1. 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 estimate 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
, 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 [11-13].
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.
2. 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 trading on the market. Let 
be the number of all agents which have ever taken part in trading. At each moment 
the agent number i, 
proposes a bid price 
and an ask price 
for a goods on the market. We assume that
. It is convenient to set 
and 
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
, where 
and
. We say that there is a trade between 
-th and 
-th agents at moment 
if 
or 
. It means that there is a trade between agents with minimal ask price 
and maximal bid price 
provided that they are equal
. In order to escape some pathological examples we always assume that at every time t there exist two different agents, say number i and j, i ≠ j, such that 
and
. In the case when more than one of the agents have the same minimal ask price and maximal bid price, say 
and
, we suppose that a trade occurs between agents with numbers 
and
, where
.
The bids and asks can be changed only by the agents. It may happen that 
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
, then the respective ask price 
will be not less then the price before the trade
. Therefore we can say that
where 
is a nonnegative random variable (it is possible to add one more value 
if the agent decides to leave the market). For the bid prices we can write similarly
with nonnegative random variable 
(with the same note about
). The random variables 
and 
are 
- and 
-adapted, respectively, where 
and 
are 
-fields containing information which the agents know before the time
, inclusively. Note that 
and 
are defined only at the moment 
of trades.
As in the previous case we can write the same equalities for a moment 
when the respective agent was not involved in a trade. Hence for any 
we have
(2.1)
where 
and
, 
are nonnegative random variables. The moment 
and the price 
of the last trade before time 
inclusively are given by
(2.2)
Set 
and
.
The purpose of present paper is to calculate the distributions of 
and 
from Equation (2.2) by using the known distributions of 
and 
from Equations (2.1).
Taking min and max in Equations (2.1) yields
(2.3)
where 
and
are nonnegative random variables. Notice that 
and 
are 
-measurable, where 
is information known to at least one agent before time
, inclusively.
Let us consider two nonnegative random processes 
and
. From Equalities (2.3) we deduce that
(2.4)
(2.5)
Since the trade occurs at the moment 
if and only if 
or, equivalently, if
, then the last moment of a trade before the time 
(2.6)
is the last moment before 
when the process 
reached the level 1. The price of the last trade before the time 
is given by
(2.7)
Now the problem is reduced to finding the law of random time 
given by (2.6) and the law of the process 
given by Equation (2.4) at the time
.
3. Simplest Behavior of Agents
Since the bid prices can be changed by the agent in even moments only, then
. Therefore from Equation (2.3) we deduce that
(3.1)
Similarly 
and
(3.2)
Then Equations (3.1), (3.2) and (2.5) imply that 
and
. Moreover, we have 
and
. Define a new sequence 
by 
for 
and 
if
,
. Then
, 
and 
. Hence the trade occurs at time 
if and only if
.
In order to obtain some result we need to have more assumptions on the behavior of the processes 
and
. The simplest assumption is that
, 
is a sequence of independent identically distributed (i.i.d.) random variables. Denote by p the probability that 
takes value zero:
. The variable 
is a last zero of the sequence 
before the moment
. We put 
if there are no zeros (no trades) before time
, inclusively. Hence 
takes values
. The probabilities of these values are given by
and for 
Let
, 
denote the number of trades before time t inclusively. Hence 
is number of zeros in the sequence
,
. Then 
has a binomial distribution with parameters 
and
, i.e.,
here 
is a binomial coefficient.
Moreover 
has a binomial distribution with the same parameters 
and
. As a consequence of independence of the variables 
we get that for any 
the random variables 
are independent.
Define the sequence
, 
of random times inductively by the following expression.
with 
and
. We adopt the convention that the infinum of empty set is equal to infinity. Then
, 
is a moment of 
-th trade (or zero of the sequence
) and
for
. Easy calculation shows that
and
.
Furthermore for all
, 
we have
and
For any 
and 
we have
and
In the same way one can obtain
Notice that
.
Hence 
and 
are not independent.
Let us consider process 
given by Equation (2.4). The solution of this equation can be written as
(3.3)
Since 
and 
then
where 
denotes the integer part of number
.
Therefore taking into account that 
one has
(3.4)
From the Equation (3.4) and definition of 
and 
we obtain the prices 
and 
of the last trade and the 
-th trade:
(3.5)
(3.6)
Now we calculate the characteristic function 
of the logarithm
. It follows from representation (3.5) that
Notice that event 
occur if and only if 
and 
if 
does not coincide with some of the
. This fact, formula (3.5), independence and the distribution of 
imply
where 
is the characteristic function of 
conditioned on
. From the relationships 
and 
we have
(3.7)
Notice that if only 
numbers of 
are even then
Therefore
where 
is a number of possibilities to choose 
even and 
odd numbers from the set
. Here
. There are only 
even and 
odd numbers among
. Hence
if 
or 
and
if 
and
. Putting this expression into the Formula (3.7) yields
Using equation (3.6) one can compute joint characteristic function 
of the moment 
of the first trade and the logarithm 
provided there was at least one trade, 
in the following way
Since 
and the random variables 
are independent then
(3.8)
where 
is defined above. The relationships 
and 
imply
Similarly we can find joint characteristic function 
of the difference 
between moments of 
-th and 
-st trades, 
and the logarithm 
of the ratio between these trades provided there were at least 
trades, ,
.
Since 
and all multipliers here are independent then
where 
as above. After the changing the order of summation and summation indexes we have
The same arguments as after Equality (3.8) lead to the following expression
Now we consider one more simplest case.
Recall the expressions for
, 
and
.
where
, 
for 
and 
if
,
.
Assume that 
is a sequence of independent random variables. Then the power of exponent in the expression for 
is a random walk and 
is a discrete analogue of geometrical Brownian motion, which is classical choice for modeling of the price process. But in our model the price process describes by
, geometrical random walk computed at random time and the distributions of 
and 
can be completely different. We show that indeed this is the case and the distribution of 
is trivial.
Denote
: then we have
Since 
and 
then
and
. Therefore
and 
which implies the following equality:
(3.9)
From the meaning of process 
we have 
for all 
hence 
for any 
a.s. satisfy the following system of inequalities
Denote the left side of the last inequality by
. Then 
and
for all
. It is evident that the random variables 
and 
are independent and 
if and only if
.
The following technical lemma will be needed.
Lemma 3.1. Let 
and 
be two independent random variables. Then
Proof. Recall the formula for distribution function of the sum of two independent random variables 
and 
where 
is the distribution function of the random variable
. Since 
for all 
then
for all
. This implies that
. Since the opposite inequality is obvious then we have the statement of the lemma.
It follows from the non-negativity of 
and lemma above that for all 
The trade occurs at time 
if and only if
, i.e. when the last inequality becomes in fact equality. In this case we have that 
for any
. Therefore
And the price of the last trade is deterministic and is equal to the following expression
In particular, if 
for all 
then 
is a last possible moment of trade. There is a trade at each time 
with the same price 
and there are no trades at all after the moment
.
4. The Connection to Continuous Time Analogue of the Model
In this section we give an example of the agents’ behavior such that the geometrical Brownian motion can be regarded as the limit of the price process 
with discrete time
. For this purpose let 
be a sequence of random variables describing the state of the real world (noise sequence). Assume that at each time 
the agents make their decisions about how to change bid or ask prices according to the history of the noise sequence before the present time
. For instance 
and
. The simplest case, with agents taking into account only the present value of noise 
was considered above.
Now we consider the case when the agents are taking into account only the present 
and previous 
information, 
and 
for even and odd moments. Assume that 
is a sequence of independent identically distributed random variables and set 
and 
, where 
and
.
For such 
and 
we can compute the distribution of
. For simplicity assume that 
. If there are no trades then
The last event happens if and only if the following condition is satisfied: for all 
at least one of the numbers 
and 
is positive and for all 
at least one of the numbers 
and 
is negative. If 
and 
have the same sign then the sign of other
,
satisfying the condition above is uniquely determined. The condition above is also satisfied if 
and 
have the different signs for all
. Hence the number of possible choices of signs of 
satisfying condition above is equal to
, where 
is a number of choices of 
such that 
and 
have the same sign and 
is number of possibilities that 
and 
have the different signs for all
. Since for any choice of signs of 
the probability is equal to 
then we get
Notice that if 
then 
and 
a.s. Indeed, for even 
we have 
and since 
then 
a.s. For odd 
the proof is the same. The fact that 
if 
can be shown in the same way. Hence for 
we get
Now consider
. From Equalities (3.3) and (3.4) we have
(4.1)
where 
if 
and 
if
, and 
if 
and 
if
. Notice that the representation (4.1) is also true in the case when the random variables 
are not necessary independent and identically distributed. Since
, then 
and from the last equation we deduce that
Let us compute joint characteristic function 
of the sum 
and
.
It has been shown above that
. Since 
depends on 
and 
only then
(4.2)
where 
is the characteristic function of
.
The expression 
can be simplified as follows. If 
then 
and
For 
we have 
. Therefore
Then the Equality (4.2) has the following form
Suppose at first that
. Then from the last equality we get
(4.3)
Similarly we have for 
(4.4)
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
of the interval
. Let 
take values
. Assume that 
and
, where
. If the noise sequence 
is Gaussian, 
, then
Hence from (4.3) and (4.4) we have
Therefore for Gaussian noise the continuous version of price process 
is a geometrical Brownian motion and
.
5. Conclusions
With this work we have set forth the structure for computing a price process from first principles of agent behavior in providing bid and ask quotes to a market. As well, we have provided some content by analyzing a basic case, that of a binomial assumption on the i.i.d. sequence 
recording the moments of trades. This assumption led to the specification of a geometric random walk computed in random time, and to the joint characteristic function 
of the difference
between moments of 
-th and 
-st trades, 
and the logarithm 
of the ratio between these trades. The study culminated with an explicit expression for
, and implications for a parallel model in continuous time.
Next on the agenda is to explore alternative hypotheses on agent behaviors, and to perform simulations and other numerical work as necessary to establish a theory of consequential price processes.
[15] NOTES
[17] *The work of Aleh L. Yablonski was supported by INTAS grant 03-55- 1861.
[18] 1For a treatment of the case wherein the duration, defined as the length of time between trades, is stochastic, see [14].
[19]