Abstract
We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
Highlights
In recent years technology innovation, deregulation policies and ubiquitous availability of Internet connections have determined the emergence of new forms of trading in the financial markets
In this paper we study a market consisting of one traded asset where a multitude of retail traders and a big trader operate
We begin the numerical study of the trading execution model presented in the previous Sections discussing the problem of the existence of solutions of the mean field game problem (5), (6), (1)-(4) of the form suggested in Propositions 3.1, 3.2
Summary
In recent years technology innovation, deregulation policies and ubiquitous availability of Internet connections have determined the emergence of new forms of trading in the financial markets. The instantaneous impact factor of the trading activity of the retail traders is assumed to be proportional to the expected value of their trading execution rate This last term is not present in the asset share price dynamic equation used in [3]. Their different behaviours determine different effects on the asset share price dynamics It follows that the optimal trading execution strategy of the big trader changes as a consequence of the fact that the retail traders are buy and hold investors or are short term investors.
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