Smile Consistent Basket Skew

Abstract

This paper presents analytical approximations for the at-the-money (ATM) skew and convexity of an option on a basket of assets following correlated local volatility models, which become exact in the short-maturity limit. The results are extracted from the short maturity asymptotics of Avellaneda, Boyer-Olson, Busca and Friz (2002) for basket options.

The basket skew is expressed in terms of the ATM volatility levels and skews of the basket components as well as their correlation. The basket convexity depends also on the convexities of the basket components. We prove that the skew of a basket of correlated Black-Scholes assets is non-negative, for any correlation.

The result can be used as an approximation for the basket implied volatility at strikes around the ATM point and sufficiently small volatility or maturity. The analytical approximations are tested by comparing with numerical pricing of basket options with n=2,3 correlated assets, following both Black-Scholes and local volatility models.

Keywords: Option asymptotics, Basket options, Local volatility model

JEL Classification: C63, G12, G13