AdaptArrival

AdaptArrival - Adaptive Arrival Price Robert Almgren and...

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Unformatted text preview: Adaptive Arrival Price Robert Almgren * and Julian Lorenz ** April 27, 2006 Abstract Electronic trading of equities and other securities makes heavy use of “arrival price” algorithms, that determine optimal trade sched- ules by balancing the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, mean-variance optimal strategies are static: they do not modify the execution speed in response to price motions observed dur- ing trading. We show that with a more realistic formulation of the mean-variance tradeoff, and even with no momentum or mean re- version in the price process, substantial improvements are possible for adaptive strategies that spend trading gains to reduce risk, by accelerating execution when the price moves in the trader’s favor. The improvement is larger for large initial positions. * Electronic Trading Services, Banc of America Securities LLC, New York; [email protected] ** Institute of Theoretical Computer Science, ETH Zürich; [email protected] Partially supported by UBS AG. 1 Almgren/Lorenz: Adaptive Arrival Price April 27, 2006 2 Contents 1 Introduction 3 1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Trading in practice . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Other adaptive strategies . . . . . . . . . . . . . . . . . . . . . . 7 2 Market Model 8 2.1 Static trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Small-portfolio limit . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Portfolio comparison . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Single Update 13 3.1 Mean and variance . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Continuous Response 18 4.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Discussion and Conclusions 20 A Detailed formulas 22 A.1 Means and variances . . . . . . . . . . . . . . . . . . . . . . . . 22 A.2 Full distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Almgren/Lorenz: Adaptive Arrival Price April 27, 2006 3 1 Introduction Algorithmic trading represents a large and growing fraction of total order flow, especially in equity markets. When the size of a requested buy or sell order is larger than the market can immediately supply or absorb, then the order must be worked across some period of time, exposing the trader to price volatility. The algorithm attempts to achieve an average execution price whose probability distribution is suited to the client’s preferences. This paper proposes a way to dramatically improve this distribution....
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AdaptArrival - Adaptive Arrival Price Robert Almgren and...

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