Abstracts

Basket options are contingent claims on a group of assets such as equities, commodities, currencies and other vanilla derivatives. From a modelling point of view, the framework to price and hedge these options ought to be multidimensional since baskets of 15 to 30 assets are frequently traded. Many pricing methods that seem to work well for single assets cannot be easily expanded to a multidimensional set-up, mainly due to computational difficulties.
In the paper, we present a general computational solution to the problem of multidimensional models which lack closed-form formulae or models that require burdensome numerical procedures, avoiding most of the strong assumptions made by recent techniques. We generalize the approach in [S. Borovkova, F. Permana and H.V. Weide. A closed form approach to valuing and hedging basket and spread options. JOD (2007) 14, 8-24] by employing the Hermite polynomial expansion that replaces the risk-neutral density implied by the model and matches exactly its first m moments. Our approach elaborates on some variants of the method in [A. Leccadito, P. Toscano and R. Tunaru. Hermite binomial trees: a novel technique for derivative pricing. IJTAF (2012) 15,1-36] to deal with baskets that may take on negative values. In particular, the moment matching is carried on three different return quantities and consequently three different methods are presented. The methods can even be applied to price and hedge multi-asset derivatives in situations when some assets follow one diffusion model and other assets follow different ones, with the only assumption being able to calculate the moments of the basket in closed form. We use, as exemplification, the shifted log-normal process with jumps in pricing basket options.
Furthermore, we propose a test to evaluate the performances of methods for pricing and hedging European-style contingent claims. Through this test, we show that no more than four moments of the underlying assets processes are needed in pricing and hedging options. Moreover, the test shows that one of the proposed methods returns the best price in 84\% of cases and has a pricing error smaller than 5\% in 95\% of the cases. On the hedging side, our methods tend to slightly sub-hedge but their average error is much smaller when compared to other methods. Consequently, the methods are shown to provide superior results not only with respect to pricing but also for hedging.