Fractality of profit landscapes and validation of time series models for stock prices

TL;DR

This paper investigates the fractal nature of profit landscapes generated by a simple trading strategy on real and artificial stock prices, revealing that fractality correlates with fat-tailed return distributions and can validate time series models.

Contribution

It demonstrates that the fractal scaling exponent of profit landscapes can serve as a new measure to validate the realism of stock price models.

Findings

Local maxima in profit landscapes follow a power-law distribution.

The scaling exponent differs across different time series.

Fat-tailed return distributions are linked to the fractality measure.

Abstract

We apply a simple trading strategy for various time series of real and artificial stock prices to understand the origin of fractality observed in the resulting profit landscapes. The strategy contains only two parameters $p$ and $q$, and the sell (buy) decision is made when the log return is larger (smaller) than $p$ ($-q$). We discretize the unit square $(p, q) \in [0, 1] \times [0, 1]$ into the $N \times N$ square grid and the profit $\Pi (p, q)$ is calculated at the center of each cell. We confirm the previous finding that local maxima in profit landscapes are scattered in a fractal-like fashion: The number M of local maxima follows the power-law form $M \sim N^{a}$, but the scaling exponent $a$ is found to differ for different time series. From comparisons of real and artificial stock prices, we find that the fat-tailed return distribution is closely related to the exponent $a \approx 1.6$ observed for real stock markets. We suggest that the fractality of profit landscape characterized by $a \approx 1.6$ can be a useful measure to validate time series model for stock prices.