Mathematical Models of Price Impact and Optimal Portfolio Management in Illiquid Markets

Nikolay Andreev

Abstract The problem of optimal portfolio liquidation under transaction costs has been widely researched recently, producing several approaches to problem formulation and solving. Obtained results can be used for decision making during portfolio selection or automatic trading on high-frequency electronic markets. This work gives a review of modern studies in this field, comparing models and tracking their evolution. The paper also presents results of applying the most recent findings in this field to real MICEX shares with high-frequency data and gives an interpretation of the results.

Keywords Market liquidity • Optimal portfolio selection • Portfolio liquidation • Price impact

JEL Classification C61, G11

1 Introduction

With the development of electronic trading platforms, the importance of high – frequency trading has become obvious. This requires the need of automatic trading algorithms or decision-making systems to help portfolio managers in choosing the best portfolios in volatile high-frequency markets. Another actual problem in portfolio management field is optimal liquidation of a position under constrained liquidity during a predefined period of time.

Mathematical theory of dynamic portfolio management has received much attention since the pioneering work of Merton (1969), who obtained a closed-form solution for optimal strategy in continuous time for a portfolio of stocks where the market consisted of risk-free bank accounts and a stock with Bachelier-Samuelson dynamics of price. Optimal criterion had the following form:

N. Andreev (H)

Financial Engineering and Risk Management Laboratory, National Research University Higher School of Economics, Moscow, Russia e-mail: nandreev@hse. ru

© Springer International Publishing Switzerland 2015

A. K. Bera et al. (eds.), Financial Econometrics and Empirical Market

Microstructure, DOI 10.1007/978-3-319-09946-0_____ 1

(С,.*,.Г, 2 Argmax Е @ e~iliU (С,) Л. + B (W, Т)

where Ct is consumption rate, Xt, Yt—portfolio wealth in riskless asset and stocks respectively, Wt = Xt + Yt is total value of portfolio and U(C) = Щ-. у < 1. or log C—a constant relative risk-aversion (CRRA) utility function, B(Wt, t) is a func­tion, increasing with wealth. This criterion formulates optimality as maximization of consumption and portfolio value at the end of a period. Merton asserted that it is optimal to keep assets in constant proportion for the whole period, that is л, = W = const. This result is known as the Merton line due to the strategy’s linear representation in (Xt, Yt) plane.

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