MARKET MICROSTRUCTURE AND PRICE DISCOVERY

The design of this study is to investigate the evolution of a stochastic price process consequent to discrete processes of bids and offers in a market microstructure setting. Under a set of flexible assumptions about agent preferences, we generate a price process to compare with observation. Specifically, we allow for both rational and irrational economic behavior, abstracting the inquiry from classical studies relying on utility theory. The goal is to provide a set of economic primitives which point inexorably to the price processes we see, rather than to assume such process from the start.


Introduction
We propose to model a price process based on microstructural activity of a market.We assume a set of agents such that each agent at any moment has both bid and ask prices present in the market.A trade occurs if and only if the bid of one agent is equal to the ask of another, this common value becoming the price of a trade.We calculate the dynamics of the resulting price process, including the moments of trades, in a discrete time setting for behavioral choices of the agents.These choices are formalized in relevant probability distributions specific to the agents' behaviors.In this way, we allow for a multitude of behavioral patterns, including, but not restricted to traditional motivations inspired by utility functions.Our model is flexible enough to allow for "marks" to a trade, ancillary data such as its time stamp, so that we may study independently such features as trade clustering and time deformation.
Recent history is rich with microstructure studies of financial markets and with associations of specific families of probability distributions to financial stochastic processes.For good reviews of the microstructure literature see these works respectively [1,2].For associations of probability distributions such as the widely applied Gaussian, normal inverse Gaussian, and more inclusively the generalized hyperbolic, see these studies [3,4].In many instances such inquiries assume at the outset various forms of stochastic processes, as defined by stochastic differential equations, and then set forth to esti-mate parameters.Popular choices are Itô diffusions and Ornstein-Uhlenbeck processes, with and without the superposition of pure jump Lévy processes.
Most studies of microstructure take an econometric approach, that is, they define some structure, assume distributions as appropriate, then estimate parameters using data.In his survey with important bibliography, Bollerslev reviews the state of financial econometrics [5].In a subsection discussing time-varying volatility, he notes that, "several challenging questions related to the proper modeling of ultra high-frequency data, longer-run dependencies, and large dimensional systems remain."Further in the text, he qualifies this remark by stating: "Not withstanding much recent progress, the formulation of a workable dynamic time series model which readily accommodates all of the high-frequency data features, yet survives under temporal aggregation, remains elusive." Engle provides just such an econometric study [6] employing the Autoregressive Conditional Duration (ACD) model developed by him with Russell [7] in the study of IBM stock transactional arrival times.In the former paper, Engle, in referring to cases of the conditional duration function, relates, "In each case, the density is assumed to be exponential."Such assumptions are typical, and necessary, for an econometric study focusing on time series of prices as the fundamental data structure.
Hasbrouck, in focusing on the refinement of bid and ask quotes, proposes and estimates an Autoregressive Conditional Heteroskedasticity (ARCH) model using Alcoa stock transactions, evenly spaced at 15 minute intervals [8].Routinely, he asks the reader to consider, "a stock with an annual log return standard deviation of 0.30" The reference "return" is of course to the price sequence, a necessary expedient in the classical econometric framework which considers a price process as fundamental, rather than consequential to a set of underlying bid and ask processes.
Other studies, such as one by Bondarenko, delve into the bid and ask series, but rather as a difference, the spread [9].The focus of this work and its principal results are in the realm of market liquidity, rather than in the estimation of the price process.Once again, the classical framework requires an assumption on the distribution of the price process, as evidenced in this remark made within the context of evaluating a price change between periods."The asset's final value is denoted v  , a normal random variable with mean and variance Yet further studies attempt to develop directly a price process from first principles.An interesting and provocative example is a paper by Schaden, which formulates conclusions from financial analogues to fundamentals of quantum physics [10].As he observes in the introduction, "At this stage it is impossible to decide whether a quantum description of finance is fundamentally more appropriate than a stochastic one, but quantum theory may well provide a simpler and more effective means of capturing some of the observed correlations."Indeed, though the basic process investigated is yet a price process, not those of bids and asks.The analysis is grounded on five at first qualitative assumptions about the market, and concludes with the assertion that the evolution of prices follows "the lognormal price distribution."In this setting it is difficult to discern how a different-and more realistic-distribution could emerge without changing substantially the assumptions, or the physics.For further background reading see [11][12][13].
In our paper we choose to move to a more basic level of explanation, to specify the market mechanisms among interacting agents, and then to let the model determine the price process and its features.In this way we derive such features as the distributions of prices, rather than assuming them ab initio.
We now proceed forthwith to present our case.

Specification of the Model
We consider for simplicity the model of the market for one stock in discrete time 1 .It is reasonable to assume that in each time there are only finite number of agents taking part in the trad-ing on the market.Let be the number of all agents which have ever taken part in trading.At each moment proposes a bid price and an ask price i t for a goods on the market.We assume that t t .It is convenient to set b  if at the moment the -th agent does not take part in the trading.Supposing the rational behavior of agents on the market we have and .We say that there is a trade between i -th and -th agents at moment i and , where The bids and asks can be changed only by the agents.It may happen that t t A B  after such changing of prices.In order to avoid such possibilities we suppose that bid prices can be changed by agents only at even moments and ask prices only at odd moments.Nevertheless the trades can occur at any moment: even or odd.
How should the bid and ask prices change?The rules of changing bid and ask prices by the agents are different for each agent and they are based on different reasons; for instance: aims of agents, interpretations of information, personal reasons, and so on.If these prices are changed at time when a trade occurs, say between the i-th and j-th agents with prices t t t t , then the respective ask price will be not less then the price before the trade .Therefore we can say that

Simplest Behavior of Agents
the agent in even Since the bid prices can be changed by moments only, then 2 1 . Therefore from Equation (2.3) we deduce that Similarly 1 and Then Equations (3.1), (3.2) and (2.5) . Moreover, we have as a binomial distribution wi e pa and .As a consequence of independence of the variables t t  we get that for any and 0 0   .We adopt the convention that th f em t is equal to infinity.Then In the same way one can obtain are not indepe nt.Let us co roces nde nsider p t X given by Equatio (2.4).The solution of this equation be written as where   m erefor denotes the integer part of number Th e taking into account that From the Equation (3.4) and definition of X and we obtain the prices an of th e and the -th trade: ulate the characteristic function where is a number of possibilities to choose even odd numbers from the set .here are only even and . Putting this expression into the Formula (3.7) s where     Similarly we can find joint characteristic function between mo--th and -st trad and the logarithm of the ratio between these trades provided there were at least trades, , and all multipliers here are independent then gve The same arguments as after Equality (3.8) lead to the following expression as above.After the chan ing the orde nd summation indexes we ha Now we consider one more simplest case.
Recall the expressions for (3.9) From the meaning of process we have for all hence hich implies the following eq   Denote the left de of the last inequality by where is the distribution function of the ran . This implies that

  
   equality is obvious then we hav  .Since the oppo- ent of the le site in e the statem mma.

It follows from the vity of and l above t non-negati emma
The trade occurs at time t if and only if 0 when the last inequal ty becomes in fact equality.In this i case we have that And the price of the last trade is deterministic and is the following exp equal to ression In particu for all 0, 1, k with the same price 0 t all after the moment t  .

S X 
and there are no tr at

The Connection to Continuous Time Analogue of the Model
In agents imit of the price process ades this section we give an example of the ' behavior such that the geometrical Brownian motion can be regarded as the l , where The last event happens if and onl if the following on is sat : for all be sho Henc wn in the same way.e for 0,1, Pr Pr 0; 0; Now consider t X .From Equalities (3.3) and (3.4) we have It has been shown above that .Since Then the Equality (4.2) has the following form The last Equalities (4.3) and (4.4) allow one to obtain the characteristic function of a continuous time model analogous the process as the limit of the discrete time model.
For instance, consider the partition the ratio between these trades.The study culminated with an explicit expression for t S , and implications for a me.theses on agent be tions and other numerical work as necessary to establish a theory of consequential price processes.