Mean Field Games and Systemic Risk

Abstract

We propose a simple model of inter-bank borrowing and lending where the evolution of the logmonetary reserves of N banks is described by a system of diffusion processes coupled through their drifts in such a way that stability of the system depends on the rate of inter-bank borrowing and lending. Systemic risk is characterized by a large number of banks reaching a default threshold by a given time horizon. Our model incorporates a game feature where each bank controls its rate of borrowing/lending to a central bank. The optimization reflects the desire of each bank to borrow from the central bank when its monetary reserve falls below a critical level or lend if it rises above this critical level which is chosen here as the average monetary reserve. Borrowing from or lending to the central bank is also subject to a quadratic cost at a rate which can be fixed by the regulator. We solve explicitly for Nash equilibria with finitely many players, and we show that in this model the central bank acts as a clearing house, adding liquidity to the system without affecting its systemic risk. We also study the corresponding Mean Field Game in the limit of large number of banks in the presence of a common noise.

Mean field games: A toy model on an Erdös-Renyi graph.

“…Although this form of interaction may be of a rather restrictive form for modeling, it turns out to be easily tractable from the mathematical point of view. We refer to [7] for a specific instance of application when f and g are linear andμ N,i t is the empirical distribution over all the players (and is thus independent of i).…”

Mean Field Games with Singular Controls of Bounded Velocity

“…For comparison purposes, we present a singular control counterpart of the MFG originally formulated with regular controls by [17] for systemic risk. We will see that our solution structure is consistent with theirs, despite the differences in problem settings.…”

“…We will see that our solution structure is consistent with theirs, despite the differences in problem settings. The basic idea behind the interbank systemic risk model of [17] is as follows. (A similar model can also be found in [22].)…”

“…In most MFGs with regular controls, a linear-quadratic problem structure is necessary for the explicit solution (Bardi [1], Bardi and Priuli [2], Bensoussan et al [6], and [17]). Our paper provides explicit optimal control policies in a general singular control setting; it analyzes a particular MFG with closed-form solutions where the dynamics are linear and the cost structure is not necessarily quadratic (see Remark 1); it also illustrates in this MFG how to explore the particular problem structure to reduce MFGs with common noise to MFGs without common noise, by a simple conditioning trick.…”

“…Earlier examples include Carmona, Fouque, and Sun [17] and Garnier, Papanicolaou, and Yang [22] for systemic risks, Lasry, Lions, and Guéant [37] for the growth theory in economics, Jaimungal and Nourian [30] and Lachapelle, Lasry, Lehalle, and Lions [34] for high frequency trading, Bauso, Tembine, and Basar [3] and Guéant, Lasry, and Lions [25] for management of non-renewable resources, Caffarelli, Markowich, and Pietschmann [11] for price formation models, Gomes, Mohr, and Souza [23] and Guéant [24] for social network dynamics, and Manjrekar, Ramaswamy, and Shakkottai [39], Wiecek, Altman, and Ghosh [47], and Bayraktar, Budhiraja, and Cohen [4] for queuing systems.…”