Volatility, Persistence, and Survival in Financial Markets

Abstract

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 $q$-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 $f_q(t)\sim t^{H_q}$. 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, $P(t)\sim t^{-\theta}$, 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.