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 [
Engle provides just such an econometric study [
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 [
Other studies, such as one by Bondarenko, delve into the bid and ask series, but rather as a difference, the spread [
, 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 [
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.
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
.
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.
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
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 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.
when the respective agent was not involved in a trade. Hence for any 
we have
and
, 
are nonnegative random variables. The moment 
and the price 
of the last trade before time 
inclusively are given by
and
.
and 
from Equation (2.2) by using the known distributions of 
and 
from Equations (2.1).
and
are nonnegative random variables. Notice that 
and 
are 
-measurable, where 
is information known to at least one agent before time
, inclusively.
and
. From Equalities (2.3) we deduce that
if and only if 
or, equivalently, if
, then the last moment of a trade before the time 
when the process 
reached the level 1. The price of the last trade before the time 
is given by
given by (2.6) and the law of the process 
given by Equation (2.4) at the time
.
. Therefore from Equation (2.3) we deduce that
and
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
.
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
, 
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.,
is a binomial coefficient.
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.
, 
of random times inductively by the following expression.
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
. Easy calculation shows that
.
, 
we have
and 
we have
.
and 
are not independent.
given by Equation (2.4). The solution of this equation can be written as
and 
then
denotes the integer part of number
.
one has
and 
we obtain the prices 
and 
of the last trade and the 
-th trade:
of the logarithm
. It follows from representation (3.5) that
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
is the characteristic function of 
conditioned on
. From the relationships 
and 
we have
numbers of 
are even then
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
of the moment 
of the first trade and the logarithm 
provided there was at least one trade, 
in the following way
and the random variables 
are independent then
is defined above. The relationships 
and 
imply
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, ,
.
and all multipliers here are independent then
as above. After the changing the order of summation and summation indexes we have
Now we consider one more simplest case.
, 
and
.
, 
for 
and 
if
,
.
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.
: then we have
and 
then
and
. Therefore
and 
we have 
for all 
hence 
for any 
a.s. satisfy the following system of inequalities
. Then 
and
for all
. It is evident that the random variables 
and 
are independent and 
if and only if
.
and 
be two independent random variables. Then
and 
is the distribution function of the random variable
. Since 
for all 
then
. This implies that
. Since the opposite inequality is obvious then we have the statement of the lemma.
and lemma above that for all 
if and only if
, i.e. when the last inequality becomes in fact equality. In this case we have that 
for any
. Therefore
for all 
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
.
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.
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
.
and 
we can compute the distribution of
. For simplicity assume that 
. If there are no trades then
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
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
. From Equalities (3.3) and (3.4) we have
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
and
.
. Since 
depends on 
and 
only then
is the characteristic function of
.
can be simplified as follows. If 
then 
and
we have 
. Therefore
. Then from the last equality we get
as the limit of the discrete time model.
of the interval
. Let 
take values
. Assume that 
and
, where
. If the noise sequence 
is Gaussian, 
, then
is a geometrical Brownian motion and
.
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 
, and implications for a parallel model in continuous time.