Deep Investing in Kyle's Single Period Model
aa r X i v : . [ q -f i n . C P ] J un Deep Investing in Kyle’s Single Period Model
Paul Friedrich and Josef TeichmannThursday 25 th June, 2020
Abstract
The Kyle model describes how an equilibrium of order sizes and secu-rity prices naturally arises between a trader with insider information andthe price providing market maker as they interact through a series of auc-tions. Ever since being introduced by Albert S. Kyle in 1985, the modelhas become important in the study of market microstructure models withasymmetric information. As it is well understood, it serves as an excellentopportunity to study how modern deep learning technology can be usedto replicate and better understand equilibria that occur in certain marketlearning problems.We model the agents in Kyle’s single period setting using deep neuralnetworks. The networks are trained by interacting following the rules andobjectives as defined by Kyle. We show how the right network architec-tures and training methods lead to the agents’ behaviour converging tothe theoretical equilibrium that is predicted by Kyle’s model.
The Kyle model, introduced by Albert S. Kyle in his 1985 paper
Continu-ous Auctions and Insider Trading [1], is a well known and often cited marketmicrostructure model. It aims to describe and understand aspects of insidertrading, liquidity and the value of private information to an insider. Our workhas two parts. We use neural networks to replicate the single period Kyle modelunder Gaussian assumptions and then alter it, using the trained networks to findKyle equilibria where tractability fails to hold. For the theoretical discussions,we will mainly be following Kyle’s original paper [1].In the model, a risky asset is traded over one period of time in a seriesof auctions. Since a risk-free asset is used as a numeraire, the risk-free rateis set to zero. The market contains three actors: a risk neutral inside trader,a random noise trader and a risk neutral market maker. The inside traderpossesses private and exclusive knowledge of the liquidation price of the asset,i.e., the value at which it can be sold at the end of the auction. Based on thisknowledge, the insider submits a market order. The noise trader also submitsa market order, but its size is random. Both orders are aggregated and sentto the market maker, who is unable to distinguish which proportion of thetotal order originates from each trader. This way, the noise trader providescamouflage for the inside trader, enabling the insider to make a profit at thenoise trader’s expense. The market maker then determines a price and clearsthe market. Kyle proved in [1] that in the situation of a single auction or1 series of independent auctions under Gaussian assumptions, an equilibriumarises between the inside trader and market maker. Market maker prices andinside trader order quantities are linear functions of their respective inputs, andtheir parameters can be derived explicitly. He also showed that the multi-auctionmodel and its equilibrium converge to a continuous model and equilibrium asthe time between auctions approaches zero.The model has several important features. A crucial one is that the insiderconsiders their own impact on the offered market prices. They need to decidehow to optimally use their private information in order to maximise profit. Onecan also use the Kyle model to understand liquidity conditions of a market inan auction equilibrium. A liquid market is, as written in Kyle [1], characterisedas one that is nearly infinitely tight (i.e., turning around a large position overa short period of time is very expensive), not infinitely deep (i.e., one does notneed extremely large order flows to impact the market price) and resilient (i.e.,prices tend to return to their underlying value over time). The Kyle modelallows one to study these dynamics in a quantitative way. Lastly, the modelshows how private trading information impacts the markets via the inside trader,who gradually uses that information and incorporates it into market prices.The market situation with its agents interacting with each other using fixedconstraints and reward functions is well suited to the application of machinelearning techniques.This paper has a two step approach. In Chapter 2, we illustrate the theoret-ical setting, show the central statements about the existence of an equilibrium,and formulate the theory in a way that makes it easier to translate into machinelearning implementations. By replacing the agents with neural networks whothen trade with each other, we can better understand the nature and dynamicsof the equilibrium. Once an equilibrium is found by the agents, we can alter theparameters and constraints of the problem and introduce new aspects. Thesecan for example include transaction costs, different price distributions or irra-tional behaviour by one or both of the agents. In Chapter 3, we apply thisapproach to the version of the model where only a single auction takes placeand illustrate our methods and results.
The single period Kyle model describes a market model where a single risky assetis traded among three agents: a risk-neutral market maker , a noise trader and arisk-neutral insider . At the beginning of the period, the noise trader and insidersubmit their market orders (positive if buy order, negative if short sell order) tothe market maker, who then determines one price for both transactions. At theend of the period, the noise trader and insider realise a profit or loss, dependingon the value of the asset and initial price. We first list the definitions andassumptions within the model.The value of the traded asset at the end of the period is denoted z andassumed to be a realisation of a normally distributed random variable Z ∼N ( µ z , σ z ). The noise trader submits a market order of size y to the marketmaker. The order is modelled as a normal random variable Y ∼ N (0 , σ y ) thatis independent of Z and any other elements of the model. It is assumed thatthe insider knows with certainty the exact value z of the asset at the end of the2eriod. They do not know the order size y of the noise trader. Based on thisinformation and with the goal to maximise his profits, the insider submits amarket order of size x = X ( z ) to the market maker. The market maker receivesthe combined total of market orders from the insider and noise trader, x + y .However, they cannot differentiate which proportion of the total order camefrom the insider or noise trader. The market maker uses a fixed pricing rule P ( · ) that depends only on the total market order amount X + Y and the marketmaker’s goal of exactly breaking even on their own trade. Based on the totalorder obtained the market price p = P ( x + y ) is set according to the marketmaker’s pricing rule. Afterwards they take the position − ( x + y ) to clear themarket.Both the market maker and insider are assumed to act risk-neutral. Theinsider’s goal is to find the value x that maximises his or her expected end-of-period profit while knowing the exact end-of-period asset price z , i.e.max x E [( Z − P ( x + Y )) x | Z = z ] = max x ( z − E [ P ( x + Y )]) x. (2.1)For the market maker, it is assumed that the pricing rule solely depends onthe total market orders. Indeed, the greater that total size is, the greater thepossibility of x being large, which would indicate that the insider knows that theend-of-period price z is higher than the expected µ . Therefore the market makerwould set a higher price. The opposite also holds, the smaller the total ordersize is, the greater the possibility of x being small, indicating that z would fallbelow µ . The market maker would then lower the offered price. Since the marketmaker’s expected end-of-period profit is given by − E [( Z − P ( x + y ))( x + y )] andthey have the goal to break even, the pricing rule is given by P ( x + y ) = E [ Z | X + Y = x + y ] . (2.2)The goal is now to find a situation where the insider and market maker arein an equilibrium . Definition 2.1.
Let p = P ( X + Y ) be the random variable describing the marketmaker’s price whose distribution depends solely on X and P . Let π = X · ( Z − p ) be the random variable describing the inside trader’s profit which depends solelyon their strategy X and the market maker’s pricing rule P . An equilibrium isgiven by X and P which satisfy the following conditions:(i) Profit Maximisation: For all trading strategies X ′ = X and end-of-periodprices z , it holds that E [ π ( X, P ) | Z = z ] ≥ E [ π ( X ′ , P ) | Z = z ] . (ii) Market Efficiency: The random variable p satisfies p ( X, P ) = E [ Z | X + Y ] . Intuitively, the insider succeeds in maximising their profits given the marketmaker’s pricing rule, the market maker succeeds in having a net profit of zerogiven the total market orders by noise trader and insider and both the insiderand market maker expect each other’s behaviour.3yle argues that under the assumption that Y and Z are independent, anequilibrium can be found where both market maker and insider use a linearpricing and trading rule respectively. The proof of this theorem can be foundin [1] as the proof of Theorem 1. Theorem 2.2.
There exists a unique equilibrium in which X and P are bothlinear functions. For the constants α = − σ y σ z µ z , β = σ y σ z and µ = µ z , λ = σ y σ z it holds that X ( z ) = α + βz, P ( x + y ) = µ + λ ( x + y ) . One important fact for our implementation is that for estimating the pricingrule P ( x + y ) = E [ Z | X + Y = x + y ], a maximum likelihood estimator is the’best’ course of action in the sense that it leads to maximum efficiency whilebeing the minimum variance unbiased estimate. Since the insider uses a linearpricing rule such as X ( Z ) = α + βZ , from the point of view of the market maker, Z and V := Y + X = Y + α + βZ are jointly normally distributed. Therefore,the maximum likelihood estimate of E [ Z | V ] is linear in V and in this case isactually the least squares one, i.e. the one that minimises E [( Z − P ( V )) ]. Thisprovides an easy loss function for the market maker model. We implement Kyle’s model in a deep learning framework in order to test if apair of neural networks is able to arrive at the linear equilibrium using only thebase assumptions of the model. We implement and train the models with
Keras using
TensorFlow as a backend. The models consist of a simple feed-forwardlayer structure where the market maker possesses two layers and the insider onelayer of ten nodes each, both times connected to a single output node. As for themodel defining parameters, we choose a configuration of µ z = 0 . , σ z = 2 , σ y =1, which we refer to as ‘non-centered’. We found that using large positive valuesof µ z for training would lead to the insider’s predictions being highly inaccuratefor negative values of z , as the likelihood of encountering such values duringtraining would be very low. µ z = 0 (a ‘centered’ configuration) or close to zeroproduced the best results.All assumptions we make are in line with Kyle’s model, namely:1. The noise trader market order Y and end-of-period price Z are indepen-dent and normally distributed with the parameters we chose.2. The insider knows the end-of-period market price z .3. The market maker receives only the combined total of market orders x + y .Next, let us describe our training procedure. We run a training loop wherein each loop we first train the market maker based on the most recently trainedinsider, then train the insider based on the market maker. The very first loopdoes not yet have an insider model initialized and we therefore require an initialinsider order function to train the market maker. We then plot the results andbegin a new training loop. In this way, we alternately train each actor with theintention of both actors gradually learning from each other until an equilibrium4s reached. Before we begin training, we thus have to choose a configuration forthe distributions of Z and Y , as well as an initial insider order function. Weusually choose an initial insider order function that returns values randomly,namely normally distributed like the noise trader Y , i.e. X ( · ) ∼ Y . Thisapproach of letting the initial insider output noise uses zero outside informationto train the market maker. In the following discussion, we refer to the insiderand market maker networks as ‘I’ and ‘M’ respectively. I ( z ) or M ( x + y ) thendenote the network outputs when given an input z or x + y .Each loop begins by training the market maker. We first sample from Y and Z independently, generating samples ( y , . . . , y N ) and ( z , . . . , z N ). We usethe initial insider order function (first loop) or the last trained insider network(subsequent loops) to generate predictions I ( z i ) := x ( z i ) out of the samples of Z.The y i are added to the corresponding insider orders I ( z i ) and result in a sampleof total market orders ( I ( z ) + y , . . . , I ( z N ) + y N ). For all i ≤ N , I ( z i ) + y i is fed into the market maker neural network together with the corresponding z i (which is only used in the loss function) with batch size one. TensorFlowminimises the loss function of a whole epoch at a time. This loss is essentiallythe mean of each batches’ individual loss. We therefore define the following lossfunction for the market maker: • Calculate a loss for each input I ( z i ) + y i : L ( z i , y i ) := ( z i − M ( I ( z i ) + y i )) . • The total loss over one epoch of training is obtained by averaging all L ( z ): L epoch = E ( Z,Y ) [( Z − M ( I ( Z ) + Y )) ] . The model is trained for a number of epochs on the sampled data. Its output,via a model.predict method, then defines a pricing function P ( · ) := M ( · ) forthe insider to use.Next, we train the insider. We again sample from Y and Z independently.The samples can be used directly for training, as the market maker appearsonly in the insider’s loss function in what is a straightforward implementationof equation (2.1): • Calculate a loss for each input z i , y i : L ( z i , y i ) := − ( z i − M ( I ( z i ) + y i )) I ( z i ) . • The total loss over one epoch of training is then given by: L epoch = − E ( Z,Y ) [( Z − M ( I ( Z ) + Y )) I ( Z )] . The insider is trained and its output defines a new order function x ( · ) := I ( · )that is used to train the market maker in the next iteration. In both cases werely on neural network generalization which determines the functions outsidethe initial training data. 5
15 −10 −5 0 5 10 15End-of-period price z−6−4−202468 I n s i d e r o r d e r x ( z ) Insider NN prediction
Predicted linear beha iorNN Output −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0Total Order x+ −10.0−7.5−5.0−2.50.02.55.07.510.0 M a r k e t P r i c e P ( x + ) Market Maker NN prediction
Predicted linear behaviorNN Output−15 −10 −5 0 5 10 15End-of-period price z−8−6−4−20246 I n s i d e r o r d e r x ( z ) Insider NN prediction
Predicted linear beha iorNN Output −10 −5 0 5 10Total Order x+y−10−50510 M a r k e t P r i c e P ( x + y ) Market Maker NN prediction
Predicted linear behaviorNN Output−15 −10 −5 0 5 10 15End-of-period price z−6−4−202468 I n s i d e r o r d e r x ( z ) Insider NN prediction
Predicted linear beha iorNN Output −10 −5 0 5 10Total Order x+y−10−50510 M a r k e t P r i c e P ( x + y ) Market Maker NN prediction
Predicted linear behaviorNN Output
Figure 1: Assuming µ z = 0, both neural networks learn the theoretical equi-librium when starting with a linear (top), approximately linear (middle) orGaussian noise (bottom) insider order function. We train the model using a centered configuration with µ z = 0. Each model istrained for three epochs within one loop, with there being a total of 20 trainingloops. While the models converge using as little as 2000 samples, we use 5000samples of z and y for both the insider and market maker. This way, conver-gence happens within fewer training loops and the convergence is more stablein the sense that model predictions do not change much after first hitting anacceptable solution. Figure 1 shows how both models converge nearly perfectlyto the predicted linear equilibrium. The figure also shows the effects of usingdifferent initial order functions. Starting with an already linear (i.e. equilib-rium fulfilling) insider function leads to fast convergence in around three loopswhile an approximately linear initial order function needs five loops to converge.However, the networks learn the equilibrium even if they start with an insiderfunction that is indistinguishable from the noise trader and they do so withinmerely eight training loops. On our hardware, this training was bottlenecked bythe CPU (likely due to the necessity of setting batch size = 1), so we opted torun it entirely on the CPU. A quad-core laptop CPU took around three to four6inutes to reach convergence in the 5000 sample situation, and merely 30-60seconds for 2000 samples. Y, Z
With our first goal, managing to train networks to learn Kyle’s equilibrium,out of the way, we turn to uncharted territory and analyze the equilibrium byaltering some key assumptions. We first drop the requirement of independent,normal distributions and later introduce transaction costs to the model. Byusing the same training regiment as above, we can reliably find the new equilibriain these situations that one could not theoretically derive otherwise.So far, we have built and tested the model using various normal distributionsto sample noise trader orders Y and market prices Z . Normal distributionspossess a defined mean and variance, they are symmetrical with one peak andno skewness. When dropping these assumptions, it is not at all clear how thesystem may behave. One key fact used in the theoretical derivation of theequilibrium is that the L -optimiser of the market maker’s pricing rule is theleast squares one. This stops being true when Y or Z are no longer normallydistributed, as X + Y is then no longer jointly normal (even if X ( Z ) is linearand Z or Y respectively remain normal). By choosing different distributionsfor Y and Z , we expect to see nonlinearities in the pricing and order functionsthat depend on the type of distributions and may correlate with how similar theused distributions are to normal ones in terms of their defining characteristics.To that end, we picked several distributions with different attributes, namelythe Laplace, Gumbel and Gamma distributions, as well as a bimodal distribu-tion given by a mixture of two normal distributions. For the first three, meanand variance are defined and we thus selected their parameters for each distri-bution family such that mean and variance would match the ones chosen so far,i.e. E [ Z ] = µ z = 0 . , Var[ Z ] = σ z = 4 , E [ Y ] = µ y = 0 and Var[ Y ] = σ y = 1.If Y and Z are chosen to both be distributed as Laplace, Gumbel or Bimodal,the model learns and outputs not only a linear equilibrium, but the one corre-sponding to the setting where Y ∼ N (0 ,
1) and Z ∼ N (0 . , Y normally distributed as Y ∼ N (0 , Z . We show results for Laplace, Gumbel,Gamma and Bimodal distributions in Figure 2 while Figure 3 shows the distri-butions of the resulting model input data as histograms. The results are moremixed now. When Z is sampled from Laplace or Bimodal distributions, weobserve the same linear equilibrium as in the normal situation. If Z is sam-pled from a Gumbel distribution, both pricing and order functions are linearfor positive and negative values with a bend around the origin. For each func-tion, its two slopes are similar to the one predicted in a normal model. Thesesignificant, but not major differences to the normally distributed situation cor-respond well to the comparison between a Gumbel and normal distribution:similar overall look, but small deciding differences (skewness). When sampling7 from a Gamma distribution, we thus expect to see larger differences to thenormal situation.We have previously seen that the model struggles with data that shows exclu-sively positive values. We thus alter the parameters of the Gamma distributionto correspond to a mean of 1 and variance of 2, which produces a sample of Z with values mostly between 0 and 2 and a significant tail. We then subtract 1from that sample. The distribution is thus centered, but still with a large skewdue to being a Gamma distribution. We expect a result similar to the Gumbelone, although the skewness is much larger here. We compare the predictions tothe equilibrium situation. Both function predictions are again piecewise linearwith a single bend around the origin. Both show a large deviation for nega-tive values while having a similar (pricing function) or nearly identical (orderfunction) slope as the normal theoretical optimum. This result further indicatesthat one, the model is very sensitive to non-centered data and two, skewnessleads to piecewise linear predictions, where the amount of skewness influenceshow closely slopes align with the normal situation. We introduce market frictions to the model. Concretely, we assume that theinsider now pays a transaction fee proportional to their order size. This isimplemented by adding a penalisation term to the insider’s loss function. Thenew version works as follows: • Sample z ∼ Z (as model input) and y ∼ Y . • Calculate a loss for each pair ( z i , y i ): L ( z i , y i ) := − [( z − M ( I ( z ) + y )) I ( z ) − ε · | I ( z ) | ] • The total loss over one epoch of training is then given by: L epoch = − E ( Z,Y ) [( Z − M ( I ( Z ) + Y )) I ( Z ) − ε · | I ( Z ) | ] . Epsilon represents the fraction of the total order size that must be paidas transaction cost. This penalty only impacts the inside trader, so we do notexpect the market maker’s behaviour to change considerably. The inside trader,however, should adjust their behaviour. One could expect them to not tradeat all in a certain price range where transaction costs would be larger than theexpected payoff. If we assume that the market maker stays unimpacted (i.e.uses Kyle’s predicted optimal pricing function), then the boundary of that pricerange would lie wherever the expected payoff is zero. Looking at the above lossfunction, for a given z this happens when:0 = E [( z − P ( x ( z ) + Y )) x ( z ) − ε · | x ( z ) | ] ⇔ ε = ± (cid:18) z − σ z σ y x ( z ) − µ z (cid:19) ⇔ x ( z ) = 2 σ y σ z · ( z − ( µ z ± ε ))8
20 −15 −10 −5 0 5 10 15 20End-of-period price z 10 50510 I n s i d e r o r d e r x ( z ) Insider NN prediction
Predicted linear behaviorNN Output −10 −5 0 5 10Total Order x+y−10−5051015 M a r k e t P r i c e P ( x + y ) Market Maker NN prediction
Predicted linear behaviorNN Output−10 −5 0 5 10 15 20End-of-period price z−7.5−5.0−2.50.02.55.07.510.0 I n s i d e r o r d e r ( z ) Insider NN prediction
Predicted linear behaviorNN Output −10 −5 0 5 10Total Order x+y−10−5051015 M a r k e t P r i c e P ( x + y ) Market Maker NN prediction
Predicted linear behaviorNN Output−4 −2 0 2 4 6 8End-of-period price z−15.0−12.5−10.0−7.5−5.0−2.50.02.55.0 I n s i d e r o r d e r x ( z ) Insider NN predic ion
Predic ed linear behaviorNN Ou pu −10 −5 0 5 10Total Order +y−8−6−4−202468 M a r k e t P r i c e P ( + y ) Market Maker NN prediction
Predicted linear behaviorNN Output−10 −5 0 5 10End-of-period price −4−2024 I n s i d e r o r d e r x ( ) Insider NN prediction
Predicted linear behaviorNN Output −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0Total Order x+ −10−50510 M a r k e t P r i c e P ( x + ) Market Maker NN prediction
Predicted linear behaviorNN Output
Figure 2: Model predictions when Y is gaussian and Z is sampled from differ-ent distributions. Top to bottom: Laplace, Gumbel, Gamma (centered) andBimodal (normal mixture) distributions.9 F r e q u e n c y Histogram of model input −10 −5 0 5 10End-of-period price z050100150200250300350 F r e q u e n c y Histogram of model input −4 −2 0 2 4Total order x+y020406080100120140160 F r e q u e n c y Histogram of model input −2.5 0.0 2.5 5.0 7.5 10.0 12.5End-of-period price z0255075100125150175200 F r e q u e n c y Histogram of model input −6 −4 −2 0 2 4 6Total order x+y0255075100125150175 F r e q u e n c y Histogram of model input −1 0 1 2 3 4End-of-period price z050100150200 F r e q u e n c y Histogram of model input −4 −2 0 2 4Total order x+y0102030405060 F r e q u e n c y Histogram of model input −4 −2 0 2 4End-of-period price z01020304050 F r e q u e n c y Histogram of model input
Figure 3: Comparing input distributions when picking different distributionfamilies for Z . In order from top to bottom: Laplace, Gumbel, Gamma (cen-tered), Bimodal. Plots on the right directly show the distribution of Z .10 z Transaction cost ( ε ) Prediction Test result0.5 0.01 [0.49, 0.51] no visible change0.5 0.1 [0.4, 0.6] no visible change0.5 0.5 [0, 1] no visible change0.5 0.75 [-0.25, 1.25] [-0.5, 1.1]0.5 1 [-0.5, 1.5] [-0.54, 1.44]0.5 1.2 [-0.7, 1.7] [-0.6, 1.7]1.5 1.2 [0.3, 2.7] [0.0, 2.7]0.5 1.5 [-1, 2] [-0.9, 2.0]Table 1: Results of our series of tests.On the other hand, the range where x ( z ) = 0 holds should then be boundedby the z for which z = µ z ± ε . In our non-centered parameter setting, thiscorresponds to z = 0 . ± ε .For our testing, we slowly increase the magnitude of the transaction cost.The results, shown in figure 4, show several interesting features. The marketmaker pricing function stays at the linear optimum through all of our tests. Forsmall transaction costs of 0 .
01 and 0 .
1, there is no visible effect on the insider’sbehaviour. Only when increasing the transaction cost to large values of 0 .
75 or1 do we start to see the aforementioned ‘plateau’ in the insider order function.The size and position of this plateau matches our hypothesis rather well, ascan be seen in table 1 and figure 4. The slight differences that are presentcould come from the fact that the market maker does adjust their behaviourslightly due to receiving an input order function from the insider that is notperfectly linear (i.e. that shows a plateau around zero). This would in turninfluence the training and result of the insider function and could explain thesmall differences.In order to further confirm our hypothesis, we set µ z = 1 .
5. The result,shown in figure 5, shows that the plateau shifts accordingly and is again roughlycentered around the new value of µ z . We have proven the viability of deep neural networks for finding and betterunderstanding equilibria of the Kyle model. Our model architecture and trainingmethod leads to quick and robust convergence to Kyle’s equilibrium. We canuse our method to alter the model, including finding equilibria in the cases ofnon-normal price distributions and transaction costs.Our next step will be to extend our approach to the multi period Kyle model,where several auctions take place over the course of a single trading day.
References [1] Albert S. Kyle. Continuous auctions and insider trading.
Econometrica ,1985. 11
10 −5 0 5 10Total Order x+y−10−50510 M a r k e t P r i c e P ( x + y ) Market Maker NN prediction
Predicted linear behaviorNN Output −15 −10 −5 0 5 10 15End-of-period price z−8−6−4−202468 I n s i d e r o r d e r x ( z ) Insider NN prediction
Predicted linear beha iorNN Output −3 −2 −1 0 1 2 3End-of-period price z−2.0−1.5−1.0−0.50.00.51.0 I n s i d e r o r d e r x ( z ) Location of "bend" in the insider order function −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0Total Order x+ −7.5−5.0−2.50.02.55.07.510.0 M a r k e t P r i c e P ( x + ) Market Maker NN prediction
Predicted linear behaviorNN Output −15.0 −10.0 −5.0 0.0 5.0 10.0 15.01.0−0.5End-of-period price z−6−4−20246 I n s i d e r o r d e r x ( z ) Insider NN prediction
Predicted linear beha iorNN Output −4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 4.01.0−0.5End-of-period price z 1.5 1.0 0.50.00.51.0 I n s i d e r o r d e r x ( z ) Location of "bend" in the insider order function −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0Total Order x+ −10.0−7.5−5.0−2.50.02.55.07.510.0 M a r k e t P r i c e P ( x + ) Market Maker NN prediction
Predicted linear behaviorNN Output −20.0 −15.0 −10.0 −5.0 0.0 5.0 10.0 15.0 20.02.0−0.9End-of-period price z−8−6−4−20246 I n s i d e r o r d e r x ( z ) Insider NN predic ion
Predic ed linear behaviorNN Ou pu −4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 4.02.0−0.9End-of-period price z−1.25−1.00−0.75−0.50−0.250.000.250.50 I n s i d e r o r d e r ( z ) Location of "bend" in the insider order function
Figure 4: Comparing model predictions when assuming different transactioncost (epsilon) values and a fixed µ z = 0 .
5. Transaction cost values from top tobottom: ε = 0 . , . , . −4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 4.01.7−0.5End-of-period price −1.00−0.75−0.50−0.250.000.250.500.75 I n s i d e r o r d e r x ( ) Location of "bend" in the insider order function −4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 4.02.70.0End-of-period price −1.50−1.25−1.00−0.75−0.50−0.250.000.25 I n s i d e r o r d e r x ( ) Location of "bend" in the insider order function
Figure 5: Left: µ z = 0 .
5, right: µ z = 1 .
5. The window where the insiderfunction is zero remains centered around µ zz