A Theory of 'Auction as a Search' in speculative markets
AA Theory of ’Auction as a Search’ in Speculative Markets
Sudhanshu PaniSaturday 4 th April, 2020
Abstract
The tatonnement process in high frequency order driven markets is modeled as a search by buyers for sellersand vice-versa. We propose a total order book model, comprising limit orders and latent orders, in the absenceof a market maker. A zero intelligence approach of agents is employed using a diffusion-drift-reaction model, toexplain the trading through continuous auctions (price and volume). The search (levy or brownian) for transactionprice is the primary diffusion mechanism with other behavioural dynamics in the model inspired from foraging,chemotaxis and robotic search. Analytic and asymptotic analysis is provided for several scenarios and examples.Numerical simulation of the model extends our understanding of the relative performance between brownian,superdiffusive and ballistic search in the model.Keywords: Market Microstructure, Levy Search, Limit order markets, Continuous Auctions, High Resolu-tion, Zero Intelligence
MODERN MARKETS are built on the foundation of low latency trading and exchange infrastructure. Dataavailable from the order driven markets today can help us to investigate the markets both in high frequency andhigh resolution. O’Hara (2015) called for the market microstructure toolbox to be enhanced in order to investigate’orders’ that were the unit of information in today’s markets. Several recent papers implement objectives thatrequire a high resolution analysis, including, modelling imperfect competition and quote cancellation (Baruch andGlosten (2019)), price discovery in high resolution (Hasbrouck (2019)), price discovery in quotes (Brogaard et al.(2019)), trading in absence of designated market maker (Kyle et al. (2018)). In the context of high resolutionanalysis, the assumption of equilibrium cannot be easily justified. However, such an assumption is fundamental toour current framework used to understand continuous double auctions in financial markets. In walrasian auctionsin these markets, the price of a good is defined as the point where the supply and demand curves intersect (amarket clearing equilibrium under perfect competition). The relationship between supply and demand and theirlinkage to price discovery may not however be effectively described by the assumption of walrasian equilibriumin high resolution of analysis. 1 a r X i v : . [ q -f i n . T R ] J un ven in quote driven markets, Beja and Hakansson (1977) had shown that the instantaneous walrasian adjust-ment imposes prohibitive demands on communication and computation, while the tatonnement iterations musttake time, and thus it typically never converges. They further show that market making was necessary to priceadjustments away from equilibrium. Modern order driven markets clear in the time scales that ranges from subseconds to nano seconds. Donier and Bouchaud (2016) report the time required for walrasian adjustments to bein the scale of an hour. Jusselin et al. (2019) in their study based on Euronext exchange data and their model(presence of market maker and price discovery criterion) compute the optimal auction duration according to theircriterion for 77 European stocks traded on Euronext. They report that the suggested durations are much larger thana few milliseconds, rather of order of 1 to 5 minutes. They reconfirm that continuous limit order book (CLOB)are in terms of their metric sub-optimal. However, the quality of the price formation process in CLOB market isnot very far from that of the auction with optimal duration.This paper explores ’Auction as a search’ as an alternative to the supply and demand paradigm for tatonnementin order driven markets under the absence of an auctioneer or designated market maker. The questions we ask in-clude: What is the modern day ’tatonnement’ in the electronic order driven markets. Can we explain the dynamicsof such markets as a ’search’ by buyers for sellers and vice-versa.The walrasian tatonnement is in effect an algorithm wherein an invisible auctioneer has all information regard-ing supply and demand schedules and comes up with the price that optimises the quantity traded. This emphasisesan important function of the market, that of optimising the traded quantity. Any alternative algorithm needs tomeet these two goals. We show in this paper that the tatonnement in continuous double auctions in modern assetmarkets under the usual price-time priority may indeed be described as a search of buyers by sellers and vice-versa. ’Auction as a search’ meets both the above stated objectives. An acceptance of this proposed alternativeshould help researchers, working on price discovery and trading models in high resolution, build models that arecloser to reality. The predominant market design used in asset markets around the world today is the continuous double auctionmechanism (although these markets can vary in terms of the trading rules or matching rules they employ). Thisimplies that an asset can theoretically be traded at any point in time. Double auctions enable all participants toprovide quotes (both buy and/or sell orders). Futher these markets are organised as electronic exchanges, where2he quotes from the traders are matched by matching engines without any official role for intermediaries. Thematching is done everytime a new order comes into the market. Typically a mismatch may exist between buyersand sellers at any given instant. Hence, the need for an order based market with two fundamental kinds of orders.Traditionally, it was believed that impatient traders submited requests to buy or sell a given number of sharesimmediately at the best available price. Such orders are known as market orders. More patient traders submitlimit orders, which also state a limit price, that correspond to the worst allowable price for the transaction. Limitorders may not result in an immediate transaction, and are stored in a queue called the continuous limit order book.Buy limit orders are termed bids, and sell limit orders offers or asks. Both limit orders and market orders playan important role in the order book. Institutional investors, large traders executing meta-orders and algorithmictraders deploy both limit orders and market orders to execute their trades.A number of approaches have been used to model limit order markets. These include perfect rationalityapproaches, zero intelligence models and agent based models (refer Gould et al. (2013)). Zero-intelligence mod-els, like the framework introduced by Bak et al. (1997), are useful to model the market activities as stochasticprocesses. Researchers have been able to reproduce a number of empirical regularities in the LOB in a zero-intelligence framework using random walk diffusion models.Given the continuous and sometimes sequential nature of the auctions, the walrasian model using supply anddemand is the accepted solution to describe the equilibrium. Our contention in this paper is that when consideringan analyis in high resolution we need to accept the existence of disequilibrium and look for an alternate mechanismfor the emergence of prices and quantity trading.
The concept of a random walk is a fundamental concept in finance used to model the stochastic evolution ofthe price of an asset. If Y ( Y ∈ R , −∞ < Y < ∞) is the log-price of an asset, Y ( t ) represents the price at time t ( ≤ t ≤ ∞) . The variance σ ( t ) = (cid:104)( Y ( t ) − (cid:104) Y ( t )(cid:105)) (cid:105) of the price series explains the dynamics of the asset price.(The brackets (cid:104)(cid:105) denotes the mean. σ is also known as Mean squared displacement (MSD) in diffusion literature).In Brownian motion, the variance varies linearly with time, σ ∝ ( t ) . Prices in well functioning markets in theabsence of arbitrage opportunities, are known to display brownian characteristics. A non linear time dependenceof the variance may also be observed in the evolution of asset prices. Non linear time dependence is observedin several other complex systems. Such a non linear time dependence is a characteristic of anomalous diffusion,where σ ( t ) ∝ ( t ) α , with α (cid:44) (Denisov et al. (2012)). Anomalous diffusion has been observed in processes withlong time correlations (such as the evolution of the transaction prices or the quoted prices for an asset). When3 > , the case of superdiffusive limit, the random walker searches his environment much faster than ’Browniansearch’, provided the velocity is constrained to a limit so that he remains within the search space. The levy walkprocess is a simple stochastic model that combines these two notions – a superdiffusive evolution and finitenessof the velocity of motion.The definition of the levy walk model is close to the random walk. A walker chooses a random direction anda random time τ and walks with a constant speed v in the selected direction. After the time has elapsed a newrandom direction and a new random time are picked and the process repeats. What characterises levy walks isthat, the duration of the walks are distributed according to a power-law density (probability density function, pdf): ψ ( τ ) ∝ ( ττ ) − γ − , where a constant τ sets the characteristic time scale and the exponent γ determines explicitlythe scaling of the corresponding MSD, namely α = when γ > (normal diffusion), α = − γ when < γ < (superdiffusion), and the choice of the exponent from the interval < γ < leads to the ballistic diffusion, α = .The levy walk formalism has been successfully applied in diverse problems such as the description of DNAnucleotide patterns, modeling the dynamics of an ion placed into an optical lattice, analysis of the evolution ofmagnetic holes in ferrofluids and of photon statistics of blinking nanodots, engineering of levy glasses, an atom inan optical lattice, a tracer in a turbulent flow, T- cell motility in the brain , a predator hunting for food, or a musselamong a bunch of peers. (Denisov et al. (2012), Zaburdaev et al. (2015), Reynolds (2018)). Raposo et al. (2009)discuss the developments in random search process that includes levy formalism and random walks. Levy search has been observed in biology (foraging and chemotaxis and others) and used extensively in robotics(a relatively nascent area of research). Foraging (the search for food) strategies of organisms have been usedfor thousands of years and optimised in this time period. Many of these search strategies (including chemotaxis)involve limited availability of information. The quest for efficient search algorithms has been influenced by behav-ioral science and ecology, where researchers try to identify the strategies used by living organisms. These providemotivation for our representation. In biology, levy walk is not confined to animal foraging and search. Movementpatterns resembling levy walks have been observed at scales ranging from the microscopic to the ecological. Theyhave been seen in the molecular machinery operating within cells during intracellular trafficking, in the movementpatterns of T cells within the brain, in DNA, in some molluscs, insects, fish, birds and mammals, in the airborneflights of spores and seeds, and in the collective movements of some animal groups. Levy walks are also evident intrace fossils (ichnofossils) – the preserved form of tracks made by organisms that occupied ancient sea beds about252-66 million years ago. And they are utilised by algae that originated around two billion years ago, and still4xist today. (Reynolds (2018) and references therein). Existence of levy walk transport predate their formulationby researchers in the last thirty years. It provides motivation to model zero-intelligence search for survival usinglevy walk.Autonomous mobile robots are required to explore the environment and locate a target. Targets could rangefrom sources of chemical contamination to people needing assistance in a disaster area. Levy walks have recentlyemerged as universal search strategy in robotics. The finite velocity in levy walks that restricts search to the searchregion is useful in the context of robotics. The foraging trajectories of organisms look like periods of localizeddiffusive-like search activity altered with ballistic relocation to a new spot. This intuition is being explored inrobotics. Performance evaluation of the robot or the search efficiency is another area where we draw ideas.Our interest in levy walks as a search strategy emerges from the following: the speed of the search, thepossibility to restrict the search space and the ability to work with limited information. These are the equipmentthat can work in high frequency double auction. We deploy a specific model of levy walks, ones where thevelocities too are a random variable along with the flight time of the particles. This enables us to represent thebehavioural dynamics of both the return expectations and time spent in the market by the traders.In section 2, we propose in our model ’search’ as a possible modern day tatonnement. The model deals withthe total order book and is explained as a diffusion-advection-reaction process in Section 2.2 that also includeseffects of imbalance between supply and demand. The diffusion part of the process is the focus in section 2.1where in there is no imbalance between supply and demand. Diffusion is considered to be the main mechanismthat can explain the price at which auctions are concluded. Several canonical models in finance have explained theevolution of price using Reaction-diffusion models and some of them extend to add jumps to the models (BrownianSemi-martingale or Brownian Semi-martingale jump process). When volume imbalances exist between buyersand sellers in the market, biases develop in the system and drift processes can emerge as additional dynamics toexplain the price evolution and volume trading. Such an imbalance could be general imbalance or point imbalancesat specific prices.
We describe a model where buyers (buy orders) search for sellers (sell orders) and sellers search for buyers. Thisis a unique way of looking at predator-prey relationships (as either the buyer or seller could take the role of thepredator or prey). Only when there is a bias in the system and an imbalance favouring either the buyer or seller,5here will emerge a true predator-prey situation. We consider the total order book, comprising the visible part (thelimit order book, LOB) and the hidden part (limit orders and market orders) that will be placed in the market indue course. The hidden part of the order book is also referred to as latent order book. We start with an LOBmodel based on the framework introduced by Bak et al. (1997), extend it to the total order book, TOB and latermake appropriate changes to include behavioral dynamics of the agents. Orders are particles on a one dimensionallattice and their location corresponds to price (refer Fig 1). When orders representing a buy and sell occupy thesame position on the one dimensional pricing grid it leads to a trade resulting into a reaction A + B → ∅ . We consider two types of agents in our model. The first can be characterised as noise traders with zero intelligence.The noise traders start their journey by placing market orders at the last transaction price. However, in the modelthey are not required to absorb losses. They are zero intelligent because once they transact through a marketbuy (sell) order, they place a subsequent limit sell (buy) order to occupy the price at the best offer (bid). In thesimulation of the model (Section 3), for example, we implement this by placing a limit offer at 0.1 USD (anarbitrary choice) over the transaction price they record as a market order or a limit buy at 0.1 USD lower than thetransaction price. Once such a limit order is transacted they can again go back to place market order and the cyclerepeats. The endevour is to remain in the market and get involved in as many transactions.The second type of agents are the strategic traders. These agents represent a heterogenous group of investorswho have a view on future transaction prices. Or they may have specific portfolio execution mandates. Theyseek specific prices to enter into transactions. The origin of pricing decision (as well as quoted volume) may bevaried: portfolio decisions, private information, liquidity concerns, meta-order execution, fundamental value etc.Strategic traders may be buyers or sellers and use limit orders or market orders to execute their transactions. Whenthey place market orders, they come into the market at the specific price and expect execution of the trade. Theyleave the market after an execution.
Order cancellation by agents
Cancellation of orders sets up useful behavioural dynamics in these markets. Benzaquen and Bouchaud (2018)use cancellations to model the supply and demand in the order book. In section 2.1, neither the noise traders northe strategic traders come back to the market in case orders are cancelled before execution. Quotes from noisetraders are cancelled in section 2.1 and 2.2 under two circumstances. First, when the particles end their designatedflight time and second in case the available budget with the trader falls short of a limit ask order. We treat it as6igure 1: One dimensional lattice showing price and orders shown as particles stacked up at the price. If a buyand sell particle occupy a single lattice point it leads to a transaction. Y0 is the last transaction price and does nothave any market particles.a default in the trading process and remove the quote and trader. Quotes from strategic traders are cancelled inboth section 2.1 and 2.2 when they end their flight time. Additionally, in the examples in section 2.2 to illustratethe traders response to biased trading conditions, we allow orders from strategic traders to be cancelled and a neworder placed again. These are quote revisions.The merit of our model is to show that in high resolution price discovery and trading of volume in financialmarkets can be achieved using a model of search.Consider a continuous auction, order driven market, in a relatively homogeneous state. A homogeneous statein the order book can be achieved by a constant gradient or uniform gradient in the quantity density. The equivalentin economics perspective would be that the demand and supply is balanced. We want to avoid perturbations due tolarger buy or sell orders causing bias in the system. We envisage a model where market buy (or sell) orders searchfor targets - sell (or buy) orders, to effect a trade. The target could be a market order or limit order. A market orderis one where a transaction can get effected immediately if a counterparty is available. The behaviour of the agentsrepresented in the form of orders is treated as particles. One may consider a single particle to be equivalent to thesmallest quantity of the order that is allowed to be placed in the market.
The Total Order Book
The total order book comprises the limit order book (visible LOB) and the latent orders. Latent orders are ordersthat are yet to be placed in the market, but the traders have already planned the same. These include the limitorders that would be placed in future and the market orders. Our conceptualisation is not based on price but on the7igure 2: Representation of the Total Order Book as a one dimensional lattice. It comprises the visible limit orderbook and the latent order book. The dark border represents the visible limit order book given in fig 1. The limitorders from strategic traders (ST) are in grey. The limit orders from noise traders (NT) are placed immediatelyafter a market order is executed by a NT. Strategic traders can place limit orders (grey) or market orders (purple)in future and hold their orders in the latent order book. Noise traders place market orders at the last transactionprice Y0. BO-Best Offer; BB- Best Bid.quoting strategy of the agents. Thus when orders are present in the latent order book, the traders have decided onthe price, time, sequence or events at which they would introduce the orders into the visible order book. A marketorder in the latent (hidden) order book is introduced as a market order necessarily at specific predecided price,time or sequence. The order does not carry a price, but is only introduced when the transaction price is aroundthe predecided price. A limit order in the latent book, in contrast, will carry a limit price and would be introducedby the traders into the visible LOB at a predecided time, sequence or event. Such a representation addresses thepractice of algorithmic trading and slicing of metaorders into child orders.Fig 2 represents the total order book. It super imposes the strategic and noise particles into the limit orderbook model depicted in fig 1.To investigate the tatonnement process, we note that any transaction necessarily requires a market order asatleast one of the counter party. We suggest that modelling the market orders should be able to explain all trans-8igure 3: Levy Search. The fate of 4 market orders particles introduced at the last transaction price Y0 is repre-sented. These particles are introduced sequentially starting time t after the last transaction occurs. The particlesdiffer in the velocity imparted to them and time spent in the market before cancellation. Both these variables arerandom variables in the model. The sign of the velocity gives the direction of flight and represents a buy orderor a sell order. The market orders can be placed by both noise traders and strategic traders. The next transactionoccurs when the first particle reaches the target at time t , in this example, the best offer BO1, which is assumedto have a limit order. At this instant three particles are still in flight. Two market particles reach targets at BB1(time t ) and BO2 (time t )and are transacted sequentially. One particle is cancelled before it can reach the target.Between time t and t many more orders may have been introduced and it is not represented in the figure.actions and help us achieve our objective. The reduction in the dimensionality is an important benefit.Let u ( x , t ) be the probability density function (pdf) of the distance travelled by market order particles in searchof trades. Thus, x is the total length of the jumps and t is the total time. We can represent u as a continuous timerandom walk. Let, Y be the last traded price and Y be the next expected transaction price. When Y occurs, it endsthe flight of some of the market order particles (those that have reached or crossed the price Y ). However, someparticles are still in flight. Between, Y and Y , a number of new quotes are placed into the book, some particlescomplete their flights through cancellations and at least one would complete their flights with the transactionending at Y . We consider particles that have completed their search with success or ended with cancellation.(refer Fig. 3 where Y = Y and Y = BO ).We first specify the price quoting and cancellation behaviour and then we shall resume with the model. Quoted Price and Cancellation
Traders display their intentions by placing market and limit orders. Theyimpart a velocity v to the particles and the orders have a flight time τ . The flight ends with a quote cancellation ortrade and the particles are believed to be at rest before entering the market. The particles enter the market eitherinto the LOB or into the latent order book as limit or market orders . The length of the walk or jump of a particlecan be represented by the coupling v τ . This is close to the model in Zaburdaev et al. (2008) that dealt with a9andom walk with random velocities.The velocity imparted to the orders from strategic traders depends on certain dynamics experienced by thetraders and is an important element that determines the quoted price. For any trader, it is governed by his expecta-tion of success. The strategic traders have specific prices that they want to quote. This price may also be viewedrelative to the return they desire from the first transaction price of the trading day. Let the expectation of this returnfor the jth trader be (cid:104) r j (cid:105) and the averages of the velocity and flight time be denoted by ¯ v j and ¯ τ j respectively. So,the following holds: (cid:104) r j (cid:105) = ¯ v j ¯ τ j (1)The above dynamics is guided by the past experience of success, current trading conditions and state of theorder book. We do not investigate these dynamics and to simplify, we treat the velocity and flight times as randomvariables drawn from distribution. The velocity imparted along with the flight time determines the quoted price.The strategic traders move out of the market after a trade execution or at the end of the flight time.Noise traders quote the last transaction price in their market orders and a small profit over the last transactionprice while quoting limit orders. This over-rules any role of velocity. The flight time in case of quotes from noisetraders determines when any quote from the noise trader is cancelled and the trader moves out of the market.The particle is assumed to be in-flight if its current price (last transaction price after which it entered the market+ the jump) is still less than the transaction price quoted and it is not cancelled. The velocity v can have positiveand negative values to include buy or sell orders (direction of motion). The inter trade time period is known asa trade duration. If (cid:48) N (cid:48) number of trades takes place in a unit of time dt , the average duration can be given by (cid:104) µ (cid:105) = dt / N . Each inter-trade period is a search for the next transaction. A number of jumps get recorded in thisperiod.Equation (2) gives the total path traversed in the search x till time t . Driven by return expectation in (1) thequote from a trader is provided with a velocity drawn from a pdf h ( v ) and the flight time (cancellation behaviour)from a pdf f ( τ ) . The distance travelled by such a particle during search is given by the coupling v τ .Flight time and velocity are the two basic and independent random variables of the model. They are normalisedto 1, ´ ∞−∞ h ( v ) d v = and ´ ∞ f ( τ ) d τ = . In the base case, to avoid a bias in the system, the velocity distribution issymmetric, h ( v ) = h (− v ) . However, interactions due to limit orders or transactions in the system can change thisassumption and we shall model this in section 2.2. u ( x , t ) = ˆ ∞−∞ d v ˆ t u ( x − v τ, t − τ ) h ( v ) f ( τ ) d τ + n ( x ) δ ( t ) (2)10he right side in (2) describes the dynamics of search where the distance of v τ is traveresed in time τ so that thetotal distance traversed in search reaches x from x − v τ and total time spent in search reaches t from t − τ . Takinginto account the possible velocity-time coupling, the joint probability h ( v ) f ( τ ) , the first term in (2) integrates overall flight time and velocity combinations. Since a coupling of velocity and time represents an order, it in effectintegrates the distance travelled for all search orders. For particles that have completed their flights, they wouldhave reached a point on the lattice grid. It may or may not have resulted in a success. For all such flight times,we sum the total distance travelled or jumps for all velocities. Particles still in flight are not included. The initialdistribution of the market particles is given by n ( x ) . This search is the tatonnement in modern markets.The success of each search is a trade transaction. The pdf of the traded particles, q ( x , t ) is simply the numberof trades that take place in a time interval ( , t ) . However, the trades being the success of the search of marketparticles in a tatonnement, we define the efficiency η of the search process as the ratio of the total path traversedin the search to the number of targets searched or trades done. This gives us a link to represent q ( x , t ) in terms of u ( x , t ) and η as in equation (3). η is again a function of the coupling of velocity imparted and the flight times. q ( x , t ) = ˆ ∞−∞ d v ˆ t u ( x − v τ, t − τ ) h ( v ) f ( τ ) η ( v τ ) d τ (3)Equations (2) and (3) fully describe the dynamics of the system with a given initial density of particles and thetwo pdf for the flight times and velocities. They establish the crucial link between the tatonnement in the auctionwith the trades. Next we solve these equations analytically. First, we determine the total path traversed by marketparticles in the search process and then introduce the result into the equation of the trade density. We apply theFourier transform with respect to the spatial coordinate in (2) and then the Laplace transform with respect to time.This yields in (4) the path traversed in search in the Fourier-Laplace domain, k , s , as: u k , s = n , k − [ h k τ f ( τ )] s (4)We can introduce the result in (4) into the Fourier-Laplace expression for equation (3) to get the analyticexpression for the density of trades in the Fourier-Laplace domain, obtained through the search process in (5). q k , s = n , k η k , s [ h k τ f ( τ )] s ( − [ h k τ f ( τ )] s ) (5)To find a solution, the next step is to take the Laplace inverse of equation (5). However, an analytic represen-tation and direct inversion of the equation (5) is not feasible. Froemberg et al. (2015) recommend an asymptoticanalysis for large space and time scales, x , t → ∞ . We use the same approach. Going to Fourier-laplace space11sing the tauberian theorem, this limit corresponds to ( k , s ) → ( , ) such that k / s = constant . This has to beperformed numerically.It is important to define the velocity and flight time distributions before we move to obtain the inverse transfor-mation. In our view this needs to come from empirical analysis. Further, velocity distribution cannot be obtaineddirectly as it is a notional quantity and needs to be interpreted from equation ( ) that describes the relationshipwith returns.As discussed in Zaburdaev et al. (2008), Froemberg et al. (2015) and Zaburdaev et al. (2015) when the velocitydistribution is Cauchy or lorentian, the density of the particles also is lorentian, independent of flight times andjump lengths. Such a lorentian velocity profile appears in real physical phenomena such as two dimensionalturbulence and is also found in model distributions of kinetic theory, statistics, plasma physics and starving amoebacells. We know that a cautchy process does not give rise to a continuous sample path for the price and it differsfrom Brownian motion as there are large jumps not infrequently. As given below, we make arbitrary choice of alorentian velocity distribution h ( v ) and intuitively a flight time distribution f ( τ ) with power tails. h ( v ) = u π ( + ( v u )) (6) f ( τ ) = γ ( + τ ) + γ (7)In equation (6) u is needed to constrain the velocities so that the particles do not go beyond the ballisticcones, else it will lead to instantaneous dispersion. The γ in equation (7) is varied to get different transport. Inequation (5), given the asymptotic limit we want to evaluate, we further set a constant efficiency, so that there isan expression for the initial density of particles for the trade density. The propagator for our model can then benoted as follows: G ( k , s ) = L[ f ( τ ) h ( k τ )] − L[ f ( τ ) h ( k τ )] (8)where k τ is the Fourier variable conjugate to v . The equation (8) retains the form of the well known Montroll-Weiss equation for the pdf of the uncoupled continuous time random walk (CTRW) to find the particle x at thetime t , modified such that it applies to random jumps in velocity. Equation (8) can be rewritten as (see AppendixC for details): G ( k , s ) = ´ ∞−∞ d v f ( s + ik v τ ) h ( v )] − ´ ∞−∞ d v f ( s + ik v τ ) h ( v )] (9)12or the flight time distribution we have chosen and in the long time limit the expansion in the Laplace space isgiven by, f ( τ ) (cid:117) − τ γ Γ ( − γ ) s γ (10)Using (10) the asymptotic version of (9) is, G ( k , s ) = s ´ ∞−∞ ( + ik v / s ) γ − h ( v ) d v ´ ∞−∞ ( + ik v / s ) γ h ( v ) d v (11) Random walk with Ballistic scaling
As the simulation in section 3 confirms, ballistic scaling may be useful in thinly traded stocks where the arrivalrates of orders, transactions and cancellations is low. A method exists for the inversion of the fourier laplaceexpression for propagators with ballistic scaling. A propagator of a random walk model has ballistic scaling if itcan be written in the form: G ( x , t ) (cid:117) t φ ( xt ) , t → ∞ ,where φ is the scaling function. In Fourier-laplace space thisis, G ( k , s ) (cid:117) s g ( iks ) . Comparing the two forms we can rewrite the scaling form of our equation as in equation (12),where ξ = iks . g ( ξ ) = ´ ∞−∞ ( + ξ v ) γ − h ( v ) d v ´ ∞−∞ ( + ξ v ) γ h ( v ) d v (12)Using a arbitrarily chosen special situation induced by Cauchy distributed velocity and taking the inverseLaplace and Fourier transform we get a form of cauchy distribution as in (13) (refer Appendix B). G ( x , t ) = u t π ( u t + x ) (13)Although u comes in due to the need to restrict the particles in the ballistic cone, it is an important feature asit slows down the diffusion process. The density of trades scales inversely with time and square of the price. Role of diffusive search and the traders
The role of diffusive levy search is to provide continuity in the auction and also volume trading. The nature of thiscontinuity can be evaluated basis the success of the search in terms of the density of trades. This is further relatedto the inter trade duration. The parameters that determine the trade density are thus critical to continuous auction.First, the presence of noise traders and the market orders placed by noise traders and strategic traders. Marketorders need to be always available to provide continuity in auctions. Else, traders need to cancel and revise theirquotes, which we shall discuss in section 2.2. Trading can slow down if less number of noise traders are presentor if they get locked away post transaction either in limit orders or leave the market. We always expect strategic13raders to be present in the market. But presence of noise traders can be impacted by market design. Second, theefficiency of the search. The efficiency is dependent on the resource density or target density. This comprises bothof the limit orders and strategic market orders that come in whenever a new price is discovered. It is these marketorders that bring in efficiency improvement by leading to greater number of transactions at already discoveredprice.
Efficiency of Search
A useful tool to judge how a trading system is performing and in the case of this model, how the search for tradesperforms is to evaluate a global efficiency of the search using the trade duration. The trade durations is the timediference between one trade and the next. Thus it is the time between any two auctions in continuous tradingset up. The reciprocal of the trade duration is rate of trades. If the searchers are taking longer flight times orlonger distance in the walk before they locate the target, the rate of trades would decrease. This measure is anindirect route to use the concepts like distance travelled per success or average flight time per success, but is moremeaningful and easier to evaluate for a trading system. The rate of trades or efficiency of search can be computedas (cid:104) τ (cid:105) , where τ is the trade duration. A similar measure for efficiency of search in a different context was alsoused by Palyulina et al. (2014). In section 3 (numerical simulation) we have used this measure (rate of trades) asthe efficiency of search to compare search in ballistic, superdiffusive and brownian regimes with different marketconditions. Need for velocity as random variable
The use of Random velocities in the modelling is unique in the context of modelling stock prices. It is not requiredwhen modelling the trade prices, but required when modelling the quotes. As discussed earlier, the velocities area notional quantity and cannot be directly observed. This technique is a useful tool to connect the random walksof quotes to the trade price series. Most quotes are cancelled sooner or later and hence do not get involved inthe reaction. The random velocities through the return expectations can connect the trade durations to the quotedurations.
In section 2.1 we dealt with a limit order market (Total Order Book) in homogeneous state where the search isdiffusive. A situation where there is no shortage of noise traders and strategic traders have placed market ordersat widely dispersed prices. The diffusive search was crucial to explain price continuity. This section combines14iffusive search with other mechanism that can explain auctions when we relax the assumption of ’no bias’. Incase of excess demand (buy or sell side) of liquidity a bias will get generated in the system, resulting in a drift.The market will move to improve the search efficiency due to higher availability of resource targets, i.e more andmore targets can be acquired at the existing transaction price. The number of targets thus go up without increasein the path travelled in the search leading to higher efficiency. The market will naturally drift from resource richregions to regions where market order particles may be available. This is because market orders from strategictraders are only placed when the relevant prices and time is reached.Further, there is a stochastic evolution of the transaction price of the asset and quantities traded. This is theresult of the strategic games between the market orders and limit orders, buyers and sellers, new order arrivals,order cancellations and transactions. We allow for such a stochastic evolution although we model the dynamicsof market order particles only. Since market order are involved with each transaction, we expect to successfullyexplain the stochastic evolution of the transaction prices and volume traded. Donier et al. (2015) use a diffusion-advection-reaction model to explain the evolution of the marginal supply and demand in the market. We set up amodel where market order particles search for targets in the double auction market. While we classify our modelas a diffusion-advection-reaction model, given the view point of search, it has been built using concepts involvingforaging, predator-prey search and chemotaxis (Grunbaum (1998) and references there in).In section 2.1, Y ( Y ∈ R + , < Y < ∞) represented the price of an asset and Y ( t ) the price at time t ( ≤ t ≤ ∞) .We denote the price by x in this section (in section 2.1 x denoted total distance travelled).Let g ( x , t ) represent the density of market particles at a price x ( x ∈ R + , < x < ∞) and time t ( ≤ t ≤ ∞) . Ourfundamental set up is given by the partial differential equation (14). ∂ g ( x , t ) ∂ t = (14) ∂∂ x ( D ( t ) ∂ g ( x , t ) ∂ x − λ ( t ) ∂ f ( x , t ) ∂ x g ( x , t ) − λ ( t ) g ( x , t )) + v ( t ) g ( x , t ) + q ( x , t ) The set up in equation (14) (14) explains the stochastic time evolution of the density of market order particles in our one dimensional grid.The first term on the right hand side represents diffusion that we discussed in section 2.1. The second and thirdterms relate to volume trading and price evolution due to drift. The fourth term is the net addition of marketparticles. The last term represents large external additions of market particles into the system.Section 2.1 can explain a walrasian tatonnement where prices are set when the quantity demanded by buyersmatches the quantity supplied by sellers. Modern order driven markets operate at a speed where there is neither15ufficient time to establish a walrasian equilibrium nor there is an official market maker who will adjust the priceinstantaneously. Hence, transactions take place at prices out of equilibrium. Traders have always observed marketsoperating out of equilibrium. Even in Quote driven markets as noted by Beja and Hakansson (1977) and Beja andGoldman (1980), set up of equilibrium needs more time to allow for the information flow required for the purpose.We look at (14) as the mechanism that can take the trading system towards equilibrium. We now explain how theterms enter the right hand side in (14) : • Diffusion (first term): This is the primary search mechanism through which price continuity and quantitytransport takes place. In the absence of bias this will remain the only mechanism (already discussed in theSection 2.1). • Drift due to target density (second term): We implement a formulation widely used in foraging studies (Grunbaum (1998)). In the set up of those studies, the drift velocity is proportional to the gradient resourcedensity. And the coefficient is the taxis coefficient. The use of resource density implies the dimensionalityof the model increases. We, however, note that the density of targets (limit orders or market orders) in ouruse case should result either into a response or consequence in the system. We model the drift resultingfrom this response or consequence in market particles. f ( x , t ) is the perceived resource density basis theactions in the system. In this innovation we conjecture that the drift resulting from a gradient of resourcedensity should have a mirror image in the searcher. This mirror image may be different than the originalgradient, but is more relevant as it is likely to be a tactical response in the absence of complete information(since the visible order book is not the complete order book). We may err in magnitude but not directionand the system will try to iteratively guess the true gradient of resource density. Again we accomodate atime evolving coefficient λ . • Drift due to movement within the market order particles (third term): There are several scenarios we en-visage where a number of market particles could get activated into directed activity resulting in a drift. Forexample, high transaction activity on one side of the market, will result in noise traders getting releasedfrom limit orders suddenly due to the reactions. This early release increases returns for noise traders. Ifthese traders sense the action to continue, they would rush to the other end to repeat the cycle. Similarly,traders yet to place orders may anticipate change in resource gradient and resource density due to news flowand react by adjusting their own positions. λ is a time varying coefficient . • Net addition to market particles density v (fourth term) : v ( t ) is a time varying composite coefficient ofthe new order additions v a , cancellations v c and transactions (reactions) v r . v ( t ) g ( x , t ) = v a ( t ) g ( x , t ) − c ( t ) g ( x , t ) − v r ( t ) g ( x , t ) . New order additions of market particles includes market particles, hitherto la-tent, coming into the order book, new traders, limit orders that are cancelled and placed as market orders asthe traders become impatient. Hence, this can be a non-trivial component. Cancellations and transactionsremove market particles of strategic traders from the order book. Noise traders move out of market only atend of their flight time. • q (fifth term) is a source of market particles much larger than what comes in during the average trading dayfor the asset. It could be a demand or a supply of liquidity. The source may be a single or few point sourcesor distributed across a price range and even in time.In summary, we build a comprehensive model of stochastic evolution of market particles. Expand (14) toobtain (15) : ∂ g ( x , t ) ∂ t = (15) D ( t ) ∂ g ( x , t ) ∂ x − λ ( t ) ∂ f ( x , t ) ∂ x ∂ g ( x , t ) ∂ x − λ ( t ) ∂ f ( x , t ) ∂ x g ( x , t )− λ ( t ) ∂ g ( x , t ) ∂ x + v ( t ) g ( x , t ) + q ( x , t ) Equation (15) can be solved analytically. We draw upon the technique used by Sanskrityayn and Kumar (2016),who used the Greens function method to solve their diffusion-advection equation in the context of pollutant solutesin the atmosphere. To solve the equation, we note that q will remain untouched and we need to reduce the equationto a known form so that we find an expression for f ( x , t ) . We do a co-ordinate transformation from the domain ( x , t ) to the domain ( X ( x , t ) , t (cid:48) ) . The domain X is essentially fixed time-snapshots of the entire lattice. We want totransform (15) to the form in equation (16). ∂ G ( X , t (cid:48) ) ∂ t (cid:48) = D ( t (cid:48) ) ∂ G ( X , t (cid:48) ) ∂ X − λ ( t (cid:48) ) ∂ G ( X , t (cid:48) ) ∂ X + v ( t (cid:48) ) G ( X , t (cid:48) )) + q ( X , t (cid:48) ) (16)Using the domain transformation, we can write equation (15) as equation (16), following which we equatethe coefficients to solve for the interim variables and transformations introduced. (Refer Appendix D for details).Using the transformations we can reduce our initial equation to the following form, ∂ G ( X , t ) ∂ t = D ( t ) β ( t ) ∂ G ( X , t ) ∂ X − λ ( t ) ∂ G ( X , t ) ∂ X + v G ( X , t ) + q ( X , t ) (17)17here the dimensionless expressions and transformation are given in (18), (19) and (20). D ( t ) D ( t ) = β ( t ) (18) β = e ´ t ( v ( s )− v ( s )) ds (19) X = x β ( t ) + ˆ t ( λ ( t ) − λ ( t ) β ( t ) − λ ( t ) φ ( t ) β ( t ) ) (20)The initial conditions for this equation are G ( X , ) = G i ω ( X ) , with −∞ < X < ∞ and t > . Next, we tryto remove the drift term and the decay term from (17) by using further transformations. We use the followingtransformation equations one after the other for this purpose. In (22) β = e ´ t v ( s ) ds is a dimensionless term and in(23) β is as per (19). In (23) T is a time variable. Equation (17) now reduces to (24). η = X − λ ( t ) t (21) K ( η, t ) = G ( η, t ) β ( t ) (22) T = ˆ t β ( s ) ds (23) ∂ K ( η, T ) ∂ T = D ∂ K ( η, T ) ∂η + Q ( η, T ) β ( T ) β (24)We now need to solve equation (24) to obtain a master equation for transport of market particles in doubleauction limit order asset markets. Haberman (1987) provides a solution for equations such as (24) using GreensFunction Method (GFM). The solution to (24) based on GFM is given in (25): K ( η, T ) = ˆ T ˆ ∞−∞ Q ( χ, τ ) β ( ζ ) (cid:112) π D ( T − ζ ) β exp (− ( η − χ ) D ( T − ζ ) ) d χ d ζ (25) + ˆ ∞−∞ √ π DT G i ω ( X ) exp (− ( η − χ ) DT ) d χ Next we sequentially trace back the transformations done earlier, in reverse order to get the solution below in1826). Here, ζ = ´ τ β ( s ) ds and the initial condition g ( x , ) = G i ω ( x ) . g ( x , t ) = β ( t ) ˆ t ˆ ∞−∞ Q ( χ, τ ) (cid:112) π D ( T − ζ ) exp ( ( x β − ´ t ( λ β + λ φ β ) ds − χ ) D ( T − ζ ) ) d χ d τ (26) + β ( t ) ˆ ∞−∞ √ π DT G i ω ( χ ) exp (− ( x β − ´ t ( λ β + λ φ β ) ds − χ ) DT ) d χ Before we move into asymptotic analysis, a few observations are due:
The significance of β . The dimensionless quantity β has a physical meaning and is not simply an artefact of domain transformation.On transforming the domain from ( x , t ) to ( X , t (cid:48) ) , each point on the one dimensional lattice is now a space-timevariable. The advantage of this construction is that X alone can explain the time evolution of the system. Wehave used this intuition to our advantage. β relates the diffusion in the new domain to the original domain in (18).Should the diffusion coefficient be different in the two domains, since we have defined diffusion coefficient as avariable only in time. Since, β is defined in (19) in terms of the relative contribution from the net addition to thedensity of market particles in the two domains, we infer that at any particular price in the domain ( X , t (cid:48) ) we areable to witness the impact of the deposition, cancellation and transaction over time. Thus the behaviour seems tobe that v already incorporates time evolution, while v has to be integrated over time. As we shall see, this makesinference from asymptotic analysis easier. Quantity transport and price auction
We are intuitively claiming quantity trading. A change in density of market particles on the grid means a reactionif particles with a buy and sell intention occupy a grid position in the LOB (not latent orders). A differencebetween physical systems and asset markets is that while in the former ensemble measures are defined on thebasis of averages, in the latter the representative measure of price is determined by the last transaction, even if itmay have involved the minimum allowed quantity. This coupled with low granularity in minimum quantity andnon-discretisation of the price grid has led to the possibility of viewing the price auction and quantity auction asseparate interconnected mechanisms. We may see auctions resulting in transactions where prices remain stablewith low quantity trading or a large amount of quantity traded without price impact or quantity traded with priceimpact or price impact with low quantity trading. 19 rift and diffusion
The more market order particles diffuse, new trade transaction prices are formed. But the noise traders canget stuck in longer waiting times in limit orders. The density of the market particles may not allow absorptionof volume / quantity of resource density. Sometimes, with targets remaining unconquered the resource particles(limit orders or market orders) have a tendency to move in opposite direction to diffusion of market order particles.We have modelled the impact of this in the drift term that opposes general diffusion. The drift is dependent on thequote revisions from the traders as discussed in example in section 2.2.2.
Bid-ask bounce
Another interesting facet is the presence of a reaction front characterised by bid-ask bounce. The diffusion throughlevy search has a tendency to pull the transaction price in opposite direction resulting in the prices moving from thebuy side to the sell side. A counter force to this would come from the fact that the participating noise traders wouldget locked in limit orders and hence the number of transactions they can enter into decreases thereby graduallyreducing the efficiency. Sell market orders introduced on the ask side of transaction prices, would need to travellonger to reach the buy side and vice-versa.We now look into specific scenarios that may operate in the market through simple examples. Such examplescan be generated in conjunction with a trading model that employs search as a tatonnement as in the IntradayTrading model of Pani (2019).
We want to find a steady state solution to our master equation. Say we have total addition of market particles otherthan usual as Q ( x , t ) . A steady state solution is an equilibrium solution when the change of market particle densityover time is zero. It generally refers to the state where the long term average volume in the asset gets traded. Insteady state the diffusion coefficient and drift velocity coefficients are not time dependent. The contribution tomarket particles coming from v stabilises. In steady state, v = v = constant i.e the instantaneous change is equalto the total change. At any time t i the net change is effected only at a particular x and not other locations. Whenprice priority rule is used for matching trades, it results in such a general condition in the limit order markets. Therate of particles released from limit orders following transactions is constant and transaction rate is constant.Thereare two ways to find the solution, first by setting ∂ g ∂ t = or second, we find the solution from the master equation(26) when t → ∞ . 20ur basic equation for the gradient function in the drift velocity expression is (refer Appendix Eq. D.12): −( v ( t ) − v ( t )) (cid:115) D ( t ) D ( t ) = ∂∂ t (cid:115) D ( t ) D ( t ) When t → ∞ . the perceived resource gradient is constant, i.e d fdx is constant and d fdx = . This implies v = v = constant . What happens after a long time post addition of Q, with sufficient time to all the three processes toplay out to reach an equilibrium state. As t → ∞ , v − v = and β = . However, T becomes very large andindeterminate value. The second term of the equation (26) disappears. T − ζ = e . β is a constant.Further, since transaction rates are constant and not high λ = . There is no drift, hence λ = . The equationreduces to (27) with the assumption that Q ( x )√ De = −( x − χ ) De . Thus we recover the square root law where χ is theimpact cost. x − χ is the price region where we will have trades to resolve Q . g ( x ) = β √ π e − Q ( x )√ De (27) What happens when a large demand M comes into the market. M could be a single large order or a few orders ina short span of time. If the demand is a new market order, it would follow a path trying to move towards a steadystate dynamics. Otherwise, after the trades hit this block, it takes a few transactions to lock up the noise particlesin the next best limit order position. The market order particles start to sense a gradient and drift begins at a laterstage when the locked noise particles are released. The following example illustrates this mechanism that we offerfor the market. Fig. 4 to Fig 8 also illustrate some of the discussion below. Example
Let Y t denote the transaction price. The grid is denoted relative to the first transaction price Y t . The conti-nuity to the left and right of Y t is the bids and offers respectively, represented by BB , BB , BB , BB ... and BO , BO , BO , BO , BO ... . We assume that Y t has exhausted the demand and supply at that price. At t on bothsides of Y t on the price axis, there will exist a visible order book and a hidden latent order book. Such orderbooks are generally known to take a V or U shape. The bid side demand is at an angle θ and the ask side offer isat an angle θ . This is a visual representation of the V or U shape and a easy way to approximate the differencebetween demand and supply if it exists. The difference between θ and θ generates a bias in the system, leadingto the drift, that we illustrate below. 21ntuitively, market orders want to take the market particles and hence transactions towards + ∞ on Ask side and −∞ on bid side. The drift opposes this movement. Let us assume the introduction of M , starts with BO + BO h , BO that is visible and BO h which is hidden. The subscript h is used for the latent / hidden orders. Participantpositions are based relative a dynamic value such as the last transaction price (one may alternatively referenceeither a fundamental value of the asset or an index etc). The introduction of M makes θ > θ , the magnitudesof which is not known. The existence of large demand is revealed only on the basis of visible order book andtransactions. At t = t , the transaction price moves to Y t BO . The block M now behaves like a sink or wall.It does not allow transactions to move towards BO , BO . After Y t BO , a series of trades can take place at BO which is the new transaction price - Y t BO , Y t BO ... Y t BO . Thus, successive transactions Y t , Y t ... Y t arerecorded at the same price BO . The search for trades is still on in both directions. Once market particles exhaustthe liquidity at BO , the trade again reaches Y t which is the new best bid. New transaction price is Y t Y t . If noliquidity is available here, the levy search can move to either BO or BB . Since, the former keeps locking upthe noise traders in limit orders, more likely the trade moves to BB within a couple of cycles. Hidden liquiditycan reappear at Y t only if there is a readjustment of positions. By this time, the market particles system has madesense of the gradient and the drift can get activated. This simply involves correcting the bias θ > θ .The correction of bias is the action based on readjustment of position on either side. The reference point Y t needs to move to BO , BO or BB , BB . This is the drift in action and usually takes place when participants revisetheir position on the basis of new information or are not satisfied with the slow rate of trades. Once transactionpace picks up on either side, there could be a new source of drift from released market particles. These particlescan get involved in the carry trade for a small margin. In the absence of new information and no adjustment onbid side, the transaction price gradually moves up from BB , BB . The best offers start to move from BO to Y t , BB . In this situation, the demand M can face the lack of liquidity forcing a readjustment as illustrated in fig 6-8.When M >> G i ω ( χ ) we can ignore the second term in the equation. The expression for density reduces to: g ( x , t ) = e v ∗ t M √ π DT exp [ − DT e − v ∗ t ( x − ˆ t ( λ + λ φ ) ds ) ] (28)In (28) , what happens if market orders are time dependent and not simply as a response to one very large order.We assume this large input of market orders is large enough to assume the initial density to be very low So, Q ( x , t ) = G f ( , t ) δ ( x ) . Here the function f ( , t ) shows the continuous availability of market orders at a price.This could be in response to liquidity demand in the order book in limit orders or the perceived hidden limitorders. This liquidity is the source of large liquidity. Further from the equation below for gradient observed by22igure 4: Visual representation of a balanced total order book, visible orders shown in red and the cumulativevisible and hidden orders in blue. Hidden orders comprise the orders that may be placed to demand or supplyliquidity. The hidden limit orders are placed at a later time and the hidden market orders are placed at the particulartransaction prices. The y-axis is the quantity. Y is the last transaction price. The bid side demand is at an angle θ and the ask side offer is at an angle θ . Here, θ = θ . Fig 4 to Fig 8 is a sequential illustration showing biasesdue to new supply demand addition and resulting transactions and adjustments due to quote revision.Figure 5: A bias develops around the last transaction price given in fig 4. Visible orders shown in red and cumula-tive with hidden demand in blue and cumulative with hidden supply in green. The y-axis is the quantity. Y is thelast transaction price.A large demand M is added at BO1 (visible+hidden), BO2 (visible+hidden), B03 (hidden).The bid side demand is at an angle θ and the ask side offer is at an angle θ . Here, θ > θ a demand supplymismatch that leads to bias in the system due to hidden orders. (Fig 4 to Fig 8 is a sequential illustration showingbiases due to new supply demand addition and resulting transactions and adjustments due to quote revision)23igure 6: After a few transactions from fig 5, the visible order book moves to new last transaction price BO Y as reference by adjustment of limit orders ( Y is the fourth tansaction price). Visible orders shown in red andlatent demand in blue and latent supply in green. The latent order book is referenced to a fundamental price orthe equilibrium price from the initial auction. Y is the first transaction price in the illustration. The y-axis isthe quantity. (Fig 4 to Fig 8 is a sequential illustration showing biases due to new supply demand addition andresulting transactions and adjustments due to quote revision)Figure 7: A few transactions afterfig 6, the visible order book moves to new transaction price BB Y as referenceby adjustment of limit orders. The movement is from BO1 to BB1. Visible orders shown in red and hiddendemand in blue and hidden supply in green. Note the bias adjusting in visible orders but the bias in hiddenorder remains with θ and θ , the relative angles with the axis remaining the same Fig 4 to Fig 8 is a sequentialillustration showing biases due to new supply demand addition and resulting transactions and adjustments due toquote revision)Figure 8: The transaction price (Y15) moves further to BB2Y15 by the levy search. Visible orders shown in redand hidden demand in blue and hidden supply in green. The bias due to supply corrects as it dissociates fromdemand through quote revision and moves the reference point to BB1 reducing θ . (Fig 4 to Fig 8 is a sequentialillustration showing biases due to new supply demand addition and resulting transactions and adjustments due toquote revision) 24iquidity providers (refer Appendix D.9): f ( x , t ) = v ( t ) − v ( t ) λ ( t ) x + φ ( t ) x + c We infer that Q = φ ( t ) . In such situations λ can be ignored and v , v − v can be reduced to v ∗ to further simplifythe equation. g ( x , t ) = (29) β ˆ t φ (cid:112) π D ( T − ζ ) exp [ − D ( T − ζ ) e − v ∗ t ( x β ( t ) − ˆ t ( λ β ( t ) + λ φ β ( t ) ) ds ) ] When the reaction rate is high, we evaluate the dynamics under the further assumption of no addition of marketorders through q ( x , t ) . We expect the density g at the transaction price to go down. This is represented as (30),which is the second term in (26). g ( x , t ) = (30) β ( t ) ˆ ∞−∞ √ π DT G i ω ( χ ) exp (− DT ( x β − ˆ t ( λ β + λ φ β ) ds − χ ) ) d χ Such a situation could arise if the market orders hit a liquidity pool in limit orders and a large number of noisetraders that were earlier locked up in limit orders here get released. The liquidity pool causing high transactionrates and release of noise traders creates a sustainable transaction rate. The contribution to v is coming only fromthe reaction rate. φ = and (30) reduces to (31). g ( x , t ) = β ( t ) ˆ ∞−∞ √ π DT ω ( χ ) exp (− DT ( x β − ˆ t ( λ β ) ds − χ ) ) d χ (31) In case, the drift is not time dependent and only spatially dependent and no self generated drift resulting fromrelease of noise traders. λ = and the drift in response to resource gradient reduces to (32) λ ( t ) ∂ f ( x , t ) ∂ x = λ ∂ f ( x , t ) ∂ x = λ ( v − v λ x ) + λ φ ( t ) (32)25 .2.5 Continuous spatial inflow of market orders that is not time dependent This is a case of liquidity available at every price in hidden market orders. In such a case we need to find theequivalent of (29) that is spatially dependent and not time dependent, g ( x , t ) = β ( t ) ˆ t ˆ ∞−∞ ( v − v ) χ (cid:112) π D ( T − ζ ) exp ( − D ( T − ζ ) ( x β − χ ) ) d χ d τ (33)If dispersion D too is assumed to be time independent D = D and we assume v , v − v = v ∗ because this is asmall value, (33) reduces to (34). ( D could take other forms too). g ( x , t ) = e v ∗ t ˆ t ˆ ∞−∞ v ∗ χ (cid:112) π D ( T − ζ ) exp ( − D ( T − ζ ) ( e v ∗ t x − χ ) ) d χ d τ (34) What happens as t → immediately after a transaction / auction has ended. Once a transaction price is establishednew market orders come in at that price. These market orders could be buy or sell orders or both. Other thansuch deposition we assume cancellation and transaction do not occur at this instance. Since, in this scenario v = v = constant , β = and β = . However, T and ( T − ζ ) does not exist and we cannot determine the densityof market order particles. We did not evaluate this limit for the levy search. However, in our model of levysearch process we know that market orders start coming in at the last transaction price with a velocity drawnfrom a distribution and that the process exists. This will provide the result to the time derivative at that point.It may be meaningful to find what happens in the inter-trade interval between two trades. The average of thisinterval is precisely the average trade duration. Let two consecutive trade transactions take place at time t and t , | t − t | = τ . What happens in this regime will depend on the dominant process occurring in the market: diffusion/ levy search, drift due to high transaction rate, drift due to perceived resource gradient. We keep mixed processesout of the scope of the current analysis.In the diffusive regime, the contributions from other processes is neglected. Further, further we assume Q = as presence of large market orders will invoke other processes, β = β = e v ∗ τ , T reduces to ( v ∗ e v ∗ τ ) − . In(35) B is the bid-ask spread. (35) reduces to (36) on simplifying. (36) gives the relationship between the initialdensity of particles, diffusion coefficient, spread and interauction time with the density of particles. To understandthe intuition in (36) we further assume that the liquidity demand at Y t comes from limit orders and the marketparticles provide liquidity. To find the maximum capacity during the inter-auction trade (we assume availability oftargets is not a constraint), let us assume the ideal condition that every particle that engages in search reaches the26arget. And that every new addition even in the period τ is only a deposition that can go in and provide liquidity.So both D and v ∗ are equal to . Fig. 9 and Fig. 10 gives multiplying factor to the intial density as a function ofthe spread and interauction times. g ( Y t ) = e v ∗ τ ω ( Y t ) (cid:112) π D ( v ∗ e v ∗ τ ) − exp ( − D ( v ∗ e v ∗ τ ) − ( Be v ∗ τ ) ) (35) g ( Y t ) = ω ( Y t ) (cid:112) π D ( v ∗ ) − exp ( v ∗ ( τ − B D )) (36)Fig. 9 confirms that in an ideal best search scenario the density exists. As the spread increases, the density isreduced. At spreads of 0.2 and below the density converges to the initial density of particles in interauction timesof 0.5 seconds. Fig. 10 magnifys the fig. 9 around the low interauction times prevalent in liquid equity markets.For the spreads of 0.2 and below it shows convergence around the constant ( /√ π ) in (36). This suggest theexponent of e is that is the expression ( τ − B D ) ≈ . Thus we expect in a fairly ideal scenario 40 percent of theinitial density of market particles is able to trade. This demonstrates how the presence of arbitrageurs and othertraders, who contribute to the market particles, can help increase the depth of the market. Note that this is therange we would obtain every time the efficiency of the search is equal to the contribution from the addition.The relationship between the spread and diffusion coefficient can be understood from the need for the followingto hold: B < √ D so that the exponent is positive. The diffusion coefficient could be calculated in the standardway as MSD upon average flight time.In the regime of drift due to high transaction rates, we ignore drift due to any resource gradient. The scenariois that a large number of market particles are already released due to high transaction rates before we arrive at t . Since this component is significantly higher than any new orders, v >> v , hence, β = e − v ∗ τ and β = e v ∗ τ and T = e v ∗ τ v ∗ . In place of (36) we arrive at (37), where we can see that if the spread is high the contribution from thisdrift will be small towards making market particles available for transaction. g ( Y t ) = ω ( Y t ) (cid:113) π D ( v ∗ ) exp (− v ∗ D ( B − λ ∗ τ ) ) (37)When under the process of drift due to resource gradient, v >> v and we neglect v . The equivalent expressionis given in (38). Again a high spread can reduce the contribution of this process in the limit under consideration. g ( Y t ) = ω ( Y t ) (cid:113) π D ( v ∗ e v ∗ τ ) exp (− D ( v ∗ ) ( B − λ ∗ φ ∗ τ ) ) (38)27igure 9: The multiplying factor to the initial density as a function of the interauction time ( τ ) and spread ( B ).The multiplying factor gives the proportion of initial density of particles that are traded.Figure 10: The multiplying factor to the initial density as a function of the interauction time ( τ ) and spread ( B ).This figure magnifys the fig 6(a) around the low interauction times prevalent in liquid equity markets.28 NUMERICAL SIMULATION
We performed a numerical simulation of the model to improve our understanding of the results. The numericalsimulation is different than the regular reaction-diffusion simulation because while orders may be present in thetotal order book, they come into the limit order book at different times. Researchers have in the past modelledthe arrival of orders into the limit order book using a poisson or a hawkes process (Gould et al. (2013)). Further,limit orders may get cancelled and leave the order book or may get cancelled and get modified, the latter is usuallytreated as a new order.
We begin with a market where the number of traders in the market is fixed at 1000. A trader places only a singleorder. An order is equivalent of a particle discussed in section 2.
There are in all two sets of simulation that differ only in one aspect: in the first set, the quantity of shares in thetrader’s order is one while in the second set, it may vary between one to five. We use this difference in orderedquantity of shares to simulate the bias. The minimum order quantity is one share.
The arrival of events (order arrival, order cancellation, transactions) is modelled following a poisson process.While transactions are not orders, this seems to be a practical alternative to represent the transactions in the tickby tick data as and when they occur. Thus, an event of order arrival / order cancellation is followed by a ordermatching to determine the possibility of transaction. If there is a transaction, it is posted with a timestamp of thesubsequent event arrival time, else the next order arrives at the timestamp. The event arrival rate is one of thedeterminants of the resource density.
The percentage of noise traders in the market is assigned randomly in each experiment drawn from a uniformdistribution. The quoting and cancellation behaviour is governed by the velocity and flight time distributions. Thesimulation assumes a cauchy velocity distribution (ref Eq. 6) and a flight time distribution with power tails (refEq. 7). The simulation is initialised with a starting price or known price for the asset and the traders assigned avelocity and a flight time, both drawn from the distribution. Traders are randomly assigned as a noise or strategic29rader. This is subject to total number of noise traders in the experiment. The sign of the velocity distinguishes thebuyers and sellers.Type of order choice – Limit or market is randomly assigned. The product of velocity and flight time providesthe deviation of price quote decision by the trader from the initialised starting price. This is relevant for thestrategic traders who can start by placing limit orders, as noise traders always start with market orders. Strategictraders exit the market after the transaction or the orders are cancelled at the end of the flight time. Noise tradersafter the initial placement, alternate between market orders and limit order placement and exist in the markettill they are able to buy the asset with available funds or till end of flight time. The pricing rule for limit orderplacement by Noise traders is with a spread of 0.1 from the last transaction price.
Event arrival rates vary for thickly versus thinly traded stocks. For each stock event arrival rates can vary withinthe trading day. The set of event arrival rates (measured as events per second) included in the experiment is{1,6,10,100,1000,10000}. The included set covers a wide spectrum of event arrival rates and hence the resourcedensity. It covers the necessary range, but is not dynamic i.e does not vary through the trading period. The includedset for γ (power of flight time distribution from (7)) is {0.25,0.5,0.75,1,1.25,1.5,1.75,2,2.25,2.5,2.75}. γ >2 is theregion of Brownian search, 1< γ <2 is the region of superdiffusive or levy search, γ <1 is the region of ballisticsearch. 30 experiments each are simulated for each pair of event arrival rate and γ . This design results in a totalof 3960 experiments (two sets of 1980 experiments each for bias and non-bias case) Fig. 11 gives the results of the numerical simulation (no-bias case) showing the rate of trades (identified as thesearch efficiency) per event arrival rate. Since the quantity quoted and traded is one unit, there are no biases arisingdue to resource density in this set of simulation.The simulation could be further extended by including a tradingmodel and responses of traders to a bias. Such a simulation could demonstrate the search efficiency of examplesdiscussed in section 2.2.1-2.2.6.Unlike ballistic search, search efficiency is dependent on the event arrival rate (resource or target density)in both brownian and superdiffusive search. This is in line with the findings in Palyulina et al. (2014). Whentrading intensity is low (event arrival at 1 to 100 per second), super-diffusive and ballistic search is more efficientthan brownian search with ballistic search being the most efficient. When the resource density is high with eventarrivals at 1000 events per second or higher, both diffusive and superdiffusive search is equally efficient. In the30igure 11: The efficiency of search depends on the search regime and the resource density. The former is setthrough the quoting and cancellation behaviour. And the latter through the event arrival rates. In low resourcedensity, superdiffusive (1< γ <2 ) and ballistic search ( γ <1) perform better. When resource density increases andin absence of any biases in the environment, brownian search is as efficient as the others.Figure 12: The efficiency of search depends on the search regime and the resource density. The former is setthrough the quoting and cancellation behaviour. And the latter through the event arrival rates. Additionally, thepresence of Bias due to resource density increases efficiency of superdiffusive search more than brownian search.In low resource density, superdiffusive (1< γ <2 ) and ballistic search ( γ <1) perform better.31igure 13: Comparison of the search regimes in different resource density and influence of bias. In the presenceof bias, superdiffusive search is more efficient than brownian. Ballistic search performs the best in low resourcedensity and is less influenced by resource densitypresence of bias (fig. 12) however, the superdiffusive search begins to perform better even when the resourcedensity is very high (event arrival rates of 10000 per second).Fig. 13 gives a comparison of the average trading rates under the three search regimes normalised to eventarrival rates. Superdiffusive search is more efficient than Brownian in the presence of bias. Brownian searchperforms better in the absence of bias and in an environment of high resource density. These are clearly related toactive trading periods of liquid stocks. We do not find the rate of trades being impacted by the relative presenceof noise traders (fig. 14). The poisson arrival of events (traders and orders) into the market could be the possiblereason.Fig. 15 gives a sample of the trade prices obtained under different event arrival rates, for the case when γ = . and traders can quote different quantities (bias in environment). The price time plots describe the behaviour ofprices. The saw-tooth property of transaction price equilibriation in double auctions markets (Plott (2008)) can beobserved (except with event arrivals at rate of 10000/sec). We have introduced and analysed a model, to show how auctions in high frequency markets without a designatedmarket maker can be described as a search by buyers for a seller and vice-versa. The model is based on a zerointelligence approach. We consider a Total order book model that includes a limit order book and the latent orderbook.Intuitively, when the principle for matching trades is based on ’price-time priority’ in a continuous doubleauction mechanism, it seems to resemble a search. The need for the model arises as existing theory based on32igure 14: The percentage of noise traders in each trial of the simulation is randomly assigned. The numberof noise traders does not affect the efficiency of search. The possible reason could be the poisson arrival of thetraders. The above is the no-bias case that plots the rate of trades per event arrival rate with the percentage of noisetraders in the market. The plot of the with-bias case is also similar.supply and demand assumes the existence or emergence of a walrasian equilibrium. The above assumption takesthe underlying model away from reality when considering intraday markets that trade assets in high frequencythrough continuous auctions. In high resolution the assumption of an equilibrium of demand and supply leadingto optimised quantity traded and emergence of price is fictional. This is especially true in case of order drivenmarkets without designated market makers. The general belief that in financial markets given that trading iscontinuous, prices can quickly adjust to clear the market is valid but the process is not instantaneous. Informationflow is not seamless and frictions exist. Traders are not always present in the market and alternate opportunitiesexist for the traders. Market makers and dealers in quote driven markets were the key to push the trading toequilibrium.Our model is a diffusion-drift-reaction model and inspired by search in biology and robotics. We analyse anumber of asymptotic relationships in the model. In the limit of continuous auctions we are unable to determinethe density of market particles or trading activity (similar to Donier and Bouchaud (2016)). We analyse theinterauction times in which the density exists and a relationship between the diffusion coefficient, interauctiontime, initial density of particles and spread exists. For the spread less than 0.2 USD and inter-auction time, 0.001seconds or lower, less than 40 percent of the existing market particles are able to trade. There exists a relationshipbetween the spread and the diffusion coefficient for an efficient market. So markets that also depend on drift33igure 15: The behaviour of prices under the model. The plots of price and time provides insight into the behaviourof prices. All samples are for trade prices were obtained in the numerical simulation under an environment of bias,i.e traders can quote different quantities. The parameter γ = . , signifying superdiffusive search regime. Theplots represent various rates of event arrival - 10, 25, 50,100,1000, 10000 per second. The event arrivals includesorder arrivals and a driver of the resource density. 34resulting from high transaction rates involving noise traders) apart from diffusion are adversely impacted whenthe spread is high.The numerical simulation run on the model brings out the search efficiency of different search regimes (ballis-tic, superdiffusive and brownian) in the simulated set up. The rate of trades that emerges from trade duration is anatural candidate to measure the efficiency of search. When biases exist due to resource density, the superdiffusivesearch performs better than brownian search. In the absence of bias brownian search is equally efficient in highresource density. In low resource density ballistic search performs the best followed by superdiffusive search.Performance of ballistic search is relatively not affected by resource density or the presence of bias.Future research can attempt to connect different elements of market design, such as spread, tick size, presenceof market maker etc, to the search regimes to understand appropriate design basis the resource density. Acknowledgement
We gratefully acknowledge valuable comments received, from an anonymous referee on an earlier version of thispaper and two anonymous referees on the current version, that has improved this manuscript.
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AppendixA The Search for trades in absence of bias
Let u ( x , t ) be the probability density function (pdf) of the distance travelled by market order particles in searchof trades. Thus, x is the total length of the jumps and t is the total time. The velocity v imparted by traders onparticles can have positive and negative values to include buy or sell orders (direction of motion). If (cid:48) N (cid:48) numberof trades takes place in a unit of time dt , the average duration can be given by (cid:104) µ (cid:105) = dt / N . Each inter-trade periodis a search for the next transaction. A number of jumps get recorded in this period. Equation (39) gives the totalpath traversed in the search till time t . The initial distribution of the market particles is given by n ( x ) . u ( x , t ) = ˆ ∞−∞ d v ˆ t u ( x − v τ, t − τ ) h ( v ) f ( τ ) d τ + n ( x ) δ ( t ) (39)37he pdf of the traded particles, q ( x , t ) is the number of trades that take place in a time interval ( , t ) . However,the trades being the success of the search of market particles in a tatonnement, we define the efficiency η of thesearch process as the ratio of the total path traversed in the search to the number of targets searched or trades done.This gives us a link to represent q ( x , t ) in terms of u ( x , t ) and η as in equation (40). η is again a function of thecoupling of velocity imparted and the flight times. q ( x , t ) = ˆ ∞−∞ d v ˆ t u ( x − v τ, t − τ ) h ( v ) f ( τ ) η ( v τ ) d τ (40)Equations (40) and (41) fully describe the dynamics of the system with a given initial density of particles andthe two pdf for the flight times and velocities. They establish the crucial link between the tatonnement in theauction with the trades. Next we solve these equations analytically. First, we determine the total path traversedby market particles in the search process and then introduce the result into the equation of the trade density. Weapply the Fourier transform with respect to the spatial coordinate in (39). Due to the shift property of the Fouriertransform, an additional exponential factor e − ikv appears under the integral. Integration with respect to v gives theFourier transform of h ( v ) with a reciprocal velocity k τ . The Fourier transform of (40) is: u k ( t ) = ˆ t u k ( t − τ ) h k τ f ( τ ) d τ + n , k δ ( t ) (41)where the indices k and k τ denote the Fourier components. Note that, F { h ( v ) = ˆ ∞−∞ e − ik τ v h ( v ) d v } = h ( k τ ) (42). Next, we apply the Laplace transform with respect to time and use its convolution property to obtain the follow-ing: u k , s = u k , s [ h k τ f ( τ )] s + n , k (43)where the index s corresponds to the Laplace component. So the path traversed in search can be given in theFourier-Laplace domain, k , s , as: u k , s = n , k − [ h k τ f ( τ )] s (44)We can introduce the result in (44) into the Fourier-Laplace expression for equation (41) (similar to above) toget the analytic expression for the density of trades in the Fourier-Laplace domain, obtained through the searchprocess. 38 k ( t ) = ˆ t u k ( t − τ ) h k τ f ( τ ) η k ( τ ) d τ (45) q k , s = u k , s [ h k τ f ( τ )] s η k , s (46) q k , s = n , k η k , s [ h k τ f ( τ )] s ( − [ h k τ f ( τ )] s ) (47)To find a solution, the next step is to take the Laplace inverse of equation (47). However, an analytic represen-tation and direct inversion of the equation (47) is not feasible. Froemberg et al. (2015) recommend an asymptoticanalysis for large space and time scales, x , t → ∞ . We use the same approach. Going to Fourier-laplace spaceusing the tauberian theorem, this limit corresponds to ( k , s ) → ( , ) such that k / s = constant . This has to beperformed numerically.It is important to define the velocity and flight time distributions before we move to obtain the inverse transfor-mation. In our view this needs to come from empirical analysis. Further, velocity distribution cannot be obtaineddirectly as it is a notional quantity and needs to be interpreted from equation (39) that describes the relationshipwith returns.As discussed in Zaburdaev et al. (2008), Froemberg et al. (2015) and Zaburdaev et al. (2015) when the velocitydistribution is Cauchy or lorentian, the density of the particles also is lorentian independent of flight times and jumplengths. Such a lorentian velocity profile appears in real physical phenomena such as two dimensional turbulenceand is also found in model distributions of kinetic theory, statistics, plasma physics and starving amoeba cells.We know that a cautchy process does not give rise to a continuous sample path for the price and it differs fromBrownian motion as there are large jumps not infrequently. As given below we make arbitrary choice of a lorentianvelocity distribution h ( v ) and intuitively a flight time distribution f ( τ ) with power tails. h ( v ) = u π ( + ( v u )) (48) f ( τ ) = γ ( + τ ) + γ (49)In equation (48) u is needed to constrain the velocities so that the particles do not go beyond the ballisticcones, else it will lead to instantaneous dispersion. The γ in equation (49) is varied to get different transport. γ = / for normal diffusion. In equation (47), given the asymptotic limit we want to evaluate, we further set39 constant efficiency, so that there is an expression for the initial density of particles for the trade density. Thepropagator for our model can then be noted as follows: G ( k , s ) = L[ f ( τ ) h ( k τ )] − L[ f ( τ ) h ( k τ )] (50)where k τ is the Fourier variable conjugate to v . The equation (50) retains the form of the well known Montroll-Weiss equation for the pdf of the uncoupled continuous time random walk (CTRW) to find the particle x at thetime t , modified such that it applies to random jumps in velocity. Equation (50) can be rewritten as (see AppendixC for details): G ( k , s ) = ´ ∞−∞ d v f ( s + ik v τ ) h ( v )] − ´ ∞−∞ d v f ( s + ik v τ ) h ( v )] (51)For the flight time distribution we have chosen and in the long time limit the expansion in the Laplace space isgiven by, f ( τ ) (cid:117) − τ γ Γ ( − γ ) s γ (52)Using (52) the asymptotic version of (51) is, G ( k , s ) = s ´ ∞−∞ ( + ik v / s ) γ − h ( v ) d v ´ ∞−∞ ( + ik v / s ) γ h ( v ) d v (53) B Inversion of the fourier laplace expression for propagators with ballis-tic scaling
A method exists for the inversion of the fourier laplace expression for propagators with ballistic scaling. Apropagator of a random walk model has ballistic scaling if it can be written in the form, G ( x , t ) (cid:117) t φ ( xt ) , t →∞ ,where φ is the scaling function. In Fourier-laplace space this is, G ( k , s ) (cid:117) s g ( iks ) . Comparing the above twoforms we can rewrite the scaling form of our equation as in equation (54), where ξ = iks . g ( ξ ) = ´ ∞−∞ ( + ξ v ) γ − h ( v ) d v ´ ∞−∞ ( + ξ v ) γ h ( v ) d v (54)Further, this can be inverted according to the general equation (55) (from Froemberg et al. (2015) based onGodreche and Luck (2001)) to obtain equation (56). The scaling function in (55) is defined as φ ( y ) = (cid:104) δ ( y − Y )(cid:105) .Here angular brackets denote the averaging with respect to a random variable X which has a pdf P ( X ) , (cid:104) F ( X )(cid:105) = ´ ∞−∞ F ( X ) P ( X ) dX and Y is the time average of the particles velocity, i.e x / t . (55) is obtained using Sokhotsky-40eirstrass theorem: lim (cid:15) → x ± i (cid:15) = x ∓ i πδ ( x ) and thus ∓ π Im lim (cid:15) → x ± i (cid:15) = δ ( x ) . φ ( y ) = − π lim (cid:15) → (cid:61)[ y + i (cid:15) g ( − y + i (cid:15) )] (55) φ ( y ) = − π lim (cid:15) → (cid:61)[ ´ ∞−∞ ( + i (cid:15) − v ) γ − h ( v ) d v ´ ∞−∞ ( + i (cid:15) − v ) γ h ( v ) d v ] (56)While the velocity distribution could be anything from a two state, or uniform distribution (Froemberg et al.(2015) discuss a number of examples) we arbitrarily choose the special situation induced by Cauchy distributedvelocity. To use this we start with the following propagator and velocity distribution . The propagator reduces to(59) without prescribing to any particular form for f ( τ ) . G ( k , s ) = { f ( τ ) h ( k τ )} − { f ( τ ) h ( k τ )} (57) h ( k τ ) = exp (− u | k | τ ) (58) G ( k , s ) = s + u | k | (59)Now taking the inverse Laplace and Fourier transform we get a form of Cauchy distribution. The scalingfunction is φ ( y ) = π ( + y ) . G ( x , t ) = u t π ( u t + x ) (60) C Explanation for equation (9)
Numerator, L[ f ( τ ) h ( k τ )] ( k , s ) = ∞ ˆ −∞ d τ ˆ ∞−∞ e − ik τ v e − s τ f ( τ ) h ( v )] = ∞ ˆ −∞ d τ ˆ ∞−∞ e −( s + ikv ) τ e − s τ f ( τ ) h ( v )] ∞ ˆ −∞ d τ ˆ ∞−∞ f ( s + ik v ) h ( v )] Similarly the denominator can be arrived at.
D Search for trades in presence of bias (The Complete Model)
We build a comprehensive model of stochastic evolution of market particles. Our fundamental set up of thecomplete model is (61) which is expanded to (62) ∂ g ( x , t ) ∂ t = (61) ∂∂ x ( D ( t ) ∂ g ( x , t ) ∂ x − λ ( t ) ∂ f ( x , t ) ∂ x g ( x , t ) − λ ( t ) g ( x , t )) + v ( t ) g ( x , t ) + q ( x , t ) ∂ g ( x , t ) ∂ t = D ( t ) ∂ g ( x , t ) ∂ x − λ ( t ) ∂ f ( x , t ) ∂ x ∂ g ( x , t ) ∂ x − λ ( t ) ∂ f ( x , t ) ∂ x g ( x , t ) (62) − λ ( t ) ∂ g ( x , t ) ∂ x + v ( t ) g ( x , t ) + q ( x , t ) (62) can be solved analytically. We draw upon the technique used by Sanskrityayn and Kumar (2016), whoused the Greens function method to solve their diffusion-advection equation in the context of pollutant solutes inthe atmosphere. To solve the equation, we note that q will remain untouched and we need to reduce the equationto a known form so that we find an expression for f ( x , t ) . We do a co-ordinate transformation from the domain ( x , t ) to the domain ( X ( x , t ) , t (cid:48) ) . The domain X is essentially fixed time-snapshots of the entire lattice. We want totransform (62) to the form in equation (63). ∂ G ( X , t (cid:48) ) ∂ t (cid:48) = D ( t (cid:48) ) ∂ G ( X , t (cid:48) ) ∂ X − λ ( t (cid:48) ) ∂ G ( X , t (cid:48) ) ∂ X + v ( t (cid:48) ) G ( X , t (cid:48) )) + q ( X , t (cid:48) ) (63)Using the domain transformation, we can write equation (62) as equation (64), following which we equate thecoefficients to obtain equation (65,66 and 67): ∂ G ( X , t (cid:48) ) ∂ t (cid:48) = D ( t (cid:48) )( ∂ X ∂ x ) ∂ G ( X , t (cid:48) ) ∂ X (64)42 [− D ( t (cid:48) ) ∂ X ∂ x + λ ( t (cid:48) ) ∂ f ( x , t ) ∂ x ∂ X ∂ x + λ ( t ) ∂ X ∂ x + ∂ X ∂ t ] ∂ G ( X , t (cid:48) ) ∂ X + [ v ( t (cid:48) ) − λ ( t (cid:48) ) ∂ f ( x , t ) ∂ x ] G ( X , t (cid:48) )) + q ( X , t (cid:48) ) D ( t (cid:48) )( ∂ X ∂ x ) = D ( t ) (65) − [− D ( t (cid:48) ) ∂ X ∂ x + λ ( t (cid:48) ) ∂ f ( x , t ) ∂ x ∂ X ∂ x + λ ( t ) ∂ X ∂ x + ∂ X ∂ t ] = − λ (66) v ( t (cid:48) ) − λ ( t (cid:48) ) ∂ f ( x , t ) ∂ x = v (67)From (65) we obtain an expression for X in (68) and from (67) an expression for f in (69). We insert thesetwo results into (66) and equate similar coefficients to obtain (70) and (71). For the sake of convenience, we use tinstead of t’ hence forth. X = (cid:114) D D x + φ ( t ) (68) f ( x , t ) = v ( t ) − v ( t ) λ ( t ) x + φ ( t ) x + c (69) − λ ( t )( v ( t ) − v ( t ) λ ( t ) x + φ ( t )) (cid:115) D ( t ) D ( t ) − λ ( t ) (cid:115) D ( t ) D ( t ) − ∂∂ t ( (cid:115) D ( t ) D ( t ) x + φ ) (70) = − λ ( t ) ∂∂ t φ ( t ) − λ ( t ) (cid:115) D ( t ) D ( t ) − λ ( t ) φ ( t ) (cid:115) D ( t ) D ( t ) = − λ ( t ) (71) − ( v ( t ) − v ( t )) (cid:115) D ( t ) D ( t ) = ∂∂ t (cid:115) D ( t ) D ( t ) (72)43rom (72), we get (73) where β is a dimensionless expression defined in (74) D ( t ) D ( t ) = β ( t ) (73) β = e ´ t ( v ( s )− v ( s )) ds (74)Using the expression for φ ( t ) , obtained after reorganising (71), we obtain the expression for X as below, X = x β ( t ) + ˆ t ( λ ( t ) − λ ( t ) β ( t ) − λ ( t ) φ ( t ) β ( t ) ) (75)Equipped with the expression we have obtained for f ( x , t ) in (69) and the transformations β ( t ) and X in (75)we can reduce our initial equation to the following form, ∂ G ( X , t ) ∂ t = D ( t ) β ( t ) ∂ G ( X , t ) ∂ X − λ ( t ) ∂ G ( X , t ) ∂ X + v G ( X , t ) + q ( X , t ) (76)The initial conditions for this equation are G ( X , ) = G i ω ( X ) , with −∞ < X < ∞ and t > . Next, we tryto remove the drift term and the decay term. We use the following transformation equations one after the otherfor this purpose. In (78) β = e ´ t v ( s ) ds is a dimensionless term and in (79) β is as per (73). In (79) T is a timevariable. Equation (76) now reduces to equation (80). η = X − λ ( t ) t (77) K ( η, t ) = G ( η, t ) β ( t ) (78) T = ˆ t β ( s ) ds (79) ∂ K ( η, T ) ∂ T = D ∂ K ( η, T ) ∂η + Q ( η, T ) β ( T ) β (80)We now need to solve equation (80) to obtain a master equation for transport of market particles in doubleauction limit order asset markets. Haberman (1987) provides a solution for equations such as (80) using GreensFunction Method (GFM). The solution to (80) based on the above is given in (81):44 ( η, T ) = ˆ T ˆ ∞−∞ Q ( χ, τ ) β ( ζ ) (cid:112) π D ( T − ζ ) β exp (− ( η − χ ) D ( T − ζ ) ) d χ d ζ (81) + ˆ ∞−∞ √ π DT G i ω ( X ) exp (− ( η − χ ) DT ) d χ Next we sequentially trace back the transformations done earlier, in reverse order to get the solution below in (82).Here, ζ = ´ τ β ( s ) ds and the initial condition g ( x , ) = G i ω ( x ) . g ( x , t ) = β ( t ) ˆ t ˆ ∞−∞ Q ( χ, τ ) (cid:112) π D ( T − ζ ) exp ( ( x β − ´ t ( λ β + λ φ β ) ds − χ ) D ( T − ζ ) ) d χ d τ (82) + β ( t ) ˆ ∞−∞ √ π DT G i ω ( χ ) exp (− ( x β − ´ t ( λ β + λ φ β ) ds − χ ) DT ) d χχ