Arbitrage-free estimation of the risk-neutral density from the implied volatility smile

Abstract

All methods for estimating the risk-neutral density from the volatility smile boil down to the completion of the implied volatility function by interpolating between available strike prices and extrapolating outside their range. In this paper we focus on the extrapolation and develop a new method, which is, under weak constraints, consistent with the absence of arbitrage. The method does not depend on a particular interpolation scheme and is therefore universally applicable. The implementation involves only straightforward numerical procedures. In an empirical study we apply the method to options on the German stock index DAX. The method turns out to be robust, accurate, and fast. Compared with the methods of Shimko (1993) and Bliss and Panigirtzoglou (2002), it tends to be superior.