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

摘要

We apply a simple trading strategy for various time series of real andartificial stock prices to understand the origin of fractality observed in theresulting 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 $\Pi (p, q)$ iscalculated at the center of each cell. We confirm the previous finding thatlocal maxima in profit landscapes are scattered in a fractal-like fashion: Thenumber M of local maxima follows the power-law form $M \sim N^{a}$, but thescaling exponent $a$ is found to differ for different time series. Fromcomparisons of real and artificial stock prices, we find that the fat-tailedreturn distribution is closely related to the exponent $a \approx 1.6$ observedfor real stock markets. We suggest that the fractality of profit landscapecharacterized by $a \approx 1.6$ can be a useful measure to validate timeseries model for stock prices.