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## Optimal trading strategies with limit orders I (q a )h) and 0 with probability 1 i (q b )h (resp. Cette microstructure peut tre rencontr?e par example sur les futures de taux d int?rets courts. In the following I (q a )h) and 0 with probability 1 i (q b )h (resp. Cette microstructure peut tre rencontrée par example sur les futures de taux d intérets courts. In the following section, we will present frameworks where the choice of this two elements makes dynamic the optimal strategies. Let us examine more precisely the outcomes of the two practical implementations of limit orders posting mentionned in the last paragraph. The principal advantage of a continuoustime model is the use of the powerful stochastic calculus theory, which provides tractable 85 84 Literature survey: quantitative methods in high-frequency trading computations. Puis, la probabilité de transition ij Pn1 j n i de la chane de Markov stationnaire (n) est estimée à partir de K échantillons n, n 1,.,K suivant un estimateur standard. Y T 0 where 0 is a penalization parameter. Therefore, a discrete model may not be suitable for building a trading agenda since in this case, the goal of the investor is to endogenously determine the optimal trading times. Nous prouvons que ce schéma est convergent, et proposons des illustrations numériques ainsi qu une analyse de performance comparée. The optimality criterion is defined by these expected execution costs, and an optimal solution exists and is unique if C is strictly definite positive. Therefore, a trading strategy is viewed as a function of state 84 83 Literature survey: quantitative methods in high-frequency trading variables S, B, and t 0,.

Interpolate z v h (t i, z, j ) on Z j loc. Convergence of the value function First, we computed a reference value function that we will denote v using a parallelized version of our algorithm with parameters shown in table.1. Security, became the 52nd-most traded stock on that day, according to Eric Hunsader, CEO of market data service Nanex. This model is strongly related to self-exciting point process models of trades, similar to those that appears in 54 and 41 for example. First, the most 51 50 Introduction common implementation of the LOB is the price/time microstructure. Let (n) a n N and (n) b n N be two independent sequences.i.d integrable random variables valued in (0, of distribution laws a and b, which represent the transacted volume of the n th execution at best ask and best bid. Based on the current state of the LOB, they are able to compute the probability of price going up or down in the next milliseconds, and they propose a HF strategy to exploit this information. (6.4.4) This algorithm is described explicitly in backward induction by the following pseudo-code: Timestep t N T: for each y Y M, set w h, Y,M (t n, y) : 0 1 y 0 For.

We adopt the perspective of inventory management, which 67 66 Introduction Figure.5: Empirical distribution of terminal wealth X T (spline interpolation). We then provide the associated numerical resolution procedure, and convergence of this computational scheme is proved. 22 finra conducts surveillance to identify cross-market and cross-product manipulation of the price of underlying equity securities. In these conditions, they are able to provide a close-form approximation formula to the optimal"s. Finally, let us give two practical examples of such limit order books. Finally, in Section 5, we show how to extend our model in the optimal liquidation case,.e. Relation (6.3.9) then holds with c P, where represents a constant information about price direction, and ( ). 150 149 Optimal high frequency trading with limit and market orders The associated state process ( X, P,S) satisfies: Xt X t, t Y t for t T, and X T L(X T, Y T, P T, S T.

This decreasing dependence on the distance to mid-price is the modern equivalent of the price-dependent Poisson process appearing in Amihud and Mendelsohn and is chosen to exponential: ( ) N a Cox. We discuss and model the main risks specific to this microstructure, which are linked to the fact the size of the HF trades is not controlled. The variation of the (reduced-form) value function w due to predictive information is very small compared to the variation of the value function due to other market events (e.g. In this section, given C 0 and Y 0 we use the notation: the regular grid on R truncated. Décrivons maintenant notre modèle de processus de trade.

Using the integration by parts formula on this integral term, the authors show, under the assumption that martingale price process S 0, that C(X) does not anymore depend on S 0, so that there is no more source. Figure.7 shows the trade realizations of the optimal strategy for the day 04/19/ Numerical methods for an optimal order execution problem Parameter Value Parameter Value Maturity 1 Day.00E-04. Finra has stated that it will assess whether firms' testing and controls related to algorithmic trading and other automated trading strategies are adequate in light of the.S. Market model and trading strategies We consider a financial market where an investor has to liquidate an initial position of y 0 shares of risky asset by time. 120 119 Numerical methods for an optimal order execution problem Parameter Value Parameter Value Maturity 1 day X 0.02.2.5 N m A Q 10 5 B Table.2: Test 0: parameters. We assume that P T is integrable, and that the predictable finite variation term A of the special semimartingale P satisfies the canonical structure: da t d P t, with a bounded density process: t. We also provide numerical tests including computations of the optimal policies and performance analysis on a simulated data backtest. On introduit les processus suivants: (P t ) t 0,T le prix de l actif (X t ) t 0,T le montant de cash dans le portefeuille (Y t ) t 0,T le nombre d unit?s d actif. 3.2 Costs optimization strategies In this first section, we propose an overview of model-based costs minimization strategies. Cette microstructure peut aussi se rencontrer sur des marchés plus rudimentaires, tels que les marchés de paris en ligne, o un teneur de marché monopolistique maintient des"tions de telle sorte que les parieurs soient approximativement au mme nombre sur toutes les pattes du jeu. Note: This chapter is adapted from the article : 38 Guilbaud. Comme illustration numérique, nous avons réalisé une analyse de performance comparée détaillée sur des données simulées, et nous reproduisons les résultats principaux à la table, 45 44 Introduction générale Figure.9: Politique optimale à la date. On a: P (P th P t F t ) h o(h) P (P th P t F t ) h o(h) P ( P th P t F t ) o(h). In a high-frequency context, this model is of practical interest as it provides a way to include a (predictive) information about price direction. Our goal is to find optimal trading schedule and associated quantities. Indeed, at each date t i the computation of Eloc N v h (t i h, x,y, P 0,t i,p t i h, j h) and sup e C M,loc (z, j ) v h (t. Unless specified otherwise, such processes will be supposed to start at zero: typically, we assume that the investor starts from zero cash and zero inventory at date t 0 in the following numerical tests. Again, we can distinguish the three zones: buy, sell and no trade. Une autre microstructure importante, quoique plus exotique, est la microstructure au pro rata. P b : P 2 ). Then the investor has to face the following trade-off: if he chooses to trade immediately, he will penalize his performance due to market impact; if he trades gradually, he is exposed to price variation on the period of the operation.

Then, problem (5.2.6) is formulated equivalently as maximize E T U(L(X T, Y T, P T, S T ) g(y t )dt (5.3.1) 0 over all limit/market order trading strategies ( make, take ). This last point is counter-intuitive since one would expect the optimal strategy to sell quickly when the price is high, and more slowly when the asset price is low. They have their own (piecewise exponential) dynamics that reads as follows: d t ( t )dt dmt d t ( t )dt dm t The final elements of the model are the dynamics of the portfolios variables, that are exactly. D abord, la détectabilité des algorithmes d exécution est un enjeu central pour les brokers et les gestionnaires de portefeuille. Moreover, similarly as for v, and by the same arguments as in Remark.2.3, we see that v h (t, z,0) is strictly increasing in x and nondecreasing in y for (z,0) (x, y,p,0). Nous construisons un schéma numérique explicite rétrograde par séparation pour résoudre ce problème, et montrons comment réduire le nombre des variables d état jusqu à un système o n interviennent que les niveau de fourchette bid/ask et d inventaire. La figure.7 décrit l appariemment d un ordre au marché avec des ordres limites actifs du LOB. Within this section, we will denote by w h the value function and by the make/take strategy associated with the backward numerical scheme (6.4.3)-(6.4.4). Cette classe de stratégie est fondée sur l idée qu en utilisant des ordres limites, on peut acheter au prix bid, revendre au prix ask, et ainsi gagner la fourchette bid/ask dans l opération. Qualitatively, we can explain this strategy by thinking of a risk/reward trade-off. In this setup, the investor is able to choose discrete-time controls in a continuous-time system: typically, a trading strategy will be the choice of a discrete number of dates n associated with trade quantities n, which control a state. Our main goal is to construct an HFT strategy, by means of optimal stochastic control, that targets the pro-rata microstructure. In such type of markets, the price formation exclusively results from operating a limit order book (LOB an order crossing mechanism where limit orders are accumulated while waiting to be matched with incoming market orders.

At P b t : P b t (resp. We will assume that the value function is sufficiently smooth, and we focus in this paragraph on the diffusive part of the QVI, so that our target equation to solve is: t Lv 0 on. Ce probl?me se pr?sente naturellement lorsque le volume trait? est grand, en raison des quantit?s finies de liquidit? offerte dans le LOB (voir la section ci-dessus en effet, une unique transaction de grand volume peut d?s?quilibrer le LOB. Firms will be required to address whether they conduct separate, independent, and robust pre-implementation testing of algorithms and trading systems. She is not allowed to purchase stock during the liquidation period, and may only 195 194 Optimal HF trading in a pro-rata microstructure with predictive information buy back the asset in case of short position.

Let us now describe our model for trade processes. D abord les ordres limites (make strategy) sont mod?lis?s comme des contr?les continus: make t (Q b t, L b t (Q a t, L a t ) o Q b t repr?sente la cotation. 1.2 Observations qualitatives et contexte Pr?sentation g?n?rale Le trading haute fr?quence (HFT) est l utilisation de strat?gies automatis?es pour r?aliser des transactions sur des instruments financiers tels que les actions au comptant, les devises ou les produits d?riv?s, avec. In recent years, there have been a number of algorithmic trading malfunctions that caused substantial market disruptions. The first and most commonly used type of argument are the socalled statistical arbitrage arguments, that are typically cross-assets. On liffe (London International Financial Futures and options Exchange) or on CME (Chicago Mercantile Exchange). The stock mid-price is driven by a general Markov process, and we model the market spread as a discrete Markov chain that jumps according to a stochastic clock. L estimation de cette intensit? revient? estimer 2m scalaires, ce qui apporte de la flexibilit? au mod?le, mais qui requiert une m?thode sp?cifique, on d?finit: N b,qb, i t T b,qb, i t t. Puis nous discrétisons et localisons les variables d espace: Y R,M l R M, l M,.,M.

We assume that S and P are independent. We focus on the strategic stakes of high-frequency trading, and we put aside the technology issues such as latency minimization, direct market access or hardware speed improvement, which are however crucial aspects of the high frequency trading practice. The solvency constraint is a key issue in portfolio choice problem. La dynamique de l inventaire et du cash sont, sous : t t n, n t n1 n1 0,. We mention the complete survey article 33 about the limit order book, optimal trading strategies with limit orders from where we adapted the following definitions. We also plotted here the empirical distribution of the performance in figure and the efficient frontier, obtained by varying the arbitrary parameter, in figure.6 66 65 Introduction C AB C F (a) near date 0 (b). For this last purpose, we introduce a new state variable, that we interpret as a predictive price indicator, that allows us to balance our position before the price changes. Note that in figure.9, the naive strategy was overperforming the optimal strategy, due to an unexpected price increase. We assume that market price of risky asset process follows a geometric Brownian motion: dp t P t (bdt dw t ) Suppose now that the investor decides to trade the quantity. If we assume that P is a Lévy process, we have: PI P c P, d P t dt, where I P is the identity, for some constants. In a first part, we put aside market modelling issues, and focus and the optimization framework developped in 5 and 7, and further extensions and observations in recent studies. #### High Frequency, trading, iII: Optimal, execution QuantStart

Moreover, the fees structure (i.e. It only affects the execution cost of tn and not any subsequent orders. Modèle de marché Soit un espace de probabilités F,P) équipé d une filtration F (F t ) t 0, satisfaisant les conditions usuelles. Practical implementation of such rule would be, for example, to send a limit order with a fixed quantity, when the corresponding control is 1, and cancel it when it turns. We study the case of an investor that wants to unwind their portfolio, and provide a strategy that maximizes the revenue of this sale. Implementation shortfall measures the distance between the realized transaction price and the pre-trade decision price. 97 96 Numerical methods for an optimal order execution problem.1 Introduction Portfolios managers define implementation shortfall as the difference in performance between a theoretical trading strategy and the implemented portfolio. These example explain why measurement and efficient management of market impact is a key issue for financial institutions, and the research of low-touch trading strategies has found a great interest among academics. #### Optimal control in limit order books - PDF

Despite this, it is satisfactory to see that there are only three trades, which is less than on April 19 and 22, 2010, and that trading occurs when price conditions are favourable Test 3: Sensitivity to Bid/Ask spread. In such mechanism,"d prices are discrete, separated by the tick size which is typically of order.01 EUR per share. For example, P is a Lévy process or an optimal trading strategies with limit orders exponential of Lévy process (including Black-Scholes-Merton model with jumps). The interesting feature in this first graph is that we see two buying decisions when the price goes down through the.5 EUR barrier, and which corresponds roughly to a daily minimum. Des discussions sur le choix de f sont disponibles dans. In order to motivate our numerical scheme proposal, let us compare it with usual finite difference scheme. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the dpqvi. Until the market digests the news spikes in prices can hit Limit Order and it would be important to consider if you find the chance of getting into a trade to be of benefit.

(z,0) S, the set of admissible optimal trading strategies with limit orders transactions C (z, 0) y, 0 (and (z, 0, e) (x, y e, p) for e C (z,0) if x, and is empty otherwise. On the other hand, ( n ) n N is an increasing sequence of stopping times, representing the times when the investor chooses to trade at market, and n, n 0 are F n -measurable random variables valued. 100 x 0 Initial cash 0 y 0 Initial shares 0 p 0 Initial price 45 (d) Backtest parameters Table.2: Parameters 158 157 Optimal high frequency trading with limit and market orders Shape of the optimal policy. We shall denote by a i (qa ) a (q a, s b i (qb ) b (q b, s) for s i, i. We also provide numerical results both that show the behavior of the numerical scheme, the typical shape of the optimal strategy, and comparative performance analysis with respect to some benchmark execution strategies. Let v and v be defined on 0, T S by v (t, z v (t, z lim sup v (h,t,z, ) (0,t,z (t,z, ) 0,T) S h (t, z, ) liminf v h (t. Optimal trading in DP is possible, see for instance Optimal split of orders order liquidity pools: And to Order exploration and the dark pool problemby Ganchev, Nevmyvaka, Kearns, and Vaughan too. Minimisation des cots indirects de trading La minimisation des cots indirect de trading consiste à obtenir le prix le plus élevé possible pour une vente, ou obtenir le prix le moins élevé pour un achat. Giovanni Cespa, Xavier Vives (February 2017). Need more precise tests to conclude.

#### TMX TSX tsxv - TSX and tsxv Dark

Ces fournisseurs de liquidité sont en concurrence dans une enchère à la fois l achat (appelé le côté bid) et à la vente (appelé le côté ask). Proposition (Consistency) 107 106 Numerical methods for an optimal order execution problem (i) For all (t, z 0, T) S and C 1,2 (0, T) S we have (t, z, ). Le trading haute fréquence est installé dans cette phase de trading continu, et ainsi l étude des mécanismes précis qui réalisent cette double enchère continue est d une importante centrale lors de la construction d une stratégie. In this paper, we do not consider this case and focus on the main mechanism used in equity market. Soit T 0 un horizon de temps fini. Nous modélisons et étudions à la fois la microstructure standard à priorité prix/date et la microstructure exotique à priorité pro-rata. A market order of size m is an order to buy (sell) m units of the asset being traded at the lowest (highest) available price in the market. You can add some usual anti gaming features, like not using the same quantity for each of your orders, but again if your sizes are optimized according to a order enough measure, you will never place two orders with the same size. 6.3 Market making optimization procedure Control problem formulation The market model in the previous section is fully determined by the state variables (X, Y, P) controlled by the limit/market orders strategies ( make, take ). In 16, they allow a several form for these function, but let us focus on the exponential form, which is closest to the Avellaneda and Stoikov model: N Cox( N Cox( ) : exp( optimal trading strategies with limit orders t : exp( t ). We next define a discrete-time stationary Markov chain (n) n N, valued in the finite state space S i m, I m : 1,.,m, m N 0, with probability transition matrix ( ij ) 1 i,j M,.e. Finally, we matched our empirical results with the work 25 that is among the few that are dedicated to such market. We also mention the recent work 24 for useful insights on modelling with self-exciting point processes. Calibration procedures are derived for fitting the market model. Assuming that we can observe the following triplet: (Q b t, N b t, S t t 0, we aim at estimating the intensity function of the Cox process. We observe the following quantities: The tick times ( n ) n defined by: n1 inf t n : S t S t,.

Notice that close to date 0, the dependence in t seems to be negligible, which indicates that a stationary regime may be attained for large. Therefore, such algorithms are very sensitive to the response of the LOB they trade onto, and therefore are less efficient when easily detected by competitors. Contrary to the previous organization, any market participant is able to act as a liquidity provider, thanks to the use of limit order trading (see below). The above boundary condition for nonpositive inventory is related to the overtrading risk, which is the risk that the investor sold too much assets via the (oversized) limit order at the best ask price. In regular time,.e. Notre objectif est donc de maximiser le cash terminal, sachant que l optimal trading strategies with limit orders on ne d?tient aucune position sur l actif risqu?? la date T, et l on p?nalise la d?tention d un inventaire non nul pendant. Therefore, the limit orders trading operated by the market-maker has two opposite goals: on one hand, they seek at maximizing the number of trades in which they participate, in order to maximize revenue from making the spread. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Impulse control formulation As seen in previous sections, there exists both continuous-time models and discrete time model to solve an optimal liquidation problem. The form (4.2.5) was suggested in several empirical studies, see 50, 60, 4, and used also in 28,. London International Financial Futures and options Exchange, or Chicago Mercantile Exchange and will be the subject of a whole chapter of this thesis Issues faced in high-frequency trading industry In this subsection, we sum up the main industrial issues where high-frequency trading applies.

#### Carteajaimungalriccihft_wbs High Frequency, trading, order

For each simulation.N MC and for, WoMO, cst, we stored the following quantities: the terminal wealth after terminal liquidation V T : L( X P called performance in what follows ; the total executed. Teneurs de marché) proposent un prix pour n importe quel volume de transaction. The price impact is modelled via a nonlinear transaction costs function, that depends both on the quantity traded, and on a lag variable tracking the time spent since the investor s last trade. First, market making is typically not directional, in the sense that it does not profit from security price going up or down. The resulting optimal strategy is dynamic in the sense that it depends both on the time, the price of the risky asset, the cash amount and the quantity of shares in the portfolio. Qualitatively speaking, the effect of increasing the inventory penalization parameter is to increase the zone T where we trade at market. A limit order can be submitted to the market, updated in price or quantity or cancelled at any time, and therefore we call: Definition. 151 150 Optimal high frequency trading with limit and market orders by Let us also consider the impulse operator associated to market order control, and defined M(t, x,y, p,s) sup (t, take (x, y,p, s, e p,s e, where take.

We consider the following processes: (P t ) t 0,T the market price of the risky asset (X t ) t 0,T the cash holdings 56 55 Introduction (Y t ) t 0,T the number. Second, market makers keep almost no overnight position, and are unwilling to hold any risky asset at the end of the trading day. "Notice to Members 04-66 finra. This choice is a simple version of the (symmetric) Hawkes process model as presented in 41. The trading objective of the investor is to liquidate Y 0 0 assets over the finite time interval 0,. Nous nous concentrons sur les enjeux stratégiques, et mettons de côté les enjeux technologiques, tels que les accès directs aux marchés ou l optimisation de la vitesse du matériel de trading, quoiqu ils soient pourtant des aspects cruciaux de cette pratique.

Models of market impact based on stylized order book dynamics were proposed in 55, 64 and. Les fournisseurs de liquidité sont des agents de marché qui offrent ces prix, attendant qu une contrepartie saisisse leur offre et crée ainsi une transaction. Ils sont de deux types: Définition. If we focus on N b for example, this process represent arrivals of markets orders matching bid". The advantages of this approach is to start from a natural modelling of the order book, and to derive a closed-form optimal strategy. Moreover, from (4.3.7 we have the inequality: U(0) v h v, which implies by (4.2.12 lim inf v (t, z, ) U(0) v (t, z (t, z 0, T D 0 (4.3.13) (t,z, ) (t,z (t,z, ) 0,T) S Thus. Thus, by the viscosity solutions arguments of 8, we obtain the convergence of v h. H a b the operator Sh, Y,M is non-decreasing in,.e. In other words, we take so that: Y 0 T Parameters We computed the strategy with parameters shown in table.8. Université Paris-Diderot - Paris VII, English. Nous fournissons la solution numérique au problème de contrôle impulsionnel correspondant à la modélisation de cette situation, et nous prenons notamment en compte les effets des frais indirects de trading et de l impact de marché qui pénalisent les transactions trop rapides ou trop volumineuses. Figure.3: Intensités d exécution sur le 18 avril, 2011,en s 1 (interpolation affine) comme fonction du spread. This pro-rata microstructure is in use in some derivatives markets (e.g.

#### Dynamic dark pool trading strategies in limit order markets

We then see that the both the shape (pattern of execution) and the nature (static or dynamic) of the optimal strategy is fundamentally related to the choice of the setup, and in particular is determined by those two elements: Dynamics of the reference price (i.e. More precisely, a trading strategy is a pair : ( make, take ) of regular/impulse controls: make : (L a t, L b t) t 0, take : ( n, n ) n N where. 1, automated trading systems and electronic trading platforms can execute repetitive tasks at speeds with orders of magnitude greater than any human equivalent. Dans ce contexte on peut calculer la politique optimale (figure.9). The market orders used in these works are hit orders, optimal trading strategies with limit orders which means that they are actually marketable limit orders,.e. Nous construisons des sch?mas num?riques originaux pour la r?solution d in?galit?s variationnelles de la programmation dynamique, qui correspondent aux contraintes pratiques du trading haute fr?quence: nous mettons en place des m?thodes de r?duction de la dimension, ainsi que des algorithmes. The minimization problem is expressed by means of the dynamic programming principle under the form of an HJB equation: denote by B T B T 2, T t and. Thanks to a variable change, the authors are able to obtain explicit approximating formulas for the optimal"s, and they perform numerical tests.

Such papers examine a direct application of high-frequency trading: an investor who continuously submit bid and ask"s in a limit order book wants to control its exposure to market risk, by keeping its position on the risky asset close to zero at all time. A growing literature is dedicated to modelling the dynamics of the limit order book itself, and its consequences for the price formation process. Join us in building a kind, collaborative learning community via our kotona tehtävä työ 2015, code of Conduct. Market model We use the following simple price model: P the mid-price (observable: lit microstructure a Markov process of generator P valued. This differs from the price-time microstructure. The role of a marketplace is to gather and to match the order to trade, originated from market participants, that can be submitted at any time during the continuous trading phase. 13 Backtesting Backtesting of a trading system involves programmers running the program by using historical market data in order to determine whether the underlying algorithm can produce the expected results. If she chooses to improve best price.e Q b n Bb (resp. Nous cherchons à fournir un traitement complet de chaque situation, depuis la modélisation des phénomènes de marché, la résolution mathématique, jusqu à la calibration et aux expériences numériques a posteriori, qui contiennent des résultats sur données réelles lorsque celles-ci sont disponibles. De vente) ait consommé plusieurs niveaux du LOB d un coup. For proving the (pointwise) consistency in the line of Proposition.3.3, we have to estimate, for any fixed t i T m, j T i, z Z j, any smooth test function, the accuracy of the approximate. Qualitatively speaking, using the 2-sigma rule, this means that the process spends most of the time in the range.6.6. Sous l horloge tick-time, la fourchette bid/ask est supposée tre une chane de Markov stationnaire (n) n N à valeurs dans S i m, I m 1,.,m, o est la taille du tick. 182 181 Optimal HF trading in a pro-rata microstructure with predictive information We then approximate the solution w to (6.3.11)-(6.3.12) by the function w h, Y,M on M solution to the computational. The optimal trading problem is studied by stochastic control and dynamic programming methods, which lead to a characterization of the value function in terms of an integro quasi-variational inequality. Cancel the bid (resp. (N b, (n) b n N ) of intensity measure a a (dz)dt (resp. Sitemap