Tesi etd-01232018-132107 |
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Tipo di tesi
Tesi di laurea magistrale
Autore
MINUTILLO MENGA, LUCA
URN
etd-01232018-132107
Titolo
Stochastic Control and Mean Field Games with an application to Optimal Trading
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Campi, Luciano
correlatore Prof. Lillo, Fabrizio
controrelatore Prof. Romito, Marco
correlatore Prof. Lillo, Fabrizio
controrelatore Prof. Romito, Marco
Parole chiave
- stochastic control
- stochastic optimal control
- optimal control
- stochastic differential games
- stochastic games
- differential games
- game theory
- mean field games
- Nash equilibrium
- best response
- optimal trading
- optimal liquidation
- risk aversion
- utility functions
- controllo stocastico
- controllo ottimo stocastico
- controllo ottimo
- giochi differenziali stocastici
- giochi stocastici
- giochi differenziali
- teoria dei giochi
- giochi a campo medio
- equilibrio di Nash
- miglior risposta
- trading ottimo
- liquidazione ottima
- avversione al rischio
- funzioni di utilità
Data inizio appello
09/02/2018
Consultabilità
Completa
Riassunto
Cardaliaguet and Lehalle (in their paper "Mean Field Game of Controls and An Application To Trade Crowding", 2016) formulate the problem of optimal trading within a mean field game: a set of investors (viewed as players) have to buy or sell in a given time interval some shares of the same stock, whose public price is influenced by their average trading speed.
They consider a simple model for the permanent market impact and prove that, under suitable conditions, this game admits a unique Nash equilibrium. Moreover, they explicitly describe it under the additional assumption that all the investors have the same risk aversion.
Our main purpose is to extend these results to a more general model for the market impact, from which many other models widely used in the literature can be derived as special cases.
More precisely, we prove the existence and uniqueness of the Nash equilibrium according to this general model; then, we address the case of the same risk aversion for (almost) all the investors, where our general framework lead to equations for the Nash Equilibrium analogous to those formulated by Cartaliaguet and Lehalle. We perform also a few numerical experiments. For simplicity, we focus on two particular cases of our general model: the one used by Cardaliaguet and Lehalle and another one, by Obizhaeva and Wang.
Finally, we reformulate the whole problem by expressing the risk aversion of the players through suitable utility functions, and we analyze how this formulation changes the results previously presented.
They consider a simple model for the permanent market impact and prove that, under suitable conditions, this game admits a unique Nash equilibrium. Moreover, they explicitly describe it under the additional assumption that all the investors have the same risk aversion.
Our main purpose is to extend these results to a more general model for the market impact, from which many other models widely used in the literature can be derived as special cases.
More precisely, we prove the existence and uniqueness of the Nash equilibrium according to this general model; then, we address the case of the same risk aversion for (almost) all the investors, where our general framework lead to equations for the Nash Equilibrium analogous to those formulated by Cartaliaguet and Lehalle. We perform also a few numerical experiments. For simplicity, we focus on two particular cases of our general model: the one used by Cardaliaguet and Lehalle and another one, by Obizhaeva and Wang.
Finally, we reformulate the whole problem by expressing the risk aversion of the players through suitable utility functions, and we analyze how this formulation changes the results previously presented.
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