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In this work, we introduce a Monte Carlo method for the dynamic hedging ofgeneral European-type contingent claims in a multidimensional Brownianarbitrage-free market. Based on bounded variation martingale approximations forGaltchouk-Kunita-Watanabe decompositions, we propose a feasible andconstructive methodology which allows us to compute pure hedging strategiesw.r.t arbitrary square-integrable claims in incomplete markets. In particular,the methodology can be applied to quadratic hedging-type strategies for fullypath-dependent options with stochastic volatility and discontinuous payoffs. Weillustrate the method with numerical examples based on generalizedFollmer-Schweizer decompositions, locally-risk minimizing and mean-variancehedging strategies for vanilla and path-dependent options written on localvolatility and stochastic volatility models.

A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility