| Abstract: | We consider a basket of options with both positive and negative weights in the case where each asset has a smile, i.e., evolves according to its own local volatility and the driving Brownian motions are correlated. In the case of positive weights, the model has been considered in a previous work by Avellaneda, Boyer‐Olson, Busca, and Friz. We derive highly accurate analytic formulas for the prices and the implied volatilities of such baskets. The relative errors are of order 10?4 (or better) for T=½, 10?3 for T=2, and 10?2 for T=10 (years). The computational time required to implement these formulas is under two seconds even in the case of a basket on 100 assets. The combination of accuracy and speed makes these formulas potentially attractive both for calibration and for pricing. In comparison, simulation‐based techniques are prohibitively slow in achieving a comparable degree of accuracy. Thus the present work opens up a new paradigm in which asymptotics may arguably be used for pricing as well as for calibration. © 2014 Wiley Periodicals, Inc. |
