Hedging Basket Options by Using a Subset of Underlying Assets

This paper proposes two-step static hedging strategies for European basket options by using only plain-vanilla options on a subset of underlying assets. The basic idea is stimulated from a static super-hedging strategy dependent on the whole basket. However, it would be too complicated to handle when there is a large number of assets in the basket. It becomes even worse when some of the underlying assets are illiquid or not available for trading. Meanwhile, this strategy completely neglects the correlation structure of the basket which has indeed a great effect on the basket option's price. To solve these problems, Principal Components Analysis is used to figure out the subset of dominant assets through a careful study on the modified covariance of the basket. On this basis, the optimal strikes of those significant assets' plain-vanilla options are obtained in the second step via optimization. The optimality criterion depends on the risk attitude of hedgers and is defined by a certain risk measure, e.g., super-replication, minimum expected shortfall given a constraint on the hedging cost. Through analyzing a numerical example, it is concluded that this static hedging portfolio captures a trade-off between reduced hedging costs and overall super-replication. Even without considering transaction costs, hedging by using only a subset of underlying assets performs well: only a reasonable small hedging error arises when investing the capital required by the super-hedging portfolio which is composed of plain-vanilla options on all underlying assets and hence is difficult to implement or even not available in the market.