April 25, 2024
Thanks to Bill Goff, Ioanis Karatzis, and Jesper Andreasen for giving feedback that helped improve the exposition, hopefully.
Consider a stochastic volatility model of stock price (S_t)_{t\ge0} satisfying dS_t/S_t = r\,dt + \Sigma_t\,dB_t where r is constant, B_t is standard Brownian motion, and (\Sigma_t)_{t\ge0} is an Ito process.
A first guess at path-dependent volatility \Sigma_t might be \Sigma^2_t = (1/t)\int_0^t (dS_s/S_s)^2 = (1/t)\int_0^t \Sigma^2_s\,ds, the average realized variance.
Exercise. Show \Sigma_t^2 is constant.
Hint: Compute d(t\Sigma^2) two ways.
Clearly d(t\Sigma^2) = \Sigma^2\,dt. We also have d(t\Sigma^2) = t\,d\Sigma^2 + \Sigma^2\,dt so d\Sigma^2 = 0.
Consider the discrete time version where \Sigma^2_n = 1/(t_n - t_0)\sum_{0\le j < n} \Sigma^2_j (t_{j+1} - t_j).
Since \Sigma^2_1 = 1/(t_1 - t_0)\Sigma^2_0(t_1 - t_0) we have \Sigma_1 = \Sigma_0. Rinse and repeat until you stop laughing.