Modelling financial time series with $φ^{4}$ quantum field theory

Abstract

We use a $φ^{4}$ quantum field theory with inhomogeneous couplings and explicit symmetry-breaking to model an ensemble of financial time series from the S$\&$P 500 index. The continuum nature of the $φ^4$ theory avoids the inaccuracies that occur in Ising-based models which require a discretization of the time series. We demonstrate this using the example of the 2008 global financial crisis. The $φ^{4}$ quantum field theory is expressive enough to reproduce the higher-order statistics such as the market kurtosis, which can serve as an indicator of possible market shocks. Accurate reproduction of high kurtosis is absent in binarized models. Therefore Ising models, despite being widely employed in econophysics, are incapable of fully representing empirical financial data, a limitation not present in the generalization of the $φ^{4}$ scalar field theory. We then investigate the scaling properties of the $φ^{4}$ machine learning algorithm and extract exponents which govern the behavior of the learned couplings (or weights and biases in ML language) in relation to the number of stocks in the model. Finally, we use our model to forecast the price changes of the AAPL, MSFT, and NVDA stocks. We conclude by discussing how the $φ^{4}$ scalar field theory could be used to build investment strategies and the possible intuitions that the QFT operations of dimensional compactification and renormalization can provide for financial modelling.

Figures (6)

• Figure 1: The disordered $\phi^{4}$ scalar field theory in its representation as a complete graph where every distinct pair of fields is connected by a unique edge. The price change of each stock is mapped to a field $\phi_{i}$, and each edge $w_{ij}$ extracts correlations between stock $\phi_{i}$ and the remaining stocks $\lbrace \phi_{j,j \in \Lambda-i}\rbrace$. Inhomogeneous external fields $\lbrace a_{i} \rbrace$, which correspond to external news affecting the price change of a stock, break explicitly and locally the $Z_{2}$ symmetry, thus biasing the price change to a positive or negative value.
• Figure 2: Market mean (left) and market kurtosis (right), using a simple moving average of $250$ trading days, versus the trading year. The binarized mean value is normalized to reside between the minimum and maximum values of the original time series.
• Figure 3: Mean values of the weights (top) and biases (bottom) versus the lattice volume $V$ of the $\phi^{4}$ theory or, equivalently, the number of modelled stocks $V$ from the S$\&$P 500 index. The axes are logarithmic.
• Figure 4: NVDA returns versus the trading day, for the original time series, predictions from the $\phi^{4}$ model and a rescaled mean calculation using the simultaneous returns of AAPL and MSFT as predictors. The x axis corresponds to dates for October 2022.
• Figure 5: Forecasting of the AAPL returns for the upcoming trading day using the $\phi^{4}$ theory with historical AAPL returns as predictor. The x axis corresponds to dates for October 2024.