The conventional formal tool to detect effects of the financial persistence is in terms of the Hurst exponent. A typical corresponding result is that its value comes out close to 0.5, as characteristic for geometric Brownian motion, with at most small departures from this value in either direction depending on the market and on the time scales involved. We study the high frequency price changes on the American and on the German stock markets. For both corresponding indices, the Dow Jones and the DAX respectively, the Hurst exponent analysis results in values close to 0.5. However, by decomposing the market dynamics into pairs of steps such that an elementary move up (down) is followed by another move up (down) and explicitely counting the resulting conditional probabilities we find values typically close to 60%. This effect of persistence is particularly visible on the short time scales ranging from 1 up to 2 minutes, decreasing gradually to 50% and even significantly below this value on the larger time scales. This indicates a subtle nature of the financial persistence whose characteristics escape detection within the conventional Hurst exponent formalism.