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
![](https://www.scirp.org/html/1-1490090\c8d38c8b-52d3-4f56-ae19-1f337412b8ac.jpg)
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
![](https://www.scirp.org/html/1-1490090\c6c3abcc-d056-46d2-a54c-3e3f633e2173.jpg)
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 ![](https://www.scirp.org/html/1-1490090\5563f8da-34a0-4137-aeb5-b6cef1966e96.jpg)
(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
![](https://www.scirp.org/html/1-1490090\b15f996f-18e6-41f1-a40d-a07e196f1a9b.jpg)
and for ![](https://www.scirp.org/html/1-1490090\d064163f-01a2-4b26-9bf7-2bb2c66c3879.jpg)
![](https://www.scirp.org/html/1-1490090\1769f55e-f45a-4014-a907-7c1b805c2c5d.jpg)
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.,
![](https://www.scirp.org/html/1-1490090\fa57cf68-5011-4481-908c-bc245e1133b5.jpg)
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.
![](https://www.scirp.org/html/1-1490090\5f2b4046-de1b-4a2a-9312-641332e4a451.jpg)
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
![](https://www.scirp.org/html/1-1490090\356b7cbb-eda9-445f-ab1a-464ab30aa937.jpg)
for
. Easy calculation shows that
![](https://www.scirp.org/html/1-1490090\6273ad6a-2aa5-48ab-8fc6-a2b13695e86b.jpg)
and
.
Furthermore for all
,
we have
![](https://www.scirp.org/html/1-1490090\d10b1b99-55a6-4ae6-a297-a69ba9c5f94b.jpg)
and
![](https://www.scirp.org/html/1-1490090\2478319e-4c4b-4ec2-a88d-d597f91620f8.jpg)
For any
and
we have
![](https://www.scirp.org/html/1-1490090\a3b4151a-fda0-4c2b-aed5-789ae1f066a2.jpg)
and
![](https://www.scirp.org/html/1-1490090\3512325b-0984-4e50-9217-07acdf614a43.jpg)
In the same way one can obtain
![](https://www.scirp.org/html/1-1490090\aab7dd8c-8f1a-4cff-b596-5978902d5b30.jpg)
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
![](https://www.scirp.org/html/1-1490090\47dc1bf3-91f7-4f61-9ada-03ed47401c1c.jpg)
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
![](https://www.scirp.org/html/1-1490090\d310a850-05da-4df3-88ca-669bab40f478.jpg)
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
![](https://www.scirp.org/html/1-1490090\cce6db99-b474-4efa-ade6-29e4269c6ae4.jpg)
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
![](https://www.scirp.org/html/1-1490090\aede710e-6a12-4628-967d-bc96365f8a94.jpg)
Therefore
![](https://www.scirp.org/html/1-1490090\c7495a92-53be-4922-bc2d-b944a9f85779.jpg)
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
![](https://www.scirp.org/html/1-1490090\454b1799-6b2a-497d-8790-27a000e8bed0.jpg)
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
![](https://www.scirp.org/html/1-1490090\1825779e-6b62-4f26-b774-5523368dbb4b.jpg)
Since
and the random variables
are independent then
![](https://www.scirp.org/html/1-1490090\2b12b40b-e720-47a4-a509-eb96eacaf19e.jpg)
(3.8)
where
is defined above. The relationships
and
imply
![](https://www.scirp.org/html/1-1490090\fac59c0d-2a69-4d3f-ab22-018b3ad9420c.jpg)
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, ,
.
![](https://www.scirp.org/html/1-1490090\7d6463e3-a1ef-4d66-b8db-adccf04052a1.jpg)
Since
and all multipliers here are independent then
![](https://www.scirp.org/html/1-1490090\e1fec92c-4deb-48e3-b370-bd52dafda9d4.jpg)
where
as above. After the changing the order of summation and summation indexes we have
![](https://www.scirp.org/html/1-1490090\787d0fa2-5140-422d-b4b8-a8c9f8efee96.jpg)
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
.
![](https://www.scirp.org/html/1-1490090\d5e6bbb1-e791-4f55-bf79-12724f77f368.jpg)
![](https://www.scirp.org/html/1-1490090\97760729-1c7a-45da-a440-8ee3b780fb92.jpg)
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
![](https://www.scirp.org/html/1-1490090\b3b60054-cb57-4311-9e5f-79971ff22125.jpg)
Since
and
then
and
. Therefore
and ![](https://www.scirp.org/html/1-1490090\c2abe486-ad69-4271-aa1c-e4b27521662b.jpg)
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
![](https://www.scirp.org/html/1-1490090\b0d4df87-44b6-4b17-9a96-22482fedea20.jpg)
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
![](https://www.scirp.org/html/1-1490090\9216cf28-f33c-4a3f-b772-3e6393aa0f07.jpg)
Proof. Recall the formula for distribution function of the sum of two independent random variables
and ![](https://www.scirp.org/html/1-1490090\e26ada9f-3290-4804-8dc1-40ee9ea7065f.jpg)
![](https://www.scirp.org/html/1-1490090\52ef46c5-60f3-424c-bc12-486b0727c16b.jpg)
where
is the distribution function of the random variable
. Since
for all
then
![](https://www.scirp.org/html/1-1490090\7a879c97-79c5-43eb-92df-3f5dbd1c14c6.jpg)
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 ![](https://www.scirp.org/html/1-1490090\6fc8fbfa-aabd-470a-a497-7a2a2c8a078e.jpg)
![](https://www.scirp.org/html/1-1490090\8a1d27e2-ce20-4c09-8f99-7f5ecf7c8f47.jpg)
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
![](https://www.scirp.org/html/1-1490090\0c15b55f-4f2d-428a-8a22-c8beb29ddb08.jpg)
And the price of the last trade is deterministic and is equal to the following expression
![](https://www.scirp.org/html/1-1490090\503996ca-f927-4550-85ff-dd7d49c61097.jpg)
In particular, if
for all ![](https://www.scirp.org/html/1-1490090\42cb18ac-e0af-4cd6-a547-f9710e660334.jpg)
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
![](https://www.scirp.org/html/1-1490090\a1d101a3-a940-4952-a434-09176edfe7d7.jpg)
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
![](https://www.scirp.org/html/1-1490090\caeda86d-1e06-47a4-976b-f7f40cbe01f6.jpg)
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
![](https://www.scirp.org/html/1-1490090\f56bcd51-f016-457f-b866-1039e93268bd.jpg)
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
![](https://www.scirp.org/html/1-1490090\022aad4a-ec70-4dc6-841e-ec4c4adcabdd.jpg)
Let us compute joint characteristic function ![](https://www.scirp.org/html/1-1490090\fd9bac11-17d0-431c-ba5d-3953ffdcb21e.jpg)
of the sum
and
.
![](https://www.scirp.org/html/1-1490090\ed720fe1-08e6-4eaa-b1f9-0bdf5121cf6d.jpg)
It has been shown above that
. Since
depends on
and
only then
![](https://www.scirp.org/html/1-1490090\52124346-f403-4f00-8aee-8238bf5b304e.jpg)
(4.2)
where
is the characteristic function of
.
The expression
can be simplified as follows. If
then
and
![](https://www.scirp.org/html/1-1490090\22631b68-3c22-46c3-9a31-f9ecdf1eb840.jpg)
For
we have
. Therefore
![](https://www.scirp.org/html/1-1490090\3ff9fad0-3f55-4e42-a564-f0e911408fe9.jpg)
Then the Equality (4.2) has the following form
![](https://www.scirp.org/html/1-1490090\d7a1a75f-bc55-4b11-9c27-91f1bc7a6563.jpg)
Suppose at first that
. Then from the last equality we get
![](https://www.scirp.org/html/1-1490090\3514125a-e0c5-4d92-9bcb-41602baa8e68.jpg)
(4.3)
Similarly we have for ![](https://www.scirp.org/html/1-1490090\f058692a-82ae-4f39-b7a0-5e2524dd3fb0.jpg)
(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
![](https://www.scirp.org/html/1-1490090\1b23ee88-e906-4969-9024-33d3e87f1747.jpg)
Hence from (4.3) and (4.4) we have
![](https://www.scirp.org/html/1-1490090\107d9865-0baa-4902-9eb8-1fd424e1151b.jpg)
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 ![](https://www.scirp.org/html/1-1490090\1570dada-c651-474c-81e2-06e92ba6b23c.jpg)
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]