March 16th, 2021, 3:17 pm

Here's an example. Say you have a random variate X that evolves as a diffusion with stochastic volatility [$]\sigma_t[$]. The evolution equation is

[$] dX_t = \mu \, dt + \sigma_t \, dW_t[$].

The terminal value is [$]X_T[$], a random variable with a probability distribution, and say you want the (expected, time-0) variance of that distribution.

Removing the mean, and integrating the SDE, we have

[$] Y_T \equiv X_T - X_0 - \mu \, T = \int_0^T \sigma_t dW_t [$].

The variance we want is

(*) [$] E_0[Y_T^2] = E_0[(\int_0^T \sigma_t dW_t)^2] = E_0[\int_0^T \sigma_t^2 dt][$],

using Ito's isometry.

For example, if [$]dX_t = dS_t/S_t[$], where [$]S_t[$] is the S&P 500 Index price, and [$]\mu = r[$],

then (*) is (up to an annualization factor) -- the square of the VIX index for horizon T.

The *really* interesting thing, which takes a little more work, is that the far right-hand-side expression of (*), and so the VIX index, can be entirely computed from the t=0 prices of vanilla put and call options with option maturity T.