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The Mathematics of Adaptive Execution

Robert Almgren

In agency execution of large transactions, the modern trend is algorithms that adapt to time-varying market parameters such as liquidity and volatility. Clearly, the algorithm should trade more when instantaneous liquidity is higher, but the mathematical challenge is to combine that instantaneous response with a more strategic point of view that optimizes an overall combination of impact cost and volatility risk. We present a formulation using dynamic programming to derive a Hamilton-Jacobi-Bellman PDE for the optimal strategy, and show how solutions of this equation lead to optimal execution algorithms.

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