A system of quadratic BSDEs arising in a price impact model

Abstract

We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand. We characterize these prices in terms of a system of BSDEs with quadratic growth. We show that this system admits a unique solution for every bounded demand if and only if the market maker's risk-aversion is sufficiently small. The uniqueness is established in the natural class of solutions, without any additional norm restrictions. To the best of our knowledge, this is the first study that proves such (global) uniqueness result for a system of fully coupled quadratic BSDEs.

Liquidity in Competitive Dealer Markets

“…This adjustment is consistent with the small-risk aversion limit obtained for the model of [14] in [22,21]. However, whereas the price impact is permanent in these models without end-users, it becomes transient in our model.…”

“…So far, the client demand was assumed to be given exogenously as in [18,14,17,22]. We now consider how to endogenize this demand for clients that behave optimally.…”

“…For liquidation problems, a first step in this direction is undertaken in [7], where dealers strategically quote bid-ask spreads for randomly arriving clients with exogenous demand curves, and a strategic client who needs to liquidate a large position. 3 Put differently, the dealers in [14,21,22] correspond to "pure internalisers" in the terminology of [6], whereas our dealers employ both internalisation and externalisation in an optimal manner. explicit but rather intricate dynamics can be better understood for highly-liquid end-user markets.…”

A stability approach for solving multidimensional quadratic BSDEs

“…Let us remark that, in addition to its intrinsic mathematical interest, this question is important due to many applications of such equations. We can mention for example following applications: nonzero-sum risk-sensitive stochastic differential games in [EKH03,HT16], financial market equilibrium problems for several interacting agents in [ET15,FDR11,Fre14,BLDR15], financial price-impact models in [KP16b,KP16a], principal agent contracting problems with competitive interacting agents in [EP16], stochastic equilibria problems in incomplete financial markets [KXŽ15,XŽ16] or existence of martingales on curved spaces with a prescribed terminal condition [Dar95]. Let us note that moving from the scalar framework to the multidimensional one is quite challenging since tools usually used when d = 1, like monotone convergence or Girsanov transform, can no longer be used when d > 1.…”

“…First of all, a quite general result was obtain by Tevzadze in [Tev08], when the bounded terminal condition is small enough, by using a fixed-point argument and the theory of BMO martingales. Some generalizations with somewhat more general terminal conditions are considered in [Fre14,KP16a]. In [CN15], Cheridito and Nam treat some quadratic BSDEs with very specific generators.…”