From Implied to Local Volatility Surface

Abstract

We describe a single parametric model for the entire volatility surface with interpolation and extrapolation technique generating a smooth and robust implied volatility surface without arbitrage in space and time. It is used for marking option prices on indices and single stocks as well as for computing analytically a proper local volatility with smooth risk-neutral density. Greeks and stress scenarios are calculated analytically in the parametric model without recalibration of the model parameters. We perform a simple expansion of the parametric model obtaining an analytic representation of its implied volatility surface along its cone of diffusion. In view of adding control to the generated volatility surface, we modify the model by adding three new parameters producing, in an independent way, a parallel shift, skew shift and curvature shift of that surface along its cone of diffusion. These parameters can be used manually to modify the entire shape of the volatility surface, and can also be used to generate analytically the new local volatility surface when computing the vegas of an option. Then, in view of defining the best possible volatility surface for non-liquid stocks where only few brokers quotes exist, we describe a method combining historical model parameters of the implied volatility surface together with parameters from other liquid stocks observed on the market.

Keywords: Implied Volatility Surface, Local Volatility Surface, Calibration, Single Parametric Model, Analytic Greeks, Analytic Stress Scenarios