Expecting the Unexpected: Entropy and Multifractal Systems in Finance

Abstract

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point's position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor's point of view, a rise in entropy is a signal of abnormal and possibly negative returns. This means he has to expect the unexpected and prepare for it. To explore this, we analyse the New York Stock Exchange (NYSE) U.S. Index as well as its constituents. Through this examination, we assess their multifractal characteristics and identify market conditions (bearish/bullish markets) using entropy, an effective method for recognizing fluctuating fractal markets. Our findings challenge conventional beliefs by demonstrating that price declines lead to increased entropy, contrary to some studies in the literature that suggest that reduced entropy in market crises implies more determinism. Instead, we propose that bear markets are likely to exhibit higher entropy, indicating a greater chance of unexpected extreme events. Moreover, our study reveals a power-law behaviour and indicates the absence of variance.

Keywords: determinism; entropy; financial time series; investments; multifractal analysis; risk management.

Figure 1

NYSE: Histogram of daily log-returns.

Figure 2

Time histories of: (a) $\mu$, (b) $\nu$ and their mean and volatility shifts.

Figure 3

Time series of: (a) Brown noise and (b) NYSE.

Figure 4

Welch Power Spectral Density Estimate for Brownian noise (blue) and the NYSE (red).

Figure 5

Log-log plot of the PSD of brown noise (a) and the NYSE (b). Interpolating line (red), log-log of the PSD (blue).

Figure 6

Multifractal analysis. The scaling exponents for the brown noise process are a linear function of the moments, while the exponents for the NYSE show a departure from linearity.

Figure 7

Multifractal spectrum. Brown noise appears to be a monofractal signal characterised by a cluster of scaling exponents around 0.48 and a support between [0.38 0.55]. The NYSE appears to be a multifractal signal characterised by a much wider range of scaling exponents between [0.19 0.59] around its peak of 0.46.

Figure 8

Time series of daily detrended NYSE data and output of $SampEn$, standard deviation, and mean computed on sub-intervals with endpoints as change points. For $SampEn$, embedded dimension was 1 and r was set as 20% of the standard deviation of tested data. In this case $corr\left(SampEn\right(\mathrm{NYSE}),Mean(\mathrm{NYSE}\left)\right)=0.4$ where Spearman’s Rho type correlation was utilised. Notably, entropy increases when there is a negative change in the mean, while changes in volatility do not appear to have a significant impact on entropy.

Figure 9

Time series data from the detrended weekly downsampled NYSE data and the results of $SampEn$, standard deviation, and mean calculations performed on sub-intervals defined by change points. For $SampEn$, embedded dimension was 1 and r was set as 20% of the standard deviation of tested data. In this case $corr\left(SampEn\right(\mathrm{NYSE}),Mean(\mathrm{NYSE}\left)\right)=0.8$ where Spearman’s Rho rank correlation was utilised. Notably, entropy increases when there is a negative change in mean, while changes in volatility do not appear to have a significant impact on entropy.

Figure 10

Spearman rank correlation between SampEn and mean (blue), and SampEn and volatility (red), calculated over 1136 constituents of the NYSE. Note the negative outliers in the mean. As shown, there is a positive correlation between entropy and changes in the mean, whereas this is not the case for changes in volatility.