Optimal hedging of path-dependent basket options with additive models

In this paper, we consider problems of hedging path-dependent basket barrier options using individual options based on our previously proposed optimal hedging strategy in Yamada (2010-2013) for European options. The optimal hedging problem is formulated as follows: Find optimal smooth functions of individual options to minimize the mean square error from the payoff of down-and-out basket option. For solving the problem, we first express the necessary and sufficient condition in terms of linear equations regarding conditional expectations of basket barrier options. Based on the Brownian bridge decomposition and the Independence Lemma, we show that the computations involving conditional expectations of basket barrier options may be reduced to those of unconditional expectations. This computation involves multiple integrations in general, but is usually executed efficiently based on the standard Monte Carlo method by generating independent Gaussian random numbers. Finally, we demonstrate numerical experiments to illustrate our proposed methodology.