Creating Tail Dependence by Rough Stochastic Correlation Satisfying a Fractional SDE; An Application in Finance

Abstract

The stochastic correlation for Brownian motions is the integrand in the formula of their quadratic covariation. The estimation of this stochastic process becomes available from the temporally localized correlation of latent price driving Brownian motions in stochastic volatility models for asset prices. By analyzing this process for Apple and Microsoft stock prices traded minute-wise, we give statistical evidence for the roughness of its paths. Moment scaling indicates fractal behavior, and both fractal dimensions (approx. 1.95) and Hurst exponent estimates (around 0.05) point to rough paths. We model this rough stochastic correlation by a suitably transformed fractional Ornstein–Uhlenbeck process and simulate artificial stock prices, which allows computing tail dependence and the Herding Behavior Index (HIX) as functions in time. The computed HIX is hardly variable in time (e.g., standard deviation of 0.003–0.006); on the contrary, tail dependence fluctuates more heavily (e.g., standard deviation approx. 0.04). This results in a higher correlation risk, i.e., more frequent sudden coincident appearance of extreme prices than a steady HIX value indicates.

1. Introduction

The interdependence or association of various financial instruments such as asset prices or credit values and the related derivatives or credit events plays a fundamental role in investment strategies and risk management. Individual prices or values are most often described by stochastic differential equations (SDEs) that, by their nature, reflect dynamic changes. In most models, however, the evolution of the dependence structure is either neglected by using constant correlations or is not compatibly described with the individual modeling (e.g., by mixing copulas and SDEs). More coherent modeling can be built regarding the short time “localized” correlations of the price driving Brownian motions as a stochastic process, i.e., by considering stochastic correlations, similarly to stochastic volatilities in nature and name. We derive the interdependence of prices from the underlying price driving forces in the mentioned SDEs of individual (marginal) models. Therefore, we consider the price driving Brownian motions and describe their interdependence by stochastic correlation, the latter being driven by a rough SDE. This way, empirically observed phenomena such as tail dependence in prices and the herding behavior of investors can be better explained. We emphasize that the presented model differs from a multivariate stochastic volatility model. Temporally changing stochastic interdependence in volatility is modeled via Wishart processes in [1], but that approach is also different from ours.

Non-linearly associated Brownian motions can be constructed by prescribing their quadratic covariation as an integral of diffusion or a rough process, called the stochastic correlation and can, in turn, be described by another SDE generated from either a classical or a fractional Brownian motion. Since the price driving Brownian motions are directly non-observable, latent processes, a preliminary model build is necessary to obtain a quasi sample for the stochastic correlation. Wiggins’ stochastic volatility model serves this purpose in the present study. Given the quasi sample of the stochastic correlation, a diffusion or rough SDE can be fitted to it. The concept of stochastic correlation and its description by diffusion models are known in the literature, and the groundbreaking work on the topic is [2]. Either the Jacobi process as in [2,3] or an Ornstein–Uhlenbeck process transformed by function composition was used previously to model stochastic correlation. The outer functions in the composition include the normalized inverse tangent [2], the normal cumulative density [4], and the hyperbolic tangent [5] functions. The comparison of the corresponding stochastic correlation models is made in [6]. To introduce jumps into correlation is also a possibility, e.g., [7] uses transformed Lévy processes in modeling stochastic correlation, but this approach is out of the scope of the present work.

The modeling of the stochastic correlation as a rough process is a novelty of the present work. A new modeling framework to describe stochastic volatility has been introduced in the seminal paper [8] by using fractional Brownian Motions (fBm) as driving forces in stochastic differential equations. The peculiarity of the approach is the small, typically of magnitude 0.1 or even lower, value of the Hurst parameter of the fBm (cf. [9]). These models are referred to as rough volatility models, see [9,10,11,12,13,14] for more details and practical applications. Such small estimated values for H (between 0.05 and 0.2) have been found in [15] when studying the volatility process of a great number of assets.

In the light of the mentioned literature, we propose to model the stochastic correlation of Brownian motions with a suitable transformation of a rough, fractional Ornstein–Uhlenbeck (fOU) process (of the first kind cf. [16]), generated from fBm as introduced in [17], with the Hurst coefficient $H<0.5$. Integration with respect to the fractional Brownian motion (fBm) is necessary to specify the fOU process. It can be introduced in several ways, and the different resultant integrals are only equal for particular sets of integrands. We refer the reader to [18] for a review of stochastic calculus with fBm. An almost complete survey of the mentioned integrals is given in [19]. All these integrals of a certain class of deterministic functions coincide. Further, for smooth integrands, they coincide with the limits of the Riemann–Stieltjes integral sums. Since we integrate only constants that belong to both cases, we consider these Riemann–Stieltjes integrals, referring to the path-wise approach. For further details on fBm and fOU processes, we refer the reader to [17,19,20].

We provide statistical evidence to justify the choice of a rough process instead of diffusion in modeling stochastic correlation. As in the case of stochastic volatility (see [8,9]), moment scaling reveals the fractal character of the back-transformed stochastic correlation that we describe by the fOU process. Although the various Hurst exponent estimations create diverse values, all of them are very well below the 0.5 bound, indicating rough paths. As a novel approach, the fractal dimension of the estimated stochastic correlation is considered, and we found that its high value is counter-intuitive to the choice of a traditional diffusion process. A conforming match with this feature can be obtained using rough correlation instead. We choose the Hurst exponent to reconstruct both the fractal dimension—and hence, the proper roughness—and the marginal distribution of the stochastic correlation process. Once the Hurst exponent is set, the estimation of the fOU model parameters is straightforward as in [21]. The complex hierarchical model allows for extensive simulation of stochastic correlations and price processes for the goodness of fit assessment and further analysis.

The strength of the association of two random variables may depend on the magnitude of their values. This is made mathematically precise in the notion of tail dependence or asymptotic dependence in [22]. It measures the association between the variables’ extremely high or low values and depends only on their copula. Asset prices in financial markets tend to exhibit tail-dependence, particularly a lower one, corresponding to increased interdependence of low values. In the context of financial risk management, the lower tail usually corresponds to losses and hence, is of utmost importance because the appearance of high losses may hit investors hard, and hence, to avoid them is of decisive importance in risk analysis. In [23], one may find more details about the intuition behind tail dependence. Stochastic correlation may result in tail dependence in asset prices, as it has been shown in [24]. The simulation of synthetic asset prices allows for calculating tail dependence in every time instant, so its temporal variability can be observed.

There are periods in the market when prices drop or grow together by synchronized co-movements, and hence, losses in one asset cannot be balanced by gains in others. This is particularly so in stressed or critical market situations. The reason behind such a phenomenon may be that all investors or brokers may expect a similar impact on the market of an occurring piece of information and, therefore, instinctively, start to act in their trade in a similar way. Such a spontaneous synchronized action without centralized direction is called herding behavior. In order to measure its presence, the herding behavior index or HIX was created in [25] (see also [26]) and used for measuring the level of possible diversification among stock prices. The HIX takes values between 0 and 1. Its high value signals a higher degree of herd behavior: positive interdependence of market prices, allowing for less diversification in investments.

It can naturally be expected that the co-movement of prices leads to the coincident appearance of extreme values. Since HIX measures the co-movement and tail dependence of the coincidence of extreme values of stock prices, a call for comparison occurs naturally, and the simulation of a high number of synthetic paths makes it possible. However, contrary to intuition, we find that while the variability of the HIX index is very low, less than 10 percent around its mean value, tail dependence may eventually double. It means that relying only on the HIX index may result in miscalculations in the frequency of simultaneously occurring extreme situations.

In summary, we provide statistical evidence for the roughness of the stochastic correlation process derived from high-frequency stock price data; propose a novel approach to model this rough stochastic correlation using a suitably transformed fractional Ornstein–Uhlenbeck (fOU) process; demonstrate that this rough stochastic correlation model can generate significant tail dependence in simulated asset prices, consistent with empirical observations; and highlight the discrepancy between the stability of the Herding Behavior Index (HIX) and the pronounced fluctuations in tail dependence, thereby revealing a potential underestimation of correlation risk when relying solely on HIX.

The remainder of the paper is organized as follows. Section 2 introduces the concept of Brownian motions with rough stochastic correlation. Section 3 discusses the modeling of individual asset prices, from which latent Brownian motions are inferred. Section 4 details the proposed rough correlation model based on a transformed fractional Ornstein–Uhlenbeck process. Section 5 describes the application of our methodology to real high-frequency stock data. Section 6 presents the empirical evidence supporting the roughness of the stochastic correlation. Section 7 focuses on assessing herding behavior and tail dependence using simulations from our model, and compares these with empirical estimates and simpler models. Finally, Section 8 concludes the paper and offers directions for future research.

2. Brownian Motions with Rough Stochastic Correlation

Stochastic differential equations are used frequently to model data series such as interest rate, asset price, and exchange rate. For diffusion processes described by SDEs, the dependence between the solutions most often originates in correlated Brownian motions driving the equations. Suppose we are given a finite time horizon $[0,T]$ and a filtered probability space ($\Omega ,\mathcal{A},{\mathcal{F}}_t,P$ ) with a filtration, satisfying the usual conditions. For example, it is the case when the filtration is the augmented filtration of a (multidimensional) Brownian motion. Suppose we are given two independent Brownian motions, $W_0$(t) and $W_1$(t) adapted to the filtration ${\mathcal{F}}_t$ (i.e., the filtration is rich enough). Considering two adapted correlated standard Brownian motions $W_1$(t) and $W_2$(t) with $\mathrm{corr}(W_1\left(t\right),W_2\left(t\right))=\rho$, their quadratic covariation,

$$[W_1,W_2]\left(t\right)=\underset{0}{\overset{t}{\int }}\rho ds=\rho ·t,\rho \in [-1,1]$$

(1)

equals the covariance and is symbolically written as $d{W}_1\left(t\right)d{W}_2\left(t\right)=\rho dt$. Instead of a constant $\left|\rho \right|\le 1$, let us be given an adapted stochastic process $\rho$(t) with $\left|\rho \right(t\left)\right|\le$ 1. By using $\rho \left(t\right)$, we define the vector process $\mathbf{W}\left(t\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{W_1\left(t\right)}{W_2\left(t\right)}}\right)$ by $\mathbf{W}\left(0\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{0}{0}}\right)$ and the differential

$$d\mathbf{W}\left(t\right)=d\left({\displaystyle \genfrac{}{}{0pt}{}{W_1\left(t\right)}{W_2\left(t\right)}}\right)=\left[\begin{array}{c}1,0\\ \rho \left(t\right),\sqrt{1-{\rho }^2\left(t\right)}\end{array}\right]d\left({\displaystyle \genfrac{}{}{0pt}{}{W_1\left(t\right)}{W_0\left(t\right)}}\right).$$

(2)

Since $\rho$(t) is bounded and the time horizon is finite $\rho$(t)$\in {\mathcal{L}}_2\left(\Omega \times [0,T]\right)$, $\mathbf{W}\left(t\right)$ is, therefore, well-defined. The second coordinate $W_2\left(t\right)$ is clearly adapted to ${\mathcal{F}}_t$, and as a sum of two stochastic integrals with regards to Brownian motions of ${\mathcal{L}}_2\left(\Omega \times [0,T]\right)$ processes, it is a continuous martingale. As a direct consequence of Itō’s formula, the quadratic (co)variation of $\mathbf{W}\left(t\right)$ is

$$[\mathbf{W},\mathbf{W}]\left(t\right)=\underset{0}{\overset{t}{\int }}\left[\begin{array}{c}1,\rho \left(s\right)\\ \rho \left(s\right),1\end{array}\right]dt,$$

(3)

and in particular, the quadratic variation of $W_2\left(t\right)$ is t. Being an adapted continuous martingale with quadratic variation t, by virtue of Lévy’s theorem (cf. e.g., theorem 8.1.6. in [27]), $W_2\left(t\right)$ is itself a Brownian motion. However, the two dimensional process $\left({\displaystyle \genfrac{}{}{0pt}{}{W_1\left(t\right)}{W_0\left(t\right)}}\right)$ is not a two dimensional Brownian motion. Furthermore, the two dimensional distribution of the vector is not Gaussian despite of the normality of the marginals. Further, the quadratic covariation of $W_1\left(t\right)$ and $W_2\left(t\right)$ is

$$[W_1,W_2]\left(t\right)=\underset{0}{\overset{t}{\int }}\rho \left(s\right)ds.$$

(4)

To the analogy of the constant $\rho$ in (1), the process $\rho \left(t\right)$ is called the stochastic correlation of $W_1\left(t\right)$ and $W_2\left(t\right)$. When, in addition, $\rho \left(t\right)$ has rough paths, we call it, in short, the rough correlation. Note here that no semimartingale property was required of $\rho \left(t\right)$ for what has been said above. We also remark that stochastic correlation is only defined here for Brownian motions. An analogue of Goldstein’s theorem for the quadratic variation of SDEs seems to be easy to obtain, and that provides the way to extend the definition, at least to diffusion processes. It is, however, out of the scope of the present paper since we intend to avoid technical discussions that are not absolutely necessary.

3. Modeling Individual Asset Prices

The generating Brownian motions driving the SDEs of observed asset prices are directly unobservable, latent processes. Therefore, to analyze their association in the case of actual data, a suitable model is needed, its discretized version has to be fitted to the price data, and estimated Brownian motions have to be retrieved. A reliable analysis requires that these estimations feature discretized Brownian motion characteristics: independence and stationarity of increments and normal distribution. Simple models like geometric Brownian motion do not live up to this expectation. The temporally and stochastically changing nature of volatility has long been noted in the mathematical finance literature, and many models have been built to incorporate it into price description. Among them, the Heston model became one of today’s industry standards. However, when fitted to the minute-wise registered trading data—the subject of the present paper—the residuals obtained from this model did not sufficiently feature the mentioned Brownian motion characteristics. This phenomenon is well-known for practitioners and experienced by us in the current situation as well. After careful examination of several possibilities, in our case, Wiggins’ model [28] turned out to be creating probably the best residuals in terms of both normality of the distribution and the independence and stationarity of the increments.

Let us consider the following continuous-time model, proposed first by Wiggins (1987) [28], where the spot volatility process is allowed to follow a diffusion process.

$${\displaystyle \frac{dS\left(t\right)}{S\left(t\right)}}=\mu dt+\sigma \left(t\right)dW\left(t\right)$$

(5)

$$log\left(\sigma {\left(t\right)}^2\right)=Y\left(t\right)$$

(6)

$$dY\left(t\right)=\alpha (\theta -Y(t\left)\right)dt+dV\left(t\right),$$

(7)

where in (5), $S\left(t\right)$ is the asset price at time t, ${\sigma }^2\left(t\right)$ is the instantaneous variance (volatility squared), $\mu$ is the expected return of the asset, and $W\left(t\right)$ is the price-driving Brownian motion. The logarithm of the instantaneous variance $Y\left(t\right)$, defined in (7) follows a mean reverting Ornstein–Uhlenbeck process, as given in (7), with $\alpha$ being the rate of mean reversion, $\theta$ the long-run mean of the variance, and $V\left(t\right)$ the volatility-driving Brownian motion.

It is known that the Euler–Maruyama discretization of Wiggins’ model turns into Taylor’s stochastic volatility (SV) time series model with proper parametrization. The fitting of the latter via an MCMC algorithm is readily available, together with its full description, in R, in package “stochvol” [29], and we use that to obtain the parameters and retrieve the residuals. The latter serve as estimations of the price driving Brownian motions, regarded as a quasi sample from its increments, and are the basis for recovering the stochastic correlation for the further analysis described in the coming sections. For clarity, we emphasize that in the coming sections, we consider two asset price processes, each following Wiggins’ stochastic volatility model, and the price’s interdependence arises from a stochastic correlation of the price-driving Brownian motions, $W_i\left(t\right),i=1,2$.

4. A Rough Correlation Model for the Dependence of Price Driving Brownian Motions

In the applications, the locally (in time) estimated correlation fluctuates around a constant value in the mean reversion sense, and the boundaries −1 and +1 of the correlation process are non-attractive and unattainable. We require these properties when modeling the correlation as a stochastic process. Evidence for the roughness of paths, when found, influence the model choice for stochastic correlation. One natural candidate would be the fractional Jacobi process. Ref. [30] proved the existence and uniqueness theorems for the fBm generated Jacobi process; however, the properties and the estimation theory of that process are not yet elaborated. Hence, it remains a theoretical possibility but is not yet a practical alternative for empirical work. Therefore, we omit its use in the present work. Instead, we shall consider a transformed fOU process for modeling rough correlation with the aforementioned properties.

Having said all this, we turn to the dynamics of the mean-reverting fOU process as given by a Langevin-type stochastic differential equation (SDE)

$$\begin{array}{cc}\hfill dX\left(t\right)& =\kappa (\theta -X\left(t\right))dt+\sigma d{W}_H\left(t\right)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill X\left(0\right)& =X_0\in \mathbb{R}\hfill \end{array}$$

(9)

driven by a fBm $W_H=\{W_H\left(t\right),t\ge 0\}$ of Hurst parameter $H\in (0,1)$, and both the parameter $\kappa$ and the volatility $\sigma$ are positive real constants.

For modeling stochastic correlation, we map the fOU values into the [−1, 1] interval by using an appropriate transformation with a smooth, bounded, and strictly monotonous real-function $g\left(x\right)$ as $\rho \left(t\right)=g\left(X\right(t\left)\right)$, preserving, this way, the rough path property. In our analysis of actual data, presented below, we shall use for $g\left(x\right)$ the hyperbolic tangent function abbreviated in the text as “tanh” and the inverse as “atanh”

$$\rho \left(t\right)=tanh\left(X\left(t\right)\right)={\displaystyle \frac{1-e^{2X\left(t\right)}}{1+e^{2X\left(t\right)}}}\mathrm{and}X\left(t\right)=atanh\left(\rho \left(t\right)\right)={\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\rho \left(t\right)}{1-\rho \left(t\right)}}\right).$$

(10)

Using different outer function $g\left(x\right)$ in the transformation would also be possible. Our earlier work [6] compared stochastic correlation models using such transformations (normalized inverse tangent, normal CDF, hyperbolic tangent) for non-fractional (diffusion-based) $X\left(t\right)$ processes. While ref. [6] does not deal with fractional processes, the general observation that the choice of transformation impacts the distributional fit holds. For fractional processes, the primary impact on roughness (H) should be minimal for smooth transformations, but the overall goodness-of-fit, including distribution, can differ. This is exactly the case in the application, presented below, when the choice of different outer function worsens the fractal dimension’s or the distribution’s fit. Fully non-parametric estimation of the transforming function may be desirable, but we do not intend to develop it in this work.

Once a sample or a prediction for $\rho \left(t\right)$ becomes available, the same for $X\left(t\right)$ can be computed, opening the way for fitting the fOU model. For parameter estimation of the fOU process, assuming the Hurst exponent is known, we refer to methods described in [31], and in general, [21]. Specifically, ref. [21] provides a least-squares estimator for the drift in their formula (5.1) and an ergodic type estimator in their Theorem 5.4 (with formula (5.22) for its discretized version), the latter of which we utilize in our analysis. The properties of this latter estimator are analyzed in detail in [32]. As for $\sigma$, we use the specific version, given for the constant case in Proposition 4.2. of [21] in terms of p-variations, and we choose p = 2, which is permitted given the low Hurst coefficient. With that, the estimate of $\sigma$ reduces to $n^H$ times the residual standard deviation.

Our analysis’s crucial goal and novelty are to present a statistical method for evidencing that the correlation process is indeed rough and to estimate its most important characteristic, the Hurst exponent. When introducing rough processes into modeling stochastic volatilities, the authors of [8] rely on moment scaling, more precisely, its linearity on the log-log scale, widely used primarily in the physics literature. Putting the theory to the test on actual data using 5-minute realized volatility data of stock indices, [33] confirms the roughness of stochastic volatility statistically. Beyond moment scaling, they estimate the Hurst coefficients by using a specific Whittle type estimator that they show to be more accurate than the one obtained by moment scaling; the latter is either heavily biased or inaccurate in their case and—as we shall see later in this paper—does not match to other estimators in our analysis as well.

A consequence of linear moment scaling—shown in both mentioned papers and the application part of the present one—is the fractal character of the process, or more precisely: of its paths. A measure of roughness for fractals is their fractal dimension, a notion dating back to [34]. Several definitions and estimation procedures are known for fractal dimensions; a survey of them is given in [35]. Hausdorff dimension is a theoretical quantity; it is intractable empirically. Box-count, the traditionally most preferred computable counterpart, is notorious for its very slow convergence and unreliability. The authors in [35] find the use of the madogram more suitable in a wide range of situations. Motivated by their finding, we shall also rely on the madogram. Although the ’fractaldim’ package of R readily gives the value of the other estimators, they—like the variogram, rodogram, and Genton’s robust estimator—either give the same results or—like the periodogram based estimators—produce trivial, and therefore useless, values in the application. It is known [35,36] that the path of an fBm, as well as the fOU process, has fractal dimension 2-H. So, it naturally lends itself to consider the fractal dimension of the transformed stochastic correlation, the measure of the roughness of the correlation paths, parallel as an estimator of the Hurst exponent. It is performed in the present work, and it turns out that the Hurst parameter obtained from the fractal dimension is in good agreement with some more conventional estimators, like the aggregated variance, R/S, and Peng’s method (also called detrended fluctuation analysis). To that fit, however, the log-log scale regressions have to be suitably truncated, making the general use of these latter, traditionally applied methods at least more circumstantial, if not dubious.

Next, an fOU model is fitted to the transformed stochastic correlation. A crucial fitting criterion is that the Hurst exponent of the driving fBm in (8) has to be chosen so that the generated transformed fOU process fits the stochastic correlation in terms of both fractal dimension and marginal distribution. Fractal dimension is a characteristic of a continuous sample, whereas we only can have discrete-time observations. Theoretically, the fractal dimensions and thus the Hurst exponents of the fOU process and its driving fBm are the same (see Chapter 8 in [36] and also [37]). However, due to the continuous nature of the notion, it is only so in the limit when sampling frequency goes to infinity. While the drift parameter’s local or short time smoothing effect is negligible in the limit, it is not so for the finite sample. The effect of smoothing is analytically intractable, so we rely on simulations to take it into account. Besides the fractal dimension, the match of the empirical and the simulated distribution in terms of mean, standard deviation, skewness, and estimated density is also a criterion for the fit.

5. Application to Real Data: Trade Price of Two Stocks

To aid comprehension before discussing applications, in Figure 1, we include a flowchart summarizing the key steps of our approach.

The present study analyzes the association of high-frequency closing prices of two stocks, Apple (APPL) and Microsoft (MSFT). The data are registered between 19 July 2016 and 3 August 2016, consisting of 5040 minute-wise closing prices. The closing price of APPL and MSFT with their log-returns are given in Figure 2, respectively. Just as in [33] and many other papers, we model the price dynamics in the business time scale, meaning that the time in the model evolves only when a market of the asset is open.

Euler–Maruyama discretization of Wiggins’ stochastic volatility model (i.e., Taylor’s stochastic volatility time series model) is fitted to the APPL and MSFT log-return data by using the ‘svsample’ command of the ‘stochvol’ package in R. As a result of the MCMC algorithm, the estimated model parameters are obtained as sample averages from the posterior distribution. The standardized residuals can also be extracted by the in-built ‘resid’ method. The powerful BDS test [38], just like the more simple difference-sign or turning point tests, rejects dependence in the residuals. Kolmogorov–Smirnov, Anderson–Darling, and Shapiro–Wilk tests reject deviation from normality of the residuals. Hence the obtained (normal and independent) residuals can be regarded as estimations of the increments of the driving Brownian motions. The volatilities can also be subtracted—the volatility plot is provided readily by the package. To obtain the object itself requires some further coding in R. The volatilities are highly correlated, by 0.965, which is probably specific to the period when the samples were taken. The high correlation explains why we use stochastic correlation modeling only for the price residuals. We create volatilities according to this constant correlation when simulating the synthetic stock prices.

The normal plots of the residuals for the two stocks, AAPL and MSFT, are shown in Figure 2. Supposing that the association of the two Brownian motions, driving the price equations of the two stocks, originates from stochastic correlation with stationary correlation process (constant correlation included as the extreme case), the pairs of residuals sample the same two-dimensional distribution, opening the way to estimate the copula of the latter. If the two Brownian motions were simply Pearson correlated, the increments would create a Gaussian copula. However, the copula of residuals deviates from that of correlated Gaussian variables with identical Pearson correlation as visible in Figure 3.

We now create estimations for the two price-driving Brownian motions by cumulating the residuals and, after that, estimate their “local” correlations in non-overlapping windows of size of 5 min. We call this estimation the (stochastic/rough) correlation of price residuals for short and denote it by $\widehat{\rho }\left(t_i\right)$, where $t_i$ grows by 5 min. Allowing for overlapping windows would work as a smoother, inflating fractal dimension and producing misleading results. This way, we receive an estimation for the stochastic correlation process (of length 1008) of the price-driving Brownian motions.

6. Evidences for Roughness of the Stochastic Correlation

The next step is to provide statistical evidence to the presence of rough correlation. The subject of the following analysis is the correlation of price residuals that is the estimated stochastic correlation process $\widehat{\rho }\left(t_i\right)$ of the Brownian motions and its back-transform by the ‘atanh’ function $\widehat{X}\left(t_i\right)$. Just as the price residuals are regarded as a quasi sample from the Brownian motions, the back-transformed correlation of price residuals can be thought of as a quasi sample from the fOU process mentioned in Section 4, taken in the discrete-time set $t_i,i=1,\dots ,1000$. The fractal dimension of both processes can be computed. The madogram values ($\widehat{M}$) are 1.9568 and 1.9520, and the corresponding Hurst estimates are 0.0432 and 0.0480, respectively, for $\widehat{\rho }\left(t_i\right)$ and $\widehat{X}\left(t_i\right)$. Other fractal dimension estimators could have also been given from the ’fractaldim’ package of R, but most give similar results, so presenting them would only hamper transparency. The box-count values are significantly different, probably due to the slow convergence, while estimators in the frequency domain indifferently give the trivial value 2.

The mentioned quasi sample opens the way to study moment scaling. It is performed similarly in the case of the volatility process in [8] and also in [33]. Aggregation allows examining scaling properties across different time horizons and can mitigate some high-frequency noise. While computing the moments, we change the level of aggregation in 2-power magnitude. This choice—although differing from [8,33]—is quite customary in the fractal literature (see e.g., [39], Figures 11–13), and results in more transparency. On the log-log scale, the moments grow linearly with aggregation. It can be seen as statistical evidence for the fractal property. The coefficients of the log-log scale regression sit nicely on a line of slope $\widehat{H}=0.0216$—as displayed in Figure 4—and give another Hurst exponent estimate.

Although the Hurst exponent estimates obtained from the fractal dimension and moment scaling do not match well, we emphasize their role in supporting the roughness hypothesis rather than using them as definitive H estimators. The unreliability of the moment scaling estimate was already noted in [33], but being very small, it points to a rough path behavior, just like the fractal dimension estimates.

In order to obtain further inference on the Hurst coefficient of the back-transformed correlation of price residuals, the ‘fArma’ package given in R is utilized, and the R/S, aggregated variance, Peng, Higuchi, and Whittle estimators are computed, as mentioned in Section 4. Their values are given in

Table 1. Hurst exponent estimations of the back-transformed correlation of price residuals.

Methods	R/S	Aggr. Var.	Higuchi	Peng’s	Whittle	Frac. Dim.
Value	0.0537	0.0447	−0.001	0.0381	0.0466	0.0480

.

The strategy in choosing the estimators was to use a diversified set of estimators operating on different principles (time-domain, frequency-domain, fractal geometry). The values by the aggregated variance method and the Whittle estimator match best to the one corresponding to fractal dimension, and the R/S and Peng’s methods produce only slightly different values; only Higuchi’s method fails to give meaningful results. Note here that the lack of distribution theory for those estimators does not permit considering the significance of the differences. The overwhelming consistency of the Hurst estimation results ($H\ll 0.5$) provides strong evidence for roughness.

With the evidence that the atanh-transformed process of stochastic correlation of price residuals has fractal paths, its estimated fractal dimension is as high as nearly 1.95, and consistently with it, various Hurst exponent estimations produce as low values as roughly 0.05, we can safely assert that this process has rough paths.

7. Assessing Herding Behavior and Tail Dependence by Simulation

Next, we model the rough process of inverse tangent hyperbolic transformation of the correlation of price residuals as an fOU process. We measure the goodness of fit (GoF) of the fOU process by the madogram fractal dimension and the marginal distribution of the process. Since no theory is available for the properties of fractal dimension estimation and we are not aimed at developing one either, we rely on simulation results only. Supposing the Hurst exponent $H_B$ of the fOU driving fBm to be known, the estimation of the drift and volatility parameters is straightforward as is given in Section 4. So, we need $H_B$.

We look for an approximate value ${\widehat{H}}_B$ by a simple grid search, in 0.005-wise steps. The 0.005 step size was found to provide a good balance between computational effort and precision in identifying the optimal $H_B$ for matching the empirical characteristics. In each step, 1000 simulations are performed with the actual ${\widehat{H}}_B$ value and the corresponding parameters creating 1000 tanh-transformed fOU processes. Their madogram fractal dimension, together with the mean, standard deviation, and skewness of the marginal distribution, are computed for assessing GoF and compared to the corresponding values obtained from $\widehat{\rho }\left(t_i\right)$. Based on their fit, in the grid search ${\widehat{H}}_B=0.105$ can be selected as the estimation of the Hurst exponent. That, in turn, determines the values of the fitted fOU parameters exhibited in the first two rows of

Table 2. Fitted parameters of the fOU model; descriptive statistics and fractal dimension of the rough correlation of observed stock log-returns and the mean and standard deviation of the same descriptive statistics for 1000 simulated rough correlations.

Parameter Estimation		${\widehat{\mathit{H}}}_{\mathit{B}}$	$\widehat{\mathit{\kappa }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\sigma }}$
Fitted values		0.115	0.8907	0.7891	1.7322
Process $\setminus$ Characteristics		Mean	Std. Dev.	Skewness	Madogram
$\widehat{\rho }\left(t_i\right)$ correlation of price residuals		0.5305	0.4067	−1.2787	0.0480
Simulation; process length 1008	mean	0.5301	0.4045	−1.1517	0.0485
	std. dev.	0.0048	0.0089	0.0522	0.0335
Simulation; process length 10,000	mean	0.5330	0.4002	−1.1556	0.0439
	std. dev.	0.0014	0.0028	0.0149	0.0170
Simulation; process length 100,000	mean	0.5342	0.4002	−1.1574	0.0461
	std. dev.	0.0004	0.0009	0.0058	0.0034

. The GoF statistics values and their standard deviations are presented in the further rows for tanh-transformed fOU processes of lengths increasing roughly 10 times by rows, simulated with these same parameters.

The simulated tanh-transformed fOU process is of short memory because ${\widehat{H}}_B=0.105\ll 0.5$, so, the asymptotic normality of the mean, standard deviation, and skewness holds. The madogram is a variogram type estimator, with absolute differences instead of the squares in sum, so for its values again, asymptotic normality holds true under quite general conditions, but it does not hold for the fractal dimension computed from it. Remember, the latter has values between 1 and 2. The sixth row in Table 2 displays the standard deviations of these GoF statistics, used now as of simulated tanh-transformed fOU processes, i.e., simulated rough correlations of the same length, i.e., 1008, as the quasi sample. So, these may be regarded as an estimation of the standard deviations of the GoF statistics of the correlation of price residuals $\widehat{\rho }\left(t_i\right)$—from that latent process so far, we only had one quasi observation. Together with the asymptotic normality, this means that at least the first three statistics presented in the fifth row are well within the 95% confidence bound, so the given Hurst exponent estimate can be accepted on that basis. Even when we increase the length of the simulated tanh-transformed fOU processes to 10,000 and 100,000—approximately 10 and 100 times the original length—and run 1000 simulations for each case, the results remain stable. The correlation structure of the price residuals, the goodness-of-fit (GoF) statistics, and the madogram-based fractal dimension all show remarkable consistency. Note here that the box-count dimensions (not presented in the table) change significantly with length as evidence of its mentioned very slow convergence. The distribution of the simulated processes also matches well to that of $\widehat{\rho }\left(t_i\right)$. It is illustrated in Figure 5 where the quantile-quantile plot of $\widehat{\rho }\left(t_i\right)$ and its simulated counterpart is displayed together with the estimated density function of $\widehat{\rho }\left(t_i\right)$ and the simulated processes. Note here that the latter estimation is obtained by taking all simulated processes as one sample, which is 100 times more data; hence, a much smoother and more accurate function could be obtained. The excellent fit is remarkable; the somewhat higher peak of the simulated data density may result from the higher number of data and the automatic bandwidth choice. Further, a beta density function fitted by MLE as implemented in the ‘fitdistr’ function in R is also displayed in dotted green. Its poor fit implies that the distribution is not beta, an argument against using a Jacobi process in modeling.

Consider two stocks with price processes $\{S_i\left(t\right),0\le t\le T;i=1,2\}$. Suppose that their dynamics are described by SDEs. Consider the prices at a certain time instant t, and for a more transparent formula, use temporarily the notations $S_1=S_1\left(t\right)$ and $S_2=S_2\left(t\right)$. The HIX index defined in [25] simplifies for this case to

$$HIX={\displaystyle \frac{{\sigma }_{S_1}^2+{\sigma }_{S_2}^2+2·corr[S_1,S_2]·{\sigma }_{S_1}·{\sigma }_{S_2}}{{\sigma }_{S_1}^2+{\sigma }_{S_2}^2+2·corr[F_{S_1}^{-1}\left(U\right),F_{S_2}^{-1}\left(U\right)]·{\sigma }_{X_1}·{\sigma }_{X_2}}},$$

(11)

where ${\sigma }_{S_1},{\sigma }_{S_2}$ are the variances, $F_{S_1}(.),F_{S_2}(.)$ the distribution functions, and U is a uniform random variable in $[0,1]$. Note that with these choices, $corr(F_{S_1}^{-1}\left(U\right),F_{S_2}^{-1}\left(U\right))$ is the maximal correlation between $S_1$ and $S_2$ (see again [25]).

Turning now to tail dependence, consider a random vector $\mathbf{Z}=(Z_1,Z_2)$ with marginals $F_{Z_1}$ and $F_{Z_2}$. The coefficient of lower tail dependence is then defined as the limit as u approaches 0 of the conditional probability that $Z_1$ is less than or equal to the quantile $F_{Z_1}^{-1}\left(u\right)$, provided that $Z_2$ is less than or equal to $F_{Z_2}^{-1}\left(u\right)$ i.e.,

$${\lambda }_L=\underset{u\to 0^{+}}{lim}P\left(X\le F_{Z_1}^{-1}\left(u\right)|Y\le F_{Z_2}^{-1}\left(u\right)\right)$$

(12)

In bivariate random vectors with dependence structures given by the copula cumulative density function ${C:[0,1[}^2\to [0,1]$, lower tail dependence turns out to be [22]

$${\lambda }_L=\underset{u\downarrow 0}{lim}{\displaystyle \frac{C(u,u)}{u}}.$$

(13)

Non-zero values of ${\lambda }_L$ suggest lower tail dependence in C.

Similarly, we may have defined upper tail dependence, but we shall only consider lower dependences in this work. From now on, any reference to tail dependence will correspond to lower tail dependence.

The empirical counterpart, i.e., the estimation of the lower tail dependence for a given sample size n is denoted by ${\widehat{L}}_n\left(u\right)$ and can be given as

$${\widehat{L}}_n\left(u\right)={\displaystyle \frac{{\sum }_{i=1}^n{\mathbf{1}}_{\{X_i\le {\widehat{F}}_X^{-1}\left(u\right),Y_i\le {\widehat{F}}_Y^{-1}\left(u\right)\}}}{{\sum }_{i=1}^n{\mathbf{1}}_{\{X_i\le {\widehat{F}}_X^{-1}\left(u\right)\}}}},$$

(14)

where $\widehat{F}$ is the empirical distribution function of the variable written in its index, see also [22].

We use the ’somebm’ package of R to generate 10,000 fractional Brownian motions of length 5040—the same as the original data—, and with them, hyperbolic tangent transformed fOU processes for rough correlations. With them, we create 10,000 rough-correlated Brownian motion pairs of the same length. These serve as driving forces in Wiggins’ models with the estimated parameters from the fit to APPL and MSFT prices, and by that, we simulate 10,000 pairs of paths of similarly associated synthetic stock price log-returns. This way, we can create an empirical copula of log returns at every time instant. A single, non-time-dependent copula is also created from the original observations, when observations in time are supposed to sample the same two-dimensional distribution, i.e., based on the principle of ergodicity, that is taken for granted. In Figure 6, we compare this latter copula with the simulated one in an arbitrarily chosen time point—we display in the figure the copula of midday log-returns in the 14th trading day, but in any time point, it looks almost absolutely the same—and with a Gaussian counterpart of the same Pearson correlation. For better comparison, we only display 5040 points in all three plots—no more is available from the original observations.

A detailed visual inspection concludes that the obtained copulas are different. Further, the first two plots indicate much stronger tail dependence than the third one, so both observations and simulations strongly indicate the presence of (non-zero) tail dependence. From the simulated paths in every time point, we have 10,000 samples from the copula of the log-returns; hence, we can calculate the tail dependence. Having 5040 time points, we could compute the same amount of tail dependence values, but it would be a tedious and time and computer-memory consuming work with doubtful advantages. Given the stationary character of our model, we perform the calculation only for the last 500 trading times. We summarize the descriptive statistics of these tail dependence values in

Table 3. Comparison of tail dependences of simulated minute-wise log-returns by their mean, median, and standard deviation and 90%, 95%, ad 99% quantiles with their counterparts from Gaussian copulas.

Model∖ Statistics	Lower Tail Dependence
Observed	0.3267
Simulated		Mean	Median	Standard	Quantiles
log-returns				deviation	90%	95%	99%
Rough Corr.		0.2948	0.2975	0.0402	0.3426	0.3581	0.3892
Constant Corr.	C. Vol.	0.1877	0.1894	0.0445	0.2350	0.2428	0.2816
	I. Vol.	0.0922	0.0973	0.0399	0.1443	0.1460	0.1487

. To control for the effect of the rough correlation on tail dependence, we also simulate log-returns with constant Pearson correlations of the price driving Brownian motions. A source of interdependence, in this case, can be the very highly correlated (almost joint) Brownian motions driving the stochastic volatility process. So, we simulate both with such volatility processes and independent volatility processes from our model. The results are given in the last two rows of Table 3. The model with rough correlation creates tail dependence with a surprisingly excellent match to the one computed from the observed log-returns by the principle of ergodicity, i.e., using temporal observations for estimating space characteristics. Stochastic volatilities themselves—although very highly correlated, almost identical—are not capable of reaching that level of tail dependence. When correlation is constant in price driving and volatilities are independent, the created tail dependence is similar to that of the finite sample from a Gaussian copula. The values present rather convincing evidence for rough correlation, being an essential component in creating tail dependence, which is as strong as the observed one.

We calculate their correlation and the HIX by the ergodicity principle for the observed log-return processes. The simulated synthetic stock log-return values allow for estimating correlation and subsequently the HIX index at every trading time. We calculate that for the last 500 trading times and present its descriptive statistics in

Table 4. On the basis of 10,000 simulated pairs of stock log-return processes, the HIX and the correlations of minute-wise log-returns are compared using the rough stochastic correlation and a constant correlation model. The mean, median, and standard deviation, and 90%, 95%, and 99% quantiles of the last 500 trading times are computed. The values of their counterparts computed by the ergodicity principle from observed stock log-returns are presented in rows 2–3.

Statistics∖Process	Observed	Simulated
Correlation in Observed	0.5896	Mean	Median	St. dev.	Quantiles
HIX, Observed	0.7967				90%	95%	99%
HIX, Rough Model		0.7645	0.7646	0.0060	0.7718	0.7743	0.7776
HIX, Const. Corr. Model		0.7632	0.7630	0.0046	0.7688	0.7712	0.7753
Corr. in Rough Model		0.5268	0.5270	0.0028	0.5302	0.5315	0.5328

.

The correlation of the observed two log-return processes slightly deviates from the temporal mean of the correlations computed in individual trading times from simulated log-return values. It is valid for both the rough correlation and the constant correlation model. The reason may probably be that temporally localized correlations are weaker and do not reflect the overall long-term association entirely. Indeed, we built up our model for temporally localized correlations, and their mean is 0.5308, which is reflected well by the model and the numbers in the last row of Table 4. As a result of this deviance, the HIX index computed from the observations slightly exceeds the mean and even the 99% quantile of the simulated ones from both models.

However, the most interesting is the very low variability in both the HIX and the correlations. It is a result of the stationarity of the model built. Correlations are almost constant due to the accurate modeling and since the distribution of the simulated values is also steady, maximal correlations are almost unchangeable. As a result, the HIX has a very low variance. That also explains why there is practically no difference between the rough and the constant correlation model in their mean HIX values—even though variability is somewhat higher in the rough model, it is still very low. Contrary to this, the tail dependence values show much greater variability.

Parallel to this, the HIX values displayed in Figure 7 against the tail dependence values show a very weak association. This circumstance calls for caution in risk management, particularly if the risk of coincident appearance of extreme values or events is of interest. Irrespective of whether this is a finite sample phenomenon or the result of the random fluctuation of local correlation, on this level of available information, the steady HIX values may disguise an existing random fluctuation in the considered risk.

8. Conclusions

Apple and Microsoft prices in frequent minute-wise trading are modeled using Wiggins’ stochastic volatility model. The latent Brownian motions driving the price dynamics in this model are estimated, and their temporally localized correlations are analyzed. Statistical evidence is found for the fractal nature of these correlations. The estimated fractal dimension of the correlation path is very high—close to that of the ambient space—and the corresponding Hurst exponent is much lower than 0.5, around 0.05. These features are consistent with the rough path property, particularly given the sample size. Consequently, localized correlations are modeled with a rough process: a hyperbolic tangent-transformed fractional Ornstein–Uhlenbeck (fOU) process, fitted to the quasi-sample obtained from the estimated correlations of the Brownian motions. This model enables extensive simulations of synthetic Apple and Microsoft stock prices.

From these simulations, fractal dimensions, Hurst exponents, and distribution functions of local correlations are computed and used as goodness-of-fit metrics. The alignment of these metrics with those derived from the empirical data supports the model choice and strengthens the evidence for roughness in the stochastic correlation.

The rough correlation model also reproduces tail dependence similar to that observed in the empirical data under the assumption of ergodicity. In contrast, a constant correlation model with highly correlated stochastic volatilities produces significantly weaker tail dependence, while a constant correlation model with independent volatilities fails to exhibit any meaningful tail dependence—contrary to empirical observations.

These findings suggest that the association between stock prices is not solely driven by their shared exposure to a random volatility environment. Rather, the internal price-driving forces—such as fundamentals—also act in a synchronized manner, substantially increasing the likelihood of joint extreme events.

The relationship of tail dependence with the HIX index describing co-movement of prices is also analyzed. Both HIX and tail dependence are designed to characterize the behavior of random variables in extreme situations typical of stressed markets. Our finding is that the HIX index exhibits very similar values under both constant and rough correlation assumptions and shows great temporal stability. In contrast, tail dependence in the rough correlation model has a significantly higher mean—nearly double that of the constant correlation model, as does the observed tail dependence in the data, too—and it exhibits greater variability. Moreover, tail dependence fluctuates more intensively in the rough model. While HIX regards the constant and rough correlation models as similarly risky, tail dependence reveals substantial differences. Considering fractal dimensions, the model choice and the associated risk become quantitatively meaningful in real-world applications such as portfolio selection, management, or other investment decisions.

The implications of our model extend to several areas of financial applications. Traditional portfolio optimization often assumes constant correlations, but our model demonstrates that correlation is both stochastic and rough. This can lead to higher tail dependence than predicted by simpler models. As a result, diversification benefits might be overestimated—especially during extreme market movements, when correlations tend to increase (a stylized fact that our model captures via tail dependence).

In the context of dynamic hedging or asset allocation, incorporating rough stochastic correlation could lead to more responsive and robust strategies. If periods of elevated tail dependence can be even partially anticipated or modeled, portfolio weights could be adjusted dynamically to mitigate risk. The ability to simulate scenarios with fluctuating tail dependence—as shown in Figure 6, where the HIX remains stable but tail dependence varies—is valuable for stress testing. It highlights that relying solely on average correlations, or even on stable HIX values, may significantly underestimate downside risk.

Moreover, the pricing of derivatives whose payoff depends on the correlation between underlying assets—such as quanto options, basket options, and correlation swaps—is highly sensitive to the chosen correlation model. A rough stochastic correlation framework, as we propose, offers potential improvements in pricing accuracy and hedging effectiveness over models assuming constant or smoothly diffusing correlation structures. Importantly, the roughness of correlation also implies that correlation itself can be highly volatile, a critical factor in the valuation of correlation-dependent derivatives. Our model thus provides a way to represent this “volatility of correlation” in a tractable yet realistic manner. Furthermore, the model’s capacity to generate significant tail dependence is particularly valuable for pricing out-of-the-money options on baskets or for assessing the risk of instruments exposed to joint extreme market movements.

The primary aim of this study is descriptive and explanatory: to demonstrate the existence of roughness in stochastic correlation, to propose a model capable of capturing this feature, and to explore its implications for contemporaneous risk measures like tail dependence and HIX. Our focus is on the model’s ability to replicate observed in-sample characteristics of financial data dependence. Assessing predictive performance (e.g., forecasting future correlation structures or tail risk) is an important but distinct research question that would require a different experimental design and set of evaluation metrics, extending significantly beyond the current scope.

Further research may address model estimation issues, derivatives-based inference, and extensions to multiple stock price processes. Given the hierarchical setup, a Bayesian approach for model identification may prove useful, while deep learning techniques could be considered in parallel with existing rough volatility estimation methods [40]. An analogous analysis based on option price data, rather than historical returns, would also be of interest.

Expanding upon our framework, future work could involve rigorous out-of-sample testing to assess the model’s predictive capabilities for correlation dynamics and tail risk. This framework may also enhance portfolio optimization and risk management practices, particularly under stressed market conditions where traditional models often underperform. In the context of pricing and hedging correlation-sensitive derivatives—such as quanto options, basket options, or correlation swaps—using rough stochastic correlation could improve accuracy and effectiveness compared to models assuming constant or simple diffusion-based correlation. Moreover, our framework can account for the high volatility of correlation itself, a crucial input in certain derivative models. The ability to capture significant tail dependence is especially relevant for pricing out-of-the-money basket options and evaluating the risk of joint extreme movements.

A deeper exploration of the micro-level trading behaviors that give rise to rough correlation dynamics could provide insights into the structural origins of these aggregate phenomena.

The current bivariate model serves as a foundational step. Extending it to the multivariate case—modeling several stock prices instead of two—is a complex but important direction, requiring careful treatment of matrix theory and stochastic processes. In the bivariate case, the condition $\left|\rho \right(t\left)\right|\le 1$, ensured by our tanh transformation, guarantees the positive definiteness of the log-return covariance matrix at any time t.

For N assets, however, one must model an $N\times N$ instantaneous correlation matrix $\mathbf{R}\left(t\right)$ that remains positive semi-definite at all times, with ones on the diagonal and off-diagonal elements ${\rho }_{ij}\left(t\right)\in [-1,1]$. Modeling each ${\rho }_{ij}\left(t\right)$ independently as a transformed fOU process does not ensure that the resulting matrix $\mathbf{R}\left(t\right)$ remains positive semi-definite. This presents a significant challenge.

Additionally, modeling all $N(N-1)/2$ pairwise correlations individually introduces a large number of parameters, complicating estimation and calibration. Factor models for correlation or matrix-valued stochastic processes may offer more tractable alternatives. A promising candidate for modeling rough stochastic covariance (and hence correlation) is the matrix-valued fractional Wishart process (fWp), although its theoretical development for $H<0.5$ is ongoing and involves considerable complexity. Other strategies might involve decomposing the correlation matrix (e.g., via Cholesky or eigenvalue decomposition) and modeling the components, though maintaining the desired roughness and interpretability in such frameworks is non-trivial.