Abstract
In this article, we consider the problem of equilibrium price formation in an incomplete securities market consisting of one major financial firm and a large number of minor firms. They carry out continuous trading via the securities exchange to minimize their cost while facing idiosyncratic and common noises as well as stochastic order flows from their individual clients. The equilibrium price process that balances demand and supply of the securities, including the functional form of the price impact for the major firm, is derived endogenously both in the market of finite population size and in the corresponding mean field limit.
Highlights
In the traditional setups for financial derivatives and portfolio theories, a security price process is given exogenously as a part of the model inputs
Carmona and Delarue [8, 9] developed a probabilistic approach to the mean field games and mean-field type control problems based on a forward-backward stochastic differential equation (FBSDE) of McKean-Vlasov type
We further developed the model studied in the two preceding works [27, 28] by including a major agent
Summary
In the traditional setups for financial derivatives and portfolio theories, a security price process is given exogenously as a part of the model inputs. See [23] for recent generalization in the linear-quadratic system, and [7, 44] which studies the master equation for the mean field games with a major agent These developments of the MFG theory have been successfully applied to various problems regarding in particular, the energy and financial markets which naturally involve a large number of agents with similar preferences. The equilibrium price process that balances demand and supply of the securities, including the functional form of the price impact for the major agent, is derived endogenously both in the market of finite population size and in the corresponding mean field limit. A general verification theorem for the optimization problem with respect to the controlled-FBSDEs is provided in Appendix
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