Market Directional Information Derived From Time Execution Price Shares Traded Sequence of Transactions. On The Impact From The Future.

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Market Directional Information Derived From (Time, Execution
Price, Shares Traded) Sequence of Transactions.
On The Impact From The Future.
Vladislav Gennadievich Malyshkin
∗
Ioffe Institute, Politekhnicheskaya 26, St Petersburg, 194021, Russia
Mikhail Gennadievich Belov
†
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics,
GSP-1, Moscow, Vorob’evy Gory, 119991, Russia
(Dated: September, 20, 2022)
$Id: ImpactFromTheFuture.tex,v 1.269 2022/10/09 10:41:55 mal Exp $
An attempt to obtain market directional information from non–stationary solution
of the dynamic equation: “future price tends to the value maximizing the number of
shares traded per unit time” is presented. A remarkable feature of the approach is an
automatic time scale selection. It is determined from the state of maximal execution
flow calculated on past transactions. Both lagging and advancing prices are calculated.
∗malyshki@ton.ioffe.ru
†mikhail.belov@tafs.pro
arXiv:2210.04223v1 [q-fin.CP] 9 Oct 2022
2
Времена Пугачёвского бунта.
Самозванец выступает перед народом,
говорит о грядущем счастье, которое
придёт в форме мужицкого царства.
Пленный офицер спрашивает:
“Откуда деньги будут на всю эту
благодать”? Пугачёв ответил: “Ты
что, дурак? Из казны жить будем!”
Народная легенда, 1774.
I. INTRODUCTION
Introduced in [
1
] the ultimate market dynamics problem: an evidence of existence (or a
proof of non–existence) of an automated trading machine, consistently making positive P&L
trading on a free market as an autonomous agent can be formulated in its weak and strong
forms[
2
]: whether such an automated trading machine can exist with legally available data
(weak form) and whether it can exist with transaction sequence triples (time, execution price,
shares traded) as the only information available (strong form); in the later case execution
flow I=dV/dt is the only available characteristic determining market dynamics.
Let us formulate the problem in the third, “superstrong”, form: Whether the future
value of price can be predicted from (time, execution price, shares traded) sequence of past
transactions? Previously[
3
,
4
] we thought this is not possible, only P&L that includes not
only price dynamics but also trader actions can be possibly predicted. Recent results changed
our opinion.
There are two types of predicted price: “lagging” (retarded) and “advancing” (future)
Lagging price
PRet
corresponds to past observations; future direction is determined by the
difference of last price
Plast
and
PRet
. An example of
PRet
is moving average. A common
problem with lagging price is that it typically assumes an existence of a time scale the
PRet
is calculated with, what gives incorrect direction for market movements with time scales
lower than the one of
PRet
; however making the time scale too low creates a large amount of
3
false signals. Advancing price
PAdv
is predicting actual value of future price; the direction is
determined by the difference of
PAdv
and
Plast
. The
PAdv
is typically calculated from limit
order book information, brokerage clients order flow timings, etc.
In this work both lagging and advancing prices are calculated from (time, execution price,
shares traded) sequence of past transactions. The key element is to determine the state
ψ[IH]
of maximal execution flow
I
=
dV/dt
(eigenvalue problem (10)), as experiments show
it’s importance for market dynamics. Found
ψ[IH]
state automatically selects the time scale
what makes the approach robust.
Found lagging price (49) is the price in
ψ[IH]
state
P[IH]
plus trending term that
suppresses false signals. The advancing price is obtained by considering density matrix state
kρJIH k
corresponding to the state “since
ψ[IH]
till now” and experimentally observed fact
that operators
pdI
dt
and
Idp
dt
have to be equal in
kρJIH k
state. This corresponds to the
result of our previous works [
3
,
5
]: execution flow
I
=
dV/dt
(the number of shares traded
per unit time), not trading volume
V
(the number of shares traded), is the driving force of
the market: asset price is much more sensitive to execution flow
I
(dynamic impact), rather
than to traded volume V(regular impact).
This paper is concerned only with obtaining directional information from a sequence of
past transaction in a “single asset universe” just for simplicity, see Section VIII below for multi
asset universe generalization. Whereas the dynamics theory of Section IV definitely requires
additional research, the lagging indicator (49) of Section VI, see Fig. 8, can be practically
applied to trading even in a single asset universe. In this work we do not implement any
trading ideas of [
3
,
4
], where a concept of liquidity deficit trading: open a position at low
I
, then close already opened position at high
I
, as this is the only strategy that avoids
eventual catastrophic P&L losses. This paper is concerned only with obtaining a directional
information that is required to determine what side the position has to be open on a liquidity
deficit event.
II. THE STATE OF MAXIMAL EXECUTION FLOW
Introduce a wavefunction ψ(x)as a linear combination of basis function Qk(x):
ψ(x) =
n−1
X
k=0
αkQk(x)(1)
4
Then an observable market–related value
f
, corresponding to probability density
ψ2
(
x
), is
calculated by averaging timeserie sample with the weight
dµ
=
ψ2
(
x
(
t
))
ω
(
t
)
dt
; the expression
corresponds to an estimation of Radon–Nikodym derivative[6]:
fψ=hψ|f|ψi
hψ|ψi(2)
fψ=
n−1
P
j,k=0
αjhQj|f|Qkiαk
n−1
P
j,k=0
αjhQj|Qkiαk
(3)
For averages we use bra–ket notation by Paul Dirac:
hψ|
and
|ψi
. The (2) is plain ratio
of two moving averages, but the weight is not regular decaying exponent
ω
(
t
)from (A3),
but exponent multiplied by wavefunction squared as
dµ
=
ψ2
(
x
(
t
))
ω
(
t
)
dt
, the
ψ2
(
x
)defines
how to average a timeserie sample. Any
ψ
(
x
)function is defined by
n
coefficients
αk
, the
value of an observable variable
f
in
ψ
(
x
)state is a ratio of two quadratic forms on
αk
(3); as an example of a wavefunction see localized state (13), it can be used for Radon–
Nikodym interpolation:
f
(
y
)
≈ hψy|f|ψyi.hψy|ψyi
; familiar least squares interpolation is
also available: f(y)≈ hψy|fiψy(y) = Pn−1
j,k=0 hQjfiG−1
jk Qk(y).
One can also consider a more general form of average,
dµ
=
P
(
x
(
t
))
ω
(
t
)
dt
, where
P
(
x
)is
an arbitrary polynomial, not just the square of a wavefunction. These states correspond to a
density matrix average:
fρP=Spur kf|ρPk
Spur kρPk(4)
This average, the same as (2), is a ratio of two moving averages. For an algorithm to convert a
polynomial
P
(
x
)to the density matrix
kρPk
see Theorem 3 of [
7
]. A useful application of the
density matrix states is to study an average “since
|ψi
”; for example if
|ψi
corresponds to a past
dV/dt
spike, then the polynomial “since
|ψi
till now” is
P
(
x
) =
J
(
ψ2
(
x
)) with
J
(
·
)defined in
(A9); price change between “now” and the time of spike is
Plast −hψ|p|ψi
=
Spur
dp
dt ρJ(ψ2)
,
similarly, total traded volume on this interval is Spur
dV
dt ρJ(ψ2)
.
The main idea of [
3
] is to consider a wavefunction (1) then to construct (3) quadratic
forms ratio. A generalized eigenvalue problem can be considered with the two matrices from
(3). The most general case corresponds to two operators
A
and
B
. Consider an eigenvalue
problem with the matrices hQj|A|Qkiand hQj|B|Qki:
Aψ[i]=λ[i]Bψ[i](5)
5
n−1
X
k=0 hQj|A|Qkiα[i]
k=λ[i]
n−1
X
k=0 hQj|B|Qkiα[i]
k(6)
ψ[i](x) =
n−1
X
k=0
α[i]
kQk(x)(7)
δij =ψ[i]Bψ[j]=
n−1
X
k,m=0
α[i]
khQk|B|Qmiα[j]
m(8)
λ[i]δij =ψ[i]Aψ[j]=
n−1
X
k,m=0
α[i]
khQk|A|Qmiα[j]
m(9)
If at least one of these two matrices is positively defined – the problem has a unique solution
(within eigenvalues degeneracy). In the found basis
ψ[i]
the two matrices are simultaneously
diagonal: (8) and (9). See (A28) to convert an operator’s matrix from
ψ[i]
to
Qj
basis and
(A29) to convert it from Qjto ψ[i]basis.
In our previous work [
1
–
3
,
5
] we considered various
A
and
B
operators, with the goal to
find operators and states that are related to market dynamics. We established, that execution
flow
I
=
dV/dt
(the number of shares traded per unit time), not trading volume
V
(the
number of shares traded), is the driving force of the market: asset price is much more sensitive
to execution flow
I
(dynamic impact), rather than to traded volume
V
(regular impact). This
corresponds to the matrices
hQj|I|Qki
=
hQj|A|Qki
and
hQj|Qki
=
hQj|B|Qki
. These
two matrices are volume- and time- averaged products of two basis functions. Generalized
eigenvalue problem for operator I=dV/dt is the equation to determine market dynamics:
Iψ[i]=λ[i]ψ[i](10)
n−1
X
k=0 hQj|I|Qkiα[i]
k=λ[i]
n−1
X
k=0 hQj|Qkiα[i]
k(11)
ψ[i](x) =
n−1
X
k=0
α[i]
kQk(x)(12)
ψy(x) =
n−1
P
i=0
ψ[i](y)ψ[i](x)
sn−1
P
i=0
[ψ[i](y)]2
=
n−1
X
i=0 ψ[i]ψyψ[i]=
n−1
P
j,k=0
Qj(x)G−1
jk Qk(y)
sn−1
P
j,k=0
Qj(y)G−1
jk Qk(y)
(13)
ψyψ[i]2=ψ[i](y)
ψy(y)2
=ψ[i](y)2
n−1
P
k=0
[ψ[k](y)]2
(14)
摘要：

MarketDirectionalInformationDerivedFrom(Time,ExecutionPrice,SharesTraded)SequenceofTransactions.OnTheImpactFromTheFuture.VladislavGennadievichMalyshkinIoeInstitute,Politekhnicheskaya26,StPetersburg,194021,RussiaMikhailGennadievichBelovyLomonosovMoscowStateUniversity,FacultyofMechanicsandMathematic...

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