Volatility, persistence, and survival in financial markets

We study the temporal fluctuations in time-dependent stock prices (both individual and composite) as a stochastic phenomenon using general techniques and methods of nonequilibrium statistical mechanics. In particular, we analyze stock price fluctuations as a non-Markovian stochastic process using the first-passage statistical concepts of persistence and survival. We report the results of empirical measurements of the normalized qth-order correlation functions f(q)(t), survival probability S(t), and persistence probability P(t) for several stock market dynamical sets. We analyze both minute-to-minute and higher-frequency stock market recordings (i.e., with the sampling time delta t of the order of days). We find that the fluctuating stock price is multifractal and the choice of delta t has no effect on the qualitative multifractal behavior displayed by the 1/q dependence of the generalized Hurst exponent H-q associated with the power-law evolution of the correlation function f(q)(t)similar to t(q)(H). The probability S(t) of the stock price remaining above the average up to time t is very sensitive to the total measurement time t(m) and the sampling time. The probability P(t) of the stock not returning to the initial value within an interval t has a universal power-law behavior P(t)similar to t(-theta), with a persistence exponent theta close to 0.5 that agrees with the prediction theta=1-H-2. The empirical financial stocks also present an interesting feature found in turbulent fluids, the extended self-similarity.