Multifractal volatility

Chapter 1

Stochastic volatility, trading volume and long memory

Reconstructed from the supplied pages; schemas and charts are indicative redrawings.

1.1 Introduction

Following the current developments of market microstructure, it is now commonly accepted that volatility is a strong "footprint" of market behavior and activity. As trading volume, it can express market activity and naturally, a positive correlation has been found between these two variables. A commonly accepted interpretation is that the information flow is the "directing" underlying process leading the two variables (see e.g. Clark [1973], Tauchen and Pitts [1983], Andersen [1996]). However, as noted by Harris [1987], trading could also be self-generating.

Various reasons may determine market participants to trade on a financial market. Agents may differ in their risk profiles, liquidity constraints, degrees of information, trading horizons or heterogeneous prior beliefs. The liquidity constraint is the most obvious and earlier introduced in the literature. Information asymmetry is another factor, largely developed to explain exchanges on the market. However, the "No Trade" Theorem introduced by Milgrom and Stokey [1982], states that information asymmetry, taken alone, can not lead to trade and then imply that agents also share heterogeneous prior beliefs. Indeed, why would one accept a trade knowing that his counterpart is better informed than he is, and hence that he would sustain a loss?

Derived from the "No-Trade" Theorem, a formal classification of information arrivals has been proposed in the literature: there can be common knowledge news and non-common knowledge news. Common knowledge news represent public information, by definition homogeneously and simultaneously interpreted. They will generate a price jump without effect on trading volume. Conversely, non-common knowledge news are public or private (asymmetric) information, differently interpreted. Consequently, they influence both volatility and volume.

The Mixture of Distribution hypothesis (MDH) is the ideal framework to study the role of information in market fluctuations as it articulates in a bivariate model trading volume and volatility with information flow and heterogeneously informed agents. Hence, it allows to bridge the gap between microstructural literature and volatility modeling as underlined by Andersen [1996].

Many statistical stylized facts on volatility have been largely documented in the empirical literature. Volatility clustering is ubiquitous in data, and even a hyperbolic decay of the volatility autocorrelation function has been studied since Taylor’s [1986] early work - the so-called "long-memory" of volatility. However, it is not yet clear that this particular feature would be generated only by information flows. In the same time, there is no consensus on how to model this information flow. Andersen [1996] uses a simple low-order ARMA parameterization; Liesenfeld [1998] and Watanabe [2000] show that autoregressive information arrivals cannot fully account for the serial dependence observed in both volume and volatility. Lux and Ausloos [2000] show that interactive and strategic behavior can explain the hyperbolic decay without explicit information-arrival modeling. Bollerslev and Jubinski [1999] model aggregate latent information arrival as a fractional long-memory process. Liesenfeld [2001] introduces a second latent variable, the sensitivity of investors’ reservation prices to information arrivals.

Dealing with long-range dependence in volatility, some econometric studies get opposite conclusions. With high-frequency data, the volatility autocorrelation can show a geometric decrease, hence short memory, while lower-frequency analysis strongly demonstrates the contrary. Public information arrivals may lead to peaks of volatility whose impact decreases quickly; market efficiency implies rapid incorporation into prices. Hence volatility clustering remains unexplained when only public information arrivals are considered.

One possible reconciliation is a multi-component volatility model with both short- and long-run components. Andersen and Bollerslev [1997] consider overall volatility as the aggregation of numerous independent components, each with its own structure. Some components present short-run dependencies, others highly persistent patterns. Aggregated volatility presents a long-run decaying pattern irrespective of sampling frequency. Liesenfeld [2001] similarly distinguishes short-term dynamics directed by information arrivals and long-term dynamics associated with traders’ sensitivities to information.

Müller et al. [1997] propose HARCH (Heterogeneous ARCH), where volatility at time t integrates conditional variances at different scales. Correlations in volatilities from large to fine temporal scales suggest a cascade, a feature well known in developed turbulence and opening the door to the multifractal paradigm. Multifractality is associated with an underlying stochastic cascade that captures long-range dependence (intermittency) at almost all scales. In a stochastic multiplicative cascade, informational shocks at different time scales generate volatility.

The aim of this paper is to introduce the multifractal paradigm into stochastic volatility modeling. We analyse both volatility and trading volume, emphasizing the multi-component structure of volatility and the scaling law characterizing its structure functions.

1.2 Stylized Facts

Empirical evidence clearly shows that almost all financial series present typical features called stylized facts. The particularly important ones are: (1) unit root in raw prices while returns are nearly uncorrelated beyond very short lags; (2) fat tails and leptokurtosis; (3) hyperbolic decay of volatility autocorrelation and volatility clustering; (4) hyperbolic decay of volume autocorrelation and long memory; (5) strong positive contemporaneous correlation between volatility and volume; and (6) asymmetry between sampling frequencies. Volatility fluctuations appear at almost all time scales, even if at large scales the pdf becomes close to Gaussian. Positive correlation from low-frequency volatility to higher-frequency volatility can be fitted by a stochastic cascade scheme.

1.3 MDH and stochastic volatility

1.3.1 A typology of information

Information is multi-form. The temporal dimension of information flow is crucial whereas often neglected. Information about the business cycle and technological innovations affects volatility differently than information about current market liquidity. A complementary typology, discussed by Evans [2002], distinguishes common knowledge from non-common knowledge according to dispersion of interpretations. Common knowledge news is characterized by simultaneous arrival to all market participants and homogeneous interpretation of its implication for equilibrium prices. Public information can take either form because news may not lead to consensus. The common-knowledge case is a limit case in which price adjusts immediately without affecting trading volume.

Private information is obviously non-common knowledge and can be especially relevant in decentralized markets such as foreign exchange, where customers’ order flow to a dealer is not publicly observable. As participants differ in trading horizons, private information is not limited to intraday horizons. The Hypothesis of Heterogeneous Market leads to MC-ARCH (Zumbach and Lynch [2001]) and stochastic cascade volatility models (Breymann et al. [2000]).

1.3.2 What does volatility mean here?

Whereas returns are uncorrelated, long-range correlations in the scale of price fluctuations suggest that an additional stochastic process characterizing price increments is required. Volatility is often referred to as this underlying process. For a volatility process σ_t predetermined and observable conditional on variables at t − 1, its measure is identical to the conditional variance and volatility is conditionally heteroskedastic. If σ_t is influenced by a contemporaneous or unobserved stochastic vector, it is not predetermined and volatility is stochastic.

Let the price process {S_t} be defined on a probability space (Ω, ℱ, P) with filtration ℱ_t containing full price history and potentially latent state variables. Volatility measures unexpected return fluctuations. Following Andersen [1992], volatility characterizes the complete structure of second-order conditional moments with respect to the relevant information sets.

σ(S_t, t) = [ lim_h→0 (1/h) V(S_t/S_t−h | ℱ_t−h) ]^1/2

dS_t = μ(S_t, t) S_t dt + σ(S_t, t) S_t dW_t

X_t = σ_t Z_t

Here W_t is a standard Wiener process and Z_t is an i.i.d. mean-zero, constant-variance sequence. σ_t represents the volatility process of the price innovation X_t and is directed by a latent stochastic process: information flow. Public and private information coexist, and models with many components at different horizons are promising for describing the complex structure of volatility and trading volume. HARCH, MC-ARCH, multi-component SARV and stochastic cascade models are three related modeling directions.

1.3.3 The model

This model relies on Andersen [1996] in bridging asymmetric-information models à la Glosten and Milgrom [1985] and stochastic volatility modeling. The Mixture of Distribution Hypothesis developed by Clark [1973] and Tauchen and Pitts [1983] is a unified framework articulating trading and econometric models of stochastic volatility. Information arrivals lead agents to revise expectations about terminal asset value and generate price fluctuations. Information is differently interpreted because traders have different private signals or "agree to disagree". Trading appears for liquidity, asymmetric information and heterogeneous interpretations.

The market

The model represents a single asset with random terminal value v at a distant future point. Agents trade one unit and there are no transaction costs.

Agents and their information set

There are three categories of risk-neutral traders: noise/liquidity traders, informed agents and the market maker. Agents arrive sequentially, anonymously and randomly to trade one unit at the bid-ask spread or not trade. Noise traders have public information, past transaction prices and the bid-ask spread. Informed agents and the market maker also have private signals, interpretable as superior skill in observing market composition or even insider information.

Transactions

Each information arrival generates a price-discovery phase followed by equilibrium. Noise/liquidity traders act for exogenous reasons such as hedging, stop-loss orders and liquidity needs, and may be trend chasers. Their presence circumvents the No-Trade theorem. Informed traders maximize expected profit conditionally on their signals. Each transaction affects the bid-ask spread; Bertrand competition forces the market maker to set prices under a zero-profit constraint. Transaction prices form a martingale with respect to the market maker’s information set, and inventory management does not matter by construction.

The bivariate process of volatility and trading volume

The MDH justifies introducing trading volume in volatility modeling. Total daily volume is the sum of noise-trader and informed-trader transaction volume:

V_t = NV_t + IV_t

Noise traders arrive randomly according to a centered Gaussian distribution N(0, m). There are I_t information arrivals during day t. Each contains a public signal v_i and a private signal s_i, j. Informed traders and the market maker disagree on signal precision and market composition.

ΔP*_i, j = P*_i, j − P*_i−1, j = v_i + s_i, j

with v_i ∼ N(0, σ_v²) and s_i, j ∼ N(0, σ_s²), independent. Trader j’s bet is:

B_i, j = α(P*_i, j − P_i)

A positive B denotes a purchase and a negative B a sale. Informed transaction volume between two consecutive equilibrium states is:

V_i = ½ Σ_j |B_i, j − B_i−1, j| = (α/2) Σ_j |ΔP*_i, j − ΔP_i|

Market clearing implies Σ_j B_i, j = 0, hence:

P_i = (1/J) Σ_j P*_i, j

ΔP_i = v_i + s̄_i

V_i = (α/2) Σ_j |s_i, j − s̄_i|

Thus informed trading volume depends only on private signals, more precisely their absolute deviations from the mean. Its first two moments are:

μ_V = E[V_i] = (α/2) √(2(J − 1)J/π) σ_s

σ_V² = V[V_i] = (α/2)²(1 − 2/π) Jσ_s² + o(J)

Large signal variances, i.e. poor precision, generate large trading volume. For large J, a Normal approximation is appropriate. Over a day:

IV_t = I_tΣi = 1 V_i

IV_t | K_t ∼ N(K_tμ_V, K_tσ_V²)

V_t | K_t ∼ N(K_tμ_V, m + K_tσ_V²)

Information flow, strategic interactions and volatility

At the end of day t, returns may be expressed as:

X_t = I_tΣi = 1 ln(P_i/P_i−1) = Σ_i η_i,  η_i ∼ N(0, σ_η²)

Regret-free bid-ask prices imply that the market maker believes prices are fair and that trading reveals information. Transaction prices form a martingale. Using the benchmark day and a Central Limit Theorem generalization of Clark [1973], returns conditional on information intensity satisfy:

X_t | K_t ∼ N(0, σ²K_t)

X_t | K_t ∼ N(0, K_t)

X_t = K_t^1/2 Z_t,  Z_t ∼ N(0, 1)

The same mixing variable drives volume and returns:

{X_t | K_t ∼ N(0, K_t)V_t | K_t ∼ N(μ_VK_t, m + σ_V²K_t)

Information flow may therefore be interpreted as a genuine latent stochastic volatility process governing both returns and volume. Multi-component specifications produce hyperbolic autocorrelation decay and long memory.

1.4 Stochastic volatility, trading volume and long memory

1.4.1 Long memory and multiple components information arrivals

Andersen and Bollerslev [1997] extend the lognormal stochastic volatility model of Andersen [1994] to a multi-component specification, distinguishing shocks from liquidity to business-cycle horizons. Bollerslev and Jubinski [1999] model the long-run information-arrival component as fractionally integrated. Lobato and Velasco [2000] test equality of fractional differencing parameters for volatility and volume and cannot reject equal hyperbolic decay rates. Multiple short- and long-term components can reconcile geometric short-horizon decay with hyperbolic aggregate decay independent of sampling frequency.

1.4.2 Evidence of long memory in volatility and volume

X_t = K_t^1/2 Z_t

If v_j, t captures N distinct information-arrival processes, the Exponential SARV model is:

v_j, t = α_j v_j, t−1 + ε_t

with v_j, t = ln(K_j, t) − E[ln(K_j, t)] and 0 ≤ α_j ≤ 1. The aggregate latent volatility autocorrelation decays as:

ρ(v_t, j) ≈ L₁(j) j^2d−1

and trading volume similarly:

ρ(V_t, j) ≈ L₂(j) j^2d−1

Volatility and trading volume therefore appear nearly indistinguishable at this level. However, Ding et al. [1993] show that absolute returns raised to different powers have different decay rates, while volume shows nearly the same hyperbolic decay across powers. This qualitative difference points toward multifractality.

1.4.3 Long memory and scale invariance

For weakly correlated returns with similar variance sizes, the Central Limit Theorem implies convergence of sums toward a Gaussian distribution. Financial returns, however, display slow convergence: apparent scaling persists from one-minute sampling to daily and even four-day frequencies, unlike simulated i.i.d. data with the same tail index. A plausible explanation is long-range dependence in volatility. Scale invariance is valid only over a range of frequencies before a Gaussian cross-over. Andersen et al. [2001] find heteroskedasticity significant even at monthly frequency for FX data. Different slopes for absolute-return moments motivate Extended Self-Similarity and multiplicative cascade modeling.

1.4.4 Multiscaling: towards a stochastic cascade model of volatility

Scaling law behavior of the moments

ρ(|X_t|^q, j) ≈ j^2d−1

Ding, Engle and Granger [1993] show that while hyperbolic decay holds for different powers, decay rates are not identical. For q from 0.25 to 2, d reaches a maximum around q=0.75 or 1. Temporally aggregated absolute returns exhibit hyperbolic autocorrelation decay across sampling frequencies. A self-similar process is generated by the same probability law at all scales; strict self-similarity requires the same scaling coefficient across absolute moments, while Extended Self-Similarity allows different values.

Quantifying intermittency: the multiscaling analysis

Mandelbrot and coauthors showed that self-similar processes such as fractional Brownian motion exhibit the same degree of detail at regular sampling frequencies. The local Hölder exponent h(t) generalizes the Hurst exponent and characterizes local regularity. A singularity spectrum D(h) gives the fractal dimension associated with sets of points sharing a Hölder exponent. Structure-function exponents ζ(q) are related to D(h) by a Legendre transformation.

|P_t − P_t−k| = |X_t^k| ≈ (k/c)^h(t)

D(h) = dim ω_k(h)

ζ(q) = min_h(qh − D(h)),  D(h) = min_q(qh − ζ(q))

This relation is the multifractal formalism. Multiscaling analysis studies sample absolute moments through structure functions:

E[|P_t − P_t−k|^q] = E[|X_t^k|^q] ≈ c(q)(k/T)^qH

For strict self-similarity the scale exponent is linear in q. In a multifractal process:

E[|P_t − P_t−k|^q] ≈ c(q)(k/T)^ζ(q)

where ζ(q) is nonlinear and concave. Hölder’s inequality yields ζ(q) ≥ ω₁ζ(q₁)+ω₂ζ(q₂), demonstrating concavity. In turbulence, intermittency denotes alternating stability and instability; in finance this is analogous to volatility clustering.

1.4.5 From stochastic cascade to stochastic volatility

Kolmogorov [1941] first exploited cascade schemes for intermittent turbulent flows. The cascade paradigm provides a framework for multiscaling in financial data and formalizes HARCH in a more rigorous hierarchical framework. Large-scale volatility can Granger-cause short-scale volatility, suggesting a hierarchy across scales. The binomial cascade illustrates the multiplicative construction: an interval is repeatedly split and probability mass multiplied by fractions p and 1-p. Generalizations use b cells or random multipliers, producing random multifractal measures.

Stochastic cascade volatility modeling

Apply the multiplicative cascade to volatility. At each instant, fluctuations at large scales are related to fine-scale fluctuations through random multipliers W_k:

σ_k = W_k σ_k−1 = W_k W_k−1 … W₀ σ₀

σ_k = σ₀ k∏i = 0 W_i

σ₀ is homoscedastic volatility at the lowest frequency. Densifying the discrete cascade leads to a continuous, log-infinitely-divisible model. Eligible generators include log-Normal, log-Levy, log-Gamma and log-Poisson families. Financial applications include Muzy et al. [2001], Lux [2001], Breymann et al. [2000], Calvet et al. [1997], Schmitt et al. [1999, 2000], and Turiel and Perez-Vicente [2002].

1.5 Empirical evidence

In this section, we test the relevance of the MDH model developed above with its extension to volatility modeling.

1.5.1 Data description

The data was provided by Datastream and consists of quoted prices and trading volumes for Alcatel, from January 1st 1991 to December 31st 2001: 2633 returns (differences between price logarithms) and 2634 trading volumes (also in logarithms).

	Market return	Volume
Sample mean	0.0019685	7.9304
Standard deviation	1.2162	0.8417
Skewness	-2.0767	0.3422
Excess kurtosis	39.342	2.6633
Jarque-Bera	147025.8	63.8456

Market returns are highly leptokurtic and negatively skewed; volume has positive skewness and strong kurtosis. Jarque-Bera rejects normality for both distributions.

1.5.2 Fractional integration and semiparametric estimation of long memory in absolute returns and trading volume

	Augmented Dickey-Fuller	Phillips-Perron	KPSS
Log-price (level)	-1.5869	-0.1810	1.680*, 0.379*
Log-price (difference)	-47.6011*	-47.5493*	0.091, 0.088
Volatility (abs returns)	-8.3049*	-29.6952*	3.206*, 0.344*
Volume (log)	-7.4934*	-46.0342*	3.534*, 0.368*

Correlograms and Box-Pierce statistics are highly significant at distant lags. The MDH long-run proposition is:

ρ(|R_t|, j) ≈ L₁(j) j^2d−1,  ρ(V_t, j) ≈ L₂(j) j^2d−1,  0 < d < ½

Cross-correlations similarly decay hyperbolically, suggesting subordination to a common latent information-arrival process rather than direct causality.

The Geweke-Porter-Hudak [1983] estimator estimates d in the frequency domain. For long memory, the spectrum has a singularity at zero frequency:

f(ω) = (2π)⁻¹ Σ_j γ_j cos(jω)

f(ω) ∼ L(1/ω) ω^−2d,  ω → 0⁺

The semiparametric approach imposes behavior only near the origin. With (1 − L)^d |R_t| = η_t, the spectral density satisfies:

f(ω) = |1 − e^−iω|^−2d f_η(ω)

ln f(ω) ≈ ln f_η(0) − 2d ln ω

Because the original series are non-Gaussian, a less restrictive semiparametric estimator based on the ratio of periodograms at two near-zero frequencies is used (Robinson [1994], Andersen and Bollerslev [1998]):

d̂_AP = ½ − ln(F(qω) / F(ω)) / (2 ln q)

	d̂_AP
Volatility	0.4327
Volume	0.4718

Both estimates lie between 0 and 1/2: volatility and volume are highly persistent but covariance stationary, with shocks dissipating at a slow hyperbolic rate. Applying a (1 − L)^d filter removes the long-run dependence.

1.5.3 Long memory for absolute returns at various powers

Powers	Volatility	Volume
0.25	0.4442	0.4215
0.50	0.4469	0.4250
0.75	0.4426	0.4272
1.00	0.4327	0.4284
1.25	0.4146	0.4288
1.50	0.3828	0.4284
1.75	0.3305	0.4272
2.00	0.2565	0.4251
2.25	0.1737	0.4221
2.50	0.1035	0.4181
2.75	0.0567	0.4131
3.00	0.0301	0.4069
3.25	0.0162	0.3996
3.50	0.0090	0.3908
3.75	0.0054	0.3806
4.00	0.0035	0.3688

The estimated d’s show long memory in power transforms of absolute moments. Volume keeps almost the same fractional differencing parameter, while volatility differs sharply: its decay rate is maximal around q=0.5 and decreases almost to zero by q=4. Repeating the test on stock indices and exchange rates gives the same qualitative conclusion. Ding et al. [1993] first reported this stylized fact. This is where volatility and volume stop behaving in the same manner. Different decay rates imply different slopes of scaling laws and cannot be matched by a strictly self-similar process: a signature of multifractality.

1.5.4 Stochastic cascade and scale dependencies

Müller et al. [1997] propose a formal test for stochastic cascade behavior by testing correlations between volatilities on large and fine time grids. Lagged correlations indicate information-flow structures in the Granger-causality sense. Symmetry at positive and negative lags suggests synchronous information flow; statistically significant asymmetry suggests directional flow. Fine volatility is the mean absolute daily return averaged over five observations, while coarse volatility is the absolute return over a weekly interval. The maximum correlation is at lag zero, but at the first two nonzero lags coarse volatility predicts fine volatility better than the reverse, with statistically significant differences. Results across many financial assets are similar.

These findings converge with Müller et al. [1997] on exchange rates: there appears to be a flux between scales, interpreted as influence of long-term traders on short-term traders. Ghashghaie et al. [1996] draw a parallel with the energy flux in turbulence, cascading from large to small scales and often modeled as a multiplicative cascade with multifractal properties.

1.5.5 Scaling and multiscaling analysis

Multifractality extends self-similarity to allow different scaling relations for different structure functions. The main application is to characterize absolute-return moments at different powers:

E[|P_t − P_t−k|^q] = E[|R_t^k|^q] ≈ c(q)(k/T)^ζ(q)

If the process is scale-invariant, ζ(q)=qH. With multiscaling, ζ(q) is nonlinear and concave. It is estimated by linear regressions in log-log form for q from 0 to 4 by increments of 0.25:

log E[|P_t − P_t−k|^q] ≈ log c(q) + ζ(q) log(k/T)

The resulting structure-function exponents are not linear but concave. Scaling relations therefore depart from the linear benchmark of strict self-similarity. Similar results are reported for numerous financial series. Curvature depends on intermittency; exchange rates during the European Monetary System show stronger curvature around realignment jumps. Dacorogna et al. [2001] discuss scaling breakdown at very high frequencies, plausibly related to microstructure effects such as seasonality, bid-ask bounce and inventory management. Multiscaling has implications for risk management and market timing.

LeBaron [2001] shows that a three-component stochastic-volatility model can exhibit apparent multiscaling, but the complexity of stochastic-volatility estimation opens the door to cascade models. For continuous cascades, the singularity spectrum has a known closed form. Lux [2001a] provides an early econometric estimation of a multifractal model using GMM.

1.6 Conclusion

In this chapter, we introduce a new paradigm for modeling volatility: the multifractal paradigm that leads to apprehend volatility as a stochastic cascade. A multi-components volatility model allows to describe intermittency (volatility clustering) at almost all data frequencies and replicate the hyperbolic decay of volatility autocorrelation. Whether components are uncorrelated or explicitly dependent through a stochastic cascade, representation of volatility across time scales is likely to be central. Volatility aggregation has direct implications for forecasting, risk management, market timing and trading models. Gençay et al. [2002] describe the true data-generating process as a complex network of layers, each corresponding to a particular frequency.

The instantaneous correlation between volatility and trading volume is robust and both variables exhibit similar hyperbolic decay rates. However, trading volume differs from volatility under the Ding, Engle and Granger [1993] test. Moreover, positive volatility-volume correlation is only globally valid; careful scrutiny shows it is time-varying and can become negative during market stress. Combining jump-detection tests with this time-varying correlation may help understand breakdowns in the usual positive relation.

Appendix: Simulated price using a log-normal random generator

We simulate a 2600 points-length path of a typical log-normal price using the Box-Muller algorithm. We then apply the same multiscaling analysis than for Alcatel. The result shows that linearity is an acceptable hypothesis graphically.

Selected notes from the original pages

Mandelbrot [1968] described volatility clustering as large changes tending to be followed by large changes, of either sign, and small changes by small changes. Absolute returns are preferred to squared returns because they are less dominated by extreme observations and their autocorrelation is more stable under sample-size changes. The empirical Alcatel study uses a large French equity because comprehensive FX trading volume was not available for the period. Holiday corrections were made for weekly intervals. The original article also cites formal tests by Lobato and Velasco [2000] and Bollerslev and Jubinski [1999] for equality of long-memory parameters.