European basket option pricing by maximizing over a subset of lower bounds

Abstract

In this paper we derive a lower bound for the price of a European basket option under the Black and Scholes framework. The bound is based on the conditioning method of [L.C.G. Rogers and Z. Shi, The value of an Asian option, J. of Appl. Prob. 32  (1995), 1077–1088] and [M. Curran, Valuing Asian and portfolio options by conditioning on the geometric mean, Management Science 40(12) (1994), 1705–1711]. We adapt the method of [K.W. Chen and Y.D. Lyuu, Accurate pricing formulas for Asian options, Applied Mathematics and Computation 188 (2007), 1711–1724] applied for Asian options to the case of the European basket options. Using an improved Monte Carlo method based on the antithetic variates, we benchmark our results. We also calculate the lower bound for various parameters involved, compare these results with those obtained by the improved Monte Carlo simulations and conclude that the proposed bound is indeed optimal.

Keywords: European basket option, Monte Carlo simulations, variance reduction, anti-thetic variables, conditioning

Quaestiones Mathematicae 35(2012), 507–520