Measuring Intra-Day Volatility Using First Crossing Time

Here I show that a stock price path crosses either threshold of a double barrier at L percent above or below the initial stock price with expected first crossing time Eqn [1]and root-mean-square variance of crossing time : Eqn [2]given normally distributed stock price returns of volatility and zero drift. The values are obtained by solving the Backward Kolmogorov Equation with particular boundary conditions in each case. This solution can be used to define an estimator of intra-day volatility by observing the average time a fixed percentage up or down re-hedge would occur given a set of continuous tick data of the security: Eqn [3]A note is made on why this expectation value is well defined while introductory textbooks note the expected (average) first crossing time to a single barrier for a normal distribution is not defined