A Finite Agent Equilibrium in an Incomplete Market and its Strong Convergence to the Mean-Field Limit

Abstract

We investigate the problem of equilibrium price formation in an incomplete securities market. Each financial firm (agent) tries to minimize its cost via continuous-time trading with a securities exchange while facing the systemic and idiosyncratic noises as well as the stochastic order-flows from its over-the-counter clients. We have shown, in the accompanying paper (Fujii & Takahashi) [19], that the solution to a certain forward backward stochastic differential equation of conditional McKean-Vlasov type gives a good approximate of the equilibrium price which clears the market in the large population limit. In this work, we prove the existence of a unique market clearing equilibrium among the heterogeneous agents of finite population size. We show the strong convergence to the corresponding mean-field limit given in [19] under suitable conditions. In particular, we provide the stability relation between the market clearing price for the heterogeneous agents and that for the homogeneous mean-field limit.

Equilibrium Price Formation with a Major Player and its Mean Field Limit

“…Unfortunately, this looks impossible in the market consisting of finite number of agents since, in general, the market-clearing price does not solely adapted to F 0 but is dependent on the idiosyncratic shocks, too. As already mentioned in [28,29], we shall observe that this ideal situation is actually realized in the large population limit. There, we can restrict the admissible strategy of the ith minor agent to A i = H 2 (F i ; R n ), and that of the major agent to…”

“…Since the existence of the common noise makes it impossible to use the fixedpoint technique, we resorted to the continuation method developed by Yong [53] and Peng & Wu [51] to solve the conditional McKean-Vlasov FBSDEs directly under the appropriate monotone conditions. In the accompanying work [29], we proved the strong convergence of the finite agent equilibrium to the corresponding mean field limit given in [28]. Note that, if we only want a short-term solution, the monotone conditions are unnecessary.…”

“…In the presence of common shocks, the price process inevitably becomes stochastic. Such a problem has been investigated in our two preceding papers [28,29], where the former treats the mean-field limit and the latter proves the strong convergence to the mean-field limit from the corresponding equilibrium of finite population. The new twist in the current paper is the presence of one major agent, a huge financial firm, who knows that her trading volume has a significant market share.…”

“…In the current paper, we extend the setup used in our two preceding works [28,29] by including a major agent. As long as we know, this is the first attempt to solve the problem of equilibrium price formation with a major agent under the market clearing condition.…”

“…As long as we know, this is the first attempt to solve the problem of equilibrium price formation with a major agent under the market clearing condition. For given order flow from the major agent, a properly functioning market is expected to produce an equilibrium price process as in [29]. Through this function of the market, the equilibrium price process of the securities becomes dependent on the trading strategy of the major agent.…”