Transient Linear Price Impact and Fredholm Integral Equations

Mathematical Finance, Forthcoming

30 Pages Posted: 5 Jan 2010 Last revised: 3 May 2011

See all articles by Jim Gatheral

Jim Gatheral

CUNY Baruch College

Alexander Schied

University of Mannheim

Alla Slynko

Technische Universität München (TUM)

Multiple version iconThere are 2 versions of this paper

Date Written: May 30, 2010

Abstract

We consider the linear-impact case in the continuous-time market impact model with transient price impact proposed by Gatheral (2008). In this model, the absence of price manipulation in the sense of Huberman and Stanzl (2004) can easily be characterized by means of Bochner's theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure-valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko (2009) on the nonexistence of transaction-triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.

Keywords: Transient price impact, market impact model, optimal order execution, price manipulation, transaction-triggered price manipulation, Fredholm integral equation

Suggested Citation

Gatheral, Jim and Schied, Alexander and Slynko, Alla, Transient Linear Price Impact and Fredholm Integral Equations (May 30, 2010). Mathematical Finance, Forthcoming. Available at SSRN: https://ssrn.com/abstract=1531466

Jim Gatheral

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Alexander Schied (Contact Author)

University of Mannheim ( email )

Department of Mathematics
A 5, 6
Mannheim, 68131
Germany
+49-621-181-2513 (Phone)

HOME PAGE: http://www.alexschied.de/

Alla Slynko

Technische Universität München (TUM) ( email )

Arcisstrasse 21
Munich, 80333
Germany

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