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Part two of the paper exploring the correlation between historical and realized volatilities. By Paul Lynch and Gilles Zumbach

Market heterogeneities and the causal structure of volatility - Part 2

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4 Theoretical processes

4.1 General consideration on volatility processes

The significance of the above empirical study requires some benchmark theoretical processes with which to compare and contrast results. The same method that we apply to search for structures can also be applied to an artificial time series generated by a data generating processes (DGP). In this way, we can also compute the "mug shot" of a DGP and compare it to the empirical results.

The processes we have investigated have the following structure

Essentially, this is a GARCH(1,1) structure, but with a general dependency on the past for the effective volatility. The last equation in 8 expresses that the effective volatility σ_eff (t + δ + t) at the time t + δ t is a function of the current information set Ω( t) , and of a set of parameters J. The parameters are chosen so as to produce figures similar to the empirical data, but also to display the salient properties of the processes. Let us remark that these parameters can be quite different from those obtained by a log-likelihood estimate. The main reason for the discrepancy between a log-likelihood estimate and a "fit by eyes" of the mug shot is that the log-likelihood optimizes the processes at the shortest time scale (7 minutes) whereas a "fit by eye" is equally sensitive to the overall volatility dependencies up to 6 weeks. As the optimization criterion is different (log-likelihood versus overall volatility dependencies), the resulting parameters differs. Our purpose in this paper is only to see at a qualitative level what are the overall volatility dependencies present in the empirical data and in a few data generating processes. For a more quantitative study, a L2 distance, for example, could be set between the empirical and simulated .mug shot. in order to have a numerical function to estimate parameters and measure "goodness of fit".

In the eq. 7, the i.i.d. random variable e(t) are drawn from a Student-t distribution with t = 4.3 degrees of freedom, normalized to have a unit variance. This value is extracted from log-likelihood estimates, which essentially give identical values for the number of degree of freedoms for all processes. Therefore, we have taken the same Student probability distribution for all simulations. Finally, the simulation length is equivalent to the length of the empirical data, and the time step δt = 7 minutes is identical to the time interval taken for the empirical analysis.

4.2 Brownian motion

The simplest process is given by a random walk with constant volatility, or Brownian motion

This process has no volatility dependence, and all correlations are nil (up to the statistical errors in a simulation).

4.3 GARCH(1,1)

The usual equation for the GARCH(1,1) volatility is:

In this form, the parameters cannot be directly interpreted. This equation can be rewritten in a form that allows for a better interpretation and for several natural extensions:

The new "internal" variable σ₁ can be seen as the volatility of the market as perceived by one group of

market participants taking decision at a time horizon τ₁= -dt/ lnµ₁. Then, the effective volatility s_effis

given by the mean volatility σ plus an error correction term given by the difference between the perceived volatility σ₁ and the mean volatility. The "coupling constant" of the error term is written as 1-w_∞. In the second form, obtained by a trivial re-arrangement of the terms, the effective volatility s_eff is a convex combination of the perceived volatility s₁²and of the mean volatility s². The coupling constant w_∞ is the factor of the volatility σ measured at an infinite time scale. For the GARCH process, the correlation of the return square decays exponentially fast, with a characteristic time.

The GARCH(1,1) process has been simulated with the parameters τ₁= 1.15 day, w_∞= 0.22 and s_ann = √1 year/ δt σ = 0.1, leading to a correlation time of τ_corr =5.27 day. The resulting "mug shot" is given in

Fig. 7, clearly showing the characteristic times τ₁ and τ_corr of the exponentials. It is clear from this image

that the complex structure present in the empirical data cannot be approximated with a simple GARCH (1,1) process.

4.4 Long Memory Aggregated ARCH

A well known empirical fact about the financial time series is the slow decay for the lagged correlation of the absolute or square returns. Essentially, the lagged correlation ρ(∆t)of the absolute return (r) decays with an hyperbolic law 1/∆t^v with an exponent v-0.2. In (Zumbach, 2000), a few simple processes that incorporate the long memory have been introduced. These processes are natural generalization of the GARCH(1,1) process as written with the formulae 11. We will use here the long memory aggregated affine process, denoted by LM-Agg-Aff-ARCH process in (Zumbach, 2000).

In this process, the set of (historical) volatilities σ_kare measured on a set of time horizons τ_kthat increase geometrically:

The index k runs from from 1 to k_max, and the cut-off k_max is chosen large enough compared to the largest analyzed time scale. The progression ρ of the geometric series is chosen close enough to 1 so that the volatilities are measured on a "continuum" of time horizons, and practically ρ = 2 is close enough.

The effective volatility σ_effis given by the mean volatility plus a sum of the error corrections between the (historical) volatilities and the mean volatility, and where the "coupling constants" decay as a power law:

Figure 7: The mug shot for the GARCH(1,1) process.

with w_k= (1-w_∞) χ_k, and the constant c is fixed by the condition ∑_k χ_k= 1. In (Zumbach, 2000), it has been computed using an argument that the memory of this process decays as a power law. Let us emphasize that the goal of this process is not to reproduce the structure of the market, but to have a "minimal" process with few parameters that incorporate the long memory of the returns. In the next subsection, we will modify this process to incorporate the observed market components, and in the subsection 4.6 the comparison with FIGARCH will be made.

The mug shot obtained by Monte Carlo simulations of this process is given in Fig. 8. The parameters are τ₀= 2 hour and λ = 0.1 with δt = 7 minutes, ρ = √2 and l_max = 1024. These values lead to an upper cut-off of τ_max = 85 business days = 4 months, corresponding to the largest τ_kin the process. The main dependencies and asymmetry of the correlations is very well reproduced by this simple process. Yet, this process is regular in its structure (up to the cut-off τ_max), and therefore cannot describe the market components observe in the empirical data.

4.5 Market-Component ARCH

It is very simple to modify the LM-Agg-Aff-ARCH process in order to incorporate the market component: instead of taking the time horizons τ_kand weights χ_kas functions depending on a few parameters, we consider them as free parameters. Otherwise, the equation for σ_kand σ_effare left unchanged. In order to restrict the number of free parameters, the number of components is chosen as small as possible, and 4 components give already very good results. The chosen time horizons 2τ_kare 3.8 hours, 1 day, 1 week and 1 month, and only the first one has been adjusted to the empirical data. This set of time scale grows roughly like a geometric series with a progression of 5. The remaining parameters are w_∞ = 0 082, χ₁ = 0 53, χ₂ = 0 16, χ₃ = 0 14 and χ₄ 0 17. The resulting mug shot is shown in Fig. 9. The overall structure is correct, the inhomogeneity is clearly present, but the inhomogeneity structure is too soft. For example, by changing the parameters in this process, it is not possible to reproduce the strong dip around 16 hours.

In order to get sharper market components, a slight improvement of the process is needed. Instead of measuring the (historical) volatilities with a simple Exponential Moving Average (EMA), which has an exponential kernel, we use an MA operator which has a more rectangular like kernel. The MA operator is defined by (Zumbach and M¨uller, 2001):

Figure 8: The mug shot for the LM-Agg-ARCH proces

Figure 9: The mug shot for the Market-Component ARCH, with the historical
volatilities sk computed with one simpleEMA.

The coefficients µ (eq. 19) is computed from the time horizon τ, so that the memory length of the MA operator is τ. The memory length is defined as the first moment of the kernel (Zumbach and M¨uller, 2001), and for a rectangular kernel of length 2τ, its value is τ. The parameters m control the shape of the kernel, and we took m = 32 which give an almost rectangular kernel. The mug shot for this process is displayed in Fig. 10, where the parameters have been adjusted on the USD/CHF empirical figure. The agreement is superb. By introducing a fifth component at a time horizon of half a week, an even better agreement is obtained by filling the depression present at this time horizon (Zumbach and Lynch, 2001).

The differences between figures 9 and 10 show that a fairly rectangular kernel must be used to describe the memory of each market component. From this difference between the processes, we can infer that the market participants forget the past quickly beyond their time frames.

4.6 FIGARCH

The FIGARCH process (Baillie et al., 1996) was introduced to incorporate the long memory in the size of the return observed empirically in the financial data. This process present a few difficulties in its practical implementation, related to the fractional difference operator (Teyssiere, 1996; Chung, 1999; Zumbach, 2000). Theoretically, the fractional difference operator is defined by a formal expansion in the lag operator, and practically this expansion has to be cut-off at some maximal value l_max. This cut-off breaks an exact relation obeyed by the fractional difference operator, and effectively introduces a mean volatility term in the process equation. Another practical difficulty of the FIGARCH process is that the computational effort grows as the maximal cut-off l_max, precluding the uses of large cut-off. This behavior must be contrasted with the LM-ARCH process where the computational effort grows as the logarithm of the cut-off l_max.

The mug shot obtained by Monte Carlo simulations of the FIGARCH process is given in Fig. 11 (more accurately, we simulate the Lin-FIGARCH(1,d,0) in the naming used in (Zumbach, 2002). The parameters are β = 0.05 hour and d= 0 2 with δt = 7 minutes and lmax 1024. These values lead to a time horizon for the upper cut-off equal to 5 business days, or 1 week. The effect of the cut-off is clearly visible on the figure, as well as the slow regular decay of the response function below the cut-off. The agreement with the empirical data is not very good, mainly due to the to small cut-off. This comparison is a bit unfair for FIGARCH, yet, pushing the cut-off to a value comparable as the one used for the LM-ARCH process would lead to unrealistic computational time (several weeks).

Figure 10: The mug shot for the Market-Component ARCH, with the historical
volatilities sk computed with a MAoperator.

Figure 11: The mug shot for the FIGARCH process.

5 The analogy with hydrodynamic turbulence

Using an analogy between hydrodynamic turbulence and price movements from foreign exchange markets over many time scales, an argument for the presence of turbulence in financial systems has been presented (Ghashghaie et al., 1996). These authors compare the probability density functions (pdf), and of their scaling properties, of the velocity increments in a turbulent fluid with the pdf of returns in foreign exchange markets. They discovered a striking similarity between the two systems. From this evidence, they propose that financial markets present a cascade of volatilities from large to short time scales. This conjecture was initially criticized for two reasons (Mantegna and Stanley, 1996; Mantegna and Stanley, 1997; Mantegna and Stanley, 2000), namely

1. financial returns have negligible autocorrelation whereas changes in the velocity of turbulent particles are anti-correlated and,

2. Turbulence is characterized by an asymmetric pdf, the foreign exchange returns that were used in the analogy had a symmetric pdf.

Further evidence of the turbulent cascade was presented in (Arneodo et al., 1998). Using a wavelet decomposition of volatility over many time frames, they found evidence of the cascade from large to small time scales. Yet, this cascade was a smooth process, and the dominant time scales of the market agents have remained undetected.

Staying at the level of a formal analogy, the velocity increments in a turbulent uid can be compared to the volatility increment in a financial market. The pdf for the volatility increment exhibits a similar scaling behavior, with an asymmetric pdf. Moreover, they are negatively correlated. However, all of these papers discuss the analogy with fully developed turbulence, where the scaling theory developed by Kolmogorov applies (Frisch, 1995). In nature, turbulence generally displays some heterogeneous structures, at Reynolds numbers lower than needed for the onset of fully developed turbulence where the full symmetries of the Navier-Stokes equations are restore in average. In this heterogeneous regime, the energy of the turbulence is unevenly spread along the scales. The direct analogy for the heterogeneity in turbulence is given by the heterogeneity in the financial market, as display in our "mug shot". Therefore, a possibly better analogy is between the velocity increment in heterogeneous turbulence and the volatility increment in a financial market.

Yet, we think that there is a major difference between the two systems in the way that their components interact. Turbulence occurs in 1 time and 3 spatial dimensions. The 3 spatial dimensions allow the formation of eddies, that can interact in time. An eddy of a given size can split into two eddies of half the size, giving rise to the energy cascade, as first proposed by L.F. Richardson in 1922. Clearly, this is a cascade from one scale to the next smaller scale. In finance, there is only 1 time dimension. Therefore, the market participants can only interact in time, and when a traders with any time horizon decides to buy or sell, he will create volatility at the shortest time scales. In other words, the participants interact only when they trade, and it is an .instantaneous. interaction. Therefore, eventhough there are some interesting analogies in the two systems, some properties of the cascades are different because the two systems have different time-space dimensions.

6 Conclusion

The correlations between the historical volatilities and the realized volatilities, as well as the correlations between the historical volatility increments and the realized volatilities, computed for time horizons ranging from a few hours to a few months, display very clearly the structure of the market in term of the characteristic time frames of the market agents. These correlations show clearly that the market is heterogeneous, with essentially intra-day, daily, weekly and monthly components.

The comparison with theoretical processes is particularly selective in term of the structure that must be included in the process equations. The broadly used GARCH(1,1) process incorporate only one time horizon, and this is not enough to provide for a good description of a multi-horizons real market. The long memory process, that includes a "continuum" of time horizons, gives a much better overall description, including the strong asymmetry between long and short time horizons. This asymmetry originates in the trades from all market participants, regardless of their time horizons, creating short term volatility. Finally, the inhomogeneity observed in the empirical data can be reproduced by putting in the process the corresponding specific time horizons. Moreover, the volatility kernel used to evaluate the historical volatilities has to decay fast enough in order to incorporate in the process that the market participants forget the past quickly beyond their own time horizons.

With the increase of computer power and the availability of high frequency data, the finance world is moving toward a broader use of high frequency models and automated real time decision making. This increases the need to understand financial data from the time-by-tick time scale to a horizon of a few months, and to be most useful, a process or a forecasting model should incorporate as many empirical properties of the real data. The new stylized properties presented in this paper can be used, for example, in short term risk management (from a few hours to a few days), or to short term portfolio management, as both evaluations require, among other ingredients, a short term volatility forecast.

We have also revisited the analogy between finance and hydrodynamic turbulence in the light of these new results. The structure of the interaction is different between the two phenomenon, due to their different time-space dimensionalities. In finance, the components interacts only when they trade, resulting in all volatility components influencing the short term volatility, whereas short term volatility is not influencing long term volatilities. This is different from turbulence where eddies interact in space with eddies at the next scales. Yet, there is a strong analogy between the properties of the volatility changes and velocity increments in heterogeneous turbulence.

Finally, what remain surprising (at least to us) is that all the figures in this paper are computed with data that strongly look like random walks. Yet, hidden in the price path, the different trading time horizons of the market participants impact on the price history, and leave a characteristic signature revealed by our "mug shots".

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