Price formation and optimal trading in intraday electricity markets
PPrice formation and optimal trading in intraday electricitymarkets ∗Olivier Féron † , Peter Tankov ‡ , Laura Tinsi † , ‡ Abstract
We develop a tractable equilibrium model for price formation in intraday electricitymarkets in the presence of intermittent renewable generation. Using stochastic controltheory we identify the optimal strategies of agents with market impact and exhibitthe Nash equilibrium in closed form for a finite number of agents as well as in theasymptotic framework of mean field games. Our model reproduces the empiricalfeatures of intraday market prices, such as increasing price volatility at the approachof the delivery date and the correlation between price and renewable infeed forecasts,and relates these features with market characteristics like liquidity, number of agents,and imbalance penalty.
Key words: Intraday electricity market, renewable energy, mean-field games
The electricity markets around the world are undergoing a major transformation drivenby the transition towards a carbon-free energy system. The increasing penetration of inter-mittent renewables puts a stronger emphasis on short-term electricity trading and balanc-ing. The intraday electricity markets are increasingly used by the renewable producers tocompensate forecast errors. This improves market liquidity and at the same time createsfeedback effects of the renewable generation on the market price, leading to increased pricevolatility and negative correlations between renewable infeed and prices. These effectshave an adverse impact on the revenues of renewable producers. They are already signif-icant in countries with high renewable penetration and will become even more importantas new renewable capacity comes online. A better understanding of the impact of inter-mittent renewable generation on intraday electricity market prices and trading volumes istherefore needed to ensure the long-term economic sustainability of the renewable energyproduction.In this paper, we build an equilibrium model for the intraday electricity market, aimingto understand the price formation and identify the optimal strategies for market partici-pants in the setting where both the strategies of the agents and the demand or generationforecasts may affect market prices. We consider an intraday electricity market, where theparticipants optimize their revenues based on imperfect forecasts of terminal demand orproduction. We place ourselves in the standard linear-quadratic setting with quadratic ∗ The authors gratefully acknowledge financial support from the ANR (project EcoREES ANR-19-CE05-0042) and from the FIME Research Initiative. † Electricité de France ‡ ENSAE, Institut Polytechnique de Paris a r X i v : . [ q -f i n . P R ] S e p rading costs and linear market impact. The actions of each agent therefore impact mar-ket prices, leading to a stochastic game where players interact through the market price.We exhibit a closed-form Nash equilibrium for this game, and provide explicit formulas forthe market price and the strategies of the agents under two different settings: · the setting of N identical agents, having complete information about the forecasts ofthe other agents (perfect information), · the setting of an infinite number of identical small agents (the mean field), whereeach agent only observes the aggregate forecast as well as its own forecast (imperfectinformation).We then show by theoretical analysis and through numerical simulations that our modelreproduces the stylized features of the market price, which we document empirically. Inparticular, · the market price becomes more volatile at the approach of the delivery time, · the market price exhibits negative correlation with the total renewable infeed forecast,which grows in absolute value at the approach of the delivery time.Furthermore, our model provides direct quantitative links between market characteristicsand market price features, as well as the gain of individual agents. For instance, · observed price volatility decreases with market depth (inverse of permanent marketimpact), · price volatility increases for lower instantaneous trading costs, which allow agents totrade more actively, · increased competition (greater number of agents in the market) limits profit oppor-tunities for individual agents and also leads to lower price volatility.Correlations between renewable infeed and intraday market prices have been studiedempirically by a number of authors. Kiesel and Paraschiv [19] perform an econometricanalysis of the German intraday market and show that a deeper penetration of renewableenergies increases market liquidity and price-infeed correlations. The wind power outputforecast errors thus turn out to be of paramount importance in explaining the price dif-ferences between the day ahead and intraday prices. Karanfil and Li [18] draw similarconclusions from an empirical study of the Danish market, and exhibit the impact of re-newable energies on prices, bid-ask spread and volatility. Gruet, Rowińska and Veraart [24]establish a negative correlation between the wind energy penetration and the day aheadmarket prices. They also show that taking into account wind energy generation improvesgoodness of fit in their multifactor model for the German and Austrian day ahead market.In addition to creating a negative correlation between the renewable infeeds and the spotprices, Jonsson, Pinson and Madsen [17] show that a deeper penetration of the intermittentenergies modifies significantly the distribution of spot prices.Optimal strategies in the intraday market for a single wind energy producer have alsobeen the object of studies both in the price-taker and price-maker context. In the price-taker setting, Garnier and Madlener [13] solve a discrete-time optimal trading problemto arbitrate between immediate and delayed trading when price and production forecastsare uncertain. In [23], Morales, Conejo and Perez-Ruiz consider a multimarket setting2o derive an optimal bidding strategy for a wind energy producer in the day ahead andadjustment markets, while minimizing the cost incurred in the balancing market. Discretedecisions are taken for each delivery period, considering a finite number of probable sce-narii. This approach has been enhanced by Zugno, Morales, Pinson and Madsen, in [26],where the wind energy producer is now price maker in the balancing market. Following thesame framework, Delikaraoglou, Papakonstantinou, Ordoudis and Pinson [11] formulate aproblem where the renewable producer is price maker in both the day ahead and balancingmarkets and assess the relevance of strategic behaviour in the context of high renewablepenetration and varying flexible capacities. Still in the price-maker setting, continuous-time approaches have also been developed. Aïd, Gruet and Pham [1], consider the optimaltrading rate and power generation of a thermal producer when the residual demand at theterminal date is random. In the same trend, Tan and Tankov [25] develop an optimal trad-ing model for a wind energy producer. They quantify the evolution of forecast incertaintyat the approach of the delivery time, and exhibit optimal strategies depending on forecastupdates.In our study, the uncertain renewable production is also a source of randomness, andthe producers’ trading decisions impact the market. Unlike the previous papers, we con-sider the equilibrium setting with many agents and determine the market price as the resultof their interaction. Explicit results for dynamic equilibria are often difficult to obtain. Inparticular, Nash equilibria often lead to systems of coupled partial differential equations.However, the linear-quadratic setting proposed by Bouchard et al. [7], allows to find ana-lytic solutions in the case of perfect information. In the imperfect information setting, theproblem may be simplified by assuming a continuum of agents and using the mean fieldgame approach.The mean field games are stochastic differential games with a large number of symmet-ric agents, which were originally studied by Lasry and Lions [22] and Huang, Caines, andMalhamé [15]. The equilibrium of such a game is characterized through a coupled systemof a Hamilton-Jacobi-Bellman and a Fokker-Planck equation. Carmona and Delarue [9]proposed an alternative way to formalize the system inspired by the Pontryagin principleand relating the mean field game solution to a McKean-Vlasov Forward Backward Stochas-tic Differential Equation (FBSDE). From the mean field game solution one can derive an ε -Nash equilibrium of the corresponding N -player game ( ε -Nash MFG). In an alternativeapproach, Lacker [21], worked on limiting properties of N -player games and proved in somecases the convergence of N -player Nash equilibria to weak mean field Equilibria (MFE).Financial markets and energy systems with many small interacting agents are a naturaldomain of applications of MFG. Alasseur, Tahar and Matoussi [2] develop a model for theoptimal management of energy storage and distribution in a smart grid system through anextended MFG. Casgrain and Jaimungal [10] recently applied the MFG theory to optimaltrade execution with price impact and terminal inventory liquidation condition. They usedthe extended mean field setting to deal with incomplete information and heterogeneoussub-populations of agents.As in [10], we distinguish between private state variables of the agents and the com-mon information shared by all participants in the market. Unlike [10], the final tradingconstraint in our paper is given by the stochastic production / demand forecast ratherthan by an inventory liquidation condition. From the mathematical point of view, linearquadratic mean-field games have been studied, e.g., in [5] in the diffusion setting. Whileour approach is naturally less general in terms of model specification since we focus ona specific model for the electricity market, it is more general in terms of the underlyinginformation structure, since we allow a fully general adapted square integrable process forthe fundamental price and general square integrable martingales for the forecast processes.3he paper is structured as follows. Section 2 describes the market and introduces somenotions common for both settings considered in the paper. In section 3 we place ourselvesin a complete information setting with a finite number of agents, where all agents observethe forecasts of the other agents. In section 4 we assume that the agents do not observe eachother’s individual demand forecasts. This leads to an incomplete information setting inwhich the N -player equilibrium is intractable. We therefore consider the associated meanfield game. To make a connection between the N -agent setting and the mean-field gamesetting, we show in section 5 that (i) the N -player equilibrium converges to the mean-fieldequilibrium as N → ∞ , and (ii) an ε -Nash equilibrium for the N -player problem may beconstructed from the mean-field equilibrium. In section 6 we perform empirical analysisof intraday market and confront it to the theoretical results obtained in the precedingsections. In this paper we consider a unique class of agents: the small market participants withidentical characteristics. These agents are assumed to have taken a position in the dayahead market, and use the intraday market to manage the volume risk associated to theimperfect production /demand forecast used for their day ahead market bid. While ourprimary interest is to study the impact of increasing renewable penetration on intradaymarket prices, the market participants may in principle represent both renewable producerswith uncertain generation forecasts and industrial consumers with uncertain demand. Tosimplify the language and notation, in the sequel, unless specified otherwise, we will referto forecasts of all agents as demand forecasts. These forecasts represent the best estimateof the additional demand compared to the position taken by the agent in the spot market:if the agent is expected to produce less, or consume more than the day ahead position, theforecast is positive. To avoid paying the imbalance penalties, the position taken by theagent in the intraday market by the delivery date must therefore be equal to the realizeddemand.Throughout the paper we place ourselves in the intraday market for a given deliveryhour starting at time T , where time corresponds to the opening time of the market (inEPEX Intraday this happens at 3PM on the previous day). In reality, trading stops afew minutes before delivery time (e.g. 5 minutes for Germany). However, for the sake ofsimplicity we assume that market participants can trade during the entire period [0 , T ] ,and that the last observed value of the demand forecast equals the realized demand. In this section we assume that in the market there are N identical agents, and we denoteby φ it the position of i -th agent at time t . As is common in optimal execution literature, weassume that the position of i -th agent is an absolutely continuous process, and we definethe rate of trading ˙ φ it . We introduce a filtered probability space (Ω , F , F := ( F t ) t ∈ [0 ,T ] , P ) to which all processes are adapted, and which models the information available in themarket to all the agents. Without loss of generality, we assume that φ i = 0 for all i sothat the position of the i -th agent at time t is given by φ it = (cid:82) t ˙ φ is ds . As the agents’strategies may impact the market price, we distinguish the price without price impact orfundamental price ( S t ) t ∈ [0 ,T ] from the market price ( P Nt ) t ∈ [0 ,T ] , where the market impact4s included. The strategies impact the market price P Nt as follows: P Nt = S t + a ¯ φ Nt , ∀ t ∈ [0 , T ] , (1)where ¯ φ Nt = N (cid:80) Ni =1 φ it is the average position of the agents and a is a constant. Theparameter N describes the size of the market (number of agents), it is therefore naturalthat the trading strategy of each agent has an effect of order of /N on the market price.The permanent component of the price impact of trades in our model is thus linear, whichis the only shape compatible with the absence of arbitrage, see [16, 14]. On the other hand,the transient component of market impact is not modelled directly. Literature on marketmicrostructure mostly shows that metaorders have a concave transient impact on prices(see Bershova and Rakhlin [6], Bacry et al. [4], and Bouchaud [8]). However, for the sake ofsimplicity and in order to provide an analytical solution for our model, we choose a linearimpact function as in the seminal paper by Almgren and Chriss [3] and more recently inAïd et al. [1], and the transient component of the market impact is taken into accountindirectly, via a trading cost penalty.The agents trading in the market at time t incur an instantaneous cost, ˙ φ it P Nt + α ( t )2 ( ˙ φ it ) , ∀ t ∈ [0 , T ] for the i -th agent. Here the first term represents the actual cost of buying the electricity,and the second term represents the cost of trading, where α ( . ) is a continuous strictlypositive function on [0 , T ] reflecting the variation of market liquidity at the approach ofthe delivery date. The instantaneous cost paid by each agent is thus independent of thesize of the market. This corresponds to a market where immediately available liquidity(market depth) is low (thus even a minor agent has to pay order book costs) but the orderbook is resilient (thus the trade of a minor agent only has a lasting impact of order of /N on the price). This is consistent with recent empirical and theoretical studies of orderbook dynamics, for example, according to [12], while the total daily volume exchange ona typical stock is around 1/200th of its market capitalisation, the volume present in theorder book at any instant in time is 1000 times smaller than this.Each agent i has a demand forecast X it and aims to maximise her gain from tradingin the market under the volume constraint φ iT = X iT , where X iT represents the differencebetween the actual realized demand and the position in the day ahead market, which mustbe compensated with the position in the intraday market. This constraint may be enforcedas a hard constraint or as a penalty. The penalty formulation may be more realistic sincein practice when there is a mismatch between the realized demand and the aggregatepositions taken in the market, the agents pay a penalty based on the imbalance price.The processes S and ( X i ) Ni =1 satisfy the following assumption. In this assumption andbelow, we say that the process ξ is square integrable if E (cid:104)(cid:82) T ξ t dt (cid:105) < ∞ . Assumption 1.
The process S is F -adapted and square integrable, and the processes ( X i ) Ni =1 are square integrable F -martingales. Considering the demand forecast as a martingale is natural since it is the best estimateat time t of what the demand will be at the delivery time T given our current knowledge F t . We say that the strategy ( ˙ φ it ) t ∈ [0 ,T ] of the i th agent is admissible if it is F -adapted andsquare integrable. 5ach agent wishes to maximize the objective function J N,i ( φ i , φ − i ) := − E (cid:20)(cid:90) T (cid:26) α ( t )2 ( ˙ φ it ) + ˙ φ it P Nt (cid:27) dt + λ φ iT − X iT ) (cid:21) , (2)where λ is the strength of the imbalance penalty and φ − i := ( φ , . . . , φ i − , φ i +1 , . . . , φ N ) isthe vector of positions of all agents except the i th one. Remark that to simplify notation,we write the objective function in terms of agents’ positions φ i rather than their tradingstrategies ˙ φ i .Because of the price impact, each agent’s gain is affected by the decisions of others andwe thus face a non-cooperative game. The optimal strategy of each player depends on theother players’ actions and we want to describe the resulting dynamical equilibrium, whichwe define formally below. Definition 2 (Nash Equilibrium) . We say that ( ˙ φ i ∗ t ) i =1 ...Nt ∈ [0 ,T ] is a Nash Equilibrium for theN-player game if it is a vector of admissible strategies, and for each i = 1 , . . . , N , J N,i ( φ i , φ − i ∗ ) ≤ J N,i ( φ i ∗ , φ − i ∗ ) (3) for any other admissible strategy ˙ φ i of player i in the market. In other words, in the situation of Nash equilibrium, the strategy ˙ φ i ∗ used by eachagent is this agent’s best response to the strategies ˙ φ − i ∗ of all other agents.The following theorem characterizes the Nash equilibrium of the N -player game. Inthe theorem and its proof, we denote the average forecast process by X Nt := N (cid:80) Ni =1 X it and use the following shorthand notation. ∆ Ns,t := (cid:90) ts η Nu,t α ( u ) du with η Ns,t = e − (cid:82) ts ( N − aNα ( u ) du (cid:101) ∆ s,t := (cid:90) ts ˜ η Nu,t α ( u ) du, with ˜ η Ns,t = e (cid:82) ts aNα ( u ) duI Nt := (cid:90) t η Ns,t α ( s ) S s ds, (cid:101) I Nt := E (cid:34)(cid:90) T η Ns,T α ( s ) S s ds (cid:12)(cid:12)(cid:12) F t (cid:35) . (4) Theorem 3.
Under Assumption 1, the unique Nash equilibrium in the complete informa-tion N -player game is given by φ i ∗ t = − I Nt + ∆ N ,t (cid:0) aN + λ (cid:1) (cid:101) I N + λX N (cid:0) aN + λ (cid:1) ∆ N ,T + (cid:90) t ∆ Ns,t (cid:0) aN + λ (cid:1) d (cid:101) I Ns + λdX Ns (cid:0) aN + λ (cid:1) ∆ Ns,T + (cid:90) t (cid:101) ∆ s,t λd ( X is − X Ns )1 + (cid:0) aN + λ (cid:1) (cid:101) ∆ s,T + (cid:101) ∆ ,t λ ( X i − X N )1 + (cid:0) aN + λ (cid:1) (cid:101) ∆ ,T . (5) The equilibrium price has the following shape: P Nt = S t − aI Nt + a ∆ N ,t (cid:0) aN + λ (cid:1) (cid:101) I N + λX N (cid:0) aN + λ (cid:1) ∆ N ,T + a (cid:90) t ∆ Ns,t (cid:0) aN + λ (cid:1) d (cid:101) I Ns + λdX Ns (cid:0) aN + λ (cid:1) ∆ Ns,T . (6) Proof.
Step 1. First order condition of optimality for a single agent. In this step, we aregoing to show that for fixed φ − i ∗ , the strategy φ i ∗ satisfies (3) if and only if there exists asquare integrable F -martingale Y i such that, almost surely, Y it + α ( t ) ˙ φ i ∗ t + S t + a ¯ φ N ∗ t − aN φ i ∗ t = 0 , ≤ t ≤ T and Y iT = aN φ i ∗ T + λ ( φ i ∗ T − X iT ) . (7)6ssume that φ i ∗ satisfies (3). Then, for any adapted square integrable process ( ν t ) ≤ t ≤ T , J N,i ( φ i ∗ + (cid:90) · ν s ds, φ − i ∗ ) ≤ J N,i ( φ i ∗ , φ − i ∗ ) . Developing the expressions, this is equivalent to E (cid:34) (cid:90) T α ( t ) ν t dt + (cid:18) aN + λ (cid:19) (cid:18)(cid:90) T ν t dt (cid:19) (cid:35) + E (cid:20)(cid:90) T ν t (cid:110) α ( t ) ˙ φ i ∗ t + S t + a ¯ φ N ∗ t − aN φ i ∗ t (cid:111) dt + (cid:16) aN φ i ∗ T + λ ( φ i ∗ T − X iT ) (cid:17) (cid:90) T ν t dt (cid:21) ≥ , and since ν is arbitrary, we see that optimality is equivalent to E (cid:20)(cid:90) T ν t (cid:110) α ( t ) ˙ φ i ∗ t + S t + a ¯ φ N ∗ t − aN φ i ∗ t (cid:111) dt + (cid:16) aN φ i ∗ T + λ ( φ i ∗ T − X iT ) (cid:17) (cid:90) T ν t dt (cid:21) = 0 , (8)for any adapted square integrable ν . Now, assume that Y i is a square integrable martingalesatisfying (7). Then, by integration by parts, the expression in the previous line equals E (cid:20) − (cid:90) T ν t Y t dt + Y T (cid:90) T ν t dt (cid:21) = E (cid:20)(cid:90) T (cid:18)(cid:90) t ν s ds (cid:19) dY t (cid:21) = 0 , and we see that the optimality condition is satisfied. Conversely, assume that (8) is satisfiedfor any adapted square integrable process ν , and let Y i be a martingale such that Y iT = aN φ i ∗ T + λ ( φ i ∗ T − X iT ) . Then, by integration by parts, (8) is equivalent to E (cid:20)(cid:90) T ν t (cid:110) α ( t ) ˙ φ i ∗ t + S t + a ¯ φ N ∗ t − aN φ i ∗ t + Y it (cid:111) dt (cid:21) = 0 , and since ν is arbitrary, we see that (7) is satisfied.Step 2. Computing the average position. Let ( φ i ∗ ) i =1 ,...,N be a Nash equilibrium. Wehave seen that this is equivalent to (7) for i = 1 , . . . , N . Summing up these expressions for i = 1 , . . . , N and denoting Y Nt = N (cid:80) Ni =1 Y it , we get Y Nt + α ( t ) ˙¯ φ N ∗ t + S t + a N − N ¯ φ N ∗ t = 0 , ≤ t ≤ T and Y NT = aN ¯ φ N ∗ T + λ ( ¯ φ N ∗ T − X NT ) . The first equation can be solved explicitly for ¯ φ N ∗ : ¯ φ N ∗ t = − (cid:90) t η N ( s, t ) Y Ns + S s α ( s ) ds. (9)Denoting ˆ φ t := ¯ φ N ∗ t + I Nt , we obtain simplified equations: ˆ φ t = − (cid:90) t η N ( s, t ) Y Ns α ( s ) ds, Y NT = ( aN + λ ) ˆ φ T − ( aN + λ ) I NT − λX NT . Substituting ˆ φ T into the second equation and taking the expectation, we obtain anotherlinear equation, this time for Y Nt : Y NT = − ( aN + λ ) (cid:90) T η N ( s, T ) Y Ns α ( s ) ds − ( aN + λ ) I T − λX NT . Nt = − (cid:16) aN + λ (cid:17) (cid:90) t Y Ns η N ( s, T ) α ( s ) ds − (cid:16) aN + λ (cid:17) ∆ Nt,T Y Nt − (cid:16) aN + λ (cid:17) E [ I NT |F t ] − λX Nt . By integration by parts, this is equivalent to Y Nt = − (cid:16) aN + λ (cid:17) (cid:90) t ∆ Ns,T dY Ns − (cid:16) aN + λ (cid:17) ∆ N ,T Y N − (cid:16) aN + λ (cid:17) E [ I NT |F t ] − λX Nt . Taking t = 0 , we get: Y N = − (cid:0) aN + λ (cid:1) E [ I NT |F ] − λX N (cid:0) aN + λ (cid:1) ∆ N ,T On the other hand, in differential form, (cid:110) (cid:16) aN + λ (cid:17) ∆ Nt,T (cid:111) dY Nt = − (cid:16) aN + λ (cid:17) d E [ I NT |F t ] − λdX Nt , which is solved therefore explicitly by Y Nt = − (cid:0) aN + λ (cid:1) E [ I NT |F ] − λX N (cid:0) aN + λ (cid:1) ∆ N ,T − (cid:90) t (cid:0) aN + λ (cid:1) d E [ I NT |F s ] + λdX Ns (cid:0) aN + λ (cid:1) ∆ Ns,T (10)Finally ¯ φ N ∗ t = − I Nt + (cid:90) t Y Ns d ∆ Ns,t = − I Nt − Y N ∆ N ,t − (cid:90) t ∆ Ns,t dY Ns = − I Nt + ∆ N ,t (cid:0) aN + λ (cid:1) E [ I NT |F ] + λX N (cid:0) aN + λ (cid:1) ∆ N ,T + (cid:90) t ∆ Ns,t (cid:0) aN + λ (cid:1) d E [ I NT |F s ] + λdX Ns (cid:0) aN + λ (cid:1) ∆ Ns,T . (11)Step 3: computing the position of the agent. Let ˇ φ i ∗ t := φ i ∗ t − ¯ φ N ∗ t , ˇ X it = X it − X Nt and ˇ Y it := Y it − Y Nt . Then, ˇ Y i is an F -martingale and satisfies ˇ Y iT = aN ˇ φ i ∗ T + λ ( ˇ φ i ∗ T − ˇ X iT ) , ˇ Y it = − α ( t ) ˙ˇ φ i ∗ t + aN ˇ φ i ∗ t . Similarly to the second part, this system admits an explicit solution: ˇ Y it = − λ ˇ X i (cid:0) aN + λ (cid:1) (cid:101) ∆ ,T − (cid:90) t λd ˇ X is (cid:0) aN + λ (cid:1) (cid:101) ∆ s,T . and ˇ φ i ∗ t = (cid:90) t (cid:101) ∆ s,t λd ˇ X is (cid:0) aN + λ (cid:1) (cid:101) ∆ s,T + (cid:101) ∆ ,t λ ˇ X i (cid:0) aN + λ (cid:1) (cid:101) ∆ ,T . Let us take a closer look at the average strategy ¯ φ Nt = − I Nt + ∆ N ,t (cid:0) aN + λ (cid:1) (cid:101) I N + λX N (cid:0) aN + λ (cid:1) ∆ N ,T + (cid:90) t ∆ Ns,t (cid:0) aN + λ (cid:1) d (cid:101) I Ns + λdX Ns (cid:0) aN + λ (cid:1) ∆ Ns,T , which also corresponds, up to the coefficient a , to the market impact component of theprice. It is a linear combination of components depending on the average forecast process8 N , and those depending on the fundamental price S , reflecting the conflicting objectivesof the agents to limit the loss due to the terminal penalty and increase the gain fromtrading. To better understand the structure and behavior of the average strategy and themarket impact process, the following corollary derives its shape under several simplifyingassumptions and limiting cases. Corollary 4. i. When the fundamental price process S is a martingale, the average strategy satisfies, ¯ φ Nt = ∆ N ,t − S + λX N (cid:0) aN + λ (cid:1) ∆ N ,T + (cid:90) t ∆ Ns,t − dS s + λdX Ns (cid:0) aN + λ (cid:1) ∆ Ns,T . ii. In the limit of zero trading costs (when α ( t ) → uniformly on t ∈ [0 , T ] ), the averagestrategy satisfies, ¯ φ Nt || α || ∞ → −−−−−−→ λa + λ X Nt − S t a + E [ S T |F t ] − S t a ( N −
1) + λ E [ S T |F t ] a ( a + λ ) , (12) almost surely for every t ∈ [0 , T ] . When the fundamental price process S is a mar-tingale, in the limit of zero trading costs, the average strategy satisfies, ¯ φ Nt || α || ∞ → −−−−−−→ λa + λ X Nt − S t a + λ , (13) almost surely for every t ∈ [0 , T ] .iii. In the limit of infinite terminal penalty (when λ → ∞ ), the average strategy satisfies, lim λ −→∞ ¯ φ Nt = − I Nt + ∆ N ,t ∆ N ,T (cid:16) (cid:101) I N + X N (cid:17) + (cid:90) t ∆ Ns,t ∆ Ns,T (cid:16) d (cid:101) I Ns + dX Ns (cid:17) , (14) almost surely for all t ∈ [0 , T ] . When the fundamental price process S is a martingale,in the limit of infinite terminal penalty (when λ → ∞ ), the average strategy does notdepend on the fundamental price and satisfies, ¯ φ Nt = X N ∆ N ,t ∆ N ,T + (cid:90) t ∆ Ns,t ∆ Ns,T dX Ns , (15) almost surely for all t ∈ [0 , T ] .iv. In the absence of terminal penalty, the average strategy satisfies ¯ φ Nt = − I Nt + aN ∆ N ,t (cid:101) I N aN ∆ N ,T + aN (cid:90) t ∆ Ns,t d (cid:101) I Ns aN ∆ Ns,T , (16) almost surely for all t ∈ [0 , T ] .Proof. Part i.
From the expressions of I N and (cid:101) I N in (4), by integration by parts under themartingale condition, we derive: I Nt = − (cid:90) t S s d ∆ Ns,t = ∆ N ,t S + (cid:90) t ∆ Ns,t dS s (cid:101) I Ns = (cid:90) t η N ( s, T ) α ( s ) S s ds + S t (cid:90) Tt η N ( s, T ) α ( s ) ds = S ∆ N ,T + (cid:90) t ∆ Ns,T dS s . Substituting these expressions into the general formula for ¯ φ Nt , we obtain the result.9 art ii. Fixing s < t ∈ [0 , T ] , we have: ∆ Ns,t = Na ( N − (cid:18) − e − (cid:82) ts a ( N − α ( l ) N dl (cid:19) −→ Na ( N −
1) := ∆ ∗ as α ( t ) → uniformly in t . Moreover, this limit does not depend on s , and the principalcontribution is made at the right end-point of the integral. Since the fundamental priceprocess S is a.s., continuous, it is easy to see that I Nt → ∆ ∗ S t almost surely, for every t .For similar reasons, using the dominated convergence theorem, (cid:101) I Nt → ∆ ∗ E [ S T |F t ] . Finally, lim (cid:107) α (cid:107)→ ¯ φ Nt = − ∆ ∗ S t + ∆ ∗ (cid:0) aN + λ (cid:1) ∆ ∗ (cid:16) λX Nt + (cid:16) aN + λ (cid:17) ∆ ∗ E [ S T |F t ] (cid:17) = λa + λ X Nt − Na ( N − S t + aN + λa + λ Na ( N − E [ S T |F t ] Part iii.
Both formulas follow by integration by parts and dominated convergence.
Part iv.
This follows by direct substitution of λ = 0 into the general formula.The term I N , which depends on the past values of the fundamental price can be seen asa "myopic" component of the strategy, while the term (cid:101) I N , which depends on the expectedfuture prices, may be seen as a "forward-looking" component, which takes into account thefuture loss from paying the penalty and the potential future profits from trading. As seenfrom equation (16), this forward-looking term is also present in the absence of penalty,however it is only present insofar as the agent can interact strategically with the otheragents. When the number of agents tends to infinity, the potential for strategic interactiondiminishes, and the forward-looking part of the strategy disappears from the formula.In addition, from (9), we notice that for nonzero trading costs, the price impact hasa finite variation. Hence, it does not directly induce additional volatility which may bea weakness of the model. However, the drift ˙¯ φ N is stochastic and thus creates additionalprice variations, which makes the effective observed volatility larger. We will investigatethis phenomenon in more details in the following paragraph.Another interesting phenomenon is that in the absence of trading costs, for N ≥ , theaggregate equilibrium strategy and the individual strategies of the agents are well definedand the gain of each agent remains bounded in expectation. This is in contrast with thesingle-agent case, where the gain may be arbitrarily large, unless the fundamental priceprocess is a martingale. Indeed, in the single-agent case, without transaction costs theobjective function writes: J ,i ( φ ) = − E (cid:20)(cid:90) T ˙ φ t ( S t + aφ t ) dt + λ X T − φ T ) (cid:21) = E (cid:20)(cid:90) T φ t dS t − φ T S T − a φ T − λ X T − φ T ) (cid:21) , and it is clear that in the non-martingale case, this expression can be made arbitrarilylarge. This means that the "price of anarchy" in this model is infinite: if the agents chosethe same strategy they could have all obtained an infinite gain, but competition betweenagents limits everybody’s gain to a finite value.10 .1 Trading costs, volatility and correlation In this section, we analyze the effect of market structure (number of participants, ter-minal penalty, trading costs and market impact) on the overall costs/gains of participantsas well as on the aggregate market parameters such as price volatility and correlation be-tween forecast and price. To simplify the computations, we make the following additionalassumptions. · The fundamental price process S is a martingale with (cid:104) S (cid:105) t = σ S t . · The trading cost parameter α is constant. · The forecast processes of agents satisfy (cid:104) X N (cid:105) t = σ X t, (cid:104) ˇ X i (cid:105) t = ˇ σ X t, and (cid:104) X N , ˇ X i (cid:105) t = 0 ∀ i, for some constants σ X and ˇ σ X , where ˇ X it = X it − X Nt .Under these assumptions, the coefficients η Ns,t and ∆ s,t depend only on t − s and not on s and t separately. We shall therefore write them as η Nt − s and ∆ Nt − s , and similarly for theother coefficients, from now and until the end of this section. The aggregate position of N agents in equilibrium therefore writes: ¯ φ Nt = ∆ Nt − S + λX N (cid:0) aN + λ (cid:1) ∆ NT + (cid:90) t ∆ Nt − s − dS s + λdX Ns (cid:0) aN + λ (cid:1) ∆ NT − s . Volatility
We have seen that since the strategy ¯ φ N is differentiable, the quadratic varia-tion of the equilibrium price P Nt coincides with the quadratic variation of the fundamentalprice. However, the actual observed volatility, which is estimated from discretely observedprices, may be larger. Since the market impact component of the price is given by a ¯ φ Nt ,we may use the expectation of the squared derivative of the aggregate strategy, E [( ˙¯ φ Nt ) ] ,as a measure of variability of the aggregate strategy, and thus as a proxy for the additionalvariance of the equilibrium price. In this section we draw conclusions about the behaviorof price volatility by analyzing this proxy, and in Section 6 we will show in numerical ex-amples that the actual volatility, estimated from discrete observations of simulated marketprice exhibits similar behavior. Under the assumption of this section, this gives: ˙¯ φ Nt = η Nt α − S + λX N (cid:0) aN + λ (cid:1) ∆ NT + (cid:90) t η Nt − s α − dS s + λdX Ns (cid:0) aN + λ (cid:1) ∆ NT − s . E [( ˙¯ φ Nt ) ] = ( η Nt ) α ( S − λX N ) (1 + (cid:0) aN + λ (cid:1) ∆ NT ) + σ S + λ σ X α (cid:90) t ( η Nt − s ) (1 + ( aN + λ )∆ NT − s ) ds. (17)The function of interest t (cid:55)→ E [( ˙¯ φ Nt ) ] has thus two parts: the first one is due to thedeterministic components of the aggregate strategy, and it decreases with t , and the secondone is due to the volatility of the fundamental price and the aggregate forecast, and itincreases with t . We expect the second part to dominate the first one close to deliverysince the initial state of the market should not play a role at this time. As a result, thevariability of the aggregate strategy (and thus, the observed price volatility) increases atthe approach of the delivery date in our model, a phenomenon, which we also documentempirically in section 6. 11ecall that in our context, η NT = e − ( N − aNα T , and ∆ NT = N ( N − a (1 − e − ( N − aNα T ) From the expression (17), one may deduce several interesting limiting regimes for the extravariance of the equilibrium price: · Small liquidity cost regime: α → . For < t < T , ∆ Nt − s → N ( N − a uniformly on s ∈ [0 , t ] , so that, E [( ˙¯ φ Nt ) ] ∼ σ S + λ σ X α N N − a , α → . As the market impact is given by a ¯ φ Nt , this shows that the extra variance of theequilibrium price grows like aα . We recall that the parameter α represents the tradingcosts while the parameter a reflects the strength of the permanent market impactdirectly related to the market depth. Both can be interpreted as liquidity signalsbut lead to distinct market effects. Decreasing transaction costs allows the agentsto follow the forecasts more closely, leading to a higher volatility of the aggregateposition and of the market price. On the contrary, when a decreases, the marketimpact of agents is weaker and so are the price variations. As a consequence, thevolatility decreases too.On the other hand, since the function N (cid:55)→ NN − is decreasing in N , we concludethat price volatility in the small liquidity cost regime is decreasing with the numberof agents: in our model, competition between agents increases market frictions andleads to reduced volatility. · Large liquidity costs regime: α → ∞ . In this case, ∆ NT ∼ Tα and the extra variancedecays like α : E [( ˙¯ φ NT ) ] ∼ ( S − λX N ) α + σ S + λ σ X α T Higher liquidity costs decrease the trading rate of agents and lead to a lower overallmarket volatility. · Large penalty limit: λ → ∞ . In this case, the extra variance grows like λ : E [( ˙¯ φ NT ) ] ∼ λα σ X . Higher imbalance penalties force the agents to follow the forecasts more closely andlead to an overall higher price volatility.
Covariance
To understand how the forecast updates influence prices, we compute thecovariance of the increment of the aggregate strategy over an interval of length h with theincrement of the aggregate forecast over the same interval. Using the explicit form of thestrategy, we easily obtain,Cov [ ¯ φ Nt + h − ¯ φ Nt , X Nt + h − X Nt ] = λ ( σ NX ) (cid:90) t + ht ∆ Nt + h − s aN + λ )∆ NT − s ds. From this expression, we conclude that the covariance of equilibrium price with forecastupdates increases when the terminal penalty λ increases, and when the time t approachesthe delivery date . 12 rading costs As our final illustration in this section, we evaluate the total expectedgain/cost of the agents and elucidate the effect of various market parameters on this gain.The total gain/cost for the i th agent in our present context is given by J N,i ( φ i , φ − i ) = − E (cid:20)(cid:90) T (cid:110) α φ it ) + ˙ φ it P Nt (cid:111) dt + λ φ iT − X iT ) (cid:21) = − E (cid:20) α (cid:90) T ( ˙¯ φ Nt ) dt + S T ¯ φ NT + a φ NT ) + λ φ NT − X NT ) (cid:21) − E (cid:20) α (cid:90) T ( ˙ˇ φ it ) dt + λ φ iT − ˇ X iT ) (cid:21) = − E α (cid:90) T ( ˙¯ φ Nt ) dt + a + λ (cid:32) ¯ φ NT − λX NT − S T a + λ (cid:33) − λ E [( X NT ) ]+ E [( λX NT − S T ) ]2( a + λ ) − E (cid:20) α (cid:90) T ( ˙ˇ φ it ) dt + λ φ iT − ˇ X iT ) (cid:21) = − E (cid:20) α (cid:90) T ( ˙¯ φ Nt ) dt + α a + λ ) (cid:16) ˙¯ φ NT (cid:17) (cid:21) − λ E [( X NT ) ] + E [( λX NT − S T ) ]2( a + λ ) − E (cid:20) α (cid:90) T ( ˙ˇ φ it ) dt + α λ ( ˙ˇ φ iT ) (cid:21) In the present context, E [( ˙¯ φ NT ) ] = ( η NT ) E [( S − λX N ) ] α (1 + ( aN + λ )∆ NT ) + (cid:90) T ( η NT − t ) ( σ S + λ σ X ) α (1 + ( aN + λ )∆ NT − t ) dt = ( η NT ) E [( S − λX N ) ] α (1 + ( aN + λ )∆ NT ) + σ S + λ σ X α (cid:90) ∆ NT (1 − aN ( N − u )(1 + ( aN + λ ) u ) du E [( ˙ˇ φ iT ) ] = λ (˜ η NT ) E [( ˇ X i ) ] α (1 + ( aN + λ ) (cid:101) ∆ NT ) + (cid:90) T λ (˜ η NT − t ) ˇ σ X α (1 + ( aN + λ ) (cid:101) ∆ NT − t ) dt = λ (˜ η NT ) E [( ˇ X i ) ] α (1 + ( aN + λ ) (cid:101) ∆ NT ) + λ ˇ σ X α (cid:90) (cid:101) ∆ NT aN u (1 + ( aN + λ ) u ) ds (cid:90) T E [( ˙¯ φ Nt ) ] dt = 12 (1 + η NT )∆ NT E [( S − λX N ) ] α (1 + ( aN + λ )∆ NT ) + σ S + λ σ X α (cid:90) T dt (1 + η NT − t )∆ NT − t (1 + ( aN + λ )∆ NT − t ) (cid:90) T E [( ˙ˇ φ it ) ] dt = 12 λ (˜ η NT + 1) (cid:101) ∆ NT E [( ˇ X i ) ] α (1 + ( aN + λ ) (cid:101) ∆ NT ) + λ ˇ σ X α (cid:90) T dt (˜ η NT − t + 1) (cid:101) ∆ NT − t (1 + ( aN + λ ) (cid:101) ∆ NT − t ) Given the complexity of these expressions, we once again consider the limiting regimesof small liquidity costs, large liquidity costs and large imbalance penalty. · In the case of small liquidity costs, α → , J N,i ( φ i , φ − i ) → −
14 ∆ N ∞ E [( S T − λX NT ) ](1 + ( aN + λ )∆ N ∞ ) − λN E [( ˇ X iT ) ]2( aN + λ ) a (cid:18) Na + λ (cid:19) + E [( λX NT − S T ) ]2( a + λ ) − λ E [( X NT ) ] , ∆ N ∞ = Na ( N − . Thus, for small liquidity costs, the gain of an agent convergesto a finite constant. As the cost per trade decreases, the agents trade more activelyso that the overall cost does not decrease as much. For a market with a single agent( N = 1 ), both terms in the first line are zero. On the other hand, when N is large,both the first term in the first line is nonzero, and the last term in the first line growsproportionally to N . We conclude that due to competition between agents, the costof each individual agent increases as N → ∞ . · In the case of large liquidity costs, α → ∞ , J N,i ( φ i , φ − i ) → − λ E [( ˇ X iT ) ] − λ E [( X NT ) ] , which corresponds to the cost of the ’do nothing’ strategy. · In the case of large terminal penalty λ , the leading term of the single agent’s cost isgiven by J N,i ( φ i , φ − i ) ∼ − σ X (cid:90) T dt λ (1 + η NT − t )∆ Nt (1 + ( aN + λ )∆ NT − t ) − ˇ σ X (cid:90) T dt λ (˜ η NT − t + 1) (cid:101) ∆ NT − t (1 + ( aN + λ ) (cid:101) ∆ NT − t ) ∼ − ασ X (cid:90) ( aN + λ )∆ NT dt udu (1 + u ) − α ˇ σ X (cid:90) ( aN + λ ) (cid:101) ∆ NT dt udu (1 + u ) ∼ − ασ X (cid:16) aN + λ )∆ NT (cid:17) − α ˇ σ X (cid:16) aN + λ ) (cid:101) ∆ NT (cid:17) ∼ − α ( σ X + ˇ σ X )2 log λ. Thus, when the terminal penalty grows, the single agent cost tends to + ∞ at alogarithmic rate.Up to now, we considered that all players have access to the common filtration F . InTheorem 3, the optimal strategy of each agent depends on the demand forecast of the otherplayers. However, one could argue that in practice the agents do not observe the individualforecasts of one another, and their strategy may only be based on their own information,and on the market price, which depends via the price impact on the aggregate forecastof all the players. In addition, strategic considerations may push the players to changetheir strategies in order to avoid disclosing information to the market. The problem ofdetermining the Nash equilibrium in this partial information setting thus becomes verycomplex. These considerations motivate us to consider the partial information problem inthe mean field setting, where the role of each individual player is negligible and strategicconsiderations do not play a role. In this section we consider that the information available to agents is no longer the same.We directly place ourselves in the mean field game limit, that is, we assume the number ofagents in the market, N tends to infinity, while the strategy of each agent remains finite.We then consider a generic agent and denote by X := ( X t ) t ∈ [0 ,T ] the demand forecast ofthis agent, by φ the agent’s position and by F the filtration which contains the informationavailable to this agent. In addition we introduce a smaller filtration, containing the commonnoise and denoted by F . This filtration contains the information about the fundamentalprice and potentially some information about the demand forecast but, in general, not the14ull individual demand forecast of the generic agent. In the following we decompose theindividual demand forecast as follows: X t = X t + ˇ X t , where X t = E (cid:2) X t |F t (cid:3) is common forall agents (it can be seen as a national demand forecast). In this mean field game setting,the average quantities of the N -agent problem are replaced with conditional expectationswith respect to the common noise filtration F .Throughout the paper and for any F -adapted process ( ζ t ) t ∈ [0 ,T ] , we will denote ¯ ζ t = E [ ζ t |F t ] = (cid:82) R xµ ζt ( dx ) where: µ ζt := L ( ζ t |F t ) . The game is now represented by theinteraction of agents through the conditional distribution flow µ φt := L ( φ t |F t ) of the stateprocess. The price impact function, defined in the previous section as an expectation withrespect to the empirical measure, is now an integral with respect to the measure flow: P t = S t + a ¯ φ t . (18)Each individual agent now has a negligible impact on the price, but the aggregate positionof all agents has a nonzero impact. Thus, in the mean-field game setting, we consider thatthe market is very large compared to the size of the individual agent, but the immediatelyavailable liquidity in the order book is small, so that even a minor agent pays a non-zerotrading cost.The objective function for the generic agent is J MF ( φ, ¯ φ ) := − E (cid:20)(cid:90) T α ( t )2 ˙ φ t + ˙ φ t ( S t + a ¯ φ t ) dt + λ φ T − X T ) (cid:21) . (19)As in the previous section, this is maximized over the set of admissible strategies ( ˙ φ t ) t ∈ [0 ,T ] ,which contains all F -adapted square integrable strategies.We now define the mean field equilibrium. Definition 5 (mean field equilibrium) . An admissible strategy ˙ φ ∗ := ( ˙ φ ∗ t ) t ∈ [0 ,T ] is a meanfield equilibrium if it maximizes (19) and ¯ φ = ¯ φ ∗ . In this section, we make the following assumption.
Assumption 6. · The process S is square integrable and adapted to the filtration F . · The process X is a square integrable martingale with respect to the filtration F . · The process X defined by X t := E [ X t |F t ] for ≤ t ≤ T is a square integrablemartingale with respect to the filtration F . Note that if X is an F -martingale, then X is by construction an F -martingale, but itmay not necessarily be a martingale in the larger filtration F .The following theorem characterizes the mean field equilibrium in our setting. Thestatement of the theorem appears similar to that of Theorem 3, modulo replacing X N with X and making N tend to infinity. Thus the main implications of Theorem 3, givenin Corollary 4 and section 3.1, hold true also in this case, except, of course for the oneswhich describe the behavior of the market price as the number of agents tends to infinity.However, the form of the strategy and the market price given in this theorem does notrequire the knowledge of the individual forecasts of the other players, but only of thecommon one. Thus, the theoretical price given by this theorem can be computed by the15egulator, and the strategy of this theorem can be computed by an individual player, whichdo not have the complete information about the forecasts of other players.In the theorem and its proof, we use the following shorthand 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:20)(cid:90) T η s,T α ( s ) S s ds (cid:12)(cid:12)(cid:12) F t (cid:21) . (20) Theorem 7.
Under Assumption 6, the unique mean field equilibrium 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 (21) + (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 shape: 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) . (22) Proof.
Step 1. First order condition for optimality for a single agent. In this step, we showthat for fixed ¯ φ ∗ , the strategy φ ∗ maximizes (19) in Definition 5 if and only if there existsa square integrable F -martingale Y such that, almost surely, Y t + α ( t ) ˙ φ ∗ t + S t + a ¯ φ ∗ t = 0 , ≤ t ≤ T and Y T = λ ( φ ∗ T − X T ) . (23)The proof follows the lines of the proof of Theorem 3, step 1.Step 2. Computing the mean field. Let φ ∗ be the optimal position of the agent. Wehave seen that this is equivalent to (23). Taking the conditional expectation of theseexpressions with respect to F t (and using the overline notation to denote F -conditionalexpectations), Y t + α ( t ) ˙¯ φ ∗ t + S t + a ¯ φ ∗ t = 0 , ≤ t ≤ T and Y T = λ ( ¯ φ ∗ T − X T ) .S is adapted to the filtration F and by construction, X and Y are F -martingales. Hencethese equations may be solved along the lines of the proof of Theorem 3, step 2, and weobtain Y t = − λ E [ I T |F ] − λX λ ∆ ,T − (cid:90) t λd E [ I T |F s ] + λdX s λ ∆ s,T Step 3: Computing the position of the agent. 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 the explicit form of Y computed in step 2 and our assumptionit follows that ˇ Y is an F -martingale and satisfies ˇ Y T = λ ( ˇ φ ∗ T − ˇ X T ) , ˇ Y t = − α ( t ) ˙ˇ φ ∗ t . This system admits an explicit solution: ˇ Y t = − λ ˇ X λ (cid:101) ∆ ,T − (cid:90) t λd ˇ X s λ (cid:101) ∆ s,T . ˇ 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 . Moreover, E [ ˇ φ ∗ t |F t ] = 0 , hence E [ φ ∗ t |F t ] = ¯ φ ∗ t , so the optimal strategy of the agent is themean field equilibrium. In this section, we study the relationship between the equilibrium strategies and pricesin the N -agent market and those of the mean field game limit, and prove the followingresults. · The market price and the agent’s strategy in the N -agent model converge to theirrespective mean field values as N → ∞ . This shows that to understand the behaviorof agents and prices in the realistic N -agent market, one can use the mean-field gamemodel, which does not require the knowledge of individual forecasts, but only thatof the aggregate forecast. · An approximate equilibrium ( ε -Nash equilibrium) in the N -player setting may beconstructed from the MFG solution. In other words, an agent trading in the N -agent market may construct a strategy whose gain is sufficiently close to the optimalequilibrium gain in the full information setting using the mean-field game solution,which does not require the knowledge of the private forecasts of the other agents.To address these questions we need to make more precise assumptions on the proba-bilistic setup of the problem. In particular, since we would like to study the convergence ofthe N -agent problem as N → ∞ , we consider an infinity of agents. In addition all N -agentproblems and the mean field problem must be defined on the same probability space. Assumption 8. · The process S is square integrable and adapted to the filtration F . · The processes ( X i ) ∞ i =1 are square integrable F -martingales. · There exists a square intergrable F -martingale X , such that for all i ≥ , and all 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 orthogonal squareintegrable F -martingales, such that the expectation E [( ˇ X iT ) ] does not depend on i . Let us fix
N < ∞ , and consider a market with N agents. For a given i ≤ N , we maydefine the "mean-field" strategy for the i th agent as follows. φ MF,i ∗ 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 + (cid:101) ∆ ,t ˇ X i λ (cid:101) ∆ ,T + (cid:90) t (cid:101) ∆ s,t d ˇ X is λ (cid:101) ∆ s,T (cid:35) . (24)Unlike the true optimal strategy of the i -th agent, this strategy is computed using only thecommon information and the individual information of the i th agent, it does not requirethe knowledge of the private forecasts of the other agents. Moreover, this strategy does17ot depend on N . The following two results show that, on the one hand, the true optimalstrategy of the i th agent in the N -player game converges to this mean-field strategy as N → ∞ , and on the other hand, that this mean-field strategy, if used by all agents in the N -player game, constitutes an ε -Nash equilibrium. Proposition 9.
Let Assumption 8 hold true, and let φ i ∗ denote the optimal position ofthe i th agent in the N -player complete information setting, given by (5) , and by φ MF,i ∗ the optimal position in the mean field setting, given by (24) . Then, for all N ≥ , thedifferences between the strategy of a single agent, the aggregate strategy and the equilibriumprice in the N -agent model and the corresponding quantities in the mean-field model canbe bounded as follows. sup ≤ t ≤ T E [( φ i ∗ t − φ MF,i ∗ t ) ] + sup ≤ t ≤ T E [( φ N ∗ t − φ ∗ t ) ] + sup ≤ t ≤ T E [( P Nt − P t ) ] ≤ CN (cid:90) T E [ S t ] dt + CN E [( X T ) ] + CN E [( ˇ X iT ) ] , where the constant C depends only on the coefficients α , λ and a . The proof of this proposition can be found in Appendix 7.1.
Proposition 10.
Under assumption 8, consider the vector of admissible strategies forthe N -player game defined by equation (24) for i = 1 , . . . , N . Then, there is a constant C < ∞ which does not depend on N , such that for any other vector of admissible strategies ( φ it ) i =1 ...,Nt ∈ [0 ,T ] for the N -player game, J N,i ( φ i , φ MF, − i ∗ ) − CN ≤ J N,i ( φ MF,i ∗ , φ MF, − i ∗ ) , ∀ i ∈ { , ..., N } , ∀ t ∈ [0 , T ] . In other words, the vector of strategies ( φ MF,i ∗ t ) i =1 ...,Nt ∈ [0 ,T ] is an ε -Nash equilibrium for the N -player game with ε = CN .Proof. To lighten notation, and since we now have only one strategy, we omit in this proofthe superscript MF in the candidate strategy φ MF,i ∗ . The "distance to optimality" for thisstrategy is estimated as follows. J N,i ( φ i , φ − i ∗ ) − J N,i ( φ i ∗ , φ − i ∗ )= J N,i ( φ i , φ − i ∗ ) − J MF ( φ i , ¯ φ ∗ ) + J MF ( φ i , ¯ φ ∗ ) − J MF ( φ i ∗ , ¯ φ ∗ ) + J MF ( φ i ∗ , ¯ φ ∗ ) − J N,i ( φ i ∗ , φ − i ∗ ) ≤ J N,i ( φ i , φ − i ∗ ) − J MF ( φ i , ¯ φ ∗ ) + J MF ( φ i ∗ , ¯ φ ∗ ) − J N,i ( φ i ∗ , φ − i ∗ ) . The second difference is estimated via Cauchy-Schwartz inequality. J MF ( φ i ∗ , ¯ φ ∗ ) − J N,i ( φ i ∗ , φ − i ∗ ) = a E (cid:20)(cid:90) T ˙ φ i ∗ t ( ¯ φ ∗ t − ¯ φ N ∗ t ) dt (cid:21) ≤ a E (cid:20)(cid:90) T ( ˙ φ i ∗ t ) dt (cid:21) E (cid:20)(cid:90) T ( ¯ φ ∗ t − ¯ φ N ∗ t ) dt (cid:21) , (25)where E (cid:20)(cid:90) T ( ¯ φ ∗ t − ¯ φ N ∗ t ) dt (cid:21) ≤ N N (cid:88) i =1 E [( ˇ X iT ) ] = 1 N E [( ˇ X T ) ] . (26)18he first difference admits the following estimate. J N,i ( φ i , φ − i ∗ ) − J MF ( φ i , ¯ φ ∗ ) = a E (cid:20)(cid:90) T ˙ φ it ( ¯ φ ∗ t − ¯ φ N ∗ t ) dt (cid:21) − aN E (cid:20)(cid:90) T ˙ φ it ( φ it − φ i ∗ t ) dt (cid:21) ≤ a E (cid:20)(cid:90) T ( ˙ φ it ) dt (cid:21) E (cid:20)(cid:90) T ( ¯ φ ∗ t − ¯ φ N ∗ t ) dt (cid:21) + 1 N E (cid:20)(cid:90) T ( φ i ∗ t ) dt (cid:21) . (27)On the other hand, the following estimate also holds true. J N,i ( φ i , φ − i ∗ ) ≤ − E (cid:20) ¯ α (cid:90) T ( ˙ φ it ) + ˙ φ it (cid:16) S t + a ¯ φ N ∗ t + aN ( φ it − φ i ∗ t ) (cid:17) dt + λ φ iT − X T ) (cid:21) ≤ − ¯ α E (cid:20)(cid:90) T ( ˙ φ it ) dt (cid:21) + C E (cid:20)(cid:90) T ( ˙ φ it ) dt (cid:21) (28)where C = E (cid:20)(cid:90) T ( S t + a ¯ φ N ∗ t − aN φ i ∗ t ) dt (cid:21) and ¯ α = min ≤ t ≤ T α ( t ) . Let K := J N,i ( φ i ∗ , φ − i ∗ ) and A := max (cid:110)(cid:0) Cα (cid:1) , | K | ¯ α (cid:111) .There are two cases. If E (cid:20)(cid:90) T ( ˙ φ it ) dt (cid:21) ≤ A, then from (25), (26) and (27), it follows that J N,i ( φ i , φ − i ∗ ) − J N,i ( φ i ∗ , φ − i ∗ ) ≤ aN E (cid:20)(cid:90) T ( ˙ φ i ∗ t ) dt (cid:21) E (cid:2) ( ˇ X T ) (cid:3) + aA N E (cid:2) ( ˇ X T ) (cid:3) + 1 N E (cid:20)(cid:90) T ( φ i ∗ t ) dt (cid:21) . (29)If the opposite inequality holds then from (28) it follows that J N,i ( φ i , φ − i ∗ ) ≤ J N,i ( φ i ∗ , φ − i ∗ ) . Thus, the estimate (29) holds for every admissible strategy φ i . Moreover, φ i ∗ does notdepend on N , and in view of (25) and (26), A can be bounded from above by a constantwhich also does not depend on N . Thus, J N,i ( φ i , φ − i ∗ ) − J N,i ( φ i ∗ , φ − i ∗ ) ≤ CN , for a constant C which does not depend on N . In this section our objective is to analyze the empirically observed features of intradaymarket prices, demonstrate that these features are reproduced by our model, and illustrateother properties of our model, such as the convergence of the N -agent model to the mean-field limit, with numerical examples.In the rest of this section, we assume that the position taken by every agent in theday-ahead market is exactly equal to the best estimate of the future demand computed attime t = 0 . Therefore, the initial values X i , i = 0 , . . . , N will be set to 0.19 .1 Stylized features of intraday electricity market prices A brief description of the intraday electricity market and of our dataset
TheEPEX intraday electricity market ( ) opens every day at 3 p.m and allows totrade all delivery hours of the following day up to 5 minutes before delivery. Each deliveryhour is a distinct product in the market. Different European geographic zones are availableand in this study we focus on the German area. It is also possible to trade in quater-hours,but in the empirical study we focus on the full hours only.To compute the empirical price analyzed in the following sections we used the limitorder book data provided by EPEX for the st quarter of 2015 and January 2017. Thisdataset contains full information about sell and buy orders recorded on any given day,whether they result in a transaction or not. From this data we reconstruct the state of theorder book, which allows us in turn to derive the mid-quote price and the bid-ask spread. Market liquidity
In Figure 1, we plot the distribution of the number of orders andtransactions as function of time to delivery computed over all orders and transactionsin February 2015. We observe that the liquidity starts to appear only 5-6 hours beforedelivery, and grows very quickly at the approach of the delivery date. This is consistentwith the assumption that the market is used by the renewable energy producers to adjusttheir positions when precise forecasts become available.Figure 1: Distribution of orders and transactions as function of the time to delivery overFebruary 2015
10 8 6 4 2 0Time to delivery, hours0.00.10.20.30.40.50.60.70.8
Distribution of orders
10 8 6 4 2 0Time to delivery, hours0.00.20.40.60.81.0
Distribution of transactions
Price volatility
To estimate the empirical volatility, we consider mid-quote prices re-constructed from the limit order book data of the Germany delivery zone for January 2017,as explained above. The mid-quote price was computed on a uniform grid with a time stepof 1 minute. In January 2017 the market was already relatively liquid: the average numberof daily price changes for a given delivery hour varied between approximately 3400 forthe least liquid delivery hour (2AM) to approximately 5800 for the most liquid deliveryhour (6PM). Given that, as we observed above, liquidity is concentrated in the last 5-6hours, a one-minute interval during this time contains many price changes and the marketmicrostructure effects are limited.The observed midquote price is denoted ( ˜ P t ) t ∈ [0 ,T ] . For the purpose of estimatingvolatility, we assume it has the following dynamics: d ˜ P t = µ t dt + σ t dW ˜ Pt , ≤ t ≤ T, (30)20here ( W ˜ Pt ) t ∈ [0 ,T ] is a Brownian motion, and ( µ t ) t ∈ [0 ,T ] and ( σ t ) t ∈ [0 ,T ] are adapted pro-cesses. We denote by n the number of observations in the data of January 2017 and by { t , . . . , t i , . . . , t n } the (uniform) time grid over which the observations are available. Incontrast with the integrated volatility whose estimator is generally given by (cid:92) (cid:82) T σ s ds = (cid:80) ni =1 ∆ ˜ P t i − , estimating the instantaneous volatility is less straightforward. Following[20], we use a kernel-based non parametric estimator of the instantaneous volatility: 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 − ) , (31)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 . hour ( ≈ minutes) after performing some cross-validation analyses and sensitivity tests. The paths of the estimated volatility as functionof time to delivery for different delivery hours are given in Figure 2. We observe thatthe volatility increases as delivery time draws near and market participants trade moreactively. Figure 2: Instantaneous market volatility for different delivery hours Correlation between price and renewable indeed forecasts
We finally study theempirical correlation between the intraday market prices and the renewable wind produc-tion forecasts. Contrary to the rest of the paper, here we use actual wind infeed forecasts,not the demand forecasts. To compute empirical correlation estimates, we use the limitorder book data from the intraday EPEX market of the first three months of 2015 for theGermany delivery zone, from which, as before, we compute the mid-quote prices. The pro-duction forecasts correspond to the same period and are updated every 15 minutes for eachdelivery hour. In Figure 3, we plot the correlation between the increments of the marketprice and the increments of the production forecasts for the delivery time 12h (averagedover 90 days in the dataset), together with the 2-standard deviation bounds. To matchthe forecast update frequency, the mid-quote price is also sampled at 15-minute intervalshere.We find that the correlation between the price increments and those of the productionforecast is negative and increases in absolute value as we approach the delivery time.
Model specification
We now define the dynamics for the fundamental price and for thedemand forecasts used in the simulations. We also give the chosen values of the different21igure 3: Correlation between the market price increments and the renewable productionforecast increments for the German delivery zone in winter 2015
Price/forecast correlation, delivery at 12h
CorrelationUpper boundLower bound parameters. Our objective here is to illustrate the features of the model and show that itreproduces the stylized facts of the market prices. Therefore the majority of the parametersare not precisely estimated, but are given plausible values.The evolution of the fundamental price is described as follows: dS t = σ S dW t (32)where σ S is a constant and ( W t ) t ∈ [0 ,T ] is Brownian motion. We also assume that theliquidity function α ( . ) is given by α ( t ) = α × ( T − t ) + β, ∀ t ∈ [0 , T ] (33)where α and β are strictly positive constants. The liquidity function is decreasing withtime. This assumption relies on the fact that, as we observed in Section 6.1, the marketbecomes more liquid as we get closer to the delivery time and it is less costly to trade whenthe market is liquid.To simulate demand forecasts we assume the following dynamics: d ¯ X Nt = σ X d ¯ B t (34) d ˇ X it = ˇ σ X dB it , i ∈ { , . . . , N } (35)where σ X and ˇ σ X are constants and ( ¯ B t ) t ∈ [0 ,T ] , ( B it ) t ∈ [0 ,T ] are independent Brownianmotions, also independent from ( W t ) t ∈ [0 ,T ] .In this illustration, we choose the same parameters for the dynamics of the common andthe individual demand forecasts (that is, σ X = ˇ σ X ). The common volatility is calibratedto wind energy forecasts in Germany over January 2015 during the last quotation hour, byusing the classical volatility estimator σ X = ˇ σ X = √ ∆ tn (cid:48) − n (cid:48) (cid:88) i =1 Y i (36)with ∆ t the time step between two observations, Y i = X t i − X t i − the increment betweentwo successive observations and n (cid:48) the total number of observed increments. As the fore-casts are updated every 15 minutes, there are three increments during the last tradinghour, available on each day from the 3 rd of January to the 31 th of January. Thus, foreach delivery hour we dispose of n (cid:48) = 87 increments points to estimate the volatility. Thevolatility, as well as the other model parameters are specified in Table 1.22arameter Value Parameter Value S e /MWh a e /MWh σ S e /MWh · h / λ e /MWh X N σ X , ˇ σ X MWh/h / α e /MW · h ˇ X i β e /MW Table 1: Parameters of the model
Price trajectories.
In Figure 4, we plot a simulated trajectory of the fundamental price S starting six hours before the delivery time (corresponding to t = 0 ), up to the time T of delivery, together with the market price P associated with the different settings studiedin this paper: the N -player Nash equilibrium with N = 100 players, the mean field andthe (cid:15) -Nash equilibrium. Graphs were all simulated with the same demand forecasts, initialvalues, volatilities and parameters as specified in Table 1.Figure 4: Model price trajectories (left) associated to a given common demand forecasttrajectory (right) in different settingsIn all settings, the model reflects the price impact of the positions taken by the agents.This price impact is influenced by the market price and the demand forecasts. If agentsanticipate to have overestimated the demand (negative values of the demand process),there is an excess of supply in the market, thus the price impact is negative and themarket price decreases. On the contrary, if they anticipate to have underestimated thedemand (positive values of the demand forecast process), there is a lack of supply and themarket price increases. Volatility and correlation
In this paragraph we compare the price volatility and thecorrelation between price and renewable infeed forecasts in our model with the empiricalones. We have already seen through theoretical analysis in section 3.1 that our modelreproduces the observed features of the volatility; the goal of this paragraph is to confirmthis using simulated prices. We once again highlight the fact that the market impact caninduce an increase in the price variations but no changes in the quadratic variation sincethe price impact, though it is stochastic, has a finite quadratic variation. However, thevolatility estimated from discrete price observations, which is the only quantity relevant inpractice, does increase in our model, as we shall see below.We focus on hourly products and on several different delivery hours: 2 a.m, 8 a.m, 12p.m and 6 p.m to include both peak (high electricity demand) and off-peak (low electricitydemand) times. The volatility of the fundamental price S is assumed to be constant,23 σ S = 10 e /MWh · h / ) to ensure that the observed volatility changes are only due tothe stochastic drift of the market price, i.e the aggregate trading rates of the agents. Thevolatility of the production forecasts for the different delivery hours has been calibratedusing the estimator defined in (36) and is shown in Table 2.Hour Volatility (MWh/h / ) Table 2: Calibrated volatility of the production forecast for different delivery hoursDuring peak hours, both market activity and liquidity are higher. To account for thisphenomenon in our model, we chose different levels of the liquidity coefficients α and β defined in (33) and presented in Table 3. Since calibrating the model to market data isHours Coefficients α ( e /h.MW ) β ( e /MW )2h00 1.2 0.58h00 0.5 0.212h00 0.7 0.318h00 0.3 0.1Table 3: Liquidity coefficients used for different delivery hoursnot the purpose of this study, we chose plausible values for these coefficients in an ad hocmanner with lower trading costs corresponding to delivery hours for which the market ismore liquid. All other model parameters are specified in Table 1.Figure 5 shows the estimated volatility of the simulated model price P in the Nash N -player game setting with N = 100 , averaged over 1000 simulations. The volatility wascomputed using the estimator (31), with the same window width and time step as in theempirical analysis. From this graph we can see that the model is able to reproduce theincreasing shape of the empirical market price volatility at the approach of the deliverytime, and that it captures the different levels of volatility corresponding to the differentdelivery hours.Figure 5: Simulated model volatility for different delivery hours Correlation between price and renewable infeed
An important stylized feature ofintraday market prices, observed empirically in [19] is the correlation between the price24nd the renewable production forecasts. Figure 6 plots the correlation between 15-minuteincrements of the simulated market price and the 15-minute increments of the simulatedrenewable production forecasts as function of time. For each time step, the correlation ρ t = corr (∆ Y t , ∆ P t ) is computed by Monte Carlo using the following estimator: 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 stands for number of simulations (we considered N sim = 50000 ), ∆ Y kt = − ( X N,kt + dt − X N,kt ) , ∆ P kt = P N,kt + dt − P N,kt and N = 100 . Notice that we use the minus signin front of the forecast increment to plot the correlation of production forecasts, whereas X stands for the demand forecast.Figure 6: Correlation between the simulated market price increments and the renewableproduction forecast increments in the model during the last six hours of tradingWe first note that the correlation is negative: an expected increase of the renewableproduction is correlated to a decrease in the market price and an expected lack of renewableproduction is correlated to an increase in the price. As we get closer to the deliverydate, the agents trade more actively as new forecast information becomes available, andthe market price becomes more strongly dependent on the forecast updates. The modeloutputs qualitatively match the results observed empirically. However, the strength of thecorrelation seems to be greater in the model than in reality. This can be explained bythe fact that the model does not take into account other renewable means of productionsuch as the solar energy. The slight increase of the correlation for longer times to delivery(the right-hand side of the graph) may be explained by the fact that the correlation iscomputed as the ratio of the covariance to the square root of the product of variances.While both quantities decrease for longer times to delivery, the denominator may decreasefaster, explaining the slight increase in the correlation values. Convergence and approximations
In Figure 7 we plot the mean field position, theaggregate N -player Nash equilibrium position and the aggregate position for the (cid:15) -Nashequilibrium (respectively given by Theorem 3, Theorem 7 and Proposition 10) for a modelwith N = 5 players and N = 100 players. The trajectories were computed with the samesimulated fundamental price, common production forecast and parameters as the Figure4 above, over the 6 hours preceding the delivery time. The left graph ( N = 5 ) shows25 big difference between the Nash equilibrium and (cid:15) -Nash approximation on one hand,and the mean field on the other hand. This is explained by the individual productionforecast taken into account in the Nash and (cid:15) -Nash equilibria. When we consider a largernumber of players, N = 100 , the three position trajectories are much closer to each other.This confirms the asymptotic convergence to the mean field discussed in section 5 for the N -player Nash equilibrium and (cid:15) -Nash equilibrium.Figure 7: Aggregate position in different settings with N = 5 (left) and N = 100 (right)agents Proof.
For all t ∈ [0 , T ] , we define: g Ns,t = ∆
Ns,t (1 + ( aN + λ )∆ Ns,T ) , g s,t = ∆ s,t (1 + λ ∆ s,T ) , ˜ g Ns,t = (cid:101) ∆ Ns,t (1 + ( aN + λ ) (cid:101) ∆ Ns,T ) , ˜ g s,t = (cid:101) ∆ s,t (1 + λ (cid:101) ∆ s,T ) , so that, for some constant C depending only on the parameters a , α and λ , but not onother ingredients of the model, | g Ns,t − g s,t | + | ˜ g Ns,t − ˜ g s,t | ≤ CN .
Now, let us consider the optimal strategies ( φ i ∗ t ) t ∈ [0 ,T ] and ( φ MF,i ∗ t ) t ∈ [0 ,T ] of the genericagent i respectively in the N -player setting and the mean field setting. Fix t ∈ [0 , T ] .26hen, φ i ∗ t − φ MF,i ∗ t = − I Nt + I t + g N ,t (cid:110)(cid:16) aN + λ (cid:17) (cid:101) I N + λX N (cid:111) − λg t (0)( (cid:101) I + X )+ (cid:90) t g Ns,t (cid:110)(cid:16) aN + λ (cid:17) d (cid:101) I Ns + λdX Ns (cid:111) − λ (cid:90) t g t ( s )( d (cid:101) I s + dX s )+ (cid:90) t λ ˜ g Ns,t d ( X is − X Ns ) + λ ˜ g N ,t ( X i − X N ) − λ (cid:20) ˜ g ,t ( X i − X ) + (cid:90) t ˜ g s,t d ( X is − X s ) (cid:21) = I t − I Nt + (cid:110) g N ,t (cid:16) aN + λ (cid:17) − λg ,t (cid:111) (cid:101) I N + λg ,t ( (cid:101) I N − (cid:101) I )+ (cid:90) t (cid:110) g Ns,t (cid:16) aN + λ (cid:17) − λg s,t (cid:111) d (cid:101) I Ns + (cid:90) t λg s,t d ( (cid:101) I Ns − (cid:101) I s )+ λ (cid:8) g N ,t − g ,t (cid:9) X N + λg ,t ( X N − X ) + (cid:90) t λ (cid:8) g Ns,t − g s,t (cid:9) dX Ns + (cid:90) t λg s,t d ( X Ns − X s )+ λ (˜ g N ,t − ˜ g ,t ) X i − λ (˜ g N ,t − ˜ g ,t ) X N + λ ˜ g ,t ( X − X N )+ λ (cid:90) t (˜ g Ns,t − ˜ g ,t ) dX is − λ (cid:90) t (˜ g Ns,t − ˜ g s,t ) dX Ns + λ (cid:90) t ˜ g s,t d ( X s − X Ns ) Therefore, for some constant C depending only on the parameters a , α and λ , but not onother ingredients of the model, E [( φ i ∗ t − φ MF,i ∗ t ) ] ≤ E [( I Nt − I t ) ] + CN E [( (cid:101) I Nt ) ] + C E [( (cid:101) I Nt − (cid:101) I t ) ]+ CN E [( X Nt ) ] + C E [( X Nt − X t ) ] + CN E [( X it ) ] ≤ E [( I Nt − I t ) ] + CN E [( I NT ) ] + C E [( I NT − I T ) ] + CN E [( X t ) ]+ CN N (cid:88) i =1 E [( ˇ X it ) ] ≤ CN (cid:90) T E [ S t ] dt + CN E [( X t ) ] + CN E [( ˇ X it ) ] . where the estimate for the first line above is obtained through Jensen’s inequality. Theother two estimates of the proposition are obtained in a similar way. References [1]
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