Non-Parametric Estimation of Stable Local Volatility Surfaces
(English)Manuscript (preprint) (Other academic)
In this paper we develop a general optimization based framework for estimation of the option implied local volatility surface. We show that our method produces local volatility surfaces with very high quality and which are consistent with observed S&P 500 index 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 for input data which contains a lot of noise. Through a time series study we show that our optimization based framework produces squared local volatility surfaces that are stable over time. Given a specic level of consistency with observed market prices there exist an innite 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 squared 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- , 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 resulting optimization problem is non-convex, but extensive empirical tests indicate that the solution does not get stuck in local optima.
Local volatility surface; Non-parametric estimation; Optimization; No-arbitrage conditions; Principal Component Analysis
Economics and Business
IdentifiersURN: urn:nbn:se:liu:diva-117102OAI: oai:DiVA.org:liu-117102DiVA: diva2:805718