Market Dynamics: On Directional Information Derived From (Time, Execution Price, Shares Traded) Transaction Sequences

TL;DR

This paper introduces a novel market analysis method that uses execution flow to identify relevant price moves, leading to a scalp-price indicator that signals market trend changes based on high-impact trading events.

Contribution

It defines the scalp-price based on high execution flow events, advancing market direction detection by focusing on dynamic impact rather than volume.

Findings

Scalp-price effectively signals market trend changes.

High execution flow correlates with significant price moves.

The method emphasizes market-relevant price dynamics.

Abstract

A new approach to obtaining market--directional information, based on a non-stationary solution to the dynamic equation "future price tends to the value that maximizes the number of shares traded per unit time" [1] is presented. In our previous work[2], we established that it is the share execution flow ($I=dV/dt$) and not the share trading volume ($V$) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow $I$ (the dynamic impact) than to the traded volume $V$ (the regular impact). In this paper, an important advancement is achieved: we define the "scalp-price" ${\cal P}$ as the sum of only those price moves that are relevant to market dynamics; the criterion of relevance is a high $I$. Thus, only "follow the market" (and not "little bounce") events are included in ${\cal P}$. Changes in the scalp-price defined this way indicate a market trend change - not a bear market rally or a bull market sell-off; the approach can be further extended to non-local price change. The software calculating the scalp--price given market observations triples (time, execution price, shares traded) is available from the authors.

Equations

$$\displaystyle\Ket{f\Big{|}\psi_{f}^{[i]}}$$

$$\sum _{k=0}^{n-1}⟨Q_j⟩f{Q}_k{\alpha }_k^{[i]}$$

$${\psi }_f^{[i]}(x)$$

$$\Vert \rho \Vert$$

$$f_{\rho }$$

$$\Vert {\rho }^{[IH]}\Vert$$

$$p^{[IH]}$$

$$V_s(t)$$

$$T_s(t)$$

$$p_v$$

$$p_t$$

$$p_V$$

$$p_T$$

$$w_{\psi }(t)=\underset{-\infty }{\overset{t}{\int }}{\psi }^2(x(t^{\prime }))\omega (t^{\prime })d{t}^{\prime }=\omega (t)J({\psi }^2(x(t)))$$

$$\underset{-\infty }{\overset{t_{now}}{\int }}F(t){\psi }^2(x(t))\omega (t)dt=-\underset{-\infty }{\overset{t_{now}}{\int }}\frac{dF}{dt}{w}_{\psi }(t)dt$$

$$F(t_{now})=0:\text{Boundary condition}$$

$$⟨Q_m⟩$$

$$⟨Q_{m}I⟩$$

$$⟨Q_{m}pI⟩$$

$$⟨Q_m\frac{dp}{dt}⟩$$

$$P\&L$$

$$0$$

$$dS$$

$$\Vert \frac{dS}{dt}\Vert$$

$$P\&L$$

$$\displaystyle\Ket{I\Big{|}\psi_{I}^{[i]}}$$

$$p_t^{[IH]}$$

$$P^{last}-p_t^{[IH]}$$

$$⟨Q_m\frac{dp}{dt}⟩$$

$$DIR\_scalped$$

$$⟨Q_m\mathcal{F}⟩$$

$${\mathcal{F}}_l$$

$$\mathcal{F}(t)$$

$$(t_l-t_{l-1}){\mathcal{F}}_l$$

$$\mathcal{S}(t)$$

$${\mathcal{S}}_l$$

$${\mathcal{F}}_l$$

Full text

A new approach to obtaining market–directional information, based on a non–stationary solution to the dynamic equation ‘‘future price tends to the value that maximizes the number of shares traded per unit time’’ Malyshkin and Bakhramov (2015) is presented. In our previous workMalyshkin (2016), we established that it is the share execution flow ($I=dV/dt$) and not the share trading volume ($V$) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow $I$ (the dynamic impact) than to the traded volume $V$ (the regular impact). In this paper, an important advancement is achieved: we define the ‘‘scalp–price’’ ${\cal P}$ as the sum of only those price moves that are relevant to market dynamics; the criterion of relevance is a high $I$. Thus, only ‘‘follow the market’’ (and not ‘‘little bounce’’) events are included in ${\cal P}$. Changes in the scalp–price defined this way indicate a market trend change — not a bear market rally or a bull market sell–off; the approach can be further extended to non–local price change. The software calculating the scalp–price given market observations triples (time, execution price, shares traded) is available from the authors.

††preprint: V.M.

I Introduction

Introduced in Malyshkin (2017), the ultimate market dynamics problem — finding evidence of existence (or proof of non–existence) of an automated trading machine consistently making positive P&L as a result of trading on a free market as an autonomous agent — can be formulated in its weak and strong forms:

• •

Weak form: Whether such an automated trading machine can exist at all using only legally available data. (It can definitely exist in an illegal form – e.g. when a brokerage uses client order flow information to frontrun their own clients. This type of strategies typically rely on using proprietary information about clients’ Supply–Demand future disbalance and on the subsequent monetization of this information.)

• •

Strong form: Whether such an automated trading machine can exist and be based solely on transaction sequences – say, the historical time series of (time, execution price, shares traded) market observations triples. This information has supply and demand matched for every observation: at time $t$ trader A sold $v$ shares of some security at price $P$ to trader B and received $v\cdot P$ dollars. Such a strategy can utilize only information about volume and execution flows.

We have shown in Malyshkin and Bakhramov (2015); Malyshkin (2016) that it is share execution flow $I=dV/dt$, not share trading volume $V$, that is the driving force of the market (see the Figs. 2 and 3 of Ref. Malyshkin (2016): the asset price shows singularity at a high $I$, but there is no price singularity at the maximal volume price, the median of price–volume distribution).

In Malyshkin and Bakhramov (2015, 2016), the concept of liquidity deficit trading was introduced: open a position at low $I$, then close already opened position at high $I$; this is the only strategy that avoids catastrophic P&L losses. This strategy is ideologically similar to a classic volatility trading strategy: buy a straddle at low volatility, sell it at high volatility, never go short volatility to avoid catastrophic P&L loss, but is different from it by incorporating asset price directional contribution: the decision is needed on whether to open a long or a short position at low $I$. In Malyshkin (2017), the first attempt at finding a non–stationary solution to the dynamic equation by linking asset price and liquidity deficit via ‘‘impact–from–the–future’’ operator (adding to execution flow a contribution from not–yet–executed trades) was presented. In this paper, a different approach is developed.

Instead of adding not–yet–executed trades (impact–from–the–future), we now consider removing from consideration already executed trades (impact–from–the–past) corresponding to high $I$ $\to$ low $I$ transitions. A liquidity deficit trading strategy assumes that only low $I$ $\to$ high $I$ transitions will be captured by the trader. The high $I$ $\to$ low $I$ transitions are not to be used, as they are a major source of catastrophic risk. A typical market behavior after a liquidity excess (high $I$) event is to ‘‘bounce a little,’’ then go in the original direction of the market. This creates an uncertainty of strategy. What does one bet on: ‘‘little bounce’’ or ‘‘follow the market’’? In contrast, after a liquidity deficit (low $I$) event, the market can only go in the direction of the market trend, eliminating this uncertainty. This shows the importance of the assymetry of dynamic impact (price sensitivity to $I$ Malyshkin (2016)): low $I$ $\to$ high $I$ and high $I$ $\to$ low $I$ transitions are to be considered separately, as they lead to very different price behaviors. This asymmetry is the topic of this study. The scalp–function (30) is introduced to comprise only those price moves relevant to market dynamics (high $I$), which allows constructing scalp–price ${\cal P}$ (Fig. 2) containing only ‘‘follow the market’’ (and not ‘‘little bounce’’) events. A change in the scalp–price indicates a market trend change, not a bear market rally or a bull market sell–off.

II Basic Mathematics

The key concept of the dynamic equation ‘‘future price tends to the value that maximizes the number of shares traded per unit time’’ Malyshkin and Bakhramov (2015); Malyshkin (2017) is to find an averaging weight from the behavior of a market dynamics operator $f$ (e.g. $dV/dt$, $V/t$, or $dI/dt$), then to estimate some directional indicator (e.g. price change, signed volume, etc.) using the obtained weight. Mathematically, the weight is considered in the form of an average depending on wavefunction $\psi(x)=\sum_{k=0}^{n-1}\alpha_{k}Q_{k}(x)$: $\psi^{2}(x(t))\omega(t)dt$, an important generalization of commonly–used parameter–independent fixed time scale averaging such as the exponential moving average : $\omega(t)dt$. The bases $Q_{m}(x(t))\omega(t)dt$ we use in this paper are listed in Section LABEL:Mu-basis of Ref. Malyshkin (2017)). The problem is then reduced to a generalized eigenvalue problem of operator $\|f\|$:

[TABLE]

The most general form of the averaging weight is a density matrix:

[TABLE]

The most promising result of Refs. Malyshkin and Bakhramov (2015); Malyshkin (2017) is averaging with the weight in the state $\Ket{\psi_{I}^{[IH]}}$ of the maximum execution rate $I=dV/dt$ on the past sample. This corresponds to the following density matrix and asset price:

[TABLE]

Given a state $\Ket{\psi}$, a number of values in this state can be calculated. Just a few examples. Let’s define

[TABLE]

Here, $V_{0}(t)=V(t_{now})-V(t)$ is traded volume, $V_{1}(t)$ is traded capital, $V_{1}(t)/V_{0}(t)$ is volume–weighted average price, $T_{0}(t)=t_{now}-t$, and $T_{1}(t)/T_{0}(t)$ is time–weighted average price; these are the values for the time interval: between $t$ and $t_{now}$. Then $p_{\{v,t\}}$ is {volume,time}–averaged price in the $\Ket{\psi}$ state, $p_{\{V,T\}}$ is {volume,time} averaged aggregated price in the $\Ket{\psi}$ state, calculated using the aggregated moments (8). If $\Ket{\psi}$ is localized at some given $t$, then, approximately, $p_{\{v,t\}}$ is the price at $t$ and $p_{\{V,T\}}$ are {volume,time}–weighted price moving average calculated for the time interval between $t$ and $t_{now}$:

[TABLE]

Moments $\Braket{Q_{m}V_{s}}$ and $\Braket{Q_{m}T_{s}}$ can be calculated from moments $\Braket{Q_{m}\,p^{s}I}$ and $\Braket{Q_{m}\,p^{s}}$ and, more generally, moments $\Braket{Q_{m}\frac{dF}{dt}}$ can be calculated from moments $\Braket{Q_{m}F}$ using integration by parts (see the Appendices D and E below). In some cases, it is more convenient to directly integrate the wavefunction rather the individual basis functions as in (82):

[TABLE]

For the bases we use, $J(\cdot)$ in (10) is analytically–known polynomial–to–polynomial mapping function111See the classes \seqsplitcom/polytechnik/trading/{WIntegratorLegendreShifted,WIntegratorLaguerre,WIntegratorMonomials}.getPsi2WIntegratedDt() for numerical implementations) . The Eq. (11) allows simultaneously to obtain the values of operator pairs: $(\|V_{0}\|,\|I\|)$, $(\|V_{1}\|,\|pI\|)$, etc. in the state of a given $\psi(x)$.

What input data is required to obtain all the results of this paper? The $n\times n$ matrices $\Braket{Q_{j}}{f}{Q_{k}}$ ($j,k=[0\dots n-1]$): are calculated from generalized moments ($m=[0\dots 2n-2]$):

[TABLE]

by applying basis functions multiplication operator (Eq. (LABEL:Mu-cmul) of Ref. Malyshkin (2017)). All the (13) are calculated from (Time, Price, Shares traded) transaction sequence.

III P&L and optimal position change

Given a directional density matrix $\|\rho\|$, how we do apply it? A naïve answer is to average a directional attribute with it, for example:

• •

Use price change operator $\|f\|=\|\frac{dp}{dt}\|$ (or $\|f\|=\|\frac{d^{2}p}{dt^{2}}\|$ with some boundary condition from the Appendix E), calculate $\mathrm{Spur}\,\|f|\rho\|$; in a pure state $\|\rho\|=\Ket{\psi}\Bra{\psi}$, hereof $\mathrm{Spur}\,\|f|\rho\|=\Braket{\psi}{f}{\psi}$. Other directional attributes (signed volume, spread multiplied by signed volume, time difference spent in the order book, etc.) can be also consideredMalyshkin and Bakhramov (2016).

• •

The state determining the dynamics often corresponds to a large $dI/dt$. Because $dI\approx I(t+dt)-I(t)>0$, $I=dV/dt$ is larger at the end of the interval. The asset price difference $p_{v}-p_{t}$, with volume $dV$ and time $dt$ averaged in a state with such an asymmetry, is proportional to the directional component, where $p_{v}=\Braket{\psi}{pI}{\psi}/\Braket{\psi}{I}{\psi}$, and $p_{t}=\Braket{\psi}{p}{\psi}/\Braket{\psi}{\psi}$. Note that such a difference between volume– and time–averaged attribute $p_{v}-p_{t}$ carries directional information only in a state of large $dI/dt$, which makes an asymmetry of price averaging with $dV$ and $dt$ correspond to $\delta p$. This is not the case in other states, e.g. trying to use the difference between volume– and time–averaged price in the $\Ket{\psi_{I}^{[IH]}}$ state was fruitless in Malyshkin (2017), see Appendix A for a demonstration. It is now clear why: only the states with large $dI/dt$ provide weight asymmetry required to obtain directional information using $dV$ vs. $dt$ averaging.

In Malyshkin and Bakhramov (2015) a P&L operator has been introduced in the Section II.E ‘‘P&L operator and trading strategy’’. Given a position change $dS$, the amount of shares bought ($dS>0$) or sold ($dS<0$) during time interval $dt$, the P&L is222While the P&L is $-\int pdS$, Eq. (14), the $\int IdS$, can be tried as a directional indicator.:

[TABLE]

The constraint (15) means: total asset position should be zero in the beginning and in the end of a trading period. Formally,

[TABLE]

where $w(t)$ is an arbitrary positive function, provides positive P&L in (14) (integrate by parts and assume $\frac{dp}{dt}=0$ at the boundary to satisfy (15)). Position increment $dS$ of optimal P&L trading has a symmetry of the second derivative of price. Note that in (16) other than $dp/dt$ attributes can be used, designate it as ${\cal F}$, for example: weighted price change ${\cal F}=\delta V\frac{dp}{dt}$ (price change multiplied by the volume traded at this price), signed volume, signed volume multiplied by spread, etc.

There is a $dS$ answer of integral type:

[TABLE]

but it’s non–local nature and the difficulty to choose integration limits to satisfy the constraint (15) make such an approach more difficult to implement. In the simplest form this approach is equivalent to buying below the median and selling above the median strategy considered in the Appendix LABEL:Mu-PnLTrading of Ref. Malyshkin (2017).

A very promising idea is a ‘‘local trading strategy’’ for $dS$ : in $\Ket{\psi_{I}^{[IH]}}$ state buy at prices below the $p^{[IH]}$ from (7), sell above the $p^{[IH]}$. Corresponding $\|dS/dt\|$ operator is then:

[TABLE]

For this $dS$, in the $\Ket{\psi_{I}^{[IH]}}$ state, the (15) condition is satisfied, and the $\mathrm{P\&L}$ has a meaning of price standard deviation (20).

IV Directional Information:

Beyond The Wavefunction

As we have discussed in Malyshkin and Bakhramov (2015); Malyshkin (2016) the most interesting market behavior is observed at large $I$, optimization problem $I=\frac{\Braket{\psi}{I}{\psi}}{\Braket{\psi}{\psi}}\xrightarrow[\psi]{\quad}\max$ can be reduced to a generalized eigenvalue problem (2) for $\|I\|$ operator:

[TABLE]

While the enter/exit conditions can be easily obtained from (21) as in (73), the directional information is a much trickier problemMalyshkin (2017). In Malyshkin and Bakhramov (2016), the importance of P&L dynamics was emphasized. In Section III above, several trading strategies ($dS$), retrospectively providing positive P&L are presented. The goal, however, is to build a strategy providing future positive P&L. Consider $p_{t}$ (9b) in the $\Ket{\psi_{I}^{[IH]}}$ state:

[TABLE]

The (23) is just $dp/dt$ integration with the weigh (10) for $\Ket{\psi_{I}^{[IH]}}$: the sum of the derivative values with the proper weights give the last price minus the average. The (23) can be expressed via the $\Braket{Q_{m}\frac{dp}{dt}}$ moments using an integration by parts of the Appendix D. The problem is reduced to calculation of the moments ($m=[0\dots 2n-1]$) from observations333 Here the “right” sum is selected to simplify the recurrence by preserving the invariance of the time–grid. One can possibly use the “middle” sum with the $(t_{l}-t_{l-1})/2$ in the weight $\omega(t)$ and the basis $Q_{m}(x(t))$ functions argument. sample $l=[1\dots M]$:

[TABLE]

Then (24) can be substituted for (23) and the best directional answer of Ref. Malyshkin and Bakhramov (2015): the last price minus the price in the $\Ket{\psi_{I}^{[IH]}}$ state is obtained (the (9a) and (9b) are almost identical in the $\Ket{\psi_{I}^{[IH]}}$ state). These answers are the most general form that can be obtained using the ‘‘pure wavefunction approach’’: all the answers are two quadratic forms ratio, possibly incoherently superposed to a density matrix (5). However, as we have discussed above, ‘‘not all observations are equal’’: only the events with a high $I$ are important for market dynamics. Consider the expression (23) for a general attribute ${\cal F}$:

[TABLE]

For

[TABLE]

the (25 is exactly the (23) and (24). Consider

[TABLE]

Now price change is multiplied by a $[0\dots 1]$ bounded scalp–function ${\cal S}(t)$ to select ‘‘the relevance to market dynamics’’ of any single observation moment $t_{l}$. This way, we can remove from consideration all ‘‘irrelevant’’ observations, as discussed in the introduction; the relevance is determined by estimating whether the current execution flow $I_{0}$ is extremely large. The answer obtained in Malyshkin and Bakhramov (2015); Bobyl et al. (2018) is: for every $t_{l}$ observation solve the (21) problem for the interval $[-\infty\dots t_{l}]$ and consider the projection (30) for time–shifted ($t_{now}=t_{l}$) problem (21). The calculations are straightforward. At time ‘‘now,’’ look back at all $[1\dots M]$ market observations, calculate the sum (25b); for every term at $t_{l}$ also ‘‘look back’’ to construct a separate set of matrices $\Braket{Q_{j}}{Q_{k}}$ and $\Braket{Q_{j}}{I}{Q_{k}}$ for the interval $l^{\prime}=[1\dots l]$ and calculate the scalp–function ${\cal S}$ from (30). This is a problem of $O(M^{2})$ complexity when approached directly, but it can be optimized using recurrence relation for the moments calculated for different observation intervals $l^{\prime}=[0\dots l]$, $l=[0\dots M]$. The major difference with the (23) is that the averaging (25a) can no longer be written in the density–matrix form (5) with the original $\Braket{Q_{m}\frac{dp}{dt}}$ moments. The integration weight in (23) is obtained from the integration of (10). Using Theorem 3 from the Appendix A of Ref. Malyshkin (2018), any polynomial $P(x)$ of $2n-2$ degree can be isomorphly mapped to a linear operator of the dimension $n$, thus the density matrix, corresponding to the $w_{\psi_{I}^{[IH]}}$ averaging (23), can be readily obtained. This is no longer the case for (25a) averaging. The scalp–function ${\cal S}$, while is easy to calculate numerically, does not allow to reduce (25a) averaging to a density matrix averaging (5) of the original moments (24); we now need the scalp–moments (28) to average them with the $w_{\psi_{I}^{[IH]}}$. This is similar to Bloch wavefunction in quantum mechanics, where the ‘‘true’’ wavefunction is considered as a product of slow and fast oscillating terms. Now we have a product of slow $w_{\psi}(t)$ and fast ${\cal S}(t)$ changing weights in (25a). The greatest advantage of such a transition from regular to scalp–moments, is that the averaging weight can be very sharp. Compare the $I_{0}$ in Fig. 1 with, calculated from the (13) input at fixed $t_{now}$, the ‘‘interpolated’’ $I(y(-\infty\leq t\leq t_{now}))$ in Fig. 6 of the Appendix A: even for the dimension $n=12$ obtained wavefunction states are not sufficiently localized to select the sharp spikes in price changes at high $I$. In the same time the dimension $n=12$ is perfectly OK for the execution flow $I$. The scalp–function (29) is a practical way to unify price and execution–flow dynamics within a single framework.

In the Fig. 1 the $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$ projection (30) along with $I_{0}$ and $\lambda_{I}^{[IH]}$ are presented. One can clearly see that the (30) is a very good indicator of market activity, the effect we have noticed back in Malyshkin and Bakhramov (2015); Bobyl et al. (2018). Now, however, we know how to apply this knowledge: the criterion of current execution flow being extremely high (such as $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$) can be used as a scalp–function ${\cal S}$ when calculating the $dp/dt$ moments in (25b): multiply each $p(t_{l})-p(t_{l-1})$ by the scalp–function. This way only the relevant (high $I$) market moves will be accounted in the scalp–moments $\Braket{Q_{m}{\cal F}}=\Braket{Q_{m}{\cal S}\frac{dp}{dt}}$. Typical scalping is price spikes (relatively some ‘‘average’’–like level) identification technique along with a set of rules to enter a trade and to take a profit/stoploss. As we have shownMalyshkin (2016) the spikes in the execution flow, not in the price, are responsible for market dynamics.

The (25b) main idea is to accumulate, with the $Q_{m}(x(t))\omega(t)dt$ weight, a directional attribute, such as $p(t_{l})-p(t_{l-1})$, (Ref. Malyshkin and Bakhramov (2015) result) multiplied by a scalp–function, such as (30) (this paper result); in practice this is just a directional attribute transform (58). Algorithmically, we need to listen for all trading events, and, for each coming event in sequence, obtain a directional attribute ${\cal F}_{l}$ from the regular moments, then calculate (25b) scalp–moments (recurrent optimization make it very efficient computationally) to obtain the directional information (25a). This can be implemented in several ways:

• •

Tick trading. As a transaction sequence consider every tick (execution or even limit order book event). For every tick $l$ calculate444 Most of ${\cal D}_{l}=p(t_{l})-p(t_{l-1})=0$ as most trading occur at the same price. Also note that $p_{M}-p_{m-1}=\sum_{l=m}^{M}{\cal D}_{l}$. For a different weight in the sum obtain (23).

[TABLE]

to obtain the ‘‘filtered by relevance’’ moments in (25b).

• •

Assuming we have all the ticks data555 In practice, for US equity market, a sub–millisecond data can be obtained at reasonable cost. For other markets, such as fixed income, every tick data cannot be practically obtained. Even for currency trading the fragmentation of the markets along with prohibitively high prices on sub–millisecond data, make any tick–trading approach practically unfeasible. However, as we have discussed in Ref. Malyshkin (2017), the time scale of market opportunities (along with liquidity available!) expand well beyond sub–millisecond time scale, maximal scale is determined by the availability of high enough fluctuations in the execution flow $I$, at least an order of magnitude in $\lambda_{I}^{[IH]}/\lambda_{I}^{[IL]}$. , instead of the price difference some average multiplied by the scalp–function can be used:

[TABLE]

The (32) uses the $\Ket{\psi_{0}}$ for the interval $t\in[-\infty\dots t_{l}]$ with $t_{now}=t_{l}$. The $\psi_{0}$ from (68) has an internal time scale $1/\psi^{2}_{0}(x_{0})$ (which is determined by the basis dimension $n$ and scale $\tau$), thus in (32) the $dp/dt$ is averaged over the time $1/\psi^{2}_{0}(x_{0})$. The result is very similar to price tick (31) approach, see the Fig. 7 below. A quite similar result can also be obtained with

[TABLE]

this corresponds to described above approach of the difference between volume and time averaged price.

• •

The (31) and (32) are calculated in the $\Ket{\psi_{0}}$ state. One can consider other states, the $\Ket{\psi_{I}^{[IH]}}$ is of special interest

[TABLE]

An important feature of (34) is that some of these ${\cal F}_{l}$ expressions (34b) and (34c) are calculated from $\|pI\|$ operator variation and have: 1. the dimension of capital 2. the $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$ factor entering due to the identity

[TABLE]

• •

An ability to use an expression, calculated from the regular moments $\Braket{\cdot}$, such as (34) is a very important generalization of price change directional attribute (26). The minimal time–scale of such an attribute is $1/\psi_{0}^{2}(x_{0})$, and the experiment shows that (31) and (32) produce very similar results. This makes promising to consider a directional attributes of more general form in the $\Ket{\psi_{0}}$ state: calculate the ${\cal F}_{l}$, and use it as it were a regular price change. All the previously considered ${\cal F}_{l}$ were some kind a price change analogue. In Ref. Malyshkin (2017) two new directional attributes have been introduced: skewness and probability correlation. Consider the skewness (Eq. (LABEL:Mu-skewness) of Ref. Malyshkin (2017)) calculated out of four input moments:

[TABLE]

The idea is to build two–point Gauss quadrature (the $p_{\min}$, $p_{\max}$ are min/max nodes of the quadrature, Eq. (LABEL:Mu-evp3nodes) of Ref. Malyshkin (2017), and $\overline{p}=\pi_{1}/\pi_{0}$) then to consider it’s weight asymmetry as the asymmetry of the distribution. The weigh asymmetry (37) is actually proportional to the difference between the median estimator $(p_{\min}+p_{\max})/2$ and the average $\overline{p}$. One can use the skewness

[TABLE]

as a directional attribute instead of price change. The (38) is calculated from the regular moments $\Braket{Q_{k}}$, $\Braket{IQ_{k}}$, $\Braket{pIQ_{k}}$, $\Braket{p^{2}IQ_{k}}$, and $\Braket{p^{3}IQ_{k}}$, then the ${\cal F}_{l}$ is used as it were observed at $t=t_{l}$. This way we substitute price change by the skewness calculated at $1/\psi_{0}^{2}(x_{0})$ scale. The scalp–function $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$ makes only relevant to market dynamics observations to contribute.

• •

Variate the $\|pI\|$ in the $\Ket{\psi_{I}^{[IH]}}$ state666The expression has the meaning of capital change due to (35) identity. For single asset consideration it is convenient to divide (39) by $\lambda_{I}^{[IH]}$.

with $\Ket{\psi_{0}}$:

[TABLE]

If $\Ket{\psi_{0}}$ is the $\Ket{\psi_{I}^{[IH]}}$ then (39) is zero and no directional information is available. The factor $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}$, which does not enter the (105), is included in (39) as a scalp–function; this factor also provides proper sign invariance for $\psi\to-\psi$ transform: $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}$ is not squared as it is in (30). One can also use a higher degree of $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}$ factor in (39) to make the peaks sharper.

• •

Similar to (33), but with $\Ket{\psi_{I}^{[IH]}}$ and $\Ket{\psi_{0}}$

[TABLE]

can be considered. This is price difference in $\Ket{\psi_{0}}$ and $\Ket{\psi_{I}^{[IH]}}$ states. Were it not for the scalp–function $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$ this would be almost Ref. Malyshkin and Bakhramov (2015) answer: the difference between the last price and $p^{[IH]}$ (7). The scalp–function makes this difference to be accumulated only for the events of extremeny high $I_{0}$. The (39) and (40) are zero for $\Ket{\psi_{I}^{[IH]}}$ being equal to $\Ket{\psi_{0}}$, thus satisfy Ref. Malyshkin (2017) Eq. (LABEL:Mu-dIfuturedir) condition of ‘‘no directional information about the future available’’.

• •

All the ${\cal F}_{l}$ considered above are some kind of price change. Tick trading (31) is last price minus previous price, the other (e.g. (32), (33), etc.) are calculated from the regular moments. It is a promising path to combine tick and moments approaches.

[TABLE]

Estimating the $p^{*}_{l}$ as (7), or (9) in the state $\Ket{\psi_{0}}$ or $\Ket{\psi_{I}^{[IH]}}$ will not provide a good answer, the (34) is a demonstration. A promising approach is to consider skweness like answer (37). Take (38) but consider it in a different basis of dimension two, replace the basis $1$, $p(t)$ by $\Ket{\psi_{0}}$, $\Ket{\psi_{I}^{[IH]}}$, as these are the states that are localized and relevant to market dynamics:

[TABLE]

The $\lambda_{p^{*}}^{[0]}$ and $\lambda_{p^{*}}^{[1]}$ eigenvalues give the min/max price estimates, that can be obtained in a state of $\Ket{\psi_{0}}$ and $\Ket{\psi_{I}^{[IH]}}$ superposition. An answer similar to the skewness (38) can be used as an estimator of $p_{l}$ being low/high:

[TABLE]

The ${\cal F}_{l}$ is proportional to the difference between $p_{l}$ and $\frac{1}{2}\left[\lambda_{p^{*}}^{[0]}+\lambda_{p^{*}}^{[1]}\right]$ is similar to Eq. (LABEL:Mu-skewnesslikeS0) of Ref. Malyshkin (2017). This is an approach generalizing tick and moments approaches. However, now the (54) is no longer $[-1:1]$ bounded (it would be if one replaces $p_{l}$ by \Braket{\psi_{0}}{pI}{\psi_{0}}\Big{/}\Braket{\psi_{0}}{I}{\psi_{0}}). A moments–only answer (without the last price used explicitly) can be also obtained:

[TABLE]

The sign of (55) is determined by which one of (52) eigenfunctions $\phi^{[0,1]}(x)$ is greater at $x_{0}$, (LABEL:Mu-projdiff) distance from Ref. Malyshkin (2017). The directional factor $\frac{\left[\phi^{[1]}(x_{0})\right]^{2}-\left[\phi^{[0]}(x_{0})\right]^{2}}{\left[\phi^{[1]}(x_{0})\right]^{2}+\left[\phi^{[0]}(x_{0})\right]^{2}}$ can be considered as probability correlation (Appendix LABEL:Mu-twoPvarcorrela of Ref. Malyshkin (2017)) between price and ‘‘distance to now’’. In (53) and (55) the scale factor $\left[\lambda_{p^{*}}^{[1]}-\lambda_{p^{*}}^{[0]}\right]$ by $D_{p}$ is replaced by more general form $z$, what makes the scalp–price to preserve the singularities for a variety of $D_{p}$ used.

As we have discussed in Malyshkin and Bakhramov (2015, 2016), price and price changes are secondary to execution flow and cannot be used to determine market direction for the reason of insufficient information. The main idea behind the scalp–moments is to replace in the sum (24)

[TABLE]

where ${\cal F}_{l}$ contains not only price changes, but also execution flow information. A good ${\cal F}_{l}$ selection allows us to accumulate much more directional information in the scalp–moments $\Braket{Q_{m}{\cal F}}$ compared to the information in the regular moments $\Braket{Q_{m}\frac{dp}{dt}}$. If one sum all the ${\cal F}_{l}$ terms, the ${\cal P}$, a generalized price can be obtained (59). The ${\cal P}$ is defined within a constant (it is convenient to take the last ‘‘price’’ ${\cal P}^{last}$ equals to zero). The transition from price $p$ to the scalp–price ${\cal P}$ makes all directional singularities expressed much more clearly. The directional information (25a) now take the (60) form, that is identical to (23), but instead of price $p$ the scalp–price ${\cal P}$ is used. If a trader wants to watch the prices — he should be watching the scalp–price ${\cal P}$, a much more informative characteristic in terms or market trend, than the regular price $p$.

IV.1 A Demonstration of Scalp–Price ${\cal P}$ Behavior

Before we go any further, let us demonstrate scalp–price (59) ${\cal P}(t)$ for a given ${\cal F}_{l}$. The results with ${\cal F}_{l}$ from (31), (32), and (34a) are very similar to each other, so we present only the scalp–price calculated from (32) terms; \seqsplit.dp_to_use=F_dpdt0_SCALP in \seqsplitScalpedMaxIProjection.java. The regular price is a sum of all price changes (26), the scalp–price is a sum of relevant to market dynamics (high $I$) price changes (59). In Fig. 2 regular and scalp–price are presented. One can clearly see, that while the regular price has an erratic behavior due to whatever market moves, the scalp–price ${\cal P}$ has a more regular type of behavior. If scalp–price changes it’s trend — the trend actually changes. The scalp–price ${\cal P}$ (59) is defined within a constant, and it is typically not a good idea to compare regular and scalp–price. However, if one takes an event in the past, where the price is equal to the last price, the change in the scalp–price gives marker direction, i.e. instead of comparing price and scalp–price, one needs to identify a situation of zero price change, then scalp–price change gives market directional information. From a market practitioner’s perspective, plain observation of the scalp–price is a good source of directional information. As we have discussed above, a typical price behavior after liquidity excess (high $I$) event is to bounce a little, then go in the original direction of the market. This gives a risk of on what to bet: ‘‘little bounce’’ or ‘‘follow the market’’. The ${\cal P}$, obtained from (32) ${\cal F}_{l}$, has no ‘‘little bounce’’ contributions; watching the ${\cal P}$ is actually watching pure market trend. If the price moves, and the scalp–price stays — this typically indicates a bear market rally or a bull market sell–off. The ${\cal P}$ is an integral attribute. The ${\cal F}=d{\cal P}/dt$ is a local attribute. One can try the

[TABLE]

attribute (not show in Fig. 2, see \seqsplit.F_IH field of \seqsplitScalpedMaxIProjection.java output), but the result is worse compared to the ${\cal P}$ result, no clear bear/bull market switch is observed. The situation is similar to the one in Fig. LABEL:Mu-figdpdtdirect of Ref. Malyshkin (2017): $dp/dt$ chart in the $\Ket{\psi_{I}^{[IH]}}$ state.

IV.2 A Demonstration of the Directional Information

The directional information should be accumulated over an interval of a substantial duration for the reason of low information available in a single price change. However, the strategies as the last price minus the average will never work for the reasons of fixed time scale of price averaging. In Malyshkin and Bakhramov (2015), the time–scale of the state of maximal past $I$, the $\Ket{\psi_{I}^{[IH]}}$, was introduced and the (23) answer was obtained. In this paper, the next critically–important step is made: instead of regular price $p$, the scalp–price ${\cal P}$ (it includes only high $I$ events: only relevant to market dynamics price moves) is introduced and the (60) answer is obtained. In the Fig. 3 (bottom) two directional answers are presented. In the top chart moving average $\frac{\Braket{p}}{\Braket{1}}$ and $p_{t}^{[IH]}$ are presented. In the bottom chart the (23) (\seqsplit.dp_to_use=F_SAMPLE_DP_NOSCALP), and (60) (\seqsplit.dp_to_use=F_dpdt0_SCALP), they are normalized to the same integral taken with all ${\cal F}_{l}$ positive in (23) and (60). One can clearly see that:

• •

When divided by the absolute variation, the non—scalped answer (23) is pretty small, and the scalped one (60) is much larger. This means that the price can be moved due to a variety of reasons, and only scalped price changes (27) are relevant to the market dynamics. Moreover, high $I$ market moves are much more consistent.

• •

Look at $t\in[9.9\dots 9.95]h$ interval. The price bounce around $p_{t}^{[IH]}$, what make it difficult to trade the direction as $P^{last}-p_{t}^{[IH]}$. In the same time the scalp–price (60) stays in the same sign, the scalp function ${\cal S}(t)$ (30) is about zero in this interval, see Fig. 2

• •

Look around $t=10h$. The scalped answers captured all the relevant price changes and switched from bear to bull market. The execution flow $I$ defines market sentiment.

IV.3 On the Selection of ${\cal F}_{l}$

The selection of ${\cal F}_{l}$ to be summed to the scalp price ${\cal P}$ (59) is the most important question for directional attribute selection. Consider two choices. In Fig. 4 top ${\cal F}_{l}=\Braket{\psi_{0}}{dp/dt}{\psi_{0}}$ from (32) is presented. One can see that $dp/dt$ (green) has rather erratic behavior, that is caused by a variety of market moves, the sum of these moves gives the regular price $p$. But when each price move is multiplied by the scalp–function ${\cal S}=\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$ from (30), this selects only high $I$ market moves, what makes the directional behavior much more clear (pink), the sum now gives the scalp price $\cal P$ from the Fig. 2. But even in this simplistic case the scalp–price selects only ‘‘relevant’’ market moves.

A much more interesting behavior can be observed with ${\cal F}_{l}$ from (55). Selecting $\Ket{\psi_{0}}$, $\Ket{\psi_{I}^{[IH]}}$ basis, solving (52), then obtaining (55) — this accumulates much more directional information in ${\cal F}_{l}$. When $\Ket{\psi_{I}^{[IH]}}$ and $\Ket{\psi_{0}}$ are not close to each other, the (55) for $z=\lambda_{p^{*}}^{[1]}-\lambda_{p^{*}}^{[0]}$ is approximately equals to last price and $p^{[IH]}$ difference multiplied by the scalp–function (that is close to zero). When $\Ket{\psi_{I}^{[IH]}}$ and $\Ket{\psi_{0}}$ are close to each other the (55) does not vanish777In this case ${\cal F}_{l}$ from (40) is almost zero, but ${\cal F}_{l}$ from (33) does not vanish, while providing a much smaller response than (55), i.e. the (55) does not vanish (like $p-p^{[IH]}$, yellow) when $I_{0}$ is extremely high. However, the ${\cal F}_{l}$ enters into the integral (60), and the selection of the $z$ is a non–trivial task. The most important feature of the charts in Fig. 4 is that once we got a spike in the $\Ket{\psi_{I}^{[IH]}}$ state — the trend is going to continue. These spikes are much greater in values (because of non–local price difference) compared to local price difference $p(t_{l})-p(t_{l-1})$ of Eq. (31). This allows to collect much more directional information, than can can be typically obtained from price changes.

We can generalize this non–local price change approach. Consider $p_{t}^{[IH]}$ in Fig. 3 ($p^{[IH]}$ is very close to it). A typical behavior for $p^{[IH]}$ is to jump from some past value to last price when the execution flow $I_{0}$ becomes large, (30) is the criteria of $I_{0}$ largeness. How often these jumps occur is the criteria to determine market direction, see Fig. 4. These non–local structural changes in $\Ket{\psi_{I}^{[IH]}}$ can be included to scalp–price, for every tick $l$ calculate:

[TABLE]

These are just Eq. (21) solution performed for every observation tick $l$ using $m=1\dots l$ previous ticks as input data. This is the result we had obtained back in Ref. Malyshkin and Bakhramov (2015). The new idea is to consider the $p^{[IH]}(t_{l})$ as if it were the last price $p(t_{l})$. This way one tick price change becomes non–local:

[TABLE]

Depending on the execution flow, the $\Ket{\psi_{I}^{[IH]}}$ may (or may not) change drastically at every tick. One tick non–local difference $p^{[IH]}(t_{l})-p^{[IH]}(t_{l-1})$ can be much greater than one tick local price difference $p(t_{l})-p(t_{l-1})$, see Fig. 3. As we discussed in the introduction only low $I$ $\to$ high $I$ to be considered:

[TABLE]

With a number of possible options for $z$:

[TABLE]

This is the ${\cal F}_{l}$ to be used in (60). The (65) considers every low $I$ $\to$ high $I$ jump in $p^{[IH]}(t_{l})$ (not in $p(t_{l})$) as the source of the directional information.

In Fig. 5 a demonstration of non–local price change (63) is presented. Only $p^{[IH]}(t_{l})-p^{[IH]}(t_{l-1})$ with positive $\lambda_{I}^{[IH]}(t_{l})-\lambda_{I}^{[IH]}(t_{l-1})$ are presented (the Eq. (65) with $z=1$ and $dt=1$). On can clearly see that the non–local directional information is:

• •

Much greater than the local price change $p(t_{l})-p(t_{l-1})$.

• •

The bull/bear market trend switch can be much better identified. The $p^{[IH]}(t_{l})-p^{[IH]}(t_{l-1})$ with (64) constaint preserves the sign during extended intervals.

• •

The ‘‘bounce back’’ interval $t\in[9.9\dots 9.95]h$ is clearly identified: it has no $I$ spikes, the $\Ket{\psi_{I}^{[IH]}}$ does not change, and the $p^{[IH]}(t_{l})-p^{[IH]}(t_{l-1})$ is close to zero even without ${\cal S}$ multiplied!

This makes us to conclude, that non–local price change (63) taken with the constraint (64) provides a very promising possible directional indicator. Fig. 5 presents a non–local price answer (63) obtained from one tick $p^{[IH]}$ price change $p^{[IH]}(t_{l})-p^{[IH]}(t_{l-1})$ as it were one tick regular price change $p(t_{l})-p(t_{l-1})$. This answer is similar (but much better) than Fig. 4 (bottom) answer, that is obtained from the regular moments by solving (52) $d=2$ eigenvalue problem. The (65) is the directional indicator. However, because it enters the integral (59), the selection of proper integration weight $z$ is required. This to be a subject of a separate study. In the simplest form a non–local answer can be obtained from (62) solution of (21) problem, then consider:

• •

Only $\lambda_{I}^{[IH]}(t_{l})\geq\lambda_{I}^{[IH]}(t_{l-1})$ events: $\theta_{I+}(t_{l})>0$, Eq. (64), field (74t).

• •

For such events consider $p^{[IH]}(t_{l})-p^{[IH]}(t_{l-1})$ as it were one tick price change $p(t_{l})-p(t_{l-1})$, Eq. (63), field (74u). In Fig. 5 an example of such a non–local price changes is presented.

V On the Directional Information Calculation

Given very interesting results of the previous section, let us formulate all the components, required to obtain directional information from (time, execution price, shares traded) market observations triples, and how these components can be improved.

• •

The state important for market dynamics. The answer we have is (6). Other states (such as considered in the Appendix C) can be also tried. In any case such a state is obtained from regular moments (13a) and (13b), solving some kind of $I\xrightarrow[\psi]{\quad}\max$ problem. The solution gives us open/close position signals and the scale for directional calculations.

• •

The problem to obtain the direction is way more complex, it requires scalp–moments (60). For the scalp–function ${\cal S}$ the bestBobyl et al. (2018) answer is (30). For ${\cal F}_{l}$ several answers (31), (32), (33), and (34a) produce good results, that are very similar to each other, the non–local answer (65) is of special interest. The ‘‘varied’’ answers (34b), and (34c) are worse with and without scalp–function multiplied. The simplest practical abswer is the scalp–moments directional answer (60), as a scale one can use absolute variation: take all ${\cal F}_{l}$ positive in (60). However, a number of non–local answers of (65) type can be obtained utilizing (62) and (63).

VI Speculations

The scalp–moments are price change moments filtered by high $I$ events: $I$ is the driving force of the market. The question arises whether a directional information can be obtained from the regular moments (13)? We are inclined to say no. A number of constrained (see Appendices F and G below) and unconstrained optimization problems have been tried (among many others) without any success at obtaining market directional information:

[TABLE]

The regular moments answers are: 1. not ‘‘sufficiently sharp’’, see Appendix A, and 2. price changes sum is small relatively total variation, see Fig. 3. In the same time, when we go to the scalp–moments (28) these problems get solved.

When, in September 1997, I joined Columbus Advisors LLC (Greenwich CT), the fund had been doing Emerging Market sovereign fixed income convergence–divergence relative value spread trades. The following year, I studied a classic technical analysis book with the goal to program some of the rules algorithmically. However, I was not able to program even a single rule from the book. The reason was simple: any rule required a time scale to apply. Time scale selection is the main criterion separating good traders from bad, and the criterion which defines a trader’s talent. The state (6) is an algorithmic criterion, that automatically determines the time scale. This criterion is actually very simple ideologically: look back to find an event of trading with maximal $I$. The time between this event and ‘‘now’’ is the time scale. The typical market practitioner’s activity is to watch the difference between the last price and moving average calculated on the time scale obtained his feel. With a proper time scale, any strategy (like return to the moving average) would work, and Ref. Malyshkin and Bakhramov (2015) answer of last price minus $p^{[IH]}$ (7) was my first successful attempt.

Besides the time scale, the most important result of this paper is that ‘‘not all price moves are equal’’. We need to select only the high $I$ price moves888 I think that the market impact concept is a dead end. . High execution rate requirement is the condition creating an asymmetry to separate the ‘‘bounce a little, then to go in the original direction of the market’’ and ‘‘go in the original direction of the market straight away’’ scenarios, such as to identify a bear market rally on steroids. The answer we obtained is the scalp–price (59). It does not have any ‘‘internal averaging’’, but in the same time it has all low $I$ price changes removed! This way, the scalp–price has no ‘‘bounce a little’’ behavior. Only hardcore. Only directional. See the Fig. 2. The software is availableMalyshkin (2014) under the GPLv3 license.

Acknowledgements.

Vladislav Malyshkin would like to thank Aleksandr Bobyl from the Ioffe Institute for many discussions of whether the activity of a typical market participant can be reduced to some ‘‘simplified automata’’ with intelligence as an aggregate function of past transactions, and, even more importantly, the ability (or inability) to apply the obtained knowledge to market activity; Emilio J. Lamar for discussions of slow/fast market transitions — especially the spread–widening effect during fast markets; and Aleksandr Maslov from the Ioffe Institute for discussions on contradistinction of operators $\|dV/dt\|$ and $\|V/t\|$.

Appendix A A demonstration of the difference between

time and volume weighted price.

To demonstrate the difference consider localized at $x=y$ the wavefunction $\psi_{y}(x)$ (68), producing Radon–Nikodym interpolating answer, Eq. (7) of Ref. Bobyl et al. (2016), Different attributes (price, execution flow, etc.) are interpolated using the $\psi^{2}_{y}(x)\omega(x)dx$ weight:

[TABLE]

One can see that:

• •

For a large $n$ (we use $n=12$) the $p_{t}$ and $p_{v}$ are very similar.

• •

The projection $\Braket{\psi_{y}(x)}{\psi_{I}^{[IH]}}^{2}$ is close to $1$ for large $I$, i.e. the $\psi_{I}^{[IH]}(x)$ is typically a localized function, this is not the case for other states. See four other eigenstates projections (yellow).

• •

The $p_{v}-p_{t}$ changes the sign at large $I$. Only the states with a large $dI/dt$ provide weight asymmetry required to obtain directional information using $dV$ vs. $dt$ averaging.

Appendix B Computer Implementation

The codebase architecture is described in the Appendix LABEL:Mu-codestr of Ref. Malyshkin (2017). Relevant to this paper functionality consists of:

• •

Conversion of the transaction sequence of an observable $f$ to a vector of moments $\Braket{fQ_{m}}$, $m=[0\dots 2n-2]$, several bases $Q_{k}(x)$ are implemented ($x=t$, $x=\exp\left(-(t_{now}-t)/{\tau}\right)$, and $x=p(t)$, see the Section LABEL:Mu-basis of Ref. Malyshkin (2017)), the integration measure is always exponential decay: $d\mu=\exp\left(-(t_{now}-t)/{\tau}\right)dt$. See the classes \seqsplitcom/polytechnik/trading/{QVMDataL,QVMDataP,QVMData}.java

• •

Using basis functions multiplication operator (Eq. (LABEL:Mu-cmul) of Ref. Malyshkin (2017)), obtain the $\Braket{Q_{j}}{f}{Q_{k}}$, $j,k=[0\dots n-1]$ matrix from the moments $\Braket{fQ_{m}}$, $m=[0\dots 2n-2]$.

• •

There are a number of observables $f$ possibly to consider (price, price change, execution flow, etc.). Depending on the approach used, a different set of observables is required. All the $\Braket{Q_{j}}{f}{Q_{k}}$ matrices we possibly use in this paper are stored in the class \seqsplitcom/polytechnik/trading/SMomentsData.java.

• •

If/When, in addition to a $\Braket{Q_{j}}{f}{Q_{k}}$ matrix, the matrix corresponding to the derivative $df/dt$ (or to the integral $\int^{t}f(t^{\prime})dt^{\prime}$), is required, then, for a basis with infinitesimal time–shift operator $D(Q(x))$, the result can be obtained using integration by parts, see Appendices D and E.

As a result of these preliminary steps the $n\times n$ matrices are obtained: $\Braket{Q_{j}}{Q_{k}}$, $\Braket{Q_{j}}{\frac{dp}{dt}}{Q_{k}}$, $\Braket{Q_{j}}{p}{Q_{k}}$, $\Braket{Q_{j}}{p^{2}}{Q_{k}}$, $\Braket{Q_{j}}{p^{3}}{Q_{k}}$, $\Braket{Q_{j}}{I}{Q_{k}}$, $\Braket{Q_{j}}{pI}{Q_{k}}$, $\Braket{Q_{j}}{p^{2}I}{Q_{k}}$, and $\Braket{Q_{j}}{p^{3}I}{Q_{k}}$. These are plain exponential moving–average of: an observable $f$ multiplied by two basis functions product $Q_{j}(x)Q_{k}(x)$; for example if $f=p$, then $\Braket{Q_{0}}{p}{Q_{0}}$ is exponential moving average of price.

B.1 The EVXData.java implementation

The class \seqsplitcom/polytechnik/utils/EVXData.java takes two matrices $\Braket{Q_{j}}{f}{Q_{k}}$, $\Braket{Q_{j}}{Q_{k}}$ and basis functions operations class (extending the \seqsplitcom/polytechnik/utils/OrthogonalPolynomialsABasis.java), solves generalized eigenvalue problem, such as (21) for $\|f\|=\|I\|$, and stores the result. The fields are:

[TABLE]

The squares $\mathtt{.wL}^{2}$ and $\mathtt{.wH}^{2}$ are bounded to $[0:1]$, and are very good indicators of whether the $I$ ‘‘now’’, the $I_{0}=\Braket{\psi_{0}}{I}{\psi_{0}}$, is large or small. Alternative estimator as the number of the eigenvalues above the $I_{0}$ can also be usedBobyl et al. (2018). The key concept of liquidity deficit tradingMalyshkin and Bakhramov (2015, 2016) is to open a position at low $I_{0}$, large $\Braket{\psi_{I}^{[IL]}}{\psi_{0}}^{2}$, then to close already opened position at high $I_{0}$, large $\Braket{\psi_{I}^{[IH]}}{\psi_{0}}^{2}$, the $\mathtt{.wL}^{2}$ and $\mathtt{.wH}^{2}$ are the indicators of these actions. The question is: whether to open a long or a short position at high $\mathtt{.wL}^{2}$?

B.2 The ScalpedMaxIProjection.java implementation

The class \seqsplitcom/polytechnik/trading/ScalpedMaxIProjection.java converts a transaction sequences to a set of $\Braket{fQ_{m}}$ vectors, then to a set of $\Braket{Q_{j}}{f}{Q_{k}}$ matrices, stored in the object of \seqsplitcom/polytechnik/trading/SMomentsData.java type. Then it calls the \seqsplitcom/polytechnik/utils/EVXData.java class that solves (21) eigenvalue problem. Having the $\Ket{\psi_{I}^{[IH]}}$ and $\Ket{\psi_{0}}$ states the scalp–function (30) and the ${\cal F}_{l}$ are obtained. Which one ${\cal F}_{l}$ to be used depends on the parameter \seqsplit.dp_to_use. The values \seqsplitF_SAMPLE_DP_NOSCALP,F_SAMPLE_DP_SCALP,F_dpdt0_SCALP,F_varpIH_0_divI_SCALP,F_SKEWNESS_at_Pl_SCALP,F_PROBABILITYCORRELATION_SCALP correspond to (26), (31), (32), (39), (53), and (55) respectively; there are several other options for \seqsplit.dp_to_use. The class \seqsplitScalpedMaxIProjection is assumed to be called on every tick, and the internal state is preserved in the object of \seqsplitcom/polytechnik/trading/StateWIScalpMomentsSaver.java class. The internal state contains an object of \seqsplitcom/polytechnik/trading/WIntegrator.java type, that calculates the moments of the observable ${\cal F}_{l}$ (recurrent shift of the basis offset ($t_{now}$) for previously calculated moments allows the calculations to be performed extremely fast). The \seqsplitWIntegrator is called on every tick with the $\texttt{\seqsplit{.Fdt}}=\left(t_{l}-t_{l-1}\right){\cal F}_{l}$ (the choice of the ${\cal F}_{l}$ depends on the \seqsplit.dp_to_use value) to accumulate scalped data. The scalp–moments are obtained by taking the \seqsplit.Fdt instead of the $p(t_{l})-p(t_{l-1})$ when calculating the (25b) sum. The directional information is then obtained as (25a). The fields are:

[TABLE]

The liquidity deficit indicator (74d) defines whether to open or to close a position. The directional indicator (74v) from (25a) defines, when opening a position, whether to open a long or a short.

B.3 The CallAMuseOfCashFlowAndLiquidityDeficitWithScalp.java implementation

The class \seqsplitcom/polytechnik/algorithms/CallAMuseOfCashFlowAndLiquidityDeficitWithScalp.java is ‘‘an interface’’ between transactions sequence input (a tab–separated file), liquidity deficit trading of the class \seqsplitcom/polytechnik/trading/ScalpedMaxIProjection.java, and data output, saved as a tab–separated file. The parameters are read by the class \seqsplitcom/polytechnik/algorithms/MuseConfig.java. This is an example of how to run the code:

java com/polytechnik/algorithms/CallAMuseOfCashFlowAndLiquidityDeficitWithScalp
--musein_file=aapl.csv
--musein_cols=15:1:4:5
--museout_file=museout.dat
--n=12
--tau=128
--measure=ScalpedMaxIProjectionLegendreShifted

The parameters are:

• •

\seqsplit

--musein_file=aapl.csv : Specify input tab–separated file with (time, execution price, shares traded) triples time series. If the file is gzip–ed and has the .gz extension, then internal decompression is performed.

• •

\seqsplit

--musein_cols=15:1:4:5 : Out of total 15 columns in the specified \seqsplit--musein_file=aapl.csv file, take the column #1 as time (nanoseconds since midnight), #4 (execution price), and #5 (shares traded), column index is base 0.

• •

\seqsplit

--museout_file=museout.dat : Output file name.

• •

\seqsplit

--n=12 : Basis dimension. Typical values are: $n\in[4\dots 16]$. The $m\in[0\dots 2n-2]$ moments (in $Q_{m}(x)$ basis) are calculated to obtain $n\times n$ matrices.

• •

\seqsplit

--tau=128 : Exponent time (in seconds) for the measure used.

• •

\seqsplit

--measure=ScalpedMaxIProjectionLegendreShifted : The measure. Possible values are: \seqsplit{ScalpedMaxIProjectionLegendreShifted,ScalpedMaxIProjectionLaguerre,ScalpedMaxIProjectionMonomials}, they correspond to the measures (LABEL:Mu-muslegendre) and (LABEL:Mu-muflaguerre) of Ref. Malyshkin (2017). The \seqsplitScalpedMaxIProjectionLaguerre and \seqsplitScalpedMaxIProjectionMonomials use the same measure, but different basis $Q_{k}(x)=L_{k}(x)$, $x=-t/\tau$ and $Q_{k}(x)=x^{k}$, $x=t/\tau$ respectively. These two results should be identical, as the measure is the same, and all the calculations are $Q_{k}(x)$–basis invariant (but the numerical stability can be drastically different).

Output file is a tab–separated file with the columns (35 columns total), corresponding to the results of this paper. Field names are printed in the first line of the output file. The data can be processed by any common plotting software (such as gnuplot or matlab). Below is the description of the most noticeable fields:

• •

T : Time in nanoseconds since midnight (copied from input).

• •

shares : Shares traded (copied from input).

• •

P_last : Last execution price (copied from input).

• •

\seqsplit

{pi_average,pt_average} : Regular exponential moving average of price with the given \seqsplit--tau=128, using volume/time as the weight.

• •

\seqsplit

I.{s0,sL,wL_squared,sH,wH_squared,Gamma0} : Correspond to (73) fields of $\Ket{I|\psi}=\lambda\Ket{\psi}$ eigenvalue problem (21), the solution with the given \seqsplit--n=12; the \seqsplitI.wL and \seqsplitI.wH are squared in the output, \mathtt{Gamma0}=\left(2I_{0}-\lambda_{I}^{[IL]}-\lambda_{I}^{[IH]}\right)\Big{/}\left(\lambda_{I}^{[IL]}-\lambda_{I}^{[IH]}\right) is the $\widetilde{\Gamma^{0}}$ skewness of $I$, Eq. (LABEL:Mu-skewnesslikeS0) of Ref. Malyshkin (2017). The \seqsplitI.wH_squared is the scalp–function ${\cal S}(t)$ (30).

• •

\seqsplit

{p_IH,pt_IH,pV_IH,pT_IH} Correspond to (9) prices, calculated in the state $\Ket{\psi_{I}^{[IH]}}$ (6), the (74) fields.

• •

\seqsplit

getFlFromRegularMoments() The ${\cal F}_{l}$ when it is calculated from regular moments, \seqsplitNaN otherwise, the field (74r).

• •

\seqsplit

getSumFdt() The scalp–price ${\cal P}$ (59), corresponding to given \seqsplitdp_to_use, the field (74s). See the Fig. 7 to compare the results for \seqsplit.dp_to_use=F_dpdt0_SCALP (32) and \seqsplit.dp_to_use=F_SAMPLE_DP_SCALP (31).

• •

\seqsplit

dIH,dp_IH The $\lambda_{I}^{[IH]}$ and $p^{[IH]}$ change per tick, the fields (74t) and (74u). This is the starting point of non–local price change (63) study, Fig. 5.

• •

\seqsplit

{DIR,DIRa} and etc. Correspond to (74) fields of an object of \seqsplitScalpedMaxIProjection.java type.

B.4 Installation and usage example

• •

Install java 1.8 or later.

• •

Download from Malyshkin (2014) the archive \seqsplitAMuseOfCashFlowAndLiquidityDeficit.zip with the source code.

• •

Decompress and recompile the program:

unzip AMuseOfCashFlowAndLiquidityDeficit.zip javac -g com/polytechnik/*/*java

• •

Run the test with the bundled file \seqsplitdataexamples/aapl_old.csv.gz data of Ref. Malyshkin (2016). The file contains only execution events, the (time, execution price, shares traded) market observations triples are in the 1:2:3 columns, column index is base 0; 28492 lines, 9 columns total.

java com/polytechnik/algorithms/CallAMuseOfCashFlowAndLiquidityDeficitWithScalp
--musein_file=dataexamples/aapl_old.csv.gz
--musein_cols=9:1:2:3
--museout_file=museout.dat
--n=12
--tau=128
--measure=ScalpedMaxIProjectionLegendreShifted

The code is run under 16 seconds, the output fields of the \seqsplitmuseout.dat are described in the Appendix B.3. The \seqsplitI.wH_squared, \seqsplitgetSumFdt(), and \seqsplitp_IH are the scalp–function (30), scalp–price (59) (has an arbitrary offset), and $p^{[IH]}$ from (7). The default \seqsplit.dp_to_use=F_PROBABILITYCORRELATION_SCALP corresponds to (55).

• •

Download NASDAQ ITCH data file \seqsplitS092012-v41.txt.gz from Malyshkin (2014), extract triples (time, execution price, shares traded) from NASDAQ ITCH data file:

java com/polytechnik/itch/DumpData2Trader
S092012-v41.txt.gz AAPL >aapl.csv

Execution data and limit order book edges are now saved to tab–separated file aapl.csv of 15 columns. The (time, execution price, shares traded) market observations triples are in the 1:4:5 columns, column index is base 0; 634205 lines, 15 columns total.

• •

Run the java command of the Appendix B.3 to obtain the \seqsplitmuseout.dat file of 634206 lines with: scalp–function (30), scalp–price (59) and $p^{[IH]}$ from (7) and \seqsplitcom/polytechnik/trading/ScalpedMaxIProjection.java fields is created. The code is run under 5 minutes, much longer than that of previous run. The \seqsplit--musein_file=aapl.csv input file now contains much more events than the file \seqsplit--musein_file=dataexamples/aapl_old.csv.gz.

Appendix C The state of maximal aggregated execution flow $V/t$

In our previous workMalyshkin and Bakhramov (2015); Malyshkin (2017) the extremal state of $I=dV/dt$ operator have been considered. This answer has two critically important features:

• •

Uses execution flow $I$, as it is the driving force of the market.

• •

Has automatic time–scale selection (eigenvalue problem), huge advantage compared to any fixed time scale approachMalyshkin and Bakhramov (2015).

While this result is very promising, it has an issue of zero first variation of $I$. Consider the same approach, but with the operator $V/t$. Here $V$ and $t$ are measured since $t_{now}$, they are volume/time between $t$ and $t_{now}$. The $V/t$ is aggregated execution flow, the $dV/dt$ is local execution flow. Put $f=V/t$ into (2) and obtain generalized eigenvalue problem to find the state $\Ket{\psi_{V/t}^{[\max]}}$ of maximal $\lambda_{V/t}^{[\max]}$:

[TABLE]

The calculation of $\Braket{Q_{j}}{V}{Q_{k}}$ and $\Braket{Q_{j}}{t}{Q_{k}}$ matrix elements is described in the Appendix D. In (76) the $V$ and $t$ have the sign changed to have positively defined right–hand–side matrix $\Braket{Q_{j}}{t}{Q_{k}}$, $V=V_{0}$ (8a), $t=T_{0}$ (8b). The multiplication by $V$ and $t$ create, for $t\leq t_{now}$, two Radau–like measures: $\left(V(t_{now})-V(t)\right)\omega(t)dt$ and $\left(t_{now}-t\right)\omega(t)dt$. The problem (76) finds the state $\psi_{V/t}^{[\max]}(x)$, corresponding to the maximal Radon–Nikodym derivative relatively two these measures, the maximal aggregated execution flow $V/t$. Previously Malyshkin and Bakhramov (2015) we have been considering the state $\psi_{I}^{[IH]}(x)$, corresponding to the maximal Radon–Nikodym derivative relatively the measures $\omega(t)dV$ and $\omega(t)dt$, the maximal local execution flow $dV/dt$. The eigenvectors $\Ket{\psi_{V/t}^{[i]}}$ of $\|V/t\|$ operator have the following remarkable features:

[TABLE]

Lemma**.**

In the state of maximal aggregated execution flow the $dI/dt$ is positive.

Proof.

In the state of maximal $V/t$ the second variation (106) is negative. Because the first $I$ variation (105) with $\delta\psi=D(\psi_{V/t}^{[i]})$ corresponds to $-dI/dt$, this provides positive $dI/dt$. ∎

This lemma makes the state $\Ket{\psi_{V/t}^{[\max]}}$ of maximal aggregated execution flow (the eigenvector of (76), corresponding to the maximal $\lambda_{V/t}^{[\max]}$), a very promising one for the market dynamics to consider. The ‘‘aggregated’’ attributes (8) have been originally introduced in the Section (LABEL:Mu-PsiAfterFutureI) ‘‘Measure: The Period After Maximal Future $I$’’ of Ref. Malyshkin (2017), but their application to skewness study was not a very successful back then.

Appendix D The calculation of the $\Braket{VQ_{m}}$

moments from the $\Braket{IQ_{m}}$ moments

The $\Braket{VQ_{m}}$ and $\Braket{tQ_{m}}$, $m=[0\dots 2n-2]$, moments, required to construct $\|V\|$ and $\|t\|$ operators in (75), can be calculated directly from the sample. However, in practical application it is more convenient to calculate the $\Braket{IQ_{m}}$ moments first, then to obtain the $\Braket{VQ_{m}}$ moments using an integration by parts. For Shifted Legendre and Laguerre bases the integration by parts gives:

[TABLE]

where $J(\cdot)$ is a polynomial to polynomial transforming function (10). The $\Braket{VQ_{m}}$ then can be expressed as $\Braket{IQ_{s}}$, $s=[0\dots m]$, linear combination. This is possible only for the bases in question, in general case an integration by parts $\int_{-\infty}^{t}Q_{m}(x(t^{\prime}))\omega(t^{\prime})dt^{\prime}$ cannot be reduced to a $J(Q_{m}(x(t)))\omega(t)$ form, and the $\Braket{VQ_{m}}$ moments cannot be expressed via a linear combination of the $\Braket{IQ_{s}}$ moments.

The boundary condition is straightforward, consider $V(t)-V(t_{now})$, that is zero at $t=t_{now}$. Use current volume $V(t_{now})$ as the starting value, then out–of–integral term in (82) vanish, and past/future volume correspond to negative/positive volume values999 It is sometimes convenient to change the sign of time and volume $V(t)-V(t_{now})$ as in (8), then past time and volume correspond to positive values and the right hand side matrix in (75) is positively defined. . See the method \seqsplitsetFMoments of \seqsplitcom/polytechnik/trading/{QVMDataLDirectAccess,QVMDataPDirectAccess,QVMDataDirectAccess}.java, that calculates the $\Braket{VQ_{m}}$ moments as a linear combination of the $\Braket{IQ_{s}}$, $s=[0\dots m]$ moments.

Appendix E The calculation of the $\|dI/dt\|$

operator matrix elements from the $\|I\|$ operator.

When we study an operators of execution rate change $\|dI/dt\|$, it’s matrix elements cannot be calculated directly from sample. In general case the $\Braket{\frac{dI}{dt}Q_{m}}$ moments can be calculated from $\Braket{IQ_{m}}$ moments using integration by parts (82) of the Appendix D, see the method \seqsplitsetDFMoments of \seqsplitcom/polytechnik/trading/{QVMDataLDirectAccess,QVMDataPDirectAccess,QVMDataDirectAccess}.java, that, for zero boundary condition, obtains $\Braket{\frac{dI}{dt}Q_{m}}$ as a linear combination of $\Braket{IQ_{s}}$, $s=[0\dots m]$. However, for $\|dI/dt\|$, the boundary condition may take a variety of forms, and direct operator approach is often more convenient. Consider e.g. generalized eigenvalue problem (2) for $\|dI/dt\|$ operator:

[TABLE]

where the \Braket{Q_{j}\left missing}{\frac{dI}{dt}}{Q_{k}} matrix cannot be calculated directly from sample. For a $Q_{k}(x)$ basis with infinitesimal time–shift operator $D(Q_{k}(x))$, the matrix can be obtained from the \Braket{Q_{j}\left missing}{I}{Q_{k}} matrix using integration by parts101010 See java classes for Shifted Legendre and Laguerre $Q_{k}(x)$ bases implementation of infinitesimal time–shift operator $D(Q_{k}(x))$: the method \seqsplitgetEDPsi of \seqsplitcom/polytechnik/trading/{WIntegratorLegendreShifted,WIntegratorLaguerre,WIntegratorMonomials}.java. Also see the \seqsplitcom/polytechnik/trading/QQdidtMatrix.java class, implementing the calculation of (84) matrix for the (85), (86), (87), and (88), boundary conditions. This class uses \seqsplitcom/polytechnik/utils/VolMatrix.java to calculate $\Braket{D(Q_{j})}{I}{Q_{k}}+\Braket{Q_{j}}{I}{D(Q_{k})}$, then adds boundary condition term $I^{f}Q_{j}(x_{0})Q_{k}(x_{0})$. , Eq. (LABEL:Mu-dIMatr) of Ref. Malyshkin (2017):

[TABLE]

This problem is an inverse one to considered in Appendix D, and requires a non–trivial boundary condition $I^{f}$. There are several options for $I^{f}$, that can be reasonably considered:

• •

The zero of $\|dI/dt\|$ in the $\Ket{\psi_{I}^{[IH]}}$ state, $\Braket{\psi_{I}^{[IH]}}{\frac{dI}{dt}}{\psi_{I}^{[IH]}}=0$:

[TABLE]

Regardless the $I^{f}$ selection, the $\|I\|$ and $\|dI/dt\|$ operators have no common eigenvectors unless the $\Ket{\psi_{0}}$ is the $\|I\|$ eigenvector, this degeneracy case was considered in Ref. Malyshkin (2017). The most critical degeneracy arise in the situation, when the state ‘‘now’’ and the state of ‘‘maximal past $I$’’ are the same:

[TABLE]

An example of such a degeneracy can be the situation of huge volume traded ‘‘now’’ (at $x=x_{0}$).

Appendix F Directional Information:

$I\xrightarrow[\psi]{\quad}\max$ Subject To the Constraint $\Braket{\psi}{C}{\psi}=0$.

Consider market dynamics split in two operators: $\|I\|$ (execution flow dynamics) and $\|C\|$ (price dynamics). The constrained $I\to\max$ problem is:

[TABLE]

The constraint (90b) is a requirement on price in the $\Ket{\psi}$ state. There are a number of choices for the constraint operator $\|C\|$ selection:

[TABLE]

The maximization problem (90a) with the quadratic constraint (90b) can no longer be reduced to a regular eigenvalue problem such as (21). The solution exists only if $\|C\|$ operator has both: positive and negative eigenvalues. Ideologically the (90b) constraint facilitate taking into account a typical market practitioner activity: look how the market used to behave in the past at prices near some level. Our previous paper Malyshkin (2017) has been mostly devoted to skewness and probability correlation study in the unconstrained $I\to\max$ state $\Ket{\psi_{I}^{[IH]}}$. The (90b) constraint allows us, within the framework of a single formalism of constrained optimization, take into account the driving force of the market $I\to\max$ (90a) and the reaction of the market participants on it (90b). For mathematical properties and numerical solution of (90) problem see Appendices F.1 and (F.2 below. Here we assume that the solution does exist, we denote it as $\Ket{\psi_{I}^{[{\cal M}]}}$, and name: the state of price–matching maximal execution flow. The found state $\Ket{\psi_{I}^{[{\cal M}]}}$ (it is just a pure state averaging weight $\left(\psi_{I}^{[{\cal M}]}(x(t))\right)^{2}\omega(t)dt$, not even a density matrix (4)) is the state to obtain market directional information.

F.1 The IstatesConditional.java implementation

The optimization problem (90a) with quadratic constraint (90b) can be solved using Lagrange multipliers technique:

[TABLE]

Were the constraint (92c) to be of a linear type, instead of a quadratic one, the constrained optimization problem (92a) can be reduced to a regular eigenvalue problem in a transformed basisGolub (1973). However, for the quadratic constraint (92c), such a one–step transform is not possible, and self–concordant procedure of iterational type is the simplest option:

• •

For an initial $\Ket{\psi}$ find the coefficient $\alpha$, such that:

[TABLE]

The (93b) is a quadratic equation with respect to $\alpha$, if no real solution exist — iterational process failed. If a success — obtain the solution, satisfying the (92c) constraint:

[TABLE]

From the two $\alpha$ solutions select the one with the maximal {\Braket{\widetilde{\psi}}{I}{\widetilde{\psi}}}\Big{/}{\Braket{\widetilde{\psi}}{\widetilde{\psi}}}. There are exist several good alternatives to (93), see \seqsplitcom/polytechnik/utils/IstatesConditional.java implementation for details.

• •

Put $\Ket{\widetilde{\psi}}$ to (92d), then left–multiply it by the vector \Bra{\widetilde{\psi}\Big{|}C}, obtain the Lagrange multiplier iteration $\mu$:

[TABLE]

• •

Construct an operator $\|{\cal I}\|$ and find all it’s eigenvectors:

[TABLE]

• •

Among all the $\Ket{\psi^{[i]}}$ found select the $\Ket{\psi}$, providing the maximal $\Braket{\psi}{I}{\psi}$.

• •

Repeat the process of above for this new $\Ket{\psi}$. If a solution exists, iterational procedure converges quickly (typically 5–7 iterations), unless $\|I\|$ and $\|C\|$ operators have several eigenvectors in common111111 Assume $\|I\|$ and $\|C\|$ operators have the identical eigenvectors. Then the (97) always produce the same eigenvectors, and the minimization problem (90) is reduced to a linear programming problem relatively the projections squares. . The result of this iterational process is the state of price–matching maximal execution flow $\Ket{\psi_{I}^{[{\cal M}]}}$, the (90) solution.

The class \seqsplitcom/polytechnik/utils/IstatesConditional.java implements this procedure. It takes three matrices $\Braket{Q_{j}}{Q_{k}}$, $\Braket{Q_{j}}{I}{Q_{k}}$, $\Braket{Q_{j}}{C}{Q_{k}}$, and basis functions operations class (extending the \seqsplitcom/polytechnik/utils/OrthogonalPolynomialsABasis.java), as constructor’s arguments. Then it solves generalized eigenvalue problem (21) using the \seqsplitEVXData.java class to obtain an initial $\Ket{\psi}$ and to reproduce the Malyshkin and Bakhramov (2015) results. Then ten iterations of above are performed to obtain the solution of (90): $\Ket{\psi_{I}^{[{\cal M}]}}$ and $\mu$. The fields are:

[TABLE]

F.2 The IstatesConditionalLocalized.java implementation

When the global maximum of constrained $I\to\max$ problem is not required, and localized answer with $\Ket{\psi}$ in (68) form is considered as good enough, optimization problem (90a) with quadratic constraint (90b) can be easily solved. Substitute (68) to (92c) and obtain:

[TABLE]

The (99b) constraint is a polynomial of $2n-2$ degree, it has exactly $2n-2$ root, possibly complex. The classes extending the \seqsplitcom/polytechnik/trading/OrthogonalPolynomialsABasis.java (see Appendix LABEL:Mu-codestr of Ref. Malyshkin (2017)) provide an implementation for solving $P(y)=0$ equation with a $P(y)$ in a given $Q_{k}(y)$ basis $P(y)=\sum_{m=0}^{2m-2}Q_{m}(y)$, the (99b) is a polynomial of this form. Among $2n-2$ roots found select only the real roots, then among them select the state $\Ket{\psi_{I}^{[{\cal M}]}}$, that provides the maximal $\Braket{\psi_{I}^{[{\cal M}]}}{I}{\psi_{I}^{[{\cal M}]}}$. The situation is similar to the one of Appendix G, below, with the difference that $\Ket{\psi_{I}^{[{\cal M}]}}$ is now selected among (68) states with $y$ from (99b) real roots ($2n-2$ maximal number), not among $n$ eigenvalues of some operator $\|{\cal C}\|$.

The class \seqsplitcom/polytechnik/utils/IstatesConditionalLocalized.java implements this procedure. It takes three matrices $\Braket{Q_{j}}{Q_{k}}$, $\Braket{Q_{j}}{I}{Q_{k}}$, $\Braket{Q_{j}}{C}{Q_{k}}$, and basis functions operations class (extending the \seqsplitcom/polytechnik/utils/OrthogonalPolynomialsABasis.java), as constructor’s arguments. Then it solves $P(y)=0$ polynomial roots problem (99b) using \seqsplitcom/polytechnik/trading/OrthogonalPolynomialsABasis:getPolynomialRootsFinderInBasis().findRoots($\cdot$) method to obtain a set of $y_{m}$ that are the roots of (99b). Then corresponding $\Ket{\psi_{y_{m}}}$ (68) are constructed, and the one with the maximal $\Braket{\psi_{y_{m}}}{I}{\psi_{y_{m}}}$ is selected: this is the ‘‘localized’’ $\Ket{\psi_{I}^{[{\cal M}]}}$ solution. The fields are:

[TABLE]

Appendix G Directional Information:

$I\xrightarrow[\psi]{\quad}\max$ in the States of Constraint Operator $\|{\cal C}\|$.

The constrained optimization of the Appendix F.1 above, while been very nice mathematically, does not provide a clear cut answer. There are two reasons: the difficulty to select an operator $\|C\|$ (91) and the difficulty with (97) Lagrange multiplier convergence, as $\|I\|$ and $\|C\|$ operators often have common eigenvectors. Consider a different, much more simplistic, constrained optimization approach:

[TABLE]

Here we also split the market dynamics in two operators: $\|I\|$ (execution flow dynamics) and $\|{\cal C}\|$ (price dynamics). But now we consider the $\|I\|$ only in the eigenstates of the operator $\|{\cal C}\|$. The operator $\|{\cal C}\|$ is selected in a way that it’s derivative gives the constraint operator $\|C\|$, thus the $\Ket{\psi}$ state of extremal $\|{\cal C}\|$ give zero of constraint operator $\|C\|$. Mathematically the problem (101) is simple: find all $n$ eigenvectors (101b) of $\|{\cal C}\|$ first, then select the one, providing the maximal $\|I\|$ (101a). The state of price–matching maximal execution flow $\Ket{\psi_{I}^{[{\cal M}]}}$ is now plain (101b) eigenvector, providing the maximal (101a). There are a number of choices for the operator $\|{\cal C}\|$, selecting the states $\Ket{\psi}$:

[TABLE]

The optimization with the constraint (102a) is actually the pure dynamic impact approximation of Ref. Malyshkin (2017): price and execution flow operators are assumed to have the same eigenvectors. The (102b) states, same as for the aggregated execution flow (80) below, selects the states with the moving average price equals the price, a typical market practitioner point of attention. The problem (101) uses the same input data moments (13) as the problem (90).

Appendix H The

$\Ket{\psi_{I}^{[IH]}}$ variation approach to positive and negative $dI/dt$ states separation.

The separation of the states with positive and negative $dI/dt$ can be developed based on $\Ket{\psi_{I}^{[IH]}}$ variation. For example, in the Eq. (LABEL:Mu-rqD1) of Ref. Malyshkin (2017), the variation of $I$ have been considered121212 See the class \seqsplitcom/polytechnik/utils/RayleighQuotient.java of provided software, implementing the calculation of 0-th, 1-st, and 2-nd variations of two quadratic forms ratio. :

[TABLE]

With $\delta\psi=-D(\psi_{I}^{[IH]}(x))$ variation (such a variation can be considered as a boundary condition alternative to (85), (86), (87), or (88)) obtain $\Delta_{\psi}P$ from the Eq. (LABEL:Mu-DPDirectRQ) of Ref. Malyshkin (2017). Any first variation (105) in a $\Ket{\psi_{I}^{[i]}}$ state is zero, any second variation (106) in the state $\Ket{\psi_{I}^{[IH]}}$ is negative. The first variation of the $\Ket{\psi_{I}^{[IH]}}$ state can be written as $P(x)$ polynomial average:

[TABLE]

In Malyshkin (2018), we have have proved, that any polynomial $P(x)$ of $2n-2$ degree can be isomorphly mapped to a linear operator of the dimension $n$, the algorithm is presented in the Appendix A of Ref. Malyshkin (2018):

[TABLE]

Then the $D1$ can be presented as a superposition of positive and negative terms:

[TABLE]

This way the $P(x)$ average can be split in positive and negative contributions. Despite being a $\Ket{\psi_{I}^{[IH]}}$ projection, the eigenvalues of (109) are typically all non–zero, and corresponding density matrix is a mixed state:

[TABLE]

For computer implementation see the class \seqsplitcom/polytechnik/trading/DIselDM.java of provided software.