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

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 Pi (-q). We discretize the unit square (p, q) is an element of [0, 1] x [0, 1] into the N x N square grid and the profit.(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 similar to 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 approximate to 1.6 observed for real stock markets. We suggest that the fractality of profit landscape characterized by a approximate to 1.6 can be a useful measure to validate time series model for stock prices.