American-type basket option pricing: a simple two-dimensional partial differential equation

Abstract

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.

Keywords:

• Black & Scholes
• Basket options
• Pricing and hedging
• Comonotonicity
• Partial differential equations
• Finite difference method
• Least-Squares Monte-Carlo

Acknowledgments

We are very grateful to Jan Dhaene, Michel Vellekoop and to conference and seminar participants at the University of Amsterdam and KU Leuven for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† we use the term ‘artificial’ to emphasize that the market described by the SIE's (4(4) $\begin{array}{rl}{S}_i^c\left(t\right)& =S_i\left(0\right)+{\int }_0^t{\mu }_i{S}_i^c\left(s\right)\mathrm{d}s+{\int }_0^t{\sigma }_i{S}_i^c\left(s\right)\mathrm{d}B\left(s\right),\mathrm{for}t>0\\ & \mathrm{and}i=1,2,\dots ,n,\end{array}$(4) ) is a fictitious market and does not intend to describe the real market situation.

† The accuracy of the lower bound when approximating prices for European-type basket derivatives using risk-neutral valuation is already studied in Deelstra et al. (2004), Hainaut and Deelstra (2014) and Linders (2013).

† We thank the referee for pointing out the need of a more detailed investigation.

Additional information

Funding

Daniël Linders acknowledges the financial support of the KU Leuven Research Fund (GOA/13/003). Hamza Hanbali acknowledges the financial support of the AG Chair in Health Insurance at KU Leuven.