Sharp Upper and Lower Bounds for Basket Options

Abstract

Given a basket option on two or more assets in a one period static hedging setting we consider the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. We derive sharp upper bounds for general n-asset case and sharp lower bounds for 2-asset case, both in closed forms, of the price of the basket option. We also derive optimal super and subreplicating strategies involving portfolios of stock and options in the super-replicating case and stock, options and cash in the subreplicating case. In the case n=2 we give examples of discrete distributions attaining our bounds. Our upper bounds are illustrated both on simulated and on real Dow Jones (DJX) data. In the case of the DJX data, we show how our approach leads to an optimal choice of strikes in which to invest for each component stock. A short version of this paper appeared in Risk Magazine's February 2004 issue.