Аннотация:
We consider a basket of stocks with both positive and negative weights, in the case where each asset has a smile, e.g., 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 [Risk, 2004]. We derive highly accurate analytic formulas for the prices and the implied volatilities of such baskets. These formulas are based on a basket Carr-Jarrow formula, a heat kernel expansion for the (multi-dimensional) density of of the asset at expiry and the Laplace approximation. The formulas are almost explicit, up to a minimization problem, which can be handled with simple Newton iteration, coupled with good initial guesses as derived in the paper. Moreover, we also provide asymptotic formulas for the greeks. Numerical experiments in the context of the CEV model indicate that the relative errors of these formulas are of order $10^{-4}$ (or better) for $T=\frac{1}{2}$, $10^{-3}$ for $T=2$, and $10^{-2}$ for $T=10$ years, for low, moderate and high dimensions. The computational time required to calculate 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. (Joint work with Peter Laurence.)