An FBSDE approach to market impact games with stochastic parameters
AAn FBSDE approach to market impact gameswith stochastic parameters
Samuel Drapeau ∗ Peng Luo † Alexander Schied ‡ Dewen Xiong § January 6, 2020
Abstract
We analyze a market impact game between n risk averse agents who compete for liquidity in a market impactmodel with permanent price impact and additional slippage. Most market parameters, including volatility anddrift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of afully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main resultprovides conditions under which this system of FBSDEs has indeed a unique solution, which in turn yields theunique Nash equilibrium. We furthermore obtain closed-form solutions in special situations and analyze themnumerically. Market impact games analyze situations in which several agents compete for liquidity in a market impact model ortry to exploit the price impact generated by competitors. In this paper, we follow Carlin et al. [6], Sch¨oneborn andSchied [23], Carmona and Yang [7], Schied and Zhang [19], Casgrain and Jaimungal [8], and others by analyzinga market impact game in the context of the Almgren–Chriss market impact model. In [6, 23], all agents are risk-neutral and market parameters are constant, which leads to deterministic Nash equilibria. Deterministic open-loopequilibrium strategies are also obtained in [19], where agents maximize mean variance functionals or CARA utility.In [7] closed-loop equilibria are studied numerically in a similar setup, and it is found by means of simulationsthat then equilibrium strategies may no longer be deterministic. The approach in [8] is the closest to ours. There,the authors analyze the infinite-agent, mean-field limit of a market impact game for heterogeneous, risk-averseagents in a model with constant coefficients and partial information, and they characterize the mean-field gamethrough a forward-backward stochastic differential equation (FBSDE). In addition, there are several papers thatstudy market impact games in other price impact models, including models with linear transient price impact; see,e.g., [14, 20, 18, 13].Our contribution to this literature is twofold. First, on the mathematical side, we completely solve the problemof determining an open-loop Nash equilibrium with stochastic model parameters and risk aversion for arbitrarynumbers of agents. Our solution relies on a characterization of the equilibrium strategies in terms of a fully coupledsystems of forward-backward stochastic differential equations (FBSDEs). This characterization is given in Theorem4.1. In the subsequent Theorem 4.2, we give sufficient conditions that guarantee the existence of a unique solution.The main restriction is a lower bound on the volatility. Then we analyze the case of constant coefficients and thecase in which all agents share the same parameters but have different initial inventories. Numerical simulations areprovided for the case of constant coefficients which work for many agents.Our second contribution consists in a modification of the traditional setup of the interaction term in a marketimpact game with Almgren–Chriss-style price impact. The Almgren–Chriss model has two price impact components,one permanent and one temporary. It is clear that permanent price impact must affect the execution prices of allagents equally, and in [6, 23, 7, 19, 8] the same is assumed of the transient price impact. This assumption cansometimes lead to counterintuitive results. For instance, if the temporary price impact is large in comparison withthe permanent price impact, then, in the presence of a large seller, it can be beneficial to build up a long position inthe stock, because a cessation of the trading activities of the large seller will lead to an immediate upwards jump of ∗ SAIF/CAFR/CMAR and School of Mathematical Sciences, Shanghai Jiao Tong University, China; Email: [email protected] † Department of Statistics and Actuarial Sciences, University of Waterloo, Canada; Email: [email protected] ‡ Department of Statistics and Actuarial Sciences, University of Waterloo, Canada; Email: [email protected] § School of Mathematical Sciences, Shanghai Jiao Tong University, China; Email: [email protected] a r X i v : . [ q -f i n . T R ] D ec he expected price [23]. In the price impact literature, it is however not consensus that “temporary price impact”is of the same nature as permanent price impact. For instance, Almgren et al. [3] write about temporary impact:This expression is a continuous-time approximation to a discrete process. A more accurate descriptionwould be to imagine that time is broken into intervals such as, say, one hour or one half-hour. Withineach interval, the average price we realise on our trades during that interval will be slightly less favorablethan the average price that an unbiased observer would measure during that time interval. The unbiasedprice is affected on previous trades that we have executed before this interval (as well as volatility), butnot on their timing. The additional concession during this time interval is strongly dependent on thenumber of shares that we execute in this interval.Likewise, Gatheral [10, p. 751] writes:The second component of the cost of trading corresponds to market frictions such as effective bid-askspread that affect only our execution price: We refer to this component of trading cost as slippage ( temporary impact in the terminology of Huberman and Stanzl).Based on these interpretations of “temporary price impact” as slippage, it appears to be more natural that onlythe trades of the executing agent and not the trades of the other market participants are affected by the resultingcost. In our paper, we therefore keep a term for “temporary price impact”, but it only affects the execution costsof the corresponding agent and not of the other agents.The paper is organized as follows. In Section 2, we set up our model on portfolio liquidation in the Almgren-Chriss framework. Single agent optimization is studied in Section 3, where the corresponding existence, uniquenessand characterization results for the optimal liquidation strategy are stated. Section 4 is dedicated to present thecharacterization result for Nash equilibrium and investigates the solvability of the characterizing FBSDE. Someexplicit solutions for Nash equilibria are analyzed in Section 5. Let W = ( W t ) t ≥ be a d -dimensional Brownian motion on a probability space (Ω , F , P ) and denote by ( F t ) t ≥ thecomplete filtration generated by W . Throughout, we fix a finite time horizon T >
0. We endow Ω × [0 , T ] withthe predictable σ -algebra P and R n with its Borel σ -algebra B ( R n ). Equalities and inequalities between randomvariables and processes are understood in the P -a.s. and P ⊗ dt -a.e. sense, respectively. The Euclidean norm isdenoted by | · | . For m ∈ [1 , ∞ ] and k ∈ N , we denote by (cid:107) · (cid:107) m denotes the L m -norm, by S m ( R k ) the set of k -dimensional continuous adapted processes Y on [0 , T ] such that (cid:107) Y (cid:107) S m ( R k ) := (cid:13)(cid:13)(cid:13)(cid:13) sup ≤ t ≤ T | Y t | m (cid:13)(cid:13)(cid:13)(cid:13) m < ∞ , and by H m ( R k ) the set of predictable R k -valued processes Z such that (cid:107) Z (cid:107) H m ( R k ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32)(cid:90) T | Z s | ds (cid:33) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m < ∞ . The space BMO( R k ) consists of all predictable R k -valued processes Z such that (cid:107) Z (cid:107) BMO( R k ) = sup τ ∈T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E (cid:32)(cid:90) Tτ | Z s | ds (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F τ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < ∞ where T is the set of all stopping times with values in [0 , T ]. We consider n financial agents who are active in a financial market of Almgren–Chriss-type and whose tradingstrategies interact via permanent price impact. More precisely, we adapt the continuous-time setting of [1], where2ach agent i has initial inventory Q i at time t = 0 and subsequently uses a trading strategy whose trading rate isgiven by a process q i ∈ H ( R ). That is, at time t ∈ [0 , T ], the inventory of agent i is given by Q q i t = Q i + (cid:90) t q is ds. This trading strategy impacts the price of the risky asset by means of permanent price impact. It is usually assumedthat this permanent price impact is linear in the traded inventory (see, e.g., the discussion in Section 3 of [11]).Thus, we assume that the price at which shares of the risky assets can be traded at time t is given by S qt = S + (cid:90) t µ s ds + a n (cid:88) i =1 (cid:90) t q is ds + (cid:90) t σ s dW s , (2.1)where µ ∈ S ∞ ( R ) is a generic drift, σ ∈ S ∞ ( R d ) is a volatility process, and, for a fixed price impact parameter a >
0, the term a (cid:80) ni =1 (cid:82) t q is ds describes the cumulative price impact generated by the strategies of all agents.At time t , the i th agent sells − q it dt shares at price S qt . The implementation shortfall, i.e., the difference betweenbook value and liquidation proceeds, is therefore given by Q i S − Q q i T S qT + (cid:82) T q it S qt dt . In addition, the tradingstrategy q i generates “slippage”, including transaction costs, instantaneous price impact effects etc., modeled bythe cost functional b (cid:82) T ( q it ) dt ; see, e.g., [1] and the discussion in the introduction. Moreover, any inventory heldat time t > α i (cid:16) Q q i T (cid:17) + (cid:90) T λ i σ t (cid:16) Q q i t (cid:17) dt (2.2)where α i and λ i are nonnegative constants. The first term in (2.2) is clearly a penalty term penalizing any inventorythat is still present at time T . As shown by Sch¨oneborn [22, 21], the expectation of the integral term in (2.2) canbe regarded as a continuously re-optimized variance functional with infinitesimal time horizon; see also [2, 9, 24]for related motivations of this risk term. It follows that the objective of agent i is to minimize the expectation offollowing cost functional over strategies q i ∈ H ( R ), C iT ( Q i , q i , q − i ) = Q i S q + (cid:90) T q it (cid:0) S qt + bq it (cid:1) dt − Q q i T S qT + α i (cid:16) Q q i T (cid:17) + (cid:90) T λ i σ t (cid:16) Q q i t (cid:17) dt ; (2.3)here, q − i := ( q , . . . , q i − , q i +1 , . . . , q n ) denotes the collection of the strategies of all other agents.Our goal in this paper is to discuss the existence, uniqueness and structure of Nash equilibria for the costcriterion described above. As usual, a collection q ∗ = ( q ∗ , . . . , q n ∗ ) ∈ H ( R n ) of strategies will be called a Nashequilibrium if, for i = 1 , . . . , n , min q i ∈H ( R ) E (cid:2) C iT ( Q i , q i , q − i ∗ ) (cid:3) = E (cid:2) C iT ( Q i , q i ∗ , q − i ∗ ) (cid:3) . In preparation for the discussion of Nash equilibria defined at the end of Section 2.2, we analyze first the optimizationproblem for a fixed agent i when the strategies of all other agents are fixed. A variety of methods has been used tosolve similar and related problems; see, e.g., [2, 9, 24, 17, 12, 4]. Here, our goal is to represent solutions in terms ofa BSDE in Theorem 3.1.First, plugging formula (2.1) for S q into our expression (2.3) of the cost-risk functional C iT ( Q i , q i , q − i ) andintegrating by parts, we obtain the alternative expression C iT ( Q i , q i , q − i ) = a (cid:0) Q i (cid:1) − (cid:90) T Q q i t µ t + a (cid:88) j (cid:54) = i q jt dt − (cid:90) T Q q i t σ t dW t + (cid:90) T b (cid:0) q it (cid:1) dt + (cid:16) α i − a (cid:17) (cid:16) Q q i T (cid:17) + (cid:90) T λ i σ t (cid:16) Q q i t (cid:17) dt. σ ∈ S ∞ ( R d ) and q i ∈ H ( R ), the stochastic integral (cid:82) T Q q i t σ t dW t is a true martingale, andso taking expectations yields E (cid:2) C iT ( Q i , q i , q − i ) (cid:3) = a (cid:0) Q i (cid:1) − E (cid:90) T Q q i t µ t + a n (cid:88) j =1 q jt dt + E (cid:34)(cid:90) T b (cid:0) q it (cid:1) dt (cid:35) + (cid:16) α i − a (cid:17) E (cid:20)(cid:16) Q q i T (cid:17) (cid:21) + E (cid:34)(cid:90) T λ i σ t (cid:16) Q q i t (cid:17) dt (cid:35) . In the following, we will denote β i = α i − a . Fixing 0 ≤ t ≤ T , let Q ¯ q i t,s := Q q i t + (cid:90) st ¯ q iu du, for t ≤ s ≤ T, and C it,T ( Q q i t , ¯ q i , q − i ) be the total cost on [ t, T ] if, at time t , agent i starts using the strategy ¯ q i with the inventory Q q i t , i.e., C it,T ( Q q i t , ¯ q i , q − i ) = a (cid:16) Q q i t (cid:17) − (cid:90) Tt Q ¯ q i t,u µ u + a (cid:88) j (cid:54) = i q ju du + (cid:90) Tt λ i σ u (cid:16) Q ¯ q i t,u (cid:17) du + (cid:90) Tt b (cid:0) ¯ q iu (cid:1) du − (cid:90) Tt Q ¯ q i t,u σ u dW u + β i (cid:16) Q ¯ q i t,T (cid:17) . LetΦ it (cid:16) Q q i t (cid:17) : = ess inf ¯ q i ∈H ( R ) E (cid:104) C it,T (cid:16) Q q i t , ¯ q i , q − i (cid:17) (cid:12)(cid:12)(cid:12) F t (cid:105) = a (cid:16) Q q i t (cid:17) + ess inf ¯ q i ∈H ( R ) E (cid:90) Tt − Q ¯ q i t,u µ u + a (cid:88) j (cid:54) = i q ju − λ i σ u Q ¯ q i t,u + b (cid:0) ¯ q iu (cid:1) du + β i (cid:16) Q ¯ q i t,T (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . Our next goal is to obtain a representation ofˆΦ it (cid:16) Q q i t (cid:17) := Φ it (cid:16) Q q i t (cid:17) − a (cid:16) Q q i t (cid:17) . in terms of component ( A i , B i , C i ) of a solution of a three-dimensional BSDE, which will be discussed in thefollowing proposition. Proposition 3.1
Suppose that β i ≥ and q j ∈ H ( R ) , j (cid:54) = i , then the following BSDE A it = β i − (cid:82) Tt (cid:16) b (cid:0) A is (cid:1) − λ i σ s (cid:17) ds − (cid:82) Tt Z A i s dW s ,B it = 0 − (cid:82) Tt (cid:16) b A is B is + µ s + a (cid:80) j (cid:54) = i q js (cid:17) ds − (cid:82) Tt Z B i s dW s admits a unique solution ( A i , B i , Z A i , Z B i ) ∈ S ∞ ( R ) × S ( R ) × BMO( R d ) × H ( R d ) . Moreover, the solution of theBSDE dC it = 14 b (cid:0) B it (cid:1) dt + Z C i t dW t , C iT = 0 , is well defined and given by C it = 0 − (cid:90) Tt b (cid:0) B is (cid:1) ds − (cid:90) Tt Z C i s dW s . Proof.
Denoting M = β i + λ i (cid:107) σ (cid:107) ∞ T , it follows from Pardoux and Peng [15] that BSDE A it = β i − (cid:90) Tt (cid:18) b (cid:0) ( − M ) ∨ A is ∧ M (cid:1) − λ i σ s (cid:19) ds − (cid:90) Tt Z A i s dW s A i , Z A i ) ∈ S ( R ) × H ( R d ). Moreover, we have the following estimate for A i , A it ≤ E (cid:34) β i − (cid:90) Tt (cid:18) b (cid:0) ( − M ) ∨ A is ∧ M (cid:1) − λ i σ s (cid:19) du |F t (cid:35) ≤ β i + λ i (cid:107) σ (cid:107) ∞ ( T − t ) . Meanwhile by denoting ξ t = ( − M ) ∨ A it ∧ Mb , it holds that e − (cid:82) t ξ s ds A it = e − (cid:82) T ξ s ds β i + (cid:90) Tt e − (cid:82) s ξ u du (cid:0) λ i σ s + ξ s A i − bξ s (cid:1) ds − (cid:90) Tt e − (cid:82) s ξ u du Z A i s dW s ≥ e − (cid:82) T ξ s ds β i − (cid:90) Tt e − (cid:82) s ξ u du Z A i s dW s . Therefore, we have A it ≥ E (cid:104) e − (cid:82) Tt ξ s ds β i (cid:12)(cid:12)(cid:12) F t (cid:105) ≥ β i e − M ( T − t ) b . Hence, ( A i , Z A i ) ∈ S ∞ ( R ) × H ( R d ) and satisfies A it = β i − (cid:90) Tt (cid:18) b (cid:0) A is (cid:1) − λ i σ s (cid:19) ds − (cid:90) Tt Z A i s dW s . It is easy to check that Z A i ∈ BMO( R d ). On the other hand, if A it = β i − (cid:90) Tt (cid:18) b (cid:0) A is (cid:1) − λ i σ s (cid:19) du − (cid:90) Tt Z A i s dW s admits a solution ( A i , Z A i ) ∈ S ∞ ( R ) × BMO( R d ), we have A it ≤ E (cid:34) β i − (cid:90) Tt (cid:18) b (cid:0) A is (cid:1) − λ i σ s (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ≤ β i + λ i (cid:107) σ (cid:107) ∞ ( T − t )and e − (cid:82) t Aisb ds A it = e − (cid:82) T Aisb ds ( β i ) + (cid:90) Tt e − (cid:82) s Aiub du λ i σ s ds − (cid:90) Tt e − (cid:82) s Aiub du Z A i s dW s ≥ e − (cid:82) T Aisb ds ( β i ) − (cid:90) Tt e − (cid:82) s Aiub du Z A i s dW s . Therefore, we have A it ≥ E (cid:20) e − (cid:82) Tt Aisb ds β i |F t (cid:21) ≥ β i e − M ( T − t ) b . Hence, ( A i , Z A i ) satisfies A it = β i − (cid:90) Tt (cid:18) b (cid:0) ( − M ) ∨ A is ∧ M (cid:1) − λ i σ s (cid:19) ds − (cid:90) Tt Z A i s dW s . Again, it follows from Pardoux and Peng [15] that B it = 0 − (cid:90) Tt b A is B is + µ s + a (cid:88) j (cid:54) = i q js du − (cid:90) Tt Z B i s dW s admits a unique solution ( B i , Z B i ) ∈ S ( R ) × H ( R d ). The rest is clear. (cid:3) heorem 3.1 Suppose that β i ≥ and q j ∈ H ( R ) , j (cid:54) = i , then ˆΦ it (cid:16) Q q i t (cid:17) is given by ˆΦ it (cid:16) Q q i t (cid:17) = A it (cid:16) Q q i t (cid:17) + B it Q q i t + C it where A i , B i , C i are given as in Proposition 3.1. The unique optimal strategy for the agent i is given in feedbackform by q i ∗ t = − b (cid:18) A it Q q i ∗ t + 12 B it (cid:19) . Proof.
By denoting V q i t = A it (cid:16) Q q i t (cid:17) + B it Q q i t + C it ,g A i t = b (cid:0) A it (cid:1) − λ i σ t ,g B i t = b A it B it + µ t + a (cid:80) j (cid:54) = i q jt ,g C i t = b (cid:0) B it (cid:1) , and applying Itˆo’s formula, we have dV q i t =2 A it Q q i t q it dt + (cid:16) Q q i t (cid:17) dA it + B it q it dt + Q q i t dB it + g C i t dt + Z C i t dW t =2 A it Q q i t q it dt + (cid:16) Q q i t (cid:17) g A i t dt + (cid:16) Q q i t (cid:17) Z A i t dW t + g C i t dt + Z C i t dW t + B it q it dt + Q q i t g B i t dt + Q q i t Z B i t dW t = (cid:18) A it Q q i t q it + (cid:16) Q q i t (cid:17) g A i t + B it q it + Q q i t g B i t + g C i t (cid:19) dt + (cid:18)(cid:16) Q q i t (cid:17) Z A i t + Q q i t Z B i t + Z C i t (cid:19) dW t . Therefore it holds dV q i t + − Q q i t µ t + a (cid:88) j (cid:54) = i q jt − λ i σ t Q q i t + b (cid:0) q it (cid:1) dt = A it Q q i t q it + (cid:16) Q q i t (cid:17) g A i t + B it q it + Q q i t g B i t + g C i t − Q q i t µ t + a (cid:88) j (cid:54) = i q jt − λ i σ t Q q i t + b (cid:0) q it (cid:1) dt + (cid:18)(cid:16) Q q i t (cid:17) Z A i t + Q q i t Z B i t + Z C i t (cid:19) dW t . and rearranging the drift terms, one can see dV q i t + − Q q i t µ t + a (cid:88) j (cid:54) = i q jt − λ i σ t Q q i t + b (cid:0) q it (cid:1) dt = (cid:18) A it Q q i t q it + 1 b (cid:16) Q q i t (cid:17) (cid:0) A it (cid:1) + B it q it + 1 b Q q i t A it B it + 14 b (cid:0) B it (cid:1) + b (cid:0) q it (cid:1) (cid:19) dt + (cid:18)(cid:16) Q q i t (cid:17) Z A i t + Q q i t Z B i t + Z C i t (cid:19) dW t = 1 b (cid:18) A it Q q i t + bq it + 12 B it (cid:19) dt + (cid:18)(cid:16) Q q i t (cid:17) Z A i t + Q q i t Z B i t + Z C i t (cid:19) dW t . Hence, it holds that V q i t = V q i T + (cid:90) Tt − Q q i s µ s + a (cid:88) j (cid:54) = i q js − λ i σ t Q q i s + b (cid:0) q is (cid:1) ds − (cid:90) Tt b (cid:18) A is Q q i s + bq is + 12 B is (cid:19) ds − (cid:90) Tt (cid:18)(cid:16) Q q i s (cid:17) Z A i s + Q q i s Z B i s + Z C i s (cid:19) dW s . q i , ¯ q i ∈ H ( R ) and t ∈ [0 , T ], by taking ˜ q is = q is { s ≤ t } + ¯ q is { s>t } for all s ∈ [0 , T ], we have V q i t = V ˜ q i t = β i Q ¯ q i t,T + (cid:90) Tt − Q ¯ q i t,s µ s + a (cid:88) j (cid:54) = i q js − λ i σ t Q ¯ q i t,s + b (cid:0) ¯ q is (cid:1) ds − (cid:90) Tt b (cid:18) A is Q ¯ q i t,s + b ¯ q is + 12 B is (cid:19) ds − (cid:90) Tt (cid:18)(cid:16) Q ¯ q i t,s (cid:17) Z A i s + Q ¯ q i t,s Z B i s + Z C i s (cid:19) dW s which implies that V q i t ≤ E β i Q ¯ q i t,T + (cid:90) Tt − Q ¯ q i t,s µ s + a (cid:88) j (cid:54) = i q js − λ i σ t Q ¯ q i t,s + b (cid:0) ¯ q is (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . Hence, it holds that V q i t ≤ ess inf ¯ q i ∈H ( R ) E β i Q ¯ q i t,T + (cid:90) Tt − Q ¯ q i t,s µ s + a (cid:88) j (cid:54) = i q js − λ i σ t Q ¯ q i t,s + b (cid:0) ¯ q is (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t = ˆΦ it (cid:16) Q q i t (cid:17) . On the other hand, for any t ∈ [0 , T ] and q i ∈ H ( R ), the following random ODE Q i ∗ s = Q q i t − b (cid:90) st (cid:18) A iu Q i ∗ u + 12 B iu (cid:19) du, s ∈ [ t, T ]admits a unique solution Q i ∗ ∈ S ( R ) on [ t, T ]. Therefore, by taking ˜ q is = q is { s ≤ t } + q i ∗ s { s>t } with q i ∗ s = − b (cid:18) A is Q i ∗ s + 12 B is (cid:19) , we have V q i t = V ˜ q i t = β i Q q i ∗ t,T + (cid:90) Tt − Q q i ∗ t,s µ s + a (cid:88) j (cid:54) = i q js − λ i σ t Q q i ∗ t,s + b (cid:0) q i ∗ s (cid:1) ds − (cid:90) Tt b (cid:18) A is Q q i ∗ t,s + bq i ∗ s + 12 B is (cid:19) ds − (cid:90) Tt (cid:18)(cid:16) Q q i ∗ t,s (cid:17) Z A i s + Q q i ∗ t,s Z B i s + Z C i s (cid:19) dW s which implies that V q i t = E β i Q q i ∗ t,T + (cid:90) Tt − Q q i ∗ t,s µ s + a (cid:88) j (cid:54) = i q js − λ i σ t Q q i ∗ t,s + b (cid:0) q i ∗ s (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t ≥ ˆΦ it (cid:16) Q q i t (cid:17) . Therefore, it holds that ˆΦ it (cid:16) Q q i t (cid:17) = A it (cid:16) Q q i t (cid:17) + B it Q q i t + C it . It is easy to verify that the unique optimal strategy (feedback form) for the agent i is given by q i ∗ t = − b (cid:18) A it Q q i ∗ t + 12 B it (cid:19) . (cid:3) .1 Characterization of the optimal strategy in terms of an FBSDE In this section, we show that the optimal strategy for agent i can be given by the unique solution of an FBSDE. Theorem 3.2
Suppose that β i ≥ and q j ∈ H ( R ) , j (cid:54) = i , then (cid:18) Q q i ∗ , q i ∗ , Q qi ∗ Z Ai b + Z Bi b (cid:19) is the unique solutionof the following FBSDE Q q i t = Q i + (cid:82) t q is ds,q it = − β i b Q q i T + (cid:82) Tt b (cid:18) − λ i σ s Q q i s + ( µ s + a (cid:80) j (cid:54) = i q js ) (cid:19) ds + (cid:82) Tt Z is dW s . (3.1) in S ( R ) × S ( R ) × H ( R d ) . Proof.
By denoting Λ it := e − (cid:82) t b A is ds , it is easy to deduce that: d (cid:16) Λ it Q q i ∗ t A it (cid:17) = Λ it Q q i ∗ t dA it + Λ it q i ∗ t A it dt + Q q i ∗ t A it d Λ it = Λ it q i ∗ t A it dt + Λ it Q q i ∗ t (cid:0) A it (cid:1) b dt − λ i σ t Λ it Q q i ∗ t dt − Λ it Q q i ∗ t (cid:0) A it (cid:1) b dt + Λ it Q q i ∗ t Z A i t dW t = Λ it q i ∗ t A it dt − λ i σ t Λ it Q q i ∗ t dt + Λ it Q q i ∗ t Z A i t dW t . Therefore, it holds thatΛ it Q q i ∗ t A it = β i Λ iT Q q i ∗ T − (cid:90) Tt Λ is q i ∗ s A is ds + (cid:90) Tt λ i σ s Λ is Q q i ∗ s ds − (cid:90) Tt Λ is Q q i ∗ s Z A i s dW s . Noting that Λ it B it = − (cid:90) Tt Λ is µ s + a (cid:88) j (cid:54) = i q js ds − (cid:90) Tt Λ is Z B i s dW s , one has Λ it q i ∗ t = − β i b Λ iT Q q i ∗ T + (cid:90) Tt b Λ is q i ∗ s A is − λ i σ s Λ is Q q i ∗ s + Λ is (cid:16) µ s + a (cid:80) j (cid:54) = i q js (cid:17) ds + (cid:90) Tt (cid:32) Λ is Q q i ∗ s Z A i s b + Λ is Z B i s b (cid:33) dW s . Therefore, it holds that dq i ∗ t = d (cid:16)(cid:0) Λ it (cid:1) − Λ it q i ∗ t (cid:17) = A it b q i ∗ t dt − (cid:0) Λ it (cid:1) − b Λ it q i ∗ t A it − λ i σ t Λ it Q q i ∗ t + Λ it (cid:16) µ t + a (cid:80) j (cid:54) = i q jt (cid:17) dt + (cid:32) Λ it Q q i ∗ t Z A i t b + Λ it Z B i t b (cid:33) dW t (cid:33) = 1 b λ i σ t Q q i ∗ t − (cid:16) µ t + a (cid:80) j (cid:54) = i q jt (cid:17) dt − (cid:32) Q q i ∗ t Z A i t b + Z B i t b (cid:33) dW t .
8t is easy to check that (cid:18) Q q i ∗ , q i ∗ , Q qi ∗ Z Ai b + Z Bi b (cid:19) is in S ( R ) × S ( R ) × H ( R d ). We now prove the uniqueness.Suppose that FBSDE (3.1) admits another solution ( Q ¯ q i , ¯ q i , ¯ Z i ) ∈ S ( R ) × S ( R ) × H ( R d ). Then, we have Q q i t − Q ¯ q i t = (cid:82) t (cid:0) q is − ¯ q is (cid:1) ds,q it − ¯ q it = − β i b (cid:16) Q q i T − Q ¯ q i T (cid:17) + (cid:82) Tt b (cid:16) − λ i σ s (cid:16) Q q i s − Q ¯ q i s (cid:17)(cid:17) ds + (cid:82) Tt (cid:0) Z is − ¯ Z is (cid:1) dW s . Therefore, it holds that (cid:0) q it − ¯ q it (cid:1) (cid:16) Q q i t − Q ¯ q i t (cid:17) = − β i b (cid:16) Q q i T − Q ¯ q i T (cid:17) + (cid:90) Tt b (cid:18) − λ i σ s (cid:16) Q q i s − Q ¯ q i s (cid:17) (cid:19) ds − (cid:90) Tt (cid:0) q is − ¯ q is (cid:1) ds + (cid:90) Tt (cid:16) Q q i t − Q ¯ q i t (cid:17) (cid:0) Z is − ¯ Z is (cid:1) dW s . Thus, it holds that0 = E (cid:34) − β i b (cid:16) Q q i T − Q ¯ q i T (cid:17) − (cid:90) T λ i σ s b (cid:16) Q q i s − Q ¯ q i s (cid:17) ds − (cid:90) T (cid:0) q is − ¯ q is (cid:1) ds (cid:35) ≤ (cid:3) We first provide a characterizing result of a Nash equilibrium in terms of a system of FBSDE.
Theorem 4.1
Suppose that β i ≥ , if the following FBSDE: Q q i t = Q i + (cid:82) t q is ds, i = 1 , . . . , nq it = − β i b Q q i T + (cid:82) Tt b (cid:18) − λ i σ s Q q i s + ( µ s + a (cid:80) j (cid:54) = i q js ) (cid:19) ds + (cid:82) Tt Z is dW s , i = 1 , . . . , n. (4.1) admits a solution ( Q q , q, Z ) ∈ S ( R n ) × S ( R n ) × H ( R n × d ) , then q is a Nash equilibrium. On the other hand, if q ∈ H ( R n ) is a Nash equilibrium, then ( Q q , q, Z ) is a solution of FBSDE 4.1 in S ( R n ) × S ( R n ) × H ( R n × d ) ,where Z is given by Z = (cid:32) Q q Z A b + Z B b , . . . , Q q n Z A n b + Z B n b (cid:33) (cid:48) , where for i = 1 , . . . , n , Z A i , Z B i are given as in Theorem 3.1 and M (cid:48) denotes the transpose of the matrix M . Proof.
The result follows directly from Theorem 3.1 and Theorem 3.2. (cid:3)
In order to get a Nash equilibrium, it is sufficient to have the existence of solution for FBSDE (4.1). In this section,we will investigate the solvability for FBSDE (4.1). An existence and uniqueness result for small time horizon isdue to Antonelli [5]. Under some assumptions, we get a unique global solution for FBSDE (4.1) which is stated inthe following theorem.
Theorem 4.2
Suppose that β i > , λ i > and λ i σ t > a b ( n − for all i = 1 , . . . , n and t ∈ [0 , T ] , then FBSDE (4.1) admits a unique solution ( Q q , q, Z ) ∈ S ( R n ) × S ( R n ) × H ( R n × d ) . Proof.
Denoting ˜ q it = − q it , we have Q q i t = Q i − (cid:82) t ˜ q is ds, i = 1 , . . . , n ˜ q it = β i b Q q i T − (cid:82) Tt b (cid:18) − λ i σ s Q q i s + µ s − a (cid:80) j (cid:54) = i ˜ q js (cid:19) ds − (cid:82) Tt Z is dW s , i = 1 , . . . , n. n (cid:88) i =1 β i b | x i | ≥ min ≤ i ≤ n β i b | x | and n (cid:88) i =1 − λ i σ t b | x i | − | y i | − a (cid:88) j (cid:54) = i y j x i ≤ n (cid:88) i =1 − λ i σ t b | x i | − | y i | + a ( n − | x i | + (cid:88) j (cid:54) = i n − | y j | = n (cid:88) i =1 (cid:18) − λ i σ t b + a ( n − (cid:19) | x i | ≤ − min ≤ i ≤ n inf ≤ t ≤ T (cid:18) λ i σ t b − a ( n − (cid:19) | x | , the monotonicity condition in Peng-Wu [16] is satisfied. Therefore, the solvability follows. (cid:3) As a direct consequence of Theorem 4.1 and Theorem 4.2, we have the following corollary on the existence anduniqueness of a Nash equilibrium.
Corollary 4.1
Suppose that β i > , λ i > and λ i σ t > a b ( n − for all i = 1 , . . . , n and t ∈ [0 , T ] , then thereexists a unique Nash equilibrium. Since FBSDE (4.1) is linear, we will investigate it’s solvability through Riccati equations. Indeed, FBSDE (4.1)could be rewritten as (cid:40) Q qt = Q + (cid:82) t q s ds,q ( t ) = GQ qT + (cid:82) Tt (cid:16) ˆ A s Q qs + ( − a b I n + a b ˆ B ) q s + ˆ C s (cid:17) ds + (cid:82) Tt Z s dW s where G is n × n diagonal matrix with diagonal elements − β i b , ˆ A s is n × n diagonal matrix with diagonal elements − λ i σ s b , I n is n × n identity matrix, ˆ B is n × n matrix whose elements are all equal to 1 and ˆ C s = (cid:0) µ s b , . . . , µ s b (cid:1) T .Suppose that the following holds: q t = P t Q qt + p t , t ∈ [0 , T ] , with ( P, Λ) and ( p, η ) being the adapted solutions of the following BSDEs respectively: (cid:40) dP t = Γ t dt + Λ t dW t ,P T = G and (cid:40) dp t = ξ t dt + η t dW t ,p T = 0where Γ and ξ will be chosen later. Applying Itˆo’s formula, we have the following(Γ t Q qt + P t q t + ξ t ) dt + (Λ t Q qt + η t ) dW t = dq t = − (cid:16) ˆ A t Q qt + (cid:16) − a b I n + a b ˆ B (cid:17) q t + ˆ C t (cid:17) dt − Z t dW t . Comparing drift and diffusion terms, we should have (cid:40) (cid:0) Γ t + P t (cid:1) Q qt + P t p t + ξ t = − (cid:16) ˆ A t + (cid:16) − a b I n + a b ˆ B (cid:17) P t (cid:17) Q qt − (cid:16)(cid:16) − a b I n + a b ˆ B (cid:17) p t + ˆ C t (cid:17) , Λ t Q qt + η t = − Z t . Therefore, we will take
Γ = − ˆ A t − (cid:16) − a b I n + a b ˆ B (cid:17) P t − P t ξ t = − (cid:16) − a b I n + a b ˆ B + P t (cid:17) p t − ˆ C t Thus, we obtain the following result. 10 roposition 4.1
Suppose that the following BSDE (cid:40) dP t = (cid:16) − ˆ A t − (cid:16) − a b I n + a b ˆ B (cid:17) P t − P t (cid:17) dt + Λ t dW t ,P T = G (4.2) admits an adapted solution ( P, Λ) ∈ S m ( R n × n ) × H m ( R n × n × d ) for all m ≥ . Suppose moreover that the followingBSDE (cid:40) dp t = (cid:16) − (cid:16) − a b I n + a b ˆ B + P t (cid:17) p t − ˆ C t (cid:17) dt + η t dW t ,p T = 0 (4.3) admits a unique adapted solution ( p, η ) ∈ S m ( R n ) × H m ( R n × d ) for all m ≥ and that the unique solution offollowing random ODE, Q qt = Q + (cid:90) t ( P s Q qs + p s ) ds (4.4) belongs to S m ( R n ) for all m ≥ . Then FBSDE (4.1) admits an adapted solution ( Q q , q, Z ) ∈ S ( R n ) × S ( R n ) ×H ( R n × d ) such that q t = P t Q qt + p t and Z t = − Λ t Q qt − η t . σ t = σ and µ t = µ In the current case, the FBSDE (4.1) takes the following form: (cid:40) Q qt = Q + (cid:82) t q s ds,q t = GQ qT + (cid:82) Tt (cid:16) ˆ AQ qs + ( − a b I n + a b ˆ B ) q s + ˆ C (cid:17) ds where G is n × n diagonal matrix with diagonal elements − β i b , ˆ A is n × n diagonal matrix with diagonal elements − λ i σ b , I n is n × n identity matrix, ˆ B is n × n matrix whose elements are all equal to 1 and ˆ C = (cid:0) µ b , . . . , µ b (cid:1) T . Bydenoting ˜ A = a b (cid:16) I n − ˆ B (cid:17) , we obtain the following equivalent second order inhomogeneous ODE Q (cid:48)(cid:48) = ˜ AQ (cid:48) − ˆ AQ − ˆ C equivalent to Λ (cid:48) = M Λ + N where Λ = (cid:20) Q (cid:48) Q (cid:21) M = (cid:20) ˜ A − ˆ AI n (cid:21) N = (cid:20) − ˆ C (cid:21) Since M is invertible, the solution is given byΛ = exp ( tM ) (cid:20) ξ ξ (cid:21) + (cid:90) t exp ( sM ) N ds = exp ( tM ) (cid:20) ξ ξ (cid:21) + (exp ( tM ) − I n ) M − N where ξ , ξ in R n is a vector to be determined by the conditions:Λ[ n + 1 , n ](0) = Q (0) = Q , and Λ[1 , n ]( T ) = GQ ( T )It follows that ξ = Q . Hence the second condition is given byexp ( T M ) (cid:20) ξ Q (cid:21) + (exp ( T M ) − I n ) M − N = (cid:20) G I n (cid:21) (cid:20) exp ( T M ) (cid:20) ξ Q (cid:21) + (exp ( T M ) − I n ) M − N (cid:21) equivalent to (cid:2) I n − G (cid:3) exp ( T M ) (cid:20) ξ Q (cid:21) = (cid:2) − I n G (cid:3) (exp ( T M ) − I n ) M − N Hence, denoting by exp (
T M ) = (cid:20) E E E E (cid:21) it follows that ξ is given by ξ = ( E − GE ) − (cid:2)(cid:2) − I n G (cid:3) (exp ( T M ) − I n ) M − N − ( E − GE ) Q (cid:3) . .2.1 Numerical results Throughout we consider the following set of parameters • Market parameters – drift: µ = 2% – vol: σ = 20% – Maturity: T = 1 – price impact: a = 1% – slippage: b = 1% • – Risk aversion: α = (1 , . , . λ = (1 , . , . – Position to liquidate: Q = (1 , , . • Dependence on the driftFigure 1: Plot of the three agents’inventory for different drift valuesAs the drift increases, the agents tend to liquidate slowly or even start buying at the beginning to benefitfrom the future mean return which will compensate to the liquidation cost. • Dependence on the volatilityFigure 2: Plot of the three agents’inventory for different volatility valuesAs the volatility increases, the agents tend to liquidate quickly at the beginning to reduce the liquidation risk. • Dependence on a • Dependence on b Figure 4: Plot of the three agents’inventory for different values of slippage effectFor small slippage, the agent with smaller initial inventory tend to vary between liquidation and purchasingto make profit which will compensate to the liquidation cost. • Dependence on α joint magnitudeFigure 5: Plot of the three agents’inventory for different values of risk aversion on terminal value • Dependence on α different for the first agent 13igure 6: Plot of the three agents’inventory for different values of risk aversion on terminal value • Dependence on λ joint magnitudeFigure 7: Plot of the three agents’inventory for different values of risk aversion on continuous trading • Dependence on λ first agentFigure 8: Plot of the three agents’inventory for different values of risk aversion on continuous trading for one agent • Dependence on Q first agentFigure 9: Plot of the three agents’inventory for different start value of fist agent’s inventory.14 Dependence on Q first agent with two arbitrageursFigure 10: Plot of the three agents’inventory for different start value of fist agent’s inventory with two arbitrageurs.When the initial position of the first agent is small, arbitrageurs tend to first buy and then liquidate to benefitfrom the future mean return. When the initial position of the first agent is high, arbitrageurs tend to firstshort sell and then buy to make profit from the price differences. Moreover, The arbitrageurs will not usevery aggressive strategies. Theorem 4.3
Suppose that β = . . . = β n = β ≥ and λ = . . . = λ n = λ ≥ . Then (cid:40) ˜ Q t = (cid:80) ni =1 Q i + (cid:82) t ˜ q s ds, ˜ q t = − βb ˜ Q T + (cid:82) Tt b (cid:16) − λσ s ˜ Q s + nµ s + ( n − a ˜ q s (cid:17) ds + (cid:82) Tt ˜ Z s dW s (4.5) admits a unique solution ( ˜ Q, ˜ q, ˜ Z ) ∈ S ( R ) × S ( R ) × H ( R d ) and Q q i t = Q i + (cid:82) t q is ds, i = 1 , . . . , nq it = − βb Q q i T + (cid:82) Tt b (cid:16) − λσ s Q q i s + µ s − aq it + a ˜ q s (cid:17) ds + (cid:82) Tt Z is dW s , i = 1 , . . . , n (4.6) admits a unique solution ( Q q , q, Z ) ∈ S ( R n ) × S ( R n ) × H ( R n × d ) . In addition, it holds that ˜ Q = (cid:80) ni =1 Q q i , ˜ q = (cid:80) ni =1 q i , ˜ Z = (cid:80) ni =1 Z i . Moreover, ( Q q , q, Z ) is the unique solution of FBSDE (4.1) . Proof.
We will divide the proof into several steps.
Step 1:
Denoting M = e ( n − aT b βb + e ( n − aT b λ (cid:107) σ (cid:107) ∞ b T , it follows from Pardoux and Peng [15] that BSDE P t = − βb + (cid:90) Tt (cid:18) − λσ s b + ( n − a b P s + (( − M ) ∨ P s ∧ M ) (cid:19) ds − (cid:90) Tt Λ s dW s admits a unique solution ( P, Λ) ∈ S ( R ) × H ( R d ). Moreover, we have the following a priori estimate for P . Since e ( n − at b P t = − e ( n − aT b βb + (cid:90) Tt (cid:18) − e ( n − as b λσ s b + e ( n − as b (( − M ) ∨ P s ∧ M ) (cid:19) ds − (cid:90) Tt e ( n − as b Λ s dW s it holds that e ( n − at b P t ≥ E (cid:34) − e ( n − aT b βb + (cid:90) Tt (cid:18) − e ( n − as b λσ s b (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ≥ − e ( n − aT b βb − e ( n − aT b λ (cid:107) σ (cid:107) ∞ b ( T − t )15ence P t ≥ − e ( n − aT b βb − e ( n − aT b λ (cid:107) σ (cid:107) ∞ b T Meanwhile by denoting ξ t = ( − M ) ∨ P t ∧ M , it holds that e (cid:82) t ( ξ s + ( n − a b ) ds P t = − e (cid:82) T ( ξ s + ( n − a b ) ds βb + (cid:90) Tt e (cid:82) s ( ξ u + ( n − a b ) du (cid:18) − λσ s b − ξ s P s + ξ s (cid:19) ds − (cid:90) Tt e (cid:82) s ( ξ u + ( n − a b ) du Λ s dW s ≤ − e (cid:82) T ( ξ s + ( n − a b ) ds βb − (cid:90) Tt e (cid:82) s ( ξ u + ( n − a b ) du Λ s dW s . Therefore, we have P t ≤ E (cid:34) − e (cid:82) Tt ( ξ s + ( n − a b ) ds βb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ≤ − βb e − M ( T − t ) . Hence, ( P, Λ) ∈ S ∞ ( R ) × H ( R d ) and satisfies P t = − βb + (cid:90) Tt (cid:18) − λσ s b + ( n − a b P s + P s (cid:19) ds − (cid:90) Tt Λ s dW s On the other hand, if P t = − βb + (cid:90) Tt (cid:18) − λσ s b + ( n − a b P s + P s (cid:19) ds − (cid:90) Tt Λ s dW s admits a solution ( P, Λ) ∈ S ∞ ( R ) × H ( R d ), we have e ( n − at b P t = − e ( n − aT b βb + (cid:90) Tt (cid:18) − e ( n − as b λσ s b + e ( n − as b P s (cid:19) ds − (cid:90) Tt e ( n − as b Λ s dW s and e (cid:82) t ( ( n − a b + P s ) ds P t = − e (cid:82) T ( ( n − a b + P s ) ds β b + (cid:90) Tt (cid:18) − e (cid:82) s ( ( n − a b + P u ) du λσ s b (cid:19) ds − (cid:90) Tt e (cid:82) s ( ( n − a b + P u ) du Λ s dW s Therefore, we have P t ≥ E (cid:34) − e ( n − a ( T − t )2 b βb + (cid:90) Tt (cid:18) − e ( n − a ( s − t )2 b λσ s b (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ≥ − e ( n − aT b βb − e ( n − aT b λ (cid:107) σ (cid:107) ∞ b T and P t ≤ E (cid:34) − e (cid:82) Tt ( P s + ( n − a b ) ds βb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ≤ − βb e − M ( T − t ) . Hence, ( P, Λ) satisfies P t = − βb + (cid:90) Tt (cid:18) − λσ s b + ( n − a b P s + (( − M ) ∨ P s ∧ M ) (cid:19) ds − (cid:90) Tt Λ s dW s p t = (cid:90) Tt (cid:18)(cid:18) ( n − a b + P s (cid:19) p s + nµ s b (cid:19) ds − (cid:90) Tt η s dW s admits a unique solution ( p, η ) ∈ S ( R ) × H ( R d ). Moreover, one could easily check that p ∈ S ∞ ( R ), η ∈ BMO( R d )and Λ ∈ BMO( R d ). Hence, from standard theory of SDEs, SDE (4.4) admits a unique strong solution ˜ Q ∈ S ∞ ( R ).Therefore, according to Proposition 4.1, FBSDE (4.5) admits a solution ( ˜ Q, ˜ q, ˜ Z ) ∈ S ( R ) × S ( R ) × H ( R d ).We now prove the uniqueness. Suppose that FBSDE (4.5) admits another solution ( ¯ Q, ¯ q, ¯ Z ) ∈ S ( R ) × S ( R ) ×H ( R d ). Then, we have (cid:40) ˜ Q t − ¯ Q t = (cid:82) t (˜ q s − ¯ q s ) ds, ˜ q t − ¯ q t = − βb (cid:16) ˜ Q T − ¯ Q T (cid:17) + (cid:82) Tt b (cid:16) − λσ s (cid:16) ˜ Q s − ¯ Q s (cid:17) + ( n − a (˜ q s − ¯ q s ) (cid:17) ds + (cid:82) Tt (cid:16) ˜ Z s − ¯ Z s (cid:17) dW s Therefore, it holds that(˜ q t − ¯ q t ) (cid:16) ˜ Q t − ¯ Q t (cid:17) = − βb (cid:16) ˜ Q T − ¯ Q T (cid:17) + (cid:90) Tt b (cid:18) − λσ s (cid:16) ˜ Q s − ¯ Q s (cid:17) + ( n − a q s − ¯ q s ) (cid:16) ˜ Q t − ¯ Q t (cid:17)(cid:19) ds − (cid:90) Tt (˜ q s − ¯ q s ) ds + (cid:90) Tt (cid:16) ˜ Q t − ¯ Q t (cid:17) (cid:16) ˜ Z s − ¯ Z s (cid:17) dW s Hence, we have e ( n − at (˜ q t − ¯ q t ) (cid:16) ˜ Q t − ¯ Q t (cid:17) = − βb e ( n − aT (cid:16) ˜ Q T − ¯ Q T (cid:17) − (cid:90) Tt λσ s b e ( n − as (cid:16) ˜ Q s − ¯ Q s (cid:17) ds − (cid:90) Tt e ( n − as (˜ q s − ¯ q s ) ds + (cid:90) Tt e ( n − as (cid:16) ˜ Q t − ¯ Q t (cid:17) (cid:16) ˜ Z s − ¯ Z s (cid:17) dW s Thus, it holds that0 = E (cid:34) − βb e ( n − aT (cid:16) ˜ Q T − ¯ Q T (cid:17) − (cid:90) T λσ s b e ( n − as (cid:16) ˜ Q s − ¯ Q s (cid:17) ds − (cid:90) T e ( n − as (˜ q s − ¯ q s ) ds (cid:35) ≤ Step 2:
Noting that ˜ q ∈ S ∞ ( R ), following from a similar technique as in Step 1 , FBSDE (4.6) admits a uniquesolution ( Q q , q, Z ) ∈ S ( R n ) × S ( R n ) × H ( R n × d ). Moreover, it holds that (cid:80) ni =1 Q q i t = (cid:80) ni =1 Q i + (cid:82) t (cid:80) ni =1 q is ds, (cid:80) ni =1 q it = − βb (cid:80) ni =1 Q q i T + (cid:82) Tt b (cid:16) − λσ s (cid:80) ni =1 Q q i s + nµ s − (cid:80) ni =1 aq it + an ˜ q s (cid:17) ds + (cid:82) Tt (cid:80) ni =1 Z is dW s Therefore, we have (cid:80) ni =1 Q q i t − ˜ Q t = (cid:82) t (cid:0)(cid:80) ni =1 q is − ˜ q s (cid:1) ds, (cid:80) ni =1 q it − ˜ q s = − βb (cid:16)(cid:80) ni =1 Q q i T − ˜ Q T (cid:17) + (cid:82) Tt b (cid:16) − λσ s (cid:16)(cid:80) ni =1 Q q i s − ˜ Q s (cid:17) − a (cid:0)(cid:80) ni =1 q is − ˜ q s (cid:1)(cid:17) ds + (cid:82) Tt (cid:16)(cid:80) ni =1 Z is − ˜ Z s (cid:17) dW s It follows from the uniqueness part of
Step 1 that ˜ Q = (cid:80) ni =1 Q q i , ˜ q = (cid:80) ni =1 q i , ˜ Z = (cid:80) ni =1 Z i . Step 3:
The last statement follows immediately from the uniqueness of solutions of FBSDEs (4.5) and (4.6). (cid:3) .3.1 Asymptotic property If we scale the permanent market impact by the number of agents n or equivalently the permanent market impactis generated by the average of liquidation strategy of all agents, the FBSDE characterizing the Nash equilibriumturns to be the following FBSDE Q q i,n t = Q i + (cid:82) t q i,ns ds, i = 1 , . . . , nq i,nt = − α i − a n b Q q i,n T + (cid:82) Tt b (cid:18) − λ i σ s Q q i,n s + ( µ s + an (cid:80) j (cid:54) = i q j,ns ) (cid:19) ds + (cid:82) Tt Z i,ns dW s , i = 1 , . . . , n. (4.7)Then we have the following theorem. Theorem 4.4
Suppose that α i = α > , λ i = λ ≥ for all i ∈ N and lim n →∞ n (cid:80) ni =1 Q i = Q ∗ ∈ R . Then (cid:40) Q ∗ t = Q ∗ + (cid:82) t q ∗ s ds,q ∗ t = − αb Q ∗ T + (cid:82) Tt b (cid:16) − λσ s Q ∗ s + ( µ s + aq ∗ s )2 (cid:17) ds + (cid:82) Tt Z ∗ s dW s . (4.8) admits a unique solution ( Q ∗ , q ∗ , Z ∗ ) ∈ S ( R ) × S ( R ) × H ( R d ) and Q ˜ q i t = Q i + (cid:82) t ˜ q is ds ˜ q it = − αb Q ˜ q i T + (cid:82) Tt b (cid:16) − λσ s Q ˜ q i s + ( µ s + aq ∗ s )2 (cid:17) ds + (cid:82) Tt ˜ Z is dW s (4.9) admits a unique solution ( Q ˜ q i , ˜ q i , ˜ Z i ) ∈ S ( R ) × S ( R ) × H ( R d ) for all i ∈ N . Let n be large enough such that α ≥ a n and ( Q q · ,n , q · ,n , Z · ,n ) ∈ S ( R n ) × S ( R n ) × H ( R n × d ) be the unique solution of FBSDE (4.7) . Then it holdsthat (cid:107) n n (cid:88) i =1 Q q i,n − Q ∗ (cid:107) S ( R ) + (cid:107) n n (cid:88) i =1 q i,n − q ∗ (cid:107) S ( R ) + (cid:107) n n (cid:88) i =1 Z i,n − Z ∗ (cid:107) H ( R d ) → as n → ∞ and (cid:107) Q q i,n − Q ˜ q i (cid:107) S ( R ) + (cid:107) q i,n − ˜ q i (cid:107) S ( R ) + (cid:107) Z i,n − ˜ Z i (cid:107) H ( R d ) → for all ≤ i ≤ n, as n → ∞ Proof.
It follows from a similar technique as in Theorem 4.3, FBSDE (4.8) admits a unique solution ( Q ∗ , q ∗ , Z ∗ ) ∈S ( R ) × S ( R ) × H ( R d ) and FBSDE (4.9) admits a unique solution ( Q ˜ q i , ˜ q i , ˜ Z i ) ∈ S ( R ) × S ( R ) × H ( R d ) for all i ∈ N .Let n be large enough such that α ≥ a n , it follows from Theorem 4.3 that FBSDE (4.7) admits a unique solution( Q q · ,n , q · ,n , Z · ,n ) ∈ S ( R n ) × S ( R n ) × H ( R n × d ). Moreover, one could check that there exists a constant M whichdoes not depend on n such that (cid:107) Q q i,n (cid:107) S ( R ) + (cid:107) q i,n (cid:107) S ( R ) + (cid:107) Z i,n (cid:107) H ( R d ) ≤ M, for all 1 ≤ i ≤ n. In addition, we have n (cid:80) ni =1 Q q i,n t = n (cid:80) ni =1 Q i + (cid:82) t n (cid:80) ni =1 q i,ns ds, n (cid:80) ni =1 q i,nt = − α − a n b n (cid:80) ni =1 Q q i,n T + (cid:82) Tt b (cid:16) − λσ s n (cid:80) ni =1 Q q i,n s + µ s + a ( n − n n (cid:80) ni =1 q i,ns (cid:17) ds + (cid:82) Tt n (cid:80) ni =1 Z i,ns dW s . Therefore, it holds that n (cid:80) ni =1 Q q i,n t − Q ∗ t = n (cid:80) ni =1 Q i − Q ∗ + (cid:82) t (cid:0) n (cid:80) ni =1 q i,ns − q ∗ s (cid:1) ds, n (cid:80) ni =1 q i,nt − q ∗ t = − α − a n b n (cid:80) ni =1 Q q i,n T + αb Q ∗ T − (cid:82) Tt λσ s b (cid:16) n (cid:80) ni =1 Q q i,n s − Q ∗ s (cid:17) ds + (cid:82) Tt b (cid:0) a (cid:0) n (cid:80) ni =1 q i,ns − q ∗ s (cid:1) − a n (cid:80) ni =1 q i,ns (cid:1) ds + (cid:82) Tt (cid:0) n (cid:80) ni =1 Z i,ns − Z ∗ s (cid:1) dW s . Thus, we get (cid:32) n n (cid:88) i =1 q i,nt − q ∗ t (cid:33) (cid:32) n n (cid:88) i =1 Q q i,n t − Q ∗ t (cid:33) (cid:32) − α − a n b n n (cid:88) i =1 Q q i,n T + αb Q ∗ T (cid:33) (cid:32) n n (cid:88) i =1 Q q i,n T − Q ∗ T (cid:33) − (cid:90) Tt λσ s b (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) ds + (cid:90) Tt b (cid:32) a (cid:32) n n (cid:88) i =1 q i,ns − q ∗ s (cid:33) − a n n (cid:88) i =1 q i,ns (cid:33) (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) ds − (cid:90) Tt (cid:32) n n (cid:88) i =1 q i,ns − q ∗ s (cid:33) ds + (cid:90) Tt (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) (cid:32) n n (cid:88) i =1 Z i,ns − Z ∗ s (cid:33) dW s . Therefore, we obtain e at b (cid:32) n n (cid:88) i =1 q i,nt − q ∗ t (cid:33) (cid:32) n n (cid:88) i =1 Q q i,n t − Q ∗ t (cid:33) = e aT b (cid:32) − α − a n b n n (cid:88) i =1 Q q i,n T + αb Q ∗ T (cid:33) (cid:32) n n (cid:88) i =1 Q q i,n T − Q ∗ T (cid:33) − (cid:90) Tt e as b λσ s b (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) ds − (cid:90) Tt e as b (cid:32) n n (cid:88) i =1 q i,ns − q ∗ s (cid:33) ds − (cid:90) Tt e as b a bn n (cid:88) i =1 q i,ns (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) ds + (cid:90) Tt e as b (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) (cid:32) n n (cid:88) i =1 Z i,ns − Z ∗ s (cid:33) dW s . Hence, it holds that0 ≥ E − e aT b ab (cid:32) n n (cid:88) i =1 Q q i,n T − Q ∗ T (cid:33) − (cid:90) T e as b λσ s b (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) ds − (cid:90) Tt e as b (cid:32) n n (cid:88) i =1 q i,ns − q ∗ s (cid:33) ds = E (cid:34)(cid:32) n n (cid:88) i =1 q i,n − q ∗ (cid:33) (cid:32) n n (cid:88) i =1 Q q i,n − Q ∗ (cid:33)(cid:35) + E (cid:34) e aT b a bn n (cid:88) i =1 Q q i,n T (cid:32) n n (cid:88) i =1 Q q i,n T − Q ∗ T (cid:33)(cid:35) − E (cid:34)(cid:90) T e as b a bn n (cid:88) i =1 q i,ns (cid:32) n n (cid:88) i =1 Q q i,n s − Q ∗ s (cid:33) ds (cid:35) which goes to 0 as n goes to infinity. Therefore, one could deduce that (cid:107) n n (cid:88) i =1 Q q i,n − Q ∗ (cid:107) S ( R ) + (cid:107) n n (cid:88) i =1 q i,n − q ∗ (cid:107) S ( R ) + (cid:107) n n (cid:88) i =1 Z i,n − Z ∗ (cid:107) H ( R d ) → n → (cid:107) Q q i,n − Q ˜ q i (cid:107) S ( R ) + (cid:107) q i,n − ˜ q i (cid:107) S ( R ) + (cid:107) Z i,n − ˜ Z i (cid:107) H ( R d ) → ≤ i ≤ n, as n → (cid:3) References [1] R. Almgren. Optimal execution with nonlinear impact functions and trading-enhanced risk.
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