Properties of Optimal Smooth Functions in Additive Models for Hedging Multivariate Derivatives

In this paper, we consider an optimal hedging problem for multivariate derivative based on the additive sum of smooth functions on individual assets that minimize the mean square error (or the variance with zero expected value) from the derivative payoff. By applying the necessary and sufficient condition with suitable discretization, we derive a set of linear equations to construct optimal smooth functions, where we show that the computations involving conditional expectations for the multivariate derivatives may be reduced to those of unconditional expectations, and thus, the total procedure can be executed efficiently. We investigate the theoretical properties for the optimal smooth functions and clarify the following three facts: (i) the value of each individual option takes an optimal trajectory to minimize the mean square hedging error under the risk neutral probability measure, (ii) optimal smooth functions for the put option may be constructed using those for the call option (and vice versa), and (iii) delta in the replicating portfolio may be computed efficiently. Numerical experiments are included to show the effectiveness of our proposed methodology.