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Optimal execution strategies in limit order books with general shape
functions [ PDF]
(with Aurélien Alfonsi and Alexander Schied)
Following Obizhaeva and Wang (2005), we consider optimal execution
strategies for block market orders placed in a limit order
book (LOB). Our main contribution is to allow for a general
shape of the LOB defined via a given density function and thus
to include the case of nonlinear price impact of market
orders. In this setting, there are now two possibilities of
modeling the resilience of the LOB after a large market order:
the exponential recovery of the number of limit orders,
i.e., of the volume of the LOB, or the exponential recovery
of the bid-ask spread. We consider both situations and, in each case,
derive explicit optimal execution strategies in
discrete time. Applying our results to a block-shaped LOB,
we obtain a new closed-form representation for the optimal
strategy,which explicitly solves the recursive scheme
given in Obizhaeva and Wang (2005). We also provide some
evidence for the robustness of optimal strategies with respect to
the choice of the shape function and the
resilience-type.
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Constrained portfolio liquidation in a limit order book model [ PDF]
(with Aurélien Alfonsi and Alexander Schied)
We consider the problem of optimally placing market orders so as to minimize the expected
liquidity costs from buying a given amount of shares. The liquidity price impact of market orders is described by an extension of a
model for a limit order book with resilience that was proposed by Obizhaeva and Wang
(2005). We extend their model by allowing for a time-dependent resilience rate, arbitrary
trading times, and general equilibrium dynamics for the unaffected bid and ask prices. Our
main results solve the problem of minimizing the expected liquidity costs within a given
convex set of predictable trading strategies by reducing it to a deterministic optimization
problem. This deterministic problem is explicitly solved for the case in which the
convex set of strategies is defined via finitely many linear constraints. A
detailed study of optimal portfolio liquidation in markets with opening and closing call
auctions is provided as an illustration. We also obtain closed-form solutions for the
unconstrained portfolio liquidation problem in our time-inhomogeneous setting and thus
extend a result from our earlier paper "Optimal execution strategies in limit order books with general shape
functions".
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