Volatility surfaces data is usually in a sparse, high dimensional space with limited data points. For example, if we have 500 underlying stocks, each single stock option has 20 maturities, and for each slice, we have 5 parameters (SVI curve). Then we have 50,000 parameters for each observation. If we calibrate the market five times a day, we have 250, 000 parameters per day. Given the fixed amount of information generated in the market per day, if we want to estimate too many parameters, each parameter's confidence is not reliable, and the parameters are not stable.
The reduction of dimension means that we have more magnificent information per parameter, which enhances the estimate of each parameter and stabilizes the parameters.
- Volatility parameters time series stabilization. We can use previous calibration parameters to improve this time calibration. Some methods, including using previous parameters as an initial guess for numerical optimizers, moving average to Bayesian update techniques. You assume the parameter follows some specific update rule and use maximum likelihood to estimate the hidden parameters and updating hyperparameters (like vol surf skew time-series volatility). Also, you can add priors to hyperparameters to make it a Bayesian Blender model.
- Volatility surface term structure. We assume total implied variance follows some term structure, if the normalized strike is constant, the implied volatility square is a function of maturity time. The function can be motivated by underlying process assumptions or simple geometric forms like the spline line model. Another angle is to assume the parameters of curves (like SVI parameters) follow specific functional form. It can come from underlying process assumptions or simple geometric forms like the spline line.
- Cross-sectional parameter stabilization. The linear factor model uses the equity factor model to explain the dynamics of the volatility surface, from 500 volatility surfaces to 10 volatility surfaces of factors, like volatility surface of SPX, volatility surface of momentum factor, vol surface of value factor, etc. We can also use PCA to build these cross-sectional factors. Peter Carr has one paper talking about the factor volatility surface model. Non-linear arbitrage-free model. Joint risk-neutral modeling of multiple derivatives, like joint modeling of VIX and SPX. Calibrating both VIX and SPX to provide more information to SPX. Jim Gatheral's Double Mean Reversion model DMR is an example.