  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.
