An FBSDE approach to market impact games with stochastic parameters

Abstract

<p style='text-indent:20px;'>In this study, we have analyzed a market impact game between <i>n</i> risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium. </p>

A two-player portfolio tracking game

“…Therefore, we can conclude that ( α1 , α2 ) ∈ A 1 ×A 2 is a Nash equilibrium in the sense of Definition 2.2. Necessity: Finally, as shown in the Proof of Theorem 3.5 below (which does not use the necessity assertion of the present lemma) the pair of controls ( α1 , α2 ) ∈ A 1 × A 2 presented in (21) below satisfies the coupled forward backward SDE system in (15). Therefore, by uniqueness of the Nash equilibrium via Lemma 3.2 the assertion is indeed also necessary.…”

“…for two suitable square integrable martingales (15). We have to show that α1 minimizes α 1 → J 1 (α 1 ; α2 ) over A 1 , and, vice versa, that α2 minimizes α 2 → J 2 (α 2 ; α1 ) over A 2 .…”

“…5 In view of Lemma 3.4 we merely have to show that ( X 1 , X 2 , α1 , α2 ) with dynamics described in Theorem 3.5, Eq. ( 21), is a solution of the FBSDE system in (15) with some suitable square integrable martingales (M 1 t ) 0≤t<T and (M 2 t ) 0≤t<T . Uniqueness of the Nash equilibrium then follows together with Lemma 3.2.…”

“…Step 1: We start with computing the dynamics of the controls α1 and α2 in (21) and verify that they satisfy the dynamics of the FBSDE system in (15). To this end, it is convenient to rewrite w 1 , w 2 in (20), as well as ξ 1 in ( 22) and ξ 2 in (23) by introducing…”

“…Other recent work on both finite-player as well as infinite-player mean field price impact games with Almgren-Chriss type price impact include, e.g., Cardaliaguet and Lehalle [8], Huang et al [23], Casgrain and Jaimungal [12,13], Fu et al [21], Fu and Horst [19], Evangelista and Thamsten [18], and Drapeau et al [15], where finitely and infinitely many agents pursue optimal liquidation of their initial positions and interact through common aggregated permanent and temporary price impact. Price impact games of liquidating agents in a market model with transient price impact are analyzed, e.g., in Luo and Schied [24], Schied and Zhang [30], Schied et al [31], Strehle [34]; and very recently in Fu et al [20] and Neuman and Voß [27].…”