History state formalism for time series with application to finance

Abstract

We present a method for analyzing general time series by employing the history state formalism of quantum mechanics. This formalism allows us to describe a complete evolution based on a single quantum state, the history state, which simultaneously includes -also as a quantum system- the reference clock. It naturally leads to the concept of system-time entanglement, with the ensuing entanglement entropy constituting a measure of the effective number of distinguishable states visited in the history. Through a quantum coherent state embedding of the time series data, it is then possible to associate a quantum history state to the series. The gaussian overlap between these coherent states provides thus a smooth measure of distinguishability between the series data. The eigenvalues of the corresponding overlap matrix determine in fact the entanglement spectrum and entropy of the history state, which provide a rigorous characterization of the evolution. As illustration, the formalism is applied to typical financial time-series data. Through the entanglement entropy and spectrum, different evolution regimes can be identified. Entanglement based volatility indicators are also derived, and compared with standard volatility measures.

Figures (9)

• Figure 1: Opening values of the SPY (in USD) as a function of the elapsed number of days $N$ along the period 17 March 2014 -- 30 December 2022 SPY.YF.
• Figure 2: Top panel: The von Neumann (solid lines) and quadratic Renyi (short dashed lines) entanglement entropies $E_1$ and $E_2$ respectively, Eqs. \ref{['EST2']}--\ref{['S2']}, of the history state associated to the SPY prices for the first $N$ days, $1\leq N\leq 2206$, starting at March 2014, for distinct values of the relative width $\sigma_r$. For reference the maximum attainable value $E_{\rm max}=\log_2 N$ (the same for both entropies, representing the $\sigma_r\rightarrow 0$ limit), is also depicted. Bottom: The corresponding effective number of distinct states visited in the same period, Eq. \ref{['NS']}, for the indicated entropies, as a function of $N$.
• Figure 3: Top panel: The entanglement entropy $E=E_1$ of the history state as a function of the number of elapsed days $N$ for $\sigma_r=1$ in the whole period, and the associated partial sums $\Sigma_{\ell}=\sum_{k=1}^{\ell}\lambda_k$ of the eigenvalues of the overlap matrix, as a function of $N$, for the indicated values of $\ell$, with $\Sigma_{T}=1$ the total sum. The vertical dashed line signals a regime change after which all partial sums tend to decrease, indicating majorization for increasing $N$ after this point. Bottom panel: The corresponding entanglement spectrum of the history state as a function of the number of elapsed days $N$. We depict the first 15 eigenvalues $\lambda_k$ of the overlap matrix O$_N$, (which are those of $\rho_S$ and $\rho_T$), with the elements \ref{['ov3']}. The inset shows in more detail the behavior of the lowest eigenvalues.
• Figure 4: Top: The entanglement entropy $E=E_1$ of the one-month history state for 120 months $m$ starting at March 17 2014, for $\sigma_r=1$, together with the associated partial sums $\Sigma_{\ell}=\sum_{i=1}^{\ell}\lambda_k$ of the entanglement spectrum for each month for $\ell=1,2,5,10$. The vertical dashed lines indicate months 73 and 40, where the monthly entropy reaches its maximum and minimum respectively, and accordingly, the monthly partial sums have their minimum and maximum. Bottom: The corresponding entanglement spectrum of the monthly history in the same period. We depict the first 10 eigenvalues $\lambda_k$ of the normalized overlap matrix for each month. The inset shows in more detail the behavior of the lowest eigenvalues around the region of highest monthly volatility ($68\alt m\alt 79$).
• Figure 5: Number of effective visited states $N_E$ for each month, in the same case of Fig. \ref{['fig4']}. The monthly VIX values for the same period are also depicted. There is a close qualitative agreement between both quantities.