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Traders are often faced with large block orders in markets with limitedliquidity and varying volatility. Executing the entire order at once usuallyincurs a large trading cost because of this limited liquidity. In order tominimize this cost traders split up large orders over time. Varying volatilityhowever implies that they now take on price risk, as the underlying assets'prices can move against the traders over the execution period. This executionproblem therefore requires a careful balancing between trading slow to reduceliquidity cost and trading fast to reduce the volatility cost. R. Almgrensolved this problem for a market with one asset and stochastic liquidity andvolatility parameters, using a mean-variance framework. This leads to anonlinear PDE that needs to be solved numerically. We propose a differentapproach using (quasi-)Monte Carlo which can handle any number of assets.Furthermore, our method can be run in real-time and allows the trader to changethe parameters of the underlying stochastic processes on-the-fly.

A Monte Carlo method for optimal portfolio executions