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16
Optimal Portfolio Liquidation with Limit Orders
, 2011
"... This paper addresses the optimal scheduling of the liquidation of a portfolio using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss in [2], or only on the liquidityconsuming orders like Obizhaeva and Wang in [31], we link the optimal tradeschedule to the pric ..."
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Cited by 21 (1 self)
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This paper addresses the optimal scheduling of the liquidation of a portfolio using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss in [2], or only on the liquidityconsuming orders like Obizhaeva and Wang in [31], we link the optimal tradeschedule to the price of the limit orders that have to be sent to the limit order book to optimally liquidate a portfolio. Most practitioners address these two issues separately: they compute an optimal trading curve and they then send orders to the markets to try to follow it. The results obtained here solve simultaneously the two problems. As in a previous paper that solved the “intraday market making problem ” [19], the interactions of limit orders with the market are modeled via a Poisson process pegged to a diffusive “fair price” and a HamiltonJacobiBellman equation is used to solve the tradeoff between execution risk and price risk. Backtests are finally carried out to exemplify the use of our results.
Optimal Trade Execution: A Mean–QuadraticVariation Approach
, 2009
"... We propose the use of a mean–quadraticvariation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motio ..."
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Cited by 8 (0 self)
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We propose the use of a mean–quadraticvariation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motion (GBM). We also derive the HJB PDE assuming that the trading horizon is small and that the underlying process can be approximated by Arithmetic Brownian Motion (ABM). The exact solution of the ABM formulation is in fact identical to the priceindependent approximate optimal control for the meanvariance objective function in [2]. The GBM mean–quadraticvariation optimal trading strategy is in general a function of the asset price. However, for short term trading horizons, the control determined under the ABM assumption is an excellent approximation.
Optimal execution and block trade pricing: a general framework
, 2014
"... Abstract In this article, we develop a general CARA framework to study optimal execution and to price block trades. We prove existence and regularity results for optimal liquidation strategies and we provide several differential characterizations. We also give two different proofs that the usual re ..."
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Cited by 8 (4 self)
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Abstract In this article, we develop a general CARA framework to study optimal execution and to price block trades. We prove existence and regularity results for optimal liquidation strategies and we provide several differential characterizations. We also give two different proofs that the usual restriction to deterministic liquidation strategies is optimal. In addition, we focus on the important topic of block trade pricing and we therefore give a price to financial (il)liquidity. In particular, we provide a closedform formula for the price a block trade when there is no time constraint to liquidate, and a differential characterization in the timeconstrained case. Numerical methods are eventually discussed.
Comparison between the mean variance optimal and the mean quadratic variation optimal trading strategies.
, 2011
"... Abstract We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). In particular, we compare the path dependent, timeconsistent meanquadraticvariation strategy with ..."
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Cited by 7 (0 self)
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Abstract We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). In particular, we compare the path dependent, timeconsistent meanquadraticvariation strategy with the pathindependent, timeinconsistent (precommitment) meanvariance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the meanquadraticvariation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the meanvariance strategy strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semiLagrangian method results in significantly better accuracy than standard axisaligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.
Continuous time mean variance asset allocation: a time consistent strategy. Working
, 2009
"... We develop a numerical scheme for determining the optimal asset allocation strategy for timeconsistent, continuous time, mean variance optimization. Any type of constraint can be applied to the investment policy. The optimal policies for timeconsistent and precommitment strategies are compared. W ..."
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Cited by 5 (2 self)
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We develop a numerical scheme for determining the optimal asset allocation strategy for timeconsistent, continuous time, mean variance optimization. Any type of constraint can be applied to the investment policy. The optimal policies for timeconsistent and precommitment strategies are compared. When realistic constraints are applied, the efficient frontiers for the precommitment and timeconsistent strategies are similar, but the optimal investment strategies are quite different.
Robust strategies for optimal order execution in the Almgren–Chriss framework’, Forthcoming in
 Applied Mathematical Finance . URL: http://ssrn.com/paper=1991097 Schöneborn
, 2011
"... ar ..."
Optimal Execution Problem with Market Impact
, 2009
"... We study the optimal execution problem in the market model in consideration of market impact. First we study the discretetime model and describe the value function with respect to the trader’s optimization problem. Then, by shortening the intervals of execution times, we derive the value function o ..."
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Cited by 3 (2 self)
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We study the optimal execution problem in the market model in consideration of market impact. First we study the discretetime model and describe the value function with respect to the trader’s optimization problem. Then, by shortening the intervals of execution times, we derive the value function of the continuoustime model and study some properties of them (continuity, semigroup property and the characterization as the viscosity solution of HJB.) We show that the properties of the continuoustime value function vary by the strength of market impact. Moreover we introduce some examples of this model, which tell us that the forms of the optimal execution strategies entirely change according to the amount of the security holdings. ∗
Regularized robust optimization: the optimal portfolio execution case
, 2013
"... An uncertainty set is a crucial component in robust optimization. Unfortunately, it is often unclear how to specify it precisely. Thus it is important to study sensitivity of the robust solution to variations in the uncertainty set, and to develop a method which improves stability of the robust solu ..."
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Cited by 1 (1 self)
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An uncertainty set is a crucial component in robust optimization. Unfortunately, it is often unclear how to specify it precisely. Thus it is important to study sensitivity of the robust solution to variations in the uncertainty set, and to develop a method which improves stability of the robust solution. In this paper, to address these issues, we focus on uncertainty in the price impact parameters in an optimal portfolio execution problem. We first illustrate that a small variation in the uncertainty set may result in a large change in the robust solution. We then propose a regularized robust optimization formulation which yields a solution with a better stability property than the classical robust solution. In this approach, the uncertainty set is regularized through a regularization constraint, defined by a linear matrix inequality using the Hessian of the objective function and a regularization parameter. The regularized The authors would like to thank anonymous referees whose comments have improved the presentation of this paper.
Studies on optimal trade execution
, 2015
"... This dissertation deals with the question of how to optimally execute orders for financial assets that are subject to transaction costs. We study the problem in a discrete–time model where the asset price processes of interest are subject to stochastic volatility and liquidity. First, we consider ..."
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This dissertation deals with the question of how to optimally execute orders for financial assets that are subject to transaction costs. We study the problem in a discrete–time model where the asset price processes of interest are subject to stochastic volatility and liquidity. First, we consider the case for the execution of a single asset. We find predictable strategies that minimize the expectation, mean–variance and expected exponential of the implementation cost. Second, we extend the single asset case to incorporate a dark pool where the orders can be crossed at the midprice depending on the liquidity available. The orders submitted to the dark pool face execution uncertainty and are assumed to be subject to adverse selection risk. We find strategies that minimize the expectation and the expected exponential of the implementation shortfall and show that one can incur less costs by also making use of the dark pool. Next chapter studies a multi asset setting in the presence of a dark pool. We