Stochastic volatility of financial markets as the fluctuating rate of trading: an empirical study

We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random walk (CTRW) framework. The probability distribution of the stock price changes (log-returns) for a given number of trades $N$ is found to be approximately Gaussian. The probability distribution of $N$ for a given time interval $\Delta t$ is non-Poissonian and has an exponential tail for large $N$ and a sharp cutoff for small $N$. Combining these two distributions produces a nontrivial distribution of log-returns for a given time interval $\Delta t$, which has exponential tails and a Gaussian central part, in agreement with empirical observations.\\ Reference: physics/0608299.