Non-parametric estimation of local variance surfaces
(English)Manuscript (preprint) (Other academic)
In this paper we develop a general optimization based framework for estimation of the option implied local variance surface. Given a specific level of consistency with observed market prices there exist an infinite number of possible surfaces. Instead of assuming shape constraints for the surface, as in many traditional methods, we seek the solution in the subset of realistic surfaces. We select local volatilities as variables in the optimization problem since it makes it easy to ensure absence of arbitrage, and realistic local volatilities imply realistic risk-neutral density- (RND), implied volatility- and price surfaces. The objective function combines a measure of consistency with market prices, and a weighted integral of the squared second derivatives of local volatility in the strike and the time-to-maturity direction. Derivatives prices in the optimization model are calculated efficiently with a finite difference scheme on a non-uniform grid. The framework has previously been successfully applied to the estimation of RND surfaces. Compared to when modeling the RND, it is for local volatility much easier to choose the parameters in the model. Modeling the RND produces a convex optimization problem which is not the case when modeling local volatility, but empirical tests indicate that the solution does not get stuck in local optima. We show that our method produces local volatility surfaces with very high quality and which are consistent with observed option quotes. Thus, unlike many methods described in the literature, our method does not produce a local volatility surface with irregular shape and many spikes or a non-smooth and multimodal RND for input data with a lot of noise.
Local volatility surface; Non-parametric estimation; Optimization; No-arbitrage conditions
Economics and Business Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:liu:diva-94358OAI: oai:DiVA.org:liu-94358DiVA: diva2:632463