Path Dependent Volatility

When a trader sets up an initial hedge for an option they usually come up with a volatility to plug into the Black-Scholes/Merton formula for delta. As time goes by they rehedge using the formula for gamma. If stock prices were geometric Brownian motion generated by a computer then the same volatility could be used over the life of the option. Of course they are not, so traders might observe underlying stock prices over time and update their volatility guess based on that. In this case the volatility is path dependent.

The B-S/M model assumes stock prices satisfy dS_t/S_t = \mu\,dt + \sigma\,dB_t where \mu and \sigma are constant and (B_t)_{t\ge0} is standard Brownian motion. The Ito calculus shows (dS_t/S_t)^2 = \sigma^2\,dt so a natural extension to path dependent volatility might be dS_t/S_t = \mu\,dt + \Sigma_t\,dB_t with {\Sigma_t^2 = (1/t)\int_0^t (dS/S)^2}. The volatility \Sigma is the average of the realized volatility. Unfortunately, this is not a good model since it implies \Sigma is constant. Since t\Sigma_t^2 = \int_0^t (dS/S)^2 = \int_0^t \Sigma_s^2\,ds we have d(t\Sigma_t^2) = \Sigma_t^2. By the Ito calculus we also have d(t\Sigma_t^2) = t\,d\Sigma^2_t + \Sigma_t^2\,dt so 0 = t\,d\Sigma_t^2 and \Sigma_t must be constant.

The average realized volatility assigns equal weight to each observation. If K is a function on the positive real numbers then we can consider \Sigma^2_t = \int_0^t K(t - s)(d\Sigma_s/\Sigma_s)^2\,ds It makes sense to assign more weight to recent observations so we can choose K to be a decreasing function, e.g., K(t) = \lambda\exp(-\lambda t), t > 0, for some positive \lambda\in\boldsymbol{R}. Note \int_0^\infty K(t)\,dt = 1. This can be generalized to integrals of the form \int_0^t K(t - s)\phi(d\Sigma_s/\Sigma_s)\,ds for some function \phi. Note this is scale-invariant in \Sigma_t. We may want to normalize K by dividing by \int_0^t K(t - s)\,ds.

Guyon and Lekeufack define {R_{j,t} = \int_0^t K(t - s)(d\Sigma_s/\Sigma_s)^j\,ds} for j = 1, 2. Their model for fitting volatility is \Sigma_t = \beta_0 + \beta_1 R_{1,t} + \beta_2 \sqrt{R_{2,t}}, \qquad \beta_0 > 0, \beta_1 < 0, \beta_2\in (0,1).

The paper is not clear on how data are used to produce the R_j. VIX data are 30-day vols and implied volatility can be obtained from option prices, but at what maturities?

A fundamental flaw in the paper is the authors seeming to be unaware that any model that fits European options automatically fits the VIX.

“Beyond the ability to produce desired spot-vol dynamics and capture spot-vol historical patterns, an important criterion to assess the quality of a PDV model should be its hedging performance on backtests, a task we leave for future work.”