components can be incorporated in a data generating process in order to replicate these empirical properties.

The article proceeds as follows; the next section introduces the required notation and definitions. Section 3 presents the empirical analysis for three foreign exchange rates. The next section shows the "mug shots" for a few theoretical processes, by order of increasing complexity, namely GARCH( 1,1) (sec. 4.3), the long memory process developed in (Zumbach, 2002) (sec. 4.4), the new Market-Component-ARCH (sec. 4.5), and FIGARCH (sec. 4.6). Finally, we present a discussion of the present results within the context of the analogy between hydrodynamic turbulence and price movements in sec. 5, before the conclusions. Part of these results have been presented in a letter published in the physics literature (Zumbach and Lynch, 2001).

 

2 Definitions

The preliminary data processing needed to obtain a homogeneous time series from the high frequency data is fairly straightforward conceptually, but already requires a fair amount of computations. Starting from tick-by-tick high frequency data obtained from Reuters, the data stream is first filtered for outliers. Using the method describe in (Corsi et al., 2001), we filter out the incoherent volatility component inherent to the price formation mechanism in the foreign exchange market. The average weekly seasonal pattern is computed on a moving windows of a few months, and a dynamical business time scale is build using this activity pattern (Breymann et al., 2000). Then, the filtered prices are sampled regularly in this business time scale with a time interval of δt = 7 minutes. This sampling rate is roughly equivalent to a 5 minutes sampling (in the physical time scale) from Monday to Friday, and no sampling during the week-end, but the dynamical time scale takes care of the changing activity pattern due to the opening and closing of the main world markets (London, New-York, Tokyo), of the day light saving time, and of the Holidays.

The volatility definition we choose is the commonly used L2measure based on a point wise price difference. The return measured at a time scale δt_r is given by the price difference

 

 

(1)

 

with δt_r = qδt the time interval used to measure the return, δt the time interval for the regular price sampling on our business time scale, and x = ln(p)the logarithm of the prices. The denominator is used to annualize the return using the Gaussian random walk scaling, and we take δT_ref = 1 year. In this way, the returns can be compared across different time scales δt_r, for example E[r[δt_r]² ≈ 10% regardless of δt_r for the 3 foreign exchanges below. The historical volatility σ_hat time t measures the price fluctuation in a window of length δt_σ in the past of t:

(1a)

 

with δt_σ = pδt_{r =} pqδt. Because the returns are annualized, the volatility is also annualized. Notice that the sum is evaluated over all time point spaced by δt.

Given the sampling rate δt = 7 minutes, these formulae involves to parameters δt_r and δt_σ, or equivalently p and q. All the figures below are computed with p = 24, namely by aggregating simultaneously the return and volatility, while keeping fixed the ratio δt_σ/δt_r = 24. This choice corresponds roughly to our intuition that short term intra-day traders use tick-by-tick data daily data while long term traders use daily data to make decisions. We have explored other choices of parameters, for example taking a microscopic definition of volatility with δt_r = δtfor all values of δt_σ. For all choices of parameters we have made, the computed figures are very similar, and therefore we present the results below only for p = 24. This choice reduces the number of parameters, and we can use the shorter notation σ_h [δt_σ ] = σ_h [δt_σ, δt_r /24].

The historical volatility increment is defined by

(2)

It is measuring the volatility changes occurring at a time scale öta. Indeed, for a stochastic variable evolving on a typical time scale öta, this is the meaningful definition to evaluate the "changes" (and not to use a continuum derivative). Another definition of the historical volatility increment is to compute relative changes

(3)

The correlations computed with both definitions 2 and 3 are almost identical, and all the correlation figures below correspond to the definition 3 for the logarithmic increment.

For a given time t, the return, volatility and volatility increment defined above are historical quantities, as they use information only in the past oft. By contrast, a realized quantity uses informations in the future of t. The realized volatility is a forward translation by öta of the historical volatility, or

(4)

We will use also the "centered" volatility increment defined by

(5)

and similarly for the logarithmic centered volatility increment. These centered volatility increment measure the change of volatility occurring at t, and uses the information in the past and the future oft.

The correlations between an historical quantity and a realized quantity (or with the centered volatility in­crement) measure the susceptibility of the market, namely the mean reaction of the market to a given set of (historical) conditions. This is similar to the susceptibility in electromagnetism with matter, namely given a set of (external) conditions applied to the system, what is the (mean) response of the system. We also use the name "market response function" for the correlation between historical and realized quantities, as these correlations measure the mean response of the market. Yet, in physics, a response function is the reaction to a external small perturbation of a system in equilibrium, and therefore this name may be less appropriate as it designates slightly different concepts.

Given the above volatility definitions, the possible correlations measuring the susceptibility of the financial market are between

(5a)

The correlations between σ_h[δt_σ] and δσ₀[δt′_σ], and between δσ_h[δt_σ] and δσ₀[δt′_σ], are dominated by the mean reverting property of the volatility. For example, the correlation between δσ is mostly negative, with a minimum for δt_σ = δt′_σ at around -40%. The correlations between δσ₀[δt_σ] and σ_r[δt′_σ], are dominated by the strong correlation between the increment and the value of the variable afterward, namely if the increment is positive (negative), the realized volatility is likely to be high (low). All these 3 correlations are fairly uniform across the domain of parameters and do not reveal a salient structure beside mean reversion. Therefore, we will focus below on the correlation between σ_h[δt_σ] and σ_r[δt′_σ], and between δσ_h[δt_σ] and σ_r[δt′_σ]. These correlations will be presented on two dimensional graphs with the time horizon δt_σ of the historical quantity (σ_h[δt_σ] or δσ_h[δt_σ]) on the horizontal axis, and the time horizon δt′_σ of the realized quantity σ_r[δt′_σ] on the vertical axis. We will call "mug shot" the two graphs for these correlations as they give a kind of anthropometric picture of the empirical data, or of the data generating processes.

 

3 Empirical analysis

We present results below mainly for the 3 foreign exchange rate USD/CHF, GBP/USD and USD/JPY. We have also investigated the gold bullion market (XAU/USD), the Dow Jones Industrial Average (DJIA) and the Swiss Market Index (SMI), with very similar results. The time series cover the period from 1.1.1989 to 1.7.2001(11.5 years). The analyzed volatilities have a characteristic time intervals that range from öta 2h 48min to 2 months, namely a factor 400 between the smallest and largest time horizons.

Figure 1 displays the empirical probability distribution function (pdf) for the volatilities at different time scale, for the USD/CHF foreign exchange. This shows that the Gaussian scaling makes the pdf stationary on a first approximation: only a change in the shape of the pdf remains, whereas the mean volatility is almost constant across the full time interval range. Then, except for very small time interval, the pdf is log-normal to a good approximation, as already found by many authors (see e.g. (Andersen et al., 1999; Andersen et al., 2001; Zumbach et al., 2000)).

Figure 2 shows the pdf for the logarithmic volatility increment, on a semi-log axises. In a first approximation, the pdf is symmetric, with exponential tail. The pdf is almost stationary with the increasing time horizon öta of the volatility, with a slightly decreasing variance when increasing time horizon. This means that longer time horizons are experiencing less extreme volatility changes. With a more attentive observation, we can see that the maximum of the pdf is slightly displaced toward the negative axis, while the decay for the negative volatility increment is faster than on the positive side (the mean of the empirical pdf has to converge to zero on a sufficiently large sample). This asymmetry originates in the external shocks that can produce large positive changes, whereas return to the mean value occurs at a slower rate.

Figure 1: For USD/CHF, the empirical pdf for the volatility at different time scale, on log-log scales. The pdf is evaluated with a binning procedure, using a linear interpolation to compute the weights falling in each bin.

 

The correlation between the historical and realized volatility for USD/CHF is shown in the top panel of Fig. 3. Two very interesting features appear on this graph: the large asymmetry around the diagonal and a set of local maxima. The asymmetry around the diagonal is caused by the short term realized volatility depending on all time horizons, whereas a long term realized volatility depends mainly on the long term historical volatiities. In general, the realized volatility at a given time horizon is mostly influenced by the historical volatiities at longer time horizons. In this sense, there is a volatility cascade from long horizons to short horizons, but as the asymmetry in the graph show, the historical volatility at horizon öt, is influencing all the realized volatility at shorter time horizon δt′_σ ≤ δt_σ. This is different from hydrodynamic turbulence, as developed in section 5.

The second interesting feature in the Fig. 3 is the set of local maxima. They are located essentially at 2 to 4 hours, 1 day, 1 week and 2 to 5 weeks. The positions of these maxima correspond to the expected behavior for the main class of market participants as described in the introduction: intra-day market makers and arbitrageur, hedge funds and active portfolio managers, passive portfolio managers, and finally the central banks and pension funds. Therefore, this figure displays essentially the structure of the market in term of the time horizons of the participants. Let us emphasize that the main point is the existence of several groups of market's participants active at different time horizons, and that these groups essentially agree with our anecdotal knowledge of the participant behaviors. Yet, the precise identification and characterization of each group is beyond the reach of our analysis which uses only historical prices.

Figure 5: The mug shot for the USD/JPY foreign exchange rate.