Price formation and optimal trading in intraday electricity markets with a major player
PPrice formation and optimal trading in intradayelectricity markets with a major player
Olivier F´eron ∗ Peter Tankov † Laura Tinsi ∗ , † Abstract
We study price formation in intraday electricity markets in the pres-ence of intermittent renewable generation. We consider the setting wherea major producer may interact strategically with a large number of smallproducers. Using stochastic control theory we identify the optimal strate-gies of agents with market impact and exhibit the Nash equilibrium inclosed form in the asymptotic framework of mean field games with a ma-jor player. This is a companion paper to [12], where a similar model isdeveloped in the setting of identical agents.
Key words: Intraday electricity market; renewable energy; mean field games;major player
The structure of electricity markets around the world has been profoundly trans-formed by the push towards liberalization in the late 90s and more recently bythe massive arrival of renewable energy production. Distribution has been sep-arated from production, and whereas in the past, a single producer could ownthe entire generation capacity of a given country or region, now a patchwork ofsmall, often renewable, generators competes with a big historical producer.The aim of this paper is to develop an equilibrium model for intraday elec-tricity markets where a big producer with a significant market share competeswith a large number of small renewable producers. Both the large producer andthe small producers use the intraday markets to compensate their productionand demand forecast errors, creating feedback effects on the market price. Thelarge producer can act strategically, anticipating the impact of its decisions onthe market prices and thus on the behavior of the small agents. The small agentsare not strategic, and each one has a negligible effect on the market, howeverthe behavior of all small agents taken together has a significant market impact.The large player has the first-mover advantage but does not observe the forecastof the minor players. These in turn have the information advantage since they ∗ Electricit´e de France † CREST, ENSAE, Institut Polytechnique de Paris a r X i v : . [ q -f i n . P R ] N ov bserve the forecast of the major player as well as their own forecast. This leadsto a stochastic leader-follower game where players interact through the marketprice. We place ourselves in the linear-quadratic setting, exhibit the uniqueNash equilibrium for this game in closed form in the framework of mean fieldgames with a major player, and provide explicit formulas for the market priceand the strategies of the agents. For a game with a finite number of players, weshow how an ε -Nash equilibrium can be constructed from the mean field gamesolution.This paper is a companion paper to [12], where a similar model is devel-oped for the case of identical agents with symmetric interactions, and we referthe readers to that paper for a detailed review of literature on the stochas-tic and econometric modeling of intraday electricity markets. Here we simplymention that a similar linear-quadratic setting with linear market impact, hasbeen used to determine optimal strategies for a single energy producer by A¨ıd,Gruet and Pham [2] and Tan and Tankov [23], while Bouchard et al. [5] foundan analytic expression for the equilibrium price in a linear-quadratic model ofthe stock market with symmetric interactions and perfect information. We alsomention the recent paper of A¨ıd, Cosso and Pham [1] where an equilibriumin complete information setting for a finite number of agents is derived in theintraday electricity market. This paper is close in spirit to the complete infor-mation framework of [12], but allows to treat the case of heterogeneous agentsin conditions of uncertain production with possible outages and uncertain de-mand. However, the complete information setting, where each agent observesall other agents’ forecasts does not seem realistic in electricity markets. Theincomplete information setting, where each agent only observes its own forecastand the aggregate forecast, may not be tractable for a finite number of agents.Nevertheless, in [12], it has been shown that explicit solutions may be foundin the mean field limit, where the number of agents is sent to infinity, and theinfluence of every single agent on the entire market becomes negligible.The mean field games (MFG) are stochastic differential games with infinitelymany players and symmetric interactions. The seminal papers of Lasry and Li-ons [19] and Huang, Caines, and Malham´e [16] characterized the Nash equilib-rium in this framework through a coupled system of a Hamilton-Jacobi-Bellman(HJB) and a Fokker-Planck (FP) equation. Carmona and Delarue [7] developedan alternative probabilistic approach inspired by the Pontryagin principle andrelated the mean field game solution to a McKean-Vlasov Forward BackwardStochastic Differential Equation (FBSDE). The asymptotic results obtained inmean field games can be used to construct approximate equilibria ( ε -Nash equi-libria) for games with a finite number of players. Alternatively, equilibria of N -player games can be shown to converge to the corresponding weak mean fieldequilibria [18].While the original MFG setting involves symmetric agents, Huang [15] in-troduced linear-quadratic mean field games with a major player. Caines andNourian [21] developed this approach in a general framework. In both papers,the mean field is exogenous to the actions of the major player. In contrast tothese two papers, Bensoussan, Chau, and Yam in [4], and Carmona and Zhu2n [8], considered the endogenous case where the major player can influencethe mean field. In [4], this leads to a leader-follower setting, also known asStackelberg game. The authors derived a HJB equation and a FP equationto characterize the solution in the general case, while the linear quadratic set-ting was tackled with a stochastic maximum principle approach. More recently,Lasry and Lions [20], introduced a master equation accounting for this kind ofmajor player model. Cardaliaguet, Cirant and Poretta [6] showed that the twoprevious approaches ([20] and [8]) lead to the same Nash equilibria.Financial markets and energy systems with many small interacting agentsare a natural domain of applications of MFG. Casgrain and Jaimungal [9] ap-plied the MFG theory to optimal trade execution with price impact and termi-nal inventory liquidation condition, Fujii and Takahashi [14], used this theoryto find an equilibrium price under market clearing conditions. In [9, 14], theauthors used the extended mean field setting to deal with heterogeneous sub-populations of agents and incomplete information for [9]. Alasseur, Tahar andMatoussi [3] developed a model for the optimal management of energy storageand distribution in a smart grid system through an extended MFG. Shrivats,Firoozi and Jaimungal [22] recently applied the theoretical setting developedin [9] to the case of trading in solar renewable energy certificate markets. Fi-nancial markets with a major player, leader-follower interactions and terminalinventory constraint were recently analyzed in [13, 11]. In [13], the authorsconsider a Brownian filtration, impose a zero terminal inventory constraint andcharacterize the equilibrium in terms of a McKean-Vlasov FBSDE. In [11], theauthors study a market with a finite number of small players and a major playerwith first-mover advantage and information asymmetry, and characterize the so-lution in terms of a McKean-Vlasov FBSDE in a more general setting than thatof [13].Among the cited papers, our methods and findings are closest in spirit to[4, 11, 13]. Compared to the article [4], which of course solves a more generalproblem, without focusing on a specific application, our paper allows a muchmore general dynamics for the driving processes (general semimartingales) anddoes not require an a priori bound on the strategies to prove the existence ofthe Nash equilibrium in the presence of a major player. Unlike the articles [11,13], which also study leader-follower games in financial markets, we consider astochastic terminal constraint, characterize the equilibrium in explicit form, andshow how an ε -Nash equilibrium for the finite-player game may be constructedfrom a mean field game solution.The paper is organized as follows. In section 2 we introduce the model andbriefly recall the mean field game solution obtained in [12] in the case of identicalagents. In section 3, we present the main results of this paper in the settingallowing for the presence of a major player, whose influence on the marketis not negligible in the MFG limit. In section 4, we show how the limitingMFG solution may be used to construct an approximate Nash equilibrium ina Stackelberg game with one major player and N minor players. Finally, insection 5, equilibrium price trajectories, and the effect of market parameters onthe price characteristics are illustrated with simulated data.3 Preliminaries
In this paper, we place ourselves in the intraday market for a given delivery hourstarting at time T , where time 0 corresponds to the opening time of the market(in EPEX Intraday this happens at 3PM on the previous day). In reality, tradingstops a few minutes before delivery time (e.g. 5 minutes for Germany). However,for the sake of simplicity we assume that market participants can trade duringthe entire period [0 , T ]. In the market, there are agents (producers or consumers)who are assumed to have taken a position in the day-ahead market and usethe intraday market to manage the volume risk associated to the imperfectdemand/production forecast. These forecasts represent the best estimate of theadditional demand compared to the position taken by the agent in the spotmarket: to avoid imbalance penalties, the intraday position of the agent at thedelivery date must therefore be equal to the realized demand, or, in other words,the last observed value of the demand forecast.We consider the case of a Stackelberg game where an agent called ”majoragent” faces a large number of smaller agents called ”minor agents”. We directlyplace ourselves in the setting of mean field games with a major player, that is,we assume that the number of small agents in the market is infinite, and theinfluence of each small agent on the market is negligible. The aggregate impactof the minor agents on the market is therefore modelled through a mean field.Each agent observes the common national demand forecast, and the demandforecast of the major player. In addition, the small agents also observe theirindividual demand forecasts, which are not observed by the other agents. Thecommon filtration of the market thus contains the information about the forecastof the major player and the common part of the forecasts of the minor players,but the small agents benefit from a private information advantage compared tothe major player.The demand forecast process and the position of the generic minor agentare given, respectively, by X := ( X t ) ≤ t ≤ T and φ := ( φ t ) ≤ t ≤ T , while theforecast process and the position of the major agent are given, respectively, by( X t ) t ∈ [0 ,T ] and ( φ t ) t ∈ [0 ,T ] . Note that the position and forecast of the minorand major agents are not expressed in the same units. Indeed, in the mean fieldgame limit considered in this paper, we assume that the market is very large, sothat the position of every minor agent compared to the market size is negligible,but the major agent takes up a nonzero share of the market, so that φ and X denote the position and forecast of the major agent normalized by the marketsize.We denote by F the filtration which contains all information available tothe generic minor agent and by F the filtration which contains all informationavailable to the major agent. This filtration contains the information aboutthe fundamental price, the information about the demand forecast of the ma-jor agent, and potentially some information about the demand forecast of thegeneric minor agent (the common noise) but, in general, not the full individualdemand forecast of the generic agent.Throughout the paper and for any F -adapted process ( ζ t ) t ∈ [0 ,T ] , we will4enote ¯ ζ t = E [ ζ t |F t ] = (cid:82) R xµ ζt ( dx ) where: µ ζt := L ( ζ t |F t ). In view of theconvergence results of [12], the (normalized) aggregate position of all minoragents is given by the expectation of φ with respect to the common noise:¯ φ t = E [ X t |F t ].We assume that the market price ( P t ) t ∈ [0 ,T ] is given by the fundamentalprice ( S t ) t ∈ [0 ,T ] plus a weighted combination of the aggregate position of theminor agents and the position of the major agent: P t = S t + a ¯ φ t + a φ t , ∀ t ∈ [0 , T ]where a , a are positive weights, which reflect the size of the major agent relativeto the combined size of all minor agents and the overall strength of the marketimpact. Thus, the impact of each minor agent on the entire market is negligible,but the aggregate position of all minor agents, and the position of the majoragent both have a nonzero impact.We say that the strategy of the generic minor agent ( ˙ φ t ) t ∈ [0 ,T ] is admissibleif it is F -adapted and square integrable. Similarly, the strategy of the majoragent ( ˙ φ t ) t ∈ [0 ,T ] is admissible if it is F -adapted and square integrable. Theinstantaneous cost of trading for the major agent and for the generic minoragent are defined, respectively, by:˙ φ t P t + α ( t )2 ( ˙ φ t ) , and ˙ φ t P t + α ( t )2 ( ˙ φ t ) , ∀ t ∈ [0 , T ] (1)In both instantaneous costs, the first term represents the actual cost of buyingthe electricity, and the second term represents the cost of trading, where α ( . ) and α ( . ) are continuous strictly positive functions on [0, T] reflecting the variationof market liquidity at the approach of the delivery date.The objective function of the minor agent has the following form: J MF ( φ, ¯ φ, φ ) := − E (cid:34)(cid:90) T α ( t )2 ˙ φ t + ( S t + a ¯ φ t + a φ t ) ˙ φ t dt + λ φ T − X T ) (cid:35) , (2)while the objective function of the major agent writes, J MF, ( φ , ¯ φ ) := − E (cid:34)(cid:90) T α ( t )2 ˙ φ t + ( S t + aφ t + a φ t ) ˙ φ t dt + λ φ T − X T ) (cid:35) . (3)Note that this formulation implies (as it is the case in real markets) thatthe major agent pays a much lower trading cost per unit traded and a muchlower imbalance penalty than the minor agents. Indeed, if the major agent paidthe same quadratic cost/penalty as the minor agents, since the position of themajor agent is very large, the quadratic trading cost/penalty would grow muchfaster than the linear part (the middle term in the formula), and the limitingformula would be degenerate, in the sense that the trading strategy would beindependent from the price. To obtain a nondegenerate expression in terms ofthe normalized trading strategy of the major agent, we must therefore assume5hat the actual trading cost and penalty are also renormalized. The quantities α ( t ) and λ are thus different from α ( t ) and λ since they are of different nature: α ( t ) and λ apply to the actual strategy of the generic agent, while α ( t ) and λ apply to the normalized strategy of the major agent. The different natureof trading costs for minor and major agents is confirmed by other authors [10]:while the minor agents post their orders immediately in the order book, themajor agent splits its orders into many small chunks to minimize trading costs.To close this introductory section, we briefly recall the main result from [12],which characterizes the mean field equilibrium in the setting of identical agents,in other words, we assume that a = 0 until the end of this section. Definition 1 (mean field equilibrium) . An admissible strategy ˙ φ ∗ := ( ˙ φ ∗ t ) t ∈ [0 ,T ] is a mean field equilibrium in the setting of identical agents if it maximizes thefunctional (2) with a = 0 and satisfies ¯ φ = ¯ φ ∗ .We make the following assumption. Assumption 1. • The process S is square integrable and adapted to the filtration F . • The process X is a square integrable martingale with respect to the filtra-tion F . • The process X defined by X t := E [ X t |F t ] for 0 ≤ t ≤ T is a squareintegrable martingale with respect to the filtration F .Note that if X is an F -martingale, then X is by construction an F -martingale,but it may not necessarily be a martingale in the larger filtration F .The following theorem characterizes the mean field equilibrium in the identi-cal agent setting. In the theorem, we decompose the individual demand forecastas follows: X t = X t + ˇ X t , where X t = E (cid:2) X t |F t (cid:3) , and we use the followingshorthand notation:∆ s,t := (cid:90) ts η ( u, t ) α ( u ) du with η ( s, t ) = e − (cid:82) ts aα ( u ) du and (cid:101) ∆ s,t := (cid:90) ts α − ( u ) duI t := (cid:90) t η ( s, t ) α ( s ) S s ds, (cid:101) I t := E (cid:34)(cid:90) T η ( s, T ) α ( s ) S s ds (cid:12)(cid:12)(cid:12) F t (cid:35) . (4) Theorem 1.
Under Assumption 1, the unique mean field equilibrium in thesetting of identical agents is given by φ ∗ t = − I t + λ (cid:34) ∆ ,t (cid:101) I + X λ ∆ ,T + (cid:90) t ∆ s,t d (cid:101) I s + dX s λ ∆ s,T (5)+ (cid:101) ∆ ,t ˇ X λ (cid:101) ∆ ,T + (cid:90) t (cid:101) ∆ s,t d ˇ X s λ (cid:101) ∆ s,T (cid:35) . The equilibrium price has the following form: P t = S t − aI t + aλ (cid:34) ∆ ,t (cid:101) I + X λ ∆ ,T + (cid:90) t ∆ s,t d (cid:101) I s + dX s λ ∆ s,T (cid:35) . (6)6 A game of a major and minor agents
In this section we proceed to characterize the Nash equilibrium in the Stackel-berg mean field game with a major player. Since a single minor agent has aninfinitesimal impact on the market and cannot influence the mean field or thestrategy of the major agent, the problem of the generic minor agent is to max-imize J MF ( φ, ¯ φ, φ ) for fixed ¯ φ and φ . On the other hand, by modifying herstrategy φ , the major agent may influence the strategies of the minor agents,and thus also the mean field ¯ φ . This leads to the following definition of meanfield equilibrium. Definition 2 (Stackelberg mean field equilibrium) . We call the triple φ ∗ , ¯ φ ∗ , φ ∗ Stackelberg mean field equilibrium for the game with a major and minorplayers if the following holds:i. φ ∗ and φ ∗ are admissible strategies for, respectively, the representativeminor and the major players, the consistency condition ¯ φ ∗ t = E [ φ ∗ t |F t ] issatisfied for all t ∈ [0 , T ] and for any other admissible strategy for therepresentative minor player φ , J MF ( φ, ¯ φ ∗ , φ ∗ ) ≤ J MF ( φ ∗ , ¯ φ ∗ , φ ∗ )ii. For any other triple ( φ, ¯ φ, φ ) satisfying condition i., J MF, ( φ , ¯ φ ) ≤ J MF, ( φ ∗ , ¯ φ ∗ ) . (7) Assumption 2.
In addition to assumption 1, we also assume that the process X is a square integrable martingale with respect to the filtration F .We start with the characterization of the optimal strategy for the minoragent. Proposition 1 (Minor representative agent) . Let ¯ φ and φ be fixed. The minoragent strategy φ maximizes (2) over the set of admissible strategies if and onlyif: ˙ φ t = − Y t + S t + aφ t + a φ t α ( t ) , ∀ t ∈ [0 , T ] , (8) where Y is a F -martingale satisfying Y T = λ ( φ T − X T ) .Proof. The proof follows from the first step of the proof of Theorem 1 (seeTheorem 7 in [12]) taking (cid:101) S = S + a φ as fundamental price instead of S .The problem of the major agent is more complex since the minor agentsobserve her actions and modify their strategies accordingly, which means thatthe mean field ¯ φ depends on the major agent’s strategy φ , and the problemof the major agent effectively becomes a stochastic control problem. We startwith a reformulation of the definition of Stackelberg equilibrium in terms of ¯ φ and φ only. 7 emma 1. Let ( ¯ φ ∗ , φ ∗ ) be F -adapted square integrable processes. There exists φ ∗ such that ( φ ∗ , ¯ φ ∗ , φ ∗ ) is a Stackelberg mean field equilibrium if and only ifthe couple ( ¯ φ ∗ , φ ∗ ) satisfies the following conditions:i. For every F -adapted square integrable process ν , E (cid:34)(cid:90) T ν t { α ( t ) ˙¯ φ ∗ t + S t + a φ ∗ t + a ¯ φ ∗ t } dt + λ ( ¯ φ ∗ T − X T ) (cid:90) T ν t dt (cid:35) = 0 . ii. For every other couple ( φ , ¯ φ ) satisfying the condition i, the inequality (7) holds true.Proof. Assume first that ( φ ∗ , ¯ φ ∗ , φ ∗ ) is a Stackelberg mean field equilibrium.Then, For every F -adapted square integrable process ν , J MF ( φ ∗ + (cid:90) · ν s ds, φ ∗ , φ ∗ ) ≤ J MF ( φ ∗ , φ ∗ , φ ∗ ) . Developing the functionals we get, E (cid:90) T α ( t ) ν t dt + λ (cid:32)(cid:90) T ν t dt (cid:33) + E (cid:34)(cid:90) T ν t (cid:110) α ( t ) ˙ φ ∗ t + S t + a ¯ φ ∗ t + a φ ∗ t (cid:111) dt + λ ( φ ∗ T − X T ) (cid:90) T ν t dt (cid:35) ≥ , and since ν is arbitrary, we see that this is equivalent to E (cid:34)(cid:90) T ν t (cid:110) α ( t ) ˙ φ ∗ t + S t + a ¯ φ ∗ t + a φ ∗ t (cid:111) dt + λ ( φ ∗ T − X T ) (cid:90) T ν t dt (cid:35) = 0 . Taking conditional expectations and using Fubini’s theorem, we then get con-dition i. of the lemma.Assume now that conditions i. and ii. of the lemma hold true, and let φ ∗ begiven by Proposition 1 applied to the couple ( ¯ φ ∗ , φ ∗ ). Define ˜ φ ∗ t := E [ φ ∗ t |F t ].It remains to show that ˜ φ ∗ t = ¯ φ ∗ t . Let Y ∗ be an F -martingale satisfying Y ∗ T = λ ( ¯ φ ∗ T − X T ). By integration by parts, condition i. of the lemma is equivalent to E (cid:34)(cid:90) T ν t (cid:110) α ( t ) ˙¯ φ ∗ t + S t a φ ∗ t + a ¯ φ ∗ t + Y ∗ t (cid:111) dt (cid:35) = 0 , and since ν is arbitrary, α ( t ) ˙¯ φ ∗ t + S t + a φ ∗ t + a ¯ φ ∗ t + Y ∗ t = 0 , for all t . On the other hand, by Lemma 1, taking the expectation with respectto F , we get that there exists a F -martingale (cid:101) Y with (cid:101) Y T = λ ( ˜ φ T − X T ), andsuch that α ( t ) ˙˜ φ ∗ t + S t + a φ ∗ t + a ¯ φ ∗ t + (cid:101) Y t = 0 . α ( t )( ˙¯ φ ∗ t − ˙˜ φ ∗ t ) + Y ∗ t − (cid:101) Y t = 0 , Y ∗ T − (cid:101) Y T = λ ( ¯ φ ∗ T − ˜ φ ∗ T )Thus, ¯ φ ∗ t − ˜ φ ∗ t = (cid:90) t (cid:101) Y s − Y ∗ s α ( s ) ds and therefore, using the terminal condition and the martingale property, (cid:101) Y t − Y ∗ t = E [ (cid:101) Y T − Y ∗ T |F t ] = λ (cid:90) t (cid:101) Y s − Y ∗ s α ( s ) ds + λ ( (cid:101) Y t − Y ∗ t ) (cid:90) Tt dsα ( s ) . The unique solution of this linear equation is (cid:101) Y t = Y ∗ t for all t , and therefore˜ φ ∗ t = ¯ φ ∗ t for all t .The following proposition provides a martingale characterization of the Stack-elberg mean field equilibrium. Proposition 2.
Let ( ¯ φ ∗ , φ ∗ ) be F -adapted square integrable processes. Thereexists φ ∗ such that ( φ ∗ , ¯ φ ∗ , φ ∗ ) is a Stackelberg mean field equilibrium if andonly if ˙ φ ∗ t = − M t + S t + aφ t − a N t α ( t ) , ∀ t ∈ [0 , T ] , (9) where M is an F -martingale and N is an absolutely continuous F -adaptedprocess, and there exists an F -martingale M , and an F -martingale Y suchthat the following system of equations is satisfied: M T = a N T + a φ T + λ ( φ T − X T ) M t − aφ t + α ( t ) ˙ N t − aN t = 0 , M T = aφ T + ( a + λ ) N T Y t + α ( t ) ˙¯ φ t + S t + a ¯ φ t + a φ t = 0 , Y T = λ ( ¯ φ T − X T ) (10) Proof.
The optimization problem of the major agent consists in maximizing theobjective function (3) under the constraint of Lemma 1, part i. Let us introducethe Lagrangian for this constrained optimization problem, which writes: L ( φ , ¯ φ, ν ) = E (cid:34)(cid:90) T α ( t )2 ˙ φ t + ( S t + aφ t + a φ t ) ˙ φ t dt + λ φ T − X T ) (cid:35) + E (cid:34)(cid:90) T ν t (cid:110) α ( t ) ˙¯ φ t + S t + a φ t + a ¯ φ t (cid:111) dt + λ ( ¯ φ T − X T ) (cid:90) T ν t dt (cid:35) , where ν is a square integrable F -adapted process. We claim that φ is thesolution of the problem (3) if and only if there exist ν and ¯ φ such that ( φ , ¯ φ )maximizes the Lagrangian L ( · , · , ν ), and ¯ φ satisfies the constraint of Lemma 1.Indeed, let ( φ , ¯ φ, ν ) be such a triple and ( φ (cid:48) , ¯ φ (cid:48) ) be another pair of strategiessatisfying the constraint of Lemma 1. Then, L ( φ , ¯ φ, ν ) ≥ L ( φ (cid:48) , ¯ φ (cid:48) , ν ) , φ and ¯ φ (cid:48) satisfy the constraint of Lemma 1, this implies thatinequality (7) holds true.We now turn to the problem of maximizing the Lagrangian. Let N t = (cid:82) t ν s ds . The first order condition for φ writes: there exists a martingale M such that M t + α ( t ) ˙ φ t + S t + a ¯ φ t − a N t = 0 , M T = a N T + a φ T + λ ( φ T − X T ) . The first order condition for ¯ φ writes: there exists a martingale M such that M t − aφ t + α ( t ) ˙ N t − aN t = 0 , M T = aφ T + ( a + λ ) N T . Finally, the last condition is given by the constraint that ¯ φ is optimal for thegeneric minor agent. Hence conditioning (8) by the common noise, there existsa martingale Y such that Y t + α ( t ) ˙¯ φ t + S t + a ¯ φ t + a φ t = 0 , Y T = λ ( ¯ φ T − X T ) . The following theorem provides an explicit characterization of the equilib-rium in the Stackelberg setting.
Theorem 2 (Explicit solution) . Let Ξ t = ( φ t , N t , ¯ φ t ) (cid:48) . The unique equilibriumof the mean field game with a major agent is characterized by the followingdifferential equation: B ( t ) − A Ξ t + ˙Ξ t = − α ( t ) − ( M t + S t ) α ( t ) − M t α ( t ) − ( Y t + S t ) , (11) where N is a F -adapted process with N = 0 and M , M , Y are F -martingalesthat satisfy: M T = a N T + a φ T + λ ( φ T − X T ) M T = aφ T + ( a + λ ) N T Y T = λ ( ¯ φ T − X T ) (12) and A = − a a − a − a a a , B ( t ) = α ( t ) 0 00 α ( t ) 00 0 α ( t ) . Denoting by Φ( t ) the fundamental matrix solution of the equation B ( t ) − A Ξ t +˙Ξ t = 0 , the solution is given in the integral form by the following expression: Ξ t = Υ t − Π ,t ( I + D Π ,T ) − ( D (cid:101) Υ − Λ X ) − (cid:90) t Π s,t ( I + D Π s,T ) − ( Dd (cid:101) Υ s − Λ d X s ) . here Υ t := − Φ( t ) (cid:90) t Φ( s ) − α ( s ) − S s α ( s ) − S s ds, X s := X s X s ,D := a + λ a a a + λ
00 0 λ , Λ := λ λ , Π s,t := Φ( t ) (cid:90) ts Φ( u ) − B ( u ) − du and (cid:101) Υ t = E [Υ T |F t ] . Remark . If α ( t ) = cα ( t ) for some constant c , the fundamental matrix solutionis given explicitly by Φ( t ) = exp (cid:18) − (cid:90) t B ( s ) − Ads (cid:19)
Proof.
From equations (8) in Proposition 1 and (10) in Proposition 2, we imme-diately deduce the expression of the characterizing differential equation of theequilibrium (11).Let Φ( t ) be the fundamental matrix solution of the equation B ( t ) − A Ξ t +˙Ξ t = 0, that is, for every C ∈ R , the solution with initial condition Ξ = C isgiven by Φ( t ) C . By variation of constants we have that the solution of (11) isgiven by: Ξ t = − Φ( t ) (cid:90) t Φ( s ) − α ( s ) − ( M s + S s ) α ( s ) − M s α ( s ) − ( Y s + S s ) ds. Letting: M s := M s M s Y s and (cid:98) Ξ t = Ξ t − Υ t , we obtain the simplified equation: (cid:98) Ξ t = − Φ( t ) (cid:90) t Φ( s ) − B ( s ) − M s ds. and finally, using (12) and the martingale property, the martingale componentssatisfy: M t = − D Φ( T ) (cid:90) t Φ( s ) − B ( s ) − M s ds − D Π t,T M t + D (cid:101) Υ t − Λ X t , From this we deduce, on the one hand, M = ( I + D Π ,T ) − ( D (cid:101) Υ − Λ X ) , I + D Π t,T ) d M t = Dd (cid:101) Υ t − Λ d X t , so that finally:Ξ t = Υ t + (cid:90) t d Π s,t · M s = Υ t − Π ,t M − (cid:90) t Π s,t d M s = Υ t − Π ,t ( I + D Π ,T ) − ( D (cid:101) Υ − Λ X ) − (cid:90) t Π s,t ( I + D Π s,T ) − ( Dd (cid:101) Υ s − Λ d X s ) . Let us make some comments on how minor and major player strategieschange when the parameters of the model vary. First, when the major playerhas no price impact, a = 0, we recover the homogeneous mean field settingoptimal strategy for the minor player from (10) and (12):¯ φ t = (cid:90) t η ( s, t ) α ( s ) ( Y s + S s ) ds, where Y satisfies the equation: (cid:40) Y t + α ( t ) ˙¯ φ ∗ t + S t + a ¯ φ ∗ t = 0 Y T = − λ ( ¯ φ ∗ T − X T ) . Second, we explore the limiting behavior of the optimal strategies for the majoragent and for the mean field in various limiting cases. In this corollary, we usethe notation of Theorem 2.
Corollary 1. i. Assume that the fundamental price process S is a martingale. Then theequilibrium mean field position of minor agents, and the position of themajor agent satisfy Ξ t = − Π ,t ( I + D Π ,T ) − S − λ X S − λX − (cid:90) t Π s,t ( I + D Π s,T ) − d S s − λ X s S s − λX s . i. In the limit of infinite terminal penalty (when λ, λ → ∞ ) the equilibriummean field position of minor agents, and the position of the major agentsatisfy, Ξ t → Υ t − Π ,t Π − ,T ( (cid:101) Υ − D ∞ X ) − (cid:90) t Π s,t Π − s,T ( d (cid:101) Υ s − D ∞ d X s ) , almost surely for all t ∈ [0 , T ] , where D ∞ = . When the fundamental price process S is a martingale, in the limit ofinfinite terminal penalty, the strategies do not depend on the fundamentalprice and we have, Ξ t → Π ,t Π − ,T X X + (cid:90) t Π s,t Π − s,T d X s X s . iii. In the absence of terminal penalties (when λ = λ = 0 ), the equilibriummean field position of minor agents, and the position of the major agentsatisfy, Ξ t = Υ t − Π ,t ( I + D Π ,T ) − D (cid:101) Υ − (cid:90) t Π s,t ( I + D Π s,T ) − D d (cid:101) Υ s . (13) Proof.
The first part is a simplification of the proof of Theorem 2. For thesecond part, we can rewrite:Ξ t = Υ t − Π ,t ( I + D Π ,T ) − ( D (cid:101) Υ − Λ X ) − (cid:90) t Π s,t ( I + D Π s,T ) − ( Dd (cid:101) Υ s − Λ d X s )= Υ t − Π ,t ( D − + Π ,T ) − ( (cid:101) Υ − D − Λ X ) − (cid:90) t Π s,t ( D − + Π s,T ) − ( d (cid:101) Υ s − D − Λ d X s )and when λ, λ −→ ∞ , D − → D − Λ → D ∞ . The third part follows bydirect substitution of λ = λ = 0 into the general formula.Interestingly, when the players don’t have a terminal penalty ( λ = λ = 0),the equilibrium positions of the agents in equation (13) still contain forwardlooking terms, which were absent in the case of the mean field game with iden-tical players (see Equation (5) with λ = 0). The presence of these terms is due13o the strategic interaction of the major player with the mean field of smallagents.In the limit of zero trading costs, the gain of the major player remainsbounded in expectation, however, contrary to the case of identical players, theoptimal strategy of the major agent cannot be determined uniquely from theoptimization problem. Indeed, assuming that the trading cost for minor agents iszero, the equilibrium price (computed from Equation (12) in [12] with N → ∞ )is given by P t = S t + a φ t + a ¯ φ t = λa + λ ( aX t + E [ S T |F t ] + a E [ φ T |F t ]) . Substituting this expression into the optimization problem for the major player,we need to minimize the following functional: E (cid:34) λa + λ (cid:90) T ˙ φ t ( aX t + E [ S T |F t ] + a E [ φ T |F t ]) dt + λ φ T − X T ) (cid:35) = E (cid:20) λa + λ ( aφ T X T + φ T S T + a ( φ T ) ) + λ φ T − X T ) (cid:21) , where the equality follows, in particular, from the martingale property of X T .Since the expression to be minimized only depends on the terminal value φ T of the major agent’s position, any strategy with the optimal terminal valuewill satisfy the condition of optimality: the Stackelberg equilibrium will not beunique in this case.To finish this section, we provide the explicit form of the strategy of theminor agents. Corollary 2 (Minor agent strategy) . Under assumption 2, the optimal genericminor agent position φ ∗ is given by: φ ∗ t = (cid:90) t (cid:101) ∆ s,t λd ˇ X s λ (cid:101) ∆ s,T + (cid:101) ∆ ,t λ ˇ X λ (cid:101) ∆ ,T + ¯ φ ∗ t where ¯ φ ∗ is the optimal aggregate position of the minor agents, as given byTheorem 2.Proof. Let ˇ φ ∗ t = φ ∗ t − ¯ φ ∗ t , ˇ X t = X t − X t and ˇ Y t := Y t − Y t . Then, from theexplicit form of Y and Y in Proposition 1 it follows that ˇ Y is an F -martingaleand satisfies ˇ Y T = − λ ( ˇ φ ∗ T − ˇ X T ) , ˇ Y t = α ( t ) ˙ˇ φ ∗ t . Then, ˇ φ ∗ t = (cid:90) t ˇ Y s α ( s ) ds, (14)14nd by the martingale property, Y t = − λ E [ ˇ φ ∗ T − ˇ X T |F t ] = − λ (cid:90) t ˇ Y s α ( s ) ds − λ ˇ Y t (cid:90) Tt dsα ( s ) + λ ˇ X t . Solving this linear equation for ˇ Y then substituting into (14), we get the result. N -playerStackelberg game In this section, we derive the (cid:15) -Nash approximation for the Stackelberg game.In the present leader-follower setting, we allow the minor agents to change theirstrategies when the major agent deviates from her optimal one.Since we would like to study the rate of convergence as N → ∞ , we assumethat there is a major player and an infinity of minor players replacing the genericagent. Their demand forecasts are respectively given by X it , i = 0 , . . . , ∞ , t ∈ [0 , T ]. The private demand forecasts of all agents are defined on the sameprobability space. We therefore impose the following assumption. Assumption 3. • The process S is square integrable and adapted to the filtration F . • The demand forecast X of the major agent is a square integrable F -martingale. • The processes ( X i ) ∞ i =1 are square integrable F -martingales. • There exists a square intergrable F -martingale X , such that for all i ≥ t ∈ [0 , T ], almost surely, E [ X it |F t ] = X t . • The processes ( ˇ X i ) ∞ i =1 defined by ˇ X it = X it − X t for t ∈ [0 , T ], are orthog-onal square integrable F -martingales, such that the expectation E [( ˇ X iT ) ]does not depend on i .The strategy ( ˙ φ i ) of agent i = 1 , . . . , ∞ is said to be admissible if it is F -adapted and square integrable; the strategy ( ˙ φ ) of the major agent is admissibleif it is F -adapted and square integrable. For a fixed N ≥
1, we denote: P N ( φ t , ..., φ Nt ) = S t + aφ Nt + a φ t P MF ( φ t , ¯ φ t ) = S t + a ¯ φ t + a φ t , where ¯ φ Nt = N (cid:80) Ni =1 φ it is the average position of the minor agents. And wedefine in the N -player game, the objective functions for the major agent: J N, ( φ , φ − ):= − E (cid:34)(cid:90) T (cid:26) α ( t )2 ( ˙ φ t ) + ˙ φ t P N ( φ t , . . . , φ Nt ) (cid:27) dt + λ φ T − X T ) (cid:35) , (15)15nd for the minor agents i = 1 , . . . , N : J N,i ( φ i , φ − i ) := − E (cid:34)(cid:90) T (cid:26) α ( t )2 ( ˙ φ it ) + ˙ φ it P N ( φ t , . . . , φ Nt ) (cid:27) dt + λ φ iT − X iT ) (cid:35) , (16)as well as the objective function for the minor agents i = 1 , . . . , N , in the meanfield setting: J MF ( φ i , ¯ φ, φ ) := − E (cid:34)(cid:90) T (cid:26) α ( t )2 ( ˙ φ it ) + ˙ φ it P MF ( φ t , ¯ φ t ) (cid:27) dt + λ φ iT − X iT ) (cid:35) . (17)We next provide a definition of the (cid:15) -Nash equilibrium in the present Stack-elberg setting. As mentioned above, the deviations of the major and minoragents must be treated differently: when the major agent deviates, we allow theminor agents to adjust their strategies to respond optimally to the new strategyof the major agent. We say that the minor agent strategies φ , . . . , φ N are anoptimal response to the major agent strategy φ if for every i = 1 , . . . , N andfor every admissible minor agent strategy ˜ φ i , J N,i ( ˜ φ i , φ − i ) ≤ J N,i ( φ i , φ − i ) . Definition 3 (Stackelberg ε -Nash equilibrium) . We say that ( φ i ∗ t ) t ∈ [0 ,T ] , ≤ i ≤ N is an (cid:15) -Nash equilibrium for the N-player game if these strategies are admissibleand the following holds.i. Deviation of a minor player:
For any other admissible strategy φ i for the minor player i , i = 1 , . . . , N , J N,i ( φ i , φ − i ∗ ) − ε ≤ J N,i ( φ i ∗ , φ − i ∗ ) . ii. Deviation of the major player:
For any other set of admissible strate-gies ( φ i ) , i = 0 , . . . , N , such that φ , . . . , φ N are optimal responses of minorplayers to the major player strategy φ , we have, J N, ( φ , φ − ) − ε ≤ J N, ( φ ∗ , φ − ∗ ) . Our definition of ε -Nash equilibrium is different from the one in [8]: while thelatter paper assumes that the major player deviates from her strategy unilater-ally (see Definition 4.2), we allow the minor players to respond to the deviationof the major player, in agreement with the leader-follower nature of the game.In addition, in [8], an a priori bound on the L p -norm of the new strategy ofthe major agent is required to establish Theorem 4.1, whereas no such bound isneeded in our setting. Proposition 3.
Assume that the strategies of the N minor agents are given by φ i ∗ t = (cid:90) t (cid:101) ∆ s,t λd ˇ X is λ (cid:101) ∆ s,T + (cid:101) ∆ ,t λ ˇ X i λ (cid:101) ∆ ,T + φ ∗ t , here ¯ φ is the third component of the mean field equilibrium defined in Theorem2. Assume that the strategy of the major agent is given by Theorem 2 as well.Let assumption 3 hold true. Then there exists a constant C < ∞ , which doesnot depend on N , such that these strategies form an ε -Nash equilibrium of theN-player game with ε = CN / .Remark . The ε -Nash equilibrium described in Proposition 3 approximates the N -player equilibrium in the complete information setting (where every playerobserves the others’ actions), but its implementation for each agent only requiresthe knowledge of the common information F as well as the agent’s individualforecast. Proof.
We need to show conditions i. and ii. of Definition 3. Condition i. isshown in the same way as in the case of homogeneous players (see proof ofProposition 2 in [12]). We therefore focus on condition ii. Assume that allagents change their strategies to new ones φ , . . . , φ N , such that φ , . . . , φ N areoptimal responses to φ . Let ¯ φ be the optimal ”mean field” response to themajor agent strategy φ . Step 1.
We first suppose that there exists a constant
A >
0, such that E (cid:104)(cid:82) T ( ˙ φ t ) dt (cid:105) < A .By Proposition 1 in [12], for some constants c and C which do not dependon N , and may change from line to line, E (cid:34)(cid:90) T ( ¯ φ t − ¯ φ Nt ) dt (cid:35) ≤ CN E (cid:34)(cid:90) T ( S s + a φ s ) ds (cid:35) + cN , and by our assumption, E (cid:34)(cid:90) T ( S s + a φ s ) ds (cid:35) = E (cid:34)(cid:90) T S s ds (cid:35) + 2 a E (cid:34)(cid:90) T S s ds (cid:35) E (cid:34)(cid:90) T ( φ s ) ds (cid:35) + E (cid:34)(cid:90) T ( a φ s ) ds (cid:35) < C (1 + A ) . J N, ( φ , φ − ) − J N, ( φ ∗ , φ − ∗ )= J N, ( φ , φ − ) − J MF ( φ , ¯ φ ) + J MF ( φ , ¯ φ ) − J MF ( φ ∗ , ¯ φ ∗ )+ J MF ( φ ∗ , ¯ φ ∗ ) − J N, ( φ ∗ , φ − ∗ ) ≤ J N, ( φ , φ − ) − J MF ( φ , ¯ φ ) + J MF ( φ ∗ , ¯ φ ∗ ) − J N, ( φ ∗ , φ − ∗ ) ≤ a (cid:110) E (cid:34)(cid:90) T ( ˙ φ t ) dt (cid:35) E (cid:34)(cid:90) T ( ¯ φ t − ¯ φ Nt ) dt (cid:35) + E (cid:34)(cid:90) T ( ˙ φ ∗ t ) dt (cid:35) E (cid:34)(cid:90) T ( ¯ φ ∗ t − ¯ φ N ∗ t ) dt (cid:35) (cid:111) = O (cid:16) N − (cid:17) Step 2.
Letting ¯ α = min ≤ t ≤ T α ( t ), and ¯ α = min ≤ t ≤ T α ( t ), we have, bydefinition, J N, ( φ , φ − ) ≤ − E (cid:34)(cid:90) T ¯ α φ t ) + ˙ φ t ( S t + a ¯ φ Nt + a φ t ) dt + λ φ T − X T ) (cid:35) On the other hand, since φ i for i = 1 , . . . , N are optimal responses to φ , weget that E (cid:34)(cid:90) T (cid:26) α ( t )2 ( ˙ φ it ) + ˙ φ it ( S t + a ¯ φ Nt + a φ t ) (cid:27) dt + λ φ iT − X iT ) (cid:35) ≤ λ E [( X iT ) ] ≤ C, where C < ∞ is defined by C := max i λ E [( X iT ) ]. Summing up the aboveinequality over i = 1 , . . . , N , dividing by N and using Jensen’s inequality, weget E (cid:34)(cid:90) T (cid:110) ¯ α φ Nt ) + ˙¯ φ Nt ( S t + a ¯ φ Nt + a φ t ) (cid:111) dt + λ φ NT − X NT ) (cid:35) ≤ C. aa , adding it to the first one, and using integrationby parts, we finally get J N, ( φ , φ − ) ≤ C − E (cid:34) (cid:90) T (cid:26) ¯ α φ t ) + a ¯ α a ( ˙¯ φ Nt ) + S t a ( a ˙ φ t + a ˙¯ φ Nt ) (cid:27) dt + 12 a ( a ¯ φ NT + a φ T ) + λ φ T − X T ) + aλ a ( ¯ φ NT − X NT ) (cid:35) ≤ C − E (cid:34)(cid:90) T ¯ α φ t ) dt (cid:35) + E (cid:34)(cid:90) T S t dt (cid:35) E (cid:34)(cid:90) T ( ˙ φ t ) dt (cid:35) − E (cid:34)(cid:90) T a ¯ α a ( ˙¯ φ Nt ) dt (cid:35) + aa E (cid:34)(cid:90) T S t dt (cid:35) E (cid:34)(cid:90) T ( ˙¯ φ Nt ) dt (cid:35) ≤ C − E (cid:34)(cid:90) T ¯ α φ t ) dt (cid:35) + E (cid:34)(cid:90) T S t dt (cid:35) E (cid:34)(cid:90) T ( ˙ φ t ) dt (cid:35) + a ¯ αa E (cid:34)(cid:90) T S t dt (cid:35) Thus, if E (cid:34)(cid:90) T ( ˙ φ t ) dt (cid:35) > A, for A sufficiently large (but not depending on N ), then from the above estimateit follows that J N, ( φ , φ − ) ≤ J N, ( φ ∗ , φ − ∗ ) , as well. In this section, our objective is to illustrate the theoretical results presented insections 3 and 4 with numerical simulations. We analyze the role of the majorproducer in the market and its impact on price characteristics such as volatilityand price-forecast correlation, and compare this situation to the homogeneousagent setting studied in [12]. Some comparisons with the empirical market char-acteristics are also performed, but we refer the reader to [12] and other paperscited therein for a more detailed description of intraday electricity markets andtheir empirical features. As empirical analysis in [12] are led on actual windinfeed forecasts, we will consider production forecasts here instead of demandforecasts as it is the case in the rest of the paper. Throughout, we consider thatthe production forecasts are the differences between actual production forecastsand the agents’ positions in the market at time 0. Therefore, the initial values X i , i = 0 , . . . , N will be set to 0. 19 odel specification We now define the dynamics for the fundamental priceand for the production forecasts used in the simulations and specify the param-eter values. The objective being to illustrate the model, the majority of theparameters are not precisely estimated, but are given ad hoc plausible values.The evolution of the fundamental price is described as follows: dS t = σ S dW t (18)where σ s is a constant and ( W t ) t ∈ [0 ,T ] is Brownian motion. We also assumethat the liquidity functions α ( . ) and α ( . ) have a specific form given by α ( t ) = α × ( T − t ) + β, ∀ t ∈ [0 , T ] (19) α ( t ) = α × ( T − t ) + β , ∀ t ∈ [0 , T ] (20)where α, β , α and β are strictly positive constants. The liquidity functions arethus decreasing with time. This assumption relies on the fact that the marketbecomes more liquid as we get closer to the delivery time and it is less costly totrade when the market is liquid.To simulate production forecasts we assume the following dynamics: d ¯ X t = ¯ σd ¯ B t (21) d ˇ X t = σ dB t , (22) d ˇ X it = σ X dB it , i ∈ { , . . . , N } (23)where ¯ σ , σ and σ X are constants and ( ¯ B t ) t ∈ [0 ,T ] , ( B it ) t ∈ [0 ,T ] , i ∈ { , . . . , N } are independent Brownian motions, also independent from ( W t ) t ∈ [0 ,T ] .In this illustration, we choose the same parameters for the dynamics of thecommon and the individual production forecasts, as well as the forecast of themajor agent. The common volatility is calibrated to wind energy forecastsin Germany over January 2015 during the last quotation hour, by using theclassical volatility estimator¯ σ = σ = σ X = ˆ σ = √ ∆ tn (cid:48) − n (cid:48) (cid:88) i =1 Y i (24)with ∆ t the time step between two observations, Y i = X t i − X t i − the incre-ment between two successive observations and n (cid:48) the total number of observedincrements. As the forecasts are updated every 15 minutes, there are three dailyvariations during the last hour of forecasts from the 3 rd of January to the 31 th of January. Thus, for each delivery hour we dispose of n (cid:48) = 87 increments toestimate the volatility.The model parameters are specified in Table 1. Equilibrium price and market impact
In Figure 1, we plot the majoragent production forecast and the common production forecast (respectively theorange and blue solid lines) together with the equilibrium position of the major20arameter Value Parameter Value S e /MWh a e /MWh σ S e /MWh.h / λ e /MWh X λ e /MWh ¯ σ
73 MWh/h / α e /h.MW ˇ X i α e /h.MW σ X , σ
73 MWh/h / β e /MW N β e /MW Table 1: Parameters of the modelagent and the aggregate position of the minor agents given by Proposition 2 andProposition 2 (respectively the orange and blue dashed lines). For comparison,we also plot the aggregate position in the identical agent case (dotted greenline). All trajectories have been computed with the same production forecasts,the same fundamental price, initial values, volatilities and parameters, specifiedin Table 1, except for the price impact coefficients of major and minor player,which differ according to model specification. In the Stackelberg game we chose a = a = 0 . e /MWh , and in the homogeneous case we kept a = 1 e /MWh .Figure 1: Stackelberg gameWe observe that the strategy in the setting of identical agents and the strat-egy of the minor player in the Stackelberg setting converge to the same terminalvalue due to the terminal penalty. However, in the Stackelberg case, the minoragent position tends to follow the one of the major player during the first partof the trading period. In the case of identical agents, the fluctuations are not asstrong since, contrary to the case when a major agent is present, the generic mi-nor agent has no incentive to modify her trajectory to follow the leader. Duringthe second half of the trading period, the minor agent position deviates furtheraway from the one of the major agent to target the same terminal position asthe mean field in the case of identical agents. We can argue that the strategy21f the minor agent becomes more sensitive to the terminal constraint as we getcloser to the delivery time: the weight of the terminal constraint in her strategyincreases due to the decrease of the instantaneous trading cost. Volatility and price-forecast correlation
In this paragraph, we illustratewith simulations the effect of the presence of the major agent on the pricecharacteristics such as the volatility and the correlation between the price andrenewable infeed forecasts. The volatility was estimated from simulated pricetrajectories using a kernel-based non parametric estimator of the instantaneousvolatility: ˆ σ t = (cid:80) ni =1 K h ( t i − − t )∆ ˜ P t i − (cid:80) ni =1 K h ( t i − − t )( t i − t i − ) , (25)where K ( . ) is the Epanechnikov kernel: K ( x ) = (1 − x ) [ − , ( x ) and K h ( x ) = h K ( xh ). The parameter h was taken equal to 0 .
08 hour ( ≈ a and a assigned, respectively, to themajor player and the mean field of minor players in the price impact function.We studied three different combinations of weights to illustrate the impact ofthe minor players and the major player in the game: a = a = 0 . e /MWh , theimpact of the major player and the minor players is the same; a = 0 . , a = 0 . e /MWh , the major player has a lot more impact than the minor players, andfinally a = 0 e /MWh , a = 1 e /MWh , equivalent to a market price withoutmajor player since she has no market impact in this case. These weights canbe seen as the respective market shares held by the major agent and the minorplayers.Figure 2, left graph, shows the estimated volatility trajectories for the threedifferent cases of market shares of the major agent averaged over 1000 simula-tions. We note that the volatility of the market price depends on the strengthof impact of the major player: the greater a , the higher the volatility. A possi-ble explanation for this phenomenon is that stronger competition in the market(when the major agent is absent or has a small market share) reduces profitopportunities in the market and the agents therefore trade less actively.For comparison, we also plot in Figure 2, right graph, the volatility esti-mated from empirical intraday electricity price data using the same estimator(25). This graph is taken from [12]. We see that the phenomenon of increas-ing volatility at the approach of the delivery date, clearly visible in the actualelectricity markets, is well reproduced by our model.An important stylized feature of intraday market prices, observed empiricallyin [17] and [12] is the correlation between the price and the renewable productionforecasts. In Figure 3, we plotted the correlation between the increments ofthe market price and the increments of the renewable production forecast ofthe major agent as function of time, in the market impact setting a = a =0 . e /MWh ; as well as the correlation between the price increments and theincrements of the total aggregate forecast of both the major and minor players.22igure 2: Left: volatility of simulated prices for different market shares of themajor agent. Right: volatility for different delivery hours, estimated empiricallyfrom EPEX spot intraday market data of January 2017 for the Germany deliveryzone.The correlation is computed over 15-minutes increments using the followingestimator: ˆ ρ t = (cid:80) N sim k =1 (∆ Y kt − ∆ Y t )(∆ P kt − ∆ P t ) (cid:113)(cid:80) N sim k =1 (∆ Y kt − ∆ Y t ) (cid:80) N sim k =1 (∆ P kt − ∆ P t ) , with N sim the number of simulations (we considered N sim = 50000) and where∆ Y kt and ∆ P kt = P MF,kt + dt − P MF,kt are the increments of, respectively, the forecastprocess and the market price.Figure 3: Correlation between the price increments and the major player re-newable production increments v.s the correlation between the price incrementsand the total renewable production incrementsFor the sake of clarity we only draw the Monte Carlo confidence interval forthe case of the correlation between the major player production and the priceconsidered on Figure 3. A similar confidence interval was obtained for the caseof total production correlation. In Figure 3, we observe that the correlationbetween the production forecast increments of the major agent and the price23s lower in absolute value than the correlation between the total productionforecast increments and the price. However, the gap between the correlationsdiminishes as we approach the delivery time.
Acknowledgements
The authors gratefully acknowledge financial support from the ANR (projectEcoREES ANR-19-CE05-0042) and from the FIME Research Initiative.
References [1]
R. A¨ıd, A. Cosso, and H. Pham , Equilibrium price in intraday electricitymarkets . arXiv preprint arXiv:2010.09285, 2020.[2]
R. A¨ıd, P. Gruet, and H. Pham , An optimal trading problem in intra-day electricity markets , Mathematics and Financial Economics, 10 (2016),pp. 49–85.[3]
C. Alasseur, I. Ben Taher, and A. Matoussi , An extended meanfield game for storage in smart grids , Journal of Optimization Theory andApplications, 184 (2020), pp. 644–670.[4]
A. Bensoussan, M. Chau, and S. Yam , Mean field games with a domi-nating player , Applied Mathematics & Optimization, 74 (2016), pp. 91–128.[5]
B. Bouchard, M. Fukasawa, M. Herdegen, and J. Muhle-Karbe , Equilibrium returns with transaction costs , Finance and Stochastics, 22(2018), pp. 569–601.[6]
P. Cardaliaguet, M. Cirant, and A. Porretta , Remarks on Nashequilibria in mean field game models with a major player , Proceedings ofthe American Mathematical Society, 148 (2020), pp. 4241–4255.[7]
R. Carmona and F. Delarue , Probabilistic Theory of Mean FieldGames with Applications I , Springer, 2018.[8]
R. Carmona, X. Zhu, et al. , A probabilistic approach to mean fieldgames with major and minor players , The Annals of Applied Probability,26 (2016), pp. 1535–1580.[9]
P. Casgrain and S. Jaimungal , Mean-field games with differing beliefsfor algorithmic trading , Mathematical Finance, 30 (2020), pp. 995–1034.[10]
J. Donier, J. Bonart, I. Mastromatteo, and J.-P. Bouchaud , Afully consistent, minimal model for non-linear market impact , Quantitativefinance, 15 (2015), pp. 1109–1121.2411]
D. Evangelista and Y. Thamsten , Finite population games of optimalexecution . arXiv preprint arXiv:2004.00790, 2020.[12]
O. F´eron, P. Tankov, and L. Tinsi , Price formation and optimal trad-ing in intraday electricity markets . arXiv preprint arXiv:2009.04786, 2020.[13]
G. Fu and U. Horst , Mean-field leader-follower games with terminalstate constraint , SIAM Journal on Control and Optimization, 58 (2020),pp. 2078–2113.[14]
M. Fujii and A. Takahashi , A mean field game approach to equilibriumpricing with market clearing condition . CARF Working Paper CARF-F-473, 2020.[15]
M. Huang , Large-population lqg games involving a major player: the nashcertainty equivalence principle , SIAM Journal on Control and Optimiza-tion, 48 (2010), pp. 3318–3353.[16]
M. Huang, R. P. Malham´e, and P. E. Caines , Large populationstochastic dynamic games: closed-loop McKean-Vlasov systems and thenash certainty equivalence principle , Communications in Information &Systems, 6 (2006), pp. 221–252.[17]
R. Kiesel and F. Paraschiv , Econometric analysis of 15-minute intra-day electricity prices , Energy Economics, 64 (2017), pp. 77–90.[18]
D. Lacker , On the convergence of closed-loop Nash equilibria to the meanfield game limit , Ann. Appl. Probab., 30 (2020), pp. 1693–1761.[19]
J.-M. Lasry and P.-L. Lions , Mean field games , Japanese journal ofmathematics, 2 (2007), pp. 229–260.[20] ,
Mean-field games with a major player , Comptes Rendus Mathema-tique, 356 (2018), pp. 886–890.[21]
M. Nourian and P. E. Caines , (cid:15) -nash mean field game theory for non-linear stochastic dynamical systems with major and minor agents , SIAMJournal on Control and Optimization, 51 (2013), pp. 3302–3331.[22] A. Shrivats, D. Firoozi, and S. Jaimungal , A mean-field game ap-proach to equilibrium pricing, optimal generation, and trading in solar re-newable energy certificate (srec) markets . arXiv preprint arXiv:2003.04938,2020.[23]