Multiscale Markowitz

We might question why there is a need at all to do multiscale portfolio optimization. Isn’t the single scale case enough?

In reality, two facts occur in stock and stock index prices. On the one hand, non-trivial self-similarity with varying critical exponents has been observed in pricing time series since the time of Mandelbrot (who noticed it for cotton prices). At lower frequencies (weekly and below), momentum appears, i.e. a Hurst exponent above $\frac{1}{2}$ while fat tails fade away. On the contrary, intraday returns display fatter tails but negative autocorrelation, i.e. a Hurst exponent below $\frac{1}{2}$.

On the other hand, periods of high and low volatility operate like an acceleration and slowdown of the time clock. Using a fixed time clock for the return series boils down to using some kind of random time sampling with respect to the market ”volatility time”. We will se that, in fact, volatility is a hidden market variable. In other words, short of considering it as a hidden variable, prices series are not Markovian, despite the Efficient Market Hypothesis, which one to needs to enter into high frequency trading to invalidate it.

The logic of all this is that investors at marginally different time horizons behave in a similar way. Dynamic phase transitions are also characterised by self similar fixed points (markets show these near crashes and regime shifts). Motivated by this, there are certain cases where Markowitz breaks, particularly when the volatility spikes. Exposures that are calibrated on a low volatility period turn out be way to large for the newly appearing high volatility. We will show that this can be addressed by the multiscale optimization.

The following are typical stylised facts observed in the stock market and in financial markets in general:

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  Volatility Clusters Market volatility varies through time in a irregular and time-asymmetric manner. Volatility may suddenly surge following an event or a news that triggers its instability. Then it will take time to progressively decrease.

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  Rough volatility: This causes the volatility of some assets to increase disproportionately at longer scales, causing over-allocation to them [2]. Globally, volatility remains in a limited range, implying a negative autocorrelation of its variations at various time scales.

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  Non-Ellipticity Asset returns exhibit varying fat-tailed distributions subject to the tail concentration effect: under extreme conditions, the number of effective variables driving the market shrinks down to just a few of them. The Hurst exponent also varies, due to market impact eg. illiquids, corporate bonds, PE/VC funds, emerging markets etc. We know that Markowitz generally overallocates to illiquids[12, 13, 14].

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  Crashes and Bubbles: During times of crashes, often Markowitz worsens drawdowns as opposed to an equally weighted portfolio [9]. This is because the volatility begins to scale non-trivially with scale. We expect that we see power laws during bubbles as well, with imaginary critical exponents.

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  Stochastic Volatility: This introduces uncertainty on the volatility of the different assets, resulting in instabilities and estimation errors, in addition to non-generizability out of sample [4].

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  Regime Changes: It is unable to handle stochastic regime changes such as shocks induced by changes in interest rates, unemployment numbers, and global macroeconomic and political factors [7].

A lot of these are instances when the volatility becomes increasingly rough, with Hursts substantially different from half, or when tails become disproportionately fat [2]. In the case of market crashes, in fact, that is what really happens. Bubbles too are characterized by Hursts, but those which are imaginary, corresponding to dynamics governed by the Log Periodic Power Law (LPPL). Markowitz is based on the condition of ellipticity, where all stocks have the same Hurst exponents. Therefore, optimal weights at one time scale generalize across scales.

We are motivated by the fact that pricing time series has been shown to exhibit self-similarity in various studies, starting with the time of Mandelbrot [12]. Also self-similar dynamics is at the endpoints of renormalization group flows. We start with the most general non-interacting PDE we can write down describing fractional diffusion in both space and time: $\\ \frac{\partial^{\beta}P(x,t)}{\partial t^{\beta}}=-K_{\alpha}(-\Delta)^{\alpha}P(x,t)\\$ Where

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  $P(x,t)$ is the probability density function

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  $\frac{\partial^{\beta}}{\partial t^{\beta}}$ is fractional Caputo derivative (with $0<\beta<=1$)

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  $(-\Delta)^{\alpha}$ is the fractional Laplacian (Riesz derivative) of order $\alpha$ (with $0<\alpha<=2$)

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  $K_{\alpha}$ is a generalised diffusion coefficient.

Here we see that there are qualitatively three types of cases:

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  Brownian motion ($\beta=1,\alpha=2$): This case equates to the usual Markowitz case. Here $|x|\propto t^{0.5}$

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  Fat Tails ($\beta=1,1<=\alpha<2$): This corresponds to no time dependence but fat tails with a power law $|x|\propto t^{1/\alpha}$

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  Gaussian distribution with time dependence ($0<\beta<2,\alpha=2$): Here we have the benefit of no fat tails but time dependence causes anomalous scaling $|x|\propto t^{\beta}$

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  Time dependence and Fat Tails($0<\beta<2,1<=\alpha<=2$): This is the most general base with time dependence and fat tails. Here we have that $|x|\propto t^{\beta/\alpha}$

Note that in most studies, the Hurst is considered to be solely due to time dependence, and $H:=\beta/2$. However, this obfuscates the fact that anomalous scaling can be due to both time depedence and fat tails. Hence thereafter in this study, when we refer to the Hurst, we refer to the standardised Hurst defined as $\\ H:=\beta/\alpha\\$ This recognises that anomalous scaling can be due to both fat tails and time dependence. In a similar vein, in the presence of nonlinearities we can in principle have multifractality due to both the effect of the probability distribution ($\beta_{n}\neq n\beta$) and the interactions ($\alpha_{n}\neq\alpha/n$). Analogously, we have a standardised generalized Hurst: $\\ H_{n}:=\beta_{n}/\alpha_{n}\\$ The direct observable here is the scaling law $|x|^{n}\propto t^{H_{n}}$, while it is more subtle to address the contributions of the iid distribution and dependence structure to it, and will not be addressed.

The discerning reader might be concerned here about the inconsistency of the Markowitz paradigm with the presence of fat tails, since the covariance matrix does not in principle converge under the presence of fat tails. The solution to that is straightforward: one can replace the covariance matrix with the $L^{1}$-modified covariance matrix and solve the optimisation problem: $\\ min_{w}w_{i}\Sigma^{L^{1}}_{ij}w_{j}\\$ Where the $L^{1}$-modified covariance matrix is written as: $\\ \Sigma^{L^{1}}_{ij}=\rho_{ij}|R_{i}-\tilde{R_{i}}||R_{j}-\tilde{R_{j}}|\\$ Where $\rho_{ij}$ are just the cross-asset correlations and $|R_{i}-\tilde{R_{i}}|$ and $|R_{j}-\tilde{R_{j}}|$ are absolute deviations from the robust central estimate (median). This will in principle have better convergence properties in the presence of tails, but should not make a substantial difference to the allocations.

Consider a simplified scenario where all assets exhibit the same standardised Hurst exponent, $H$, so $|x|=|t|^{H}$. Here, although we have anomalous scaling, the optimization problem can be solved at any given frequency, and the resulting solution will be applicable across all frequencies. This is due to the fact that when the scale is varied through a scaling transformation, the portfolio variance is modified by a factor of $(\Delta t_{1}/\Delta t_{2})^{H}$ [1], thus rescaling our optimisation function by a constant.

However, in practice, assets often exhibit diverse scaling behaviors with frequency and are typically not self-similar. In such cases, it is necessary to compute the optimization independently at each frequency to account for these differences [6].

In a more generalized scenario, the variances and covariances of different assets, denoted as $\sigma_{i}(\Delta t)$ and $\rho_{i}(\Delta t)$, respectively, can vary arbitrarily with scale and are derived from empirical data. Similarly, the target variance $\sigma_{\text{target}}(\Delta t)$ can be expressed as an arbitrary function of $\Delta t$. Under these circumstances, the optimization problem is formulated as follows:

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  Minimize the portfolio variance: $\sum_{i}w_{i}\Sigma^{MS}_{ij}w_{j}$

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  Here $\Sigma^{MS}_{ij}=<\Sigma_{ij}(\Delta t)/\Delta t>$

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  Ensure portfolio weights across scales sum to one: $\sum_{i}w_{i}(\Lambda)=1$

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  Maintain non-negative weights: $w_{i}(\Lambda)\geq 0\,\forall\,i,\Lambda$

This general case highlights the complexity of portfolio optimization when dealing with assets that exhibit non-uniform scaling behavior across different time scales.

Multifractality in financial time series reflects the idea that different parts of the data may exhibit different scaling behaviors. Unlike a monofractal process, which is characterized by a single Hurst exponent $H$, a multifractal process is described by a spectrum of exponents $H(q)$, where $q$ is a moment order. This spectrum captures the complex, heterogeneous nature of financial markets, where the roughness of returns can vary depending on the time scale and the statistical moment being considered.

In a multifractal framework, the variance scaling law is generalized to:

where $H(q)$ is the multifractal scaling function, which depends on the moment $q$. For example, $H(1)$ corresponds to the traditional Hurst exponent used in variance scaling. Multifractality corresponds to $H(q)\neq qH(1)$

To incorporate multifractality into portfolio optimization, the variance of each asset $i$ at a given time scale $\Delta t$ is modeled using the multifractal formalism:

where $H_{i}(q)$ is the multifractal scaling exponent for asset $i$ at moment $q$. For Markowitz, we will be interested in $q=2$ where multifractality implies that $H(2)\neq 2H(1)$. Similarly the covariance between stocks $i$ and $j$ scales as:

Where multifractality ensures $H_{ij}(2)\neq(H_{i}(1)+H_{j}(1))/2$. In fact from the above expression, if we assume a the scaling exponent for the correlation:

We find that:

This is well known to be non-zero and positive as a result of the Epps effect [15], where correlation between any two pairs of stocks is expected to increase with increasing length scale. Empirically from studies of the Epps effect, we find that $H^{\rho}_{ij}\approx 0.3$, which implies that the variation of cross correlation with scale decreases as one moves to lower and lower frequencies. The portfolio variance at time scale $\Delta t$ then becomes:

This expression now accounts for multifractality by incorporating the $q$-dependent scaling behavior of the assets, in terms of both the variance and covariance.

The optimization problem in a multifractal setting aims to minimize the portfolio variance across both time scales and moments. The optimization problem can be formulated as:

Here:

We here will expand the variances and covariances with the correct scaling properties. This is subject to:

Here, the optimization takes into account not only the scaling behavior across different time scales $\Delta t_{k}$ but also the varying roughness of the time series as captured by the multifractal spectrum $H_{i}(q)$. This leads to a more complex, yet more accurate, description of portfolio risk that better reflects the true nature of financial markets.

To implement the multifractal optimization framework, it is necessary to estimate the multifractal scaling exponents $H_{i}(q)$ for each asset. This can be done using methods such as the multifractal detrended fluctuation analysis (MF-DFA) or the wavelet transform modulus maxima (WTMM) method. These techniques allow for the extraction of the multifractal spectrum from historical return data.

The estimated multifractal spectrum can then be used to calculate the portfolio variance at different time scales and moments, which serves as the input for the optimization problem.

To demonstrate that increasing the variance $\sigma_{k}^{2}$ of a specific asset $k$ leads to a decrease in its optimal weight $w_{k}$, while keeping all other parameters constant.

We want to now express the weight of asset $k$. From the optimal weights expression:

where $s_{k}=(\Sigma^{-1}\mathbf{1})_{k}$ and $S=\mathbf{1}^{\top}\Sigma^{-1}\mathbf{1}$.

Now we want to analyze the effect of increasing $\sigma_{k}^{2}$. We aim to compute the derivative $\frac{\partial w_{k}}{\partial\sigma_{k}^{2}}$ and show that it is negative.

Computing $\frac{\partial s_{k}}{\partial\sigma_{k}^{2}}$ and $\frac{\partial S}{\partial\sigma_{k}^{2}}$

Since $\mathbf{s}=\Sigma^{-1}\mathbf{1}$, we have:

Given that $\Sigma$ only depends on $\sigma_{k}^{2}$ through the $(k,k)$-element:

where $\mathbf{e}_{k}$ is the $k$-th standard basis vector. Thus:

Specifically, the derivative of $s_{k}$ is:

Similarly, the derivative of $S$ is:

Lets get the final expression for the derivative. Substituting back:

Given that $\Sigma^{-1}$ is positive definite, $(\Sigma^{-1})_{kk}>0$, and $S>s_{k}^{2}$, it follows that:

We know that $\frac{\partial\sigma}{\partial H_{i}}=|\Delta t|^{H_{i}}ln(\Delta t)$

It follows that

Next, we consider the effect of increasing the correlation between two assets, say $i$ and $j$, on their combined weight $w_{i}+w_{j}$. The covariance between assets $i$ and $j$ is given by:

where $\rho_{ij}$ is the correlation between assets $i$ and $j$. Increasing $\rho_{ij}$ increases the covariance $\sigma_{ij}$.

The weights $w_{i}$ and $w_{j}$ depend on the inverse of the covariance matrix $\Sigma^{-1}$. To compute the effect of increasing $\rho_{ij}$, we differentiate the weights $w_{i}$ and $w_{j}$ with respect to $\rho_{ij}$:

where $s_{i}=(\Sigma^{-1}\mathbf{1})_{i}$ and $s_{j}=(\Sigma^{-1}\mathbf{1})_{j}$. The derivative $\frac{\partial\Sigma}{\partial\rho_{ij}}$ has nonzero entries only at positions $(i,j)$ and $(j,i)$. Here we can have that $w_{i}$ and $w_{j}$ might increase or decrease. However, if we do an optimisation over the combined asset $w_{i}+w_{j}$ we can conclude that the combined weight of the synthetic asset decreases with an increase in correlation. This is because in this modified picture, all you are doing is changing the variance of the two assets: $\\ <lr_{synthetic,ij}^{2}>=<(lr_{i}+lr_{j})^{2}>=<(lr_{i})^{2}>+<(lr_{j})^{2}>+<lr_{i}lr_{j}>\\$ So all the other variances and covariances are the same, except that of the synthetic combination of $i$ and $j$. Thus if we freeze $w_{ij}$, we get from the previous result:

Thus:

Here note that although empirically $H_{ij}>0$ which causes a positive multifractality, one could have had $H_{ij}<0$ as well in principle. In that case the correlations would decay with scale, and hence the weight of the synthetic asset would increase with an increase in the multifractal parameter $H_{ij}$.

To summarize our results we get as expected:

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  If you increase volatility of a stock, its portfolio weight goes down

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  If you increase Hurst ($\beta/2$) of a stock,$H_{i}$ goes up, its portfolio weight goes down

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  If you increase the fat tailedness of a stock ($1/\alpha$),$H_{i}$ goes up, its portfolio weight goes down

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  If you increase correlation between two stocks, their combined weight goes down.

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  If you introduce positive multifractality in two stocks’ correlation (increased Epps effect),$H_{ij}$ goes up, their combined weight goes down; and vice versa.