market_profile_paper.tex

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\title{mBA-Profile: Market Profile Construction from Microsecond Bid-Ask Unit Data Using The Path-weighted  Gap-filling Approach}

\author{
    \IEEEauthorblockN{
        Rembrant Oyangoren Albeos~%
        \href{https://orcid.org/0009-0006-8743-4419}{%
            \includegraphics[height=8pt]{ORCID_icon.png}%
        }%
        \textsuperscript{\hyperref[sec:author_info]{$\dagger$}}
    }
    \IEEEauthorblockA{%
        \includegraphics[height=7pt]{ContinualQuasars_icon.png}\hspace{0.4em}Continual Quasars\\
    }
}

\begin{document}
\maketitle

% ════════════════════════════════════════════════════════
\begin{abstract}
Conventional market profile construction collapses raw price data
directly into a Y-distribution histogram, recording only the price
levels that were explicitly quoted by the exchange or broker feed.
This paper presents an alternative approach---termed
\emph{path-weighted gap-filling}---in which synthetic trail-datapoints
are inserted at every intermediate unit-level price between
consecutive observations, producing an extended dataset that yields a
substantially denser and more continuous market profile.  The
modelling is grounded in microsecond-resolution raw bid/ask unit data
rather than aggregated TOHLC (time, open, high, low, close) bars or
volume figures, thereby preserving the highest available fidelity of
the underlying price process.  We demonstrate the approach on a full
trading day of \texttt{XAUUSDc} data collected from a live trading
environment, and show that the gap-filled profile eliminates the empty
bins and sparse regions that afflict raw-unit profiles during fast
directional moves, producing a more representative picture of
intraday price dynamics. All resources and code used in this work are available on GitHub at \url{https://github.com/ContinualQuasars/mBA-Profile}.

\end{abstract}

\begin{IEEEkeywords}
Market profile, unit data, microsecond bid--ask, market microstructure, path-weighted, gap-filling.
\end{IEEEkeywords}

% ════════════════════════════════════════════════════════
\section{Introduction}
\label{sec:intro}

\subsection{Market Microstructure and Unit Data}

At the most granular level of market data, financial instruments are
quoted through discrete data updates: each update represents a change in
the best bid price, the best ask price, or both
simultaneously~\cite{hasbrouck2007,ohara1995}.  Modern trading
platforms such as MetaTrader~5 (MT5) record these events with
millisecond-resolution timestamps, and the data is made available
through a Python API~\cite{mt5docs}.

The instrument studied in this paper is \texttt{XAUUSDc}, a
gold CFD (Contract for Difference) traded on a
standard cent live trading account provided by the Exness broker,
accessed through the MT5 platform.  The \texttt{c} suffix in
\texttt{XAUUSDc} is an Exness broker account-type indicator
(denoting a standard cent live account) and has no bearing on the
XAU price data itself---extracting data from \texttt{XAUUSDc}
(cent account) or \texttt{XAUUSDm} (dollar account) yields the
same XAUUSD price data with three decimal places.  The minimum price
increment for this instrument is exactly \$0.001.
Because the standard lot size for gold is 100 troy ounces, a single
price movement of \$0.001 corresponds to a profit-or-loss change of
\$0.10 per standard lot.  In this study, the market profile
bin size is set to \$0.01 (one unit, where 0.01~unit = \$0.01 XAU
price change), to produce a more
stable and interpretable distribution.

\subsection{The Market Profile Concept}

A market profile is a rotated histogram of price over a defined time
window.  The concept was introduced by J.~Peter Steidlmayer at the
Chicago Board of Trade in the 1980s~\cite{steidlmayer1986}.
Traditionally, a market profile uses 30-minute ``Time Price
Opportunity'' (TPO) letters stacked at each price level to show where
price spent the most time during a trading session~\cite{dalton2007}.
The horizontal axis represents frequency or time density, while the
vertical axis represents price.

In this study, the concept is adapted to microsecond unit data.
Instead of 30-minute TPO letters, each histogram bar represents the
number of data updates (or interpolated price levels, in the
gap-filled approach) observed at that price.  The construction is
based exclusively on raw bid/ask unit data---not on TOHLC candles or
volume bars---ensuring that no information is lost to
aggregation~\cite{ane2000,engle2000}.

\subsection{Paper Outline}

Section~\ref{sec:data} describes the data acquisition pipeline and
the dataset used.  Section~\ref{sec:raw} details the raw unit
approach.  Section~\ref{sec:filled} introduces the gap-filled
(path-weighted) approach, including a detailed explanation of why
path-weighting is used.  Section~\ref{sec:comparison} provides a
comprehensive comparison of the two approaches.
Section~\ref{sec:conclusion} concludes.


% ════════════════════════════════════════════════════════
\section{Data Acquisition}
\label{sec:data}

\subsection{Trading Environment}

The unit data used in this study was collected from a standard cent
live trading account on the Exness broker, accessed through MetaTrader~5.
MT5 is a multi-asset trading platform developed by MetaQuotes Software
Corp.\ that is widely used for forex and CFD
trading~\cite{mt5docs,metaquotes2024}.  Its Python integration exposes
the function \texttt{copy\_ticks\_range()}, which returns every data
update within a specified time window as a structured NumPy
array~\cite{numpy2020}.  Each data record contains the following
fields: a Unix timestamp in seconds, a millisecond-precision timestamp
providing sub-second resolution, the best bid price, the best ask
price, and additional metadata including flags indicating which fields
changed on that particular update.

Although the exposed timestamp has millisecond granularity, the MT5
documentation describes the system as operating at microsecond
internal resolution~\cite{mt5docs}; the millisecond field is what is
exposed through the Python API.

\subsection{Dataset Summary}

The symbol is \texttt{XAUUSDc}.  The time range covers the full UTC
day of February~12, 2026, from 00:00:00 to 23:59:59.  The flag used
retrieves all data updates regardless of whether the bid, ask, or last
price changed.

The query returned exactly \textbf{393,252~data points}.  The first data point was
recorded at \textbf{2026-02-12 00:00:00.149~UTC} and the last data point at
\textbf{2026-02-12 23:59:57.820~UTC}.  The bid price ranged from a
low of \textbf{\$4,878.380} to a high of \textbf{\$5,083.750}, a span
of \textbf{\$205.370} (20,537~units).  The ask price ranged from
\textbf{\$4,878.620} to \textbf{\$5,083.990}, a span of
\textbf{\$205.370} (20,537~units).

\subsection{Unit Size}

The unit size for \texttt{XAUUSDc} is \textbf{\$0.010} (0.01~unit,
where 0.01~unit = \$0.01 XAU price change).
This value is determined by the broker's symbol specification and is
not configurable by the user.  The lowest price resolution of XAU is
three decimal places: a change of 0.001 corresponds to a
\$0.001 price movement.  Throughout this paper, $\delta = 0.010$
denotes the unit size, and the bin width used for histogram
construction equals 0.01~unit ($w = \delta = 0.010$).


% ════════════════════════════════════════════════════════
\section{Approach~1: Raw Unit Y-Distribution}
\label{sec:raw}

\subsection{Methodology}

The raw unit approach constructs a market profile histogram directly
from the 393,252 observed unit prices without any interpolation or
modification.  The procedure begins by extracting the bid and ask
columns as separate arrays from the dataset.  Histogram bin edges are
computed starting from
$\lfloor p_{\min}/\delta \rfloor \cdot \delta - \delta$ up to
$\lceil p_{\max}/\delta \rceil \cdot \delta + \delta$, spaced by
exactly $\delta = 0.010$.  This ensures that every observed price
falls cleanly within a bin whose width is exactly 0.01~unit.  Bin edges
are rounded to avoid floating-point precision
artefacts~\cite{goldberg1991}.

A standard frequency histogram is then computed---the count of data points
whose price falls within each bin---separately for bid and ask.  The
histogram is plotted horizontally, with price on the vertical axis and
count on the horizontal axis, creating the conventional market-profile
appearance where the thickest region corresponds to the price level
that received the most data updates.

\subsection{Feature Engineering}

The feature engineering pipeline for the raw approach consists of the
following stages.  First, the raw unit data from MT5 (a structured
array) is converted into a tabular format.  The millisecond-precision
timestamp column is transformed into a UTC-aware datetime
representation.  Next, the datetime values are converted to
floating-point date numbers suitable for high-performance
plotting~\cite{matplotlib2007}.  This pre-conversion is performed once
before plotting because passing raw datetime objects to the plotting
library triggers an internal per-element conversion that is extremely
slow for arrays of 393,252 elements---the pre-conversion reduces
plotting time from several minutes to under one minute for the full
dataset.

The histogram bin edges are constructed using a range function with
step size equal to $\delta$ and then rounded.  For the observed data
range of \$4,878.380 to \$5,083.990, this produces 20,563 bin edges
defining 20,562 bins, each exactly \$0.010 wide (0.01~unit).

\subsection{Output}

The output is a $2 \times 2$ subplot figure.  The top row displays the
bid data: a horizontal histogram on the left (blue) and a time-series
line chart on the right (blue).  The bottom row displays the ask data
in the same layout using red.  The two rows share their respective
Y-axes so that price levels align horizontally between the histogram
and the line chart.

\begin{figure*}[t]
    \centering
    \includegraphics[width=\textwidth]{raw_ticks_4panel.png}
    \caption{Raw unit Y-distribution histograms (left column) and
    time-series line charts (right column) for bid (top, blue) and ask
    (bottom, red) prices of \texttt{XAUUSDc} on February~12, 2026.
    The dataset contains 393,252 data points.  Bin size = 0.01~unit (\$0.010).
    The histogram X-axis shows the count of data points observed at each
    price level.}
    \label{fig:raw_4panel}
\end{figure*}

\subsection{Interpretation}

In the raw histogram (Figure~\ref{fig:raw_4panel}), the count at each
price level reflects how many times the market's best bid or best ask
was updated to that exact price.  Levels where the market
consolidated---spending extended time with many small quote
updates---accumulate high counts and form the thick horizontal bars in
the profile~\cite{dalton2007}.

However, when the market jumps from price $A$ to price $B$ in a single
update without quoting any intermediate level, those intermediate levels
receive zero counts in the histogram.  The raw profile therefore
contains \emph{gaps}---entire price levels with no
representation---that correspond to fast directional moves.  This is a
fundamental limitation: the profile faithfully records only what was
quoted, but it does not capture the price path traversed between
observations.  This motivates the gap-filled approach presented in
Section~\ref{sec:filled}.


% ════════════════════════════════════════════════════════
\section{Approach~2: Gap-Filled (Path-Weighted) Y-Distribution}
\label{sec:filled}

\subsection{Motivation}

Consider a scenario where the bid price moves from \$5,060.000 to
\$5,060.100 in a single update.  In the raw approach, only two price
levels---\$5,060.000 and \$5,060.100---register a count, while the
eight intermediate levels (\$5,060.010 through \$5,060.090) receive no
representation at all.  Yet, under the assumption that price is a
continuous process sampled at discrete intervals, the price must have
traversed those eight levels to arrive at
\$5,060.100~\cite{cont2001,bacry2012}.  The gap-filled approach
addresses this by inserting synthetic trail-datapoints at every
intermediate unit-level price between consecutive observations,
thereby constructing a profile that reflects the full path traversed
by the market rather than only the endpoints of each move.

\subsection{Why Path-Weighting?}
\label{sec:whypathweight}

The term \emph{path-weighted} refers to the fact that each price
level's histogram count is weighted by the number of times the price
path crossed that level, not merely the number of times it was
explicitly quoted.  The rationale for this weighting rests on three
observations:

\begin{enumerate}[leftmargin=*]
    \item \textbf{Continuity of the price process.}  Financial prices
    are fundamentally continuous stochastic processes sampled at
    discrete intervals by the exchange or broker
    feed~\cite{cont2001,bacry2012}.  Between any two consecutive
    observations at prices $p_A$ and $p_B$, the underlying price
    process must have traversed every intermediate level.  The raw
    profile discards this traversal information; the path-weighted
    profile recovers it.

    \item \textbf{Elimination of empty bins.}  In the raw profile,
    fast directional moves produce stretches of price levels with zero
    counts, creating discontinuities in the histogram that can mislead
    visual interpretation.  Path-weighting ensures that every price
    level between $p_{\min}$ and $p_{\max}$ receives a non-zero count,
    producing a continuous and visually coherent
    profile~\cite{steidlmayer1986}.

    \item \textbf{Traversal as a proxy for significance.}  A price
    level that is crossed repeatedly---even by fast-moving price
    swings that do not dwell there---is a level that the market
    revisits often.  Such levels frequently correspond to support,
    resistance, or areas of high liquidity~\cite{dalton2007,
    steidlmayer1986}.  Path-weighting captures this repeated-traversal
    signal, which raw unit counting misses entirely.
\end{enumerate}

In summary, path-weighting transforms the market profile from a
histogram of \emph{quoting intensity} into a histogram of
\emph{traversal frequency}, which is a richer and more informative
representation of where the market has been.

\subsection{Algorithm}

The gap-filling algorithm operates on pairs of consecutive data points.  For
each pair $(A, B)$ with prices $p_A$ and $p_B$ and timestamps $t_A$
and $t_B$ (represented as nanosecond integers for computational
efficiency), the algorithm first computes the signed unit difference
$\Delta n = \text{round}((p_B - p_A) / \delta)$.  If
$|\Delta n| \le 1$, no interpolation is needed because the two prices
are adjacent or identical, and the pair is left unchanged.  If
$|\Delta n| > 1$, the algorithm inserts $|\Delta n| - 1$ intermediate
rows.  Each intermediate row $k$ (where $1 \le k < |\Delta n|$)
receives a price of
$p_A + k \cdot \text{sgn}(\Delta n) \cdot \delta$ and a timestamp of
$t_A + \frac{k}{|\Delta n|} \cdot (t_B - t_A)$.  The timestamp
interpolation is linear, distributing the intermediate points evenly
across the time interval between data points $A$ and
$B$~\cite{dacorogna2001}.

The implementation is fully vectorised using array operations rather
than interpreted loops~\cite{numpy2020}.  The key operations are
element repetition (to repeat each source index by the number of units
in its segment), cumulative summation (to compute segment start
positions), and element-wise arithmetic for price and timestamp
interpolation.  This vectorised approach processes the entire
393,252-point dataset in under 2~seconds on a consumer-grade machine.

The gap-filling is applied independently to the bid series and the ask
series because the bid and ask prices can move by different amounts on
the same data update.  After gap-filling, the bid series expands from
393,252 rows to exactly \textbf{4,614,400~rows} (an expansion factor
of $11.73\times$), and the ask series expands from 393,252 rows to
exactly \textbf{4,619,918~rows} (an expansion factor of
$11.75\times$).

\subsection{Feature Engineering}

The feature engineering pipeline for the gap-filled approach shares the
initial stages with the raw approach: data fetching, tabular
conversion, and datetime derivation are identical.  The additional
stage is the gap-filling itself, which produces two new arrays of
expanded prices and their corresponding interpolated timestamps.

For plotting, the expanded nanosecond timestamps must be converted to
floating-point date numbers.  Because the expanded arrays contain
approximately 4.6 million elements, calling a datetime conversion
function on individual objects would be prohibitively slow.  Instead,
the conversion is performed arithmetically: the nanosecond integer is
divided by $10^9$ to get seconds, then by 86,400 to get fractional
days since the Unix epoch, and finally offset by the appropriate
constant to align with the plotting library's date
system~\cite{matplotlib2007}.  This bypasses all object-level datetime
creation and processes the 4.6 million timestamps in a single
vectorised operation.

The histogram bins are constructed identically to the raw approach,
using 0.01-unit (\$0.010) bin widths.  Because the gap-filled data has
the same price range as the raw data (\$4,878.380 to \$5,083.990),
the number of bins is also 20,562.

\subsection{Output}

The output figure has the identical $2 \times 2$ subplot layout as
Figure~\ref{fig:raw_4panel}.

\begin{figure*}[t]
    \centering
    \includegraphics[width=\textwidth]{filled_ticks_4panel.png}
    \caption{Gap-filled (path-weighted) Y-distribution histograms
    (left column) and time-series line charts (right column) for bid
    (top, blue) and ask (bottom, red) prices of \texttt{XAUUSDc} on
    February~12, 2026.  The bid series contains 4,614,400 data points
    and the ask series contains 4,619,918 data points after
    gap-filling.  Bin size = 0.01~unit (\$0.010).  The histogram X-axis
    shows the path-weighted count: the number of times each price
    level was traversed between consecutive data points, including synthetic
    intermediate points.}
    \label{fig:filled_4panel}
\end{figure*}

\subsection{Interpretation}

The gap-filled histogram (Figure~\ref{fig:filled_4panel}) answers a
fundamentally different question than the raw histogram.  Where the
raw profile asks ``how many times was price \emph{quoted} at this
level,'' the gap-filled profile asks ``how many times did the price
\emph{path} cross this level.''  The practical consequence is visible
in the histogram scale: the raw histogram peaks at counts near 120,
while the gap-filled histogram peaks at counts near 1,200 (consistent
with the $\approx 11.7\times$ average expansion factor).

Price regions that were traversed frequently---even if the market did
not dwell there long enough to generate many raw tick
updates---accumulate higher counts in the gap-filled profile.  The
large sell-off visible around 16:00~UTC, where the bid price dropped
from the \$5,050.000 region to the \$4,878.000 region in a
concentrated burst of activity, produces substantial counts at every
intermediate price level in the gap-filled profile, whereas those same
levels appear sparse or empty in the raw profile because the market
jumped through them in large increments.


% ════════════════════════════════════════════════════════
\section{Raw vs.\ Gap-Filled: Comprehensive Comparison}
\label{sec:comparison}

\subsection{What Each Approach Measures}

The raw approach counts only actual tick updates from the broker's
data feed.  When a price level receives a high count, it means the
market's best bid or ask was actively updated to that level many
times.  This is a direct measurement of \emph{quoting
intensity}~\cite{ohara1995}: how frequently market participants were
placing or modifying orders at that price.

The gap-filled approach counts every tick-level price between
consecutive updates, including synthetic intermediate points that were
never explicitly quoted.  When a price level receives a high count in
the gap-filled profile, it means the price \emph{path} crossed that
level many times---either through actual quoting or through
interpolation during price jumps.  This is a measurement of
\emph{traversal frequency}.

\subsection{Detailed Comparison}

Table~\ref{tab:comparison} presents a comprehensive side-by-side
comparison of the two approaches across all relevant variables.

\begin{table*}[t]
\centering
\caption{Comprehensive comparison of raw tick vs.\ gap-filled
(path-weighted) market profile construction for \texttt{XAUUSDc} on
February~12, 2026.}
\label{tab:comparison}
\small
\begin{tabular}{@{}p{3.8cm}p{5.8cm}p{5.8cm}@{}}
\toprule
\textbf{Variable} & \textbf{Raw Tick Profile} & \textbf{Gap-Filled (Path-Weighted) Profile} \\
\midrule
Data source &
Microsecond bid/ask ticks from MT5 &
Same raw ticks, plus synthetic trail-datapoints \\
\midrule
Bid data points &
393,252 &
4,614,400 ($11.73\times$ expansion) \\
\midrule
Ask data points &
393,252 &
4,619,918 ($11.75\times$ expansion) \\
\midrule
Price range &
\$4,878.380 -- \$5,083.990 &
\$4,878.380 -- \$5,083.990 (identical) \\
\midrule
Bin width &
$\delta = \$0.010$ (1 tick) &
$\delta = \$0.010$ (1 tick, identical) \\
\midrule
Number of bins &
20,562 &
20,562 (identical) \\
\midrule
Avg.\ count per bin (bid) &
$393{,}252 / 20{,}537 \approx 19.15$ &
$4{,}614{,}400 / 20{,}537 \approx 224.7$ \\
\midrule
Peak histogram count &
$\sim$120 &
$\sim$1,200 \\
\midrule
Empty bins in profile &
Many (fast moves leave gaps) &
None (all intermediate levels filled) \\
\midrule
Profile continuity &
Discontinuous; sparse in trending regions &
Continuous; no gaps across entire price range \\
\midrule
What is measured &
Quoting intensity (how often each level was quoted) &
Traversal frequency (how often price path crossed each level) \\
\midrule
Consolidation zones &
High counts---dense, well-represented &
Similar to raw (few gaps to fill when moves are small) \\
\midrule
Fast directional moves &
Sparse or empty---underrepresented &
Well-represented with interpolated traversals \\
\midrule
Support/resistance detection &
Based on quoting density only &
Enhanced: repeated traversals indicate revisited levels \\
\midrule
Interpolation method &
None &
Linear timestamp interpolation, tick-step price fill \\
\midrule
Computational cost &
Minimal (direct histogram of raw data) &
Higher ($\sim$11.7$\times$ more data to process) \\
\bottomrule
\end{tabular}
\end{table*}

\subsection{Superiority of the Gap-Filled Approach}

The gap-filled approach produces a fundamentally more representative
market profile than the raw tick approach.  Its advantages are
threefold:

\begin{enumerate}[leftmargin=*]
    \item \textbf{Complete price coverage.}  The gap-filled profile
    assigns a non-zero count to every price level within the day's
    range, eliminating the misleading empty bins that appear in the
    raw profile during fast moves.  This provides a structurally
    complete picture of where the market traded.

    \item \textbf{Traversal information.}  By counting path crossings
    rather than only explicit quotes, the gap-filled profile captures
    information about how frequently the market revisited each price
    level---information that is entirely absent from the raw profile.
    This traversal signal is directly relevant to identifying dynamic
    support and resistance~\cite{dalton2007}.

    \item \textbf{Robustness to feed granularity.}  Different brokers
    and feed providers update tick data at different rates.  A slower
    feed produces larger jumps between consecutive ticks, which
    creates more gaps in the raw profile.  The gap-filled approach is
    robust to this variation because it reconstructs the intermediate
    path regardless of the feed's update frequency.
\end{enumerate}

The primary trade-off is computational cost: the gap-filling process
multiplies the dataset by a factor of approximately $11.7\times$ in
this study, which proportionally increases the time required for
histogram computation and rendering compared to a typical raw-data
market profile.  For very long time horizons or very volatile
instruments, this expansion factor could be significantly larger.

\subsection{Interaction with Bin Size}

At the 1-tick bin width ($w = 0.010$) used throughout this study,
the difference between the raw and gap-filled profiles is maximal
because gaps in the raw profile (empty bins where no tick was
observed) are filled in by the gap-filling process.  As the bin width
increases, the practical difference between the two approaches
diminishes because larger bins tend to capture at least some ticks
even in the raw profile, and the synthetic intermediate points are
absorbed into the same bins as the observed ticks.  At sufficiently
large bin widths, the raw and gap-filled histograms become nearly
indistinguishable~\cite{scott1979}.


% ════════════════════════════════════════════════════════
\section{Conclusion}
\label{sec:conclusion}

This paper presented two approaches to constructing market profiles
from 393,252 microsecond-resolution bid/ask tick updates of
\texttt{XAUUSDc} on February~12, 2026, collected from a standard cent
live trading account on the Exness broker via MetaTrader~5.  The raw
approach counted only observed tick levels, producing a profile that
reflects quoting intensity.  The gap-filled (path-weighted) approach
interpolated every intermediate price level between consecutive ticks,
expanding the dataset to 4,614,400 bid rows and 4,619,918 ask rows,
and producing a profile that reflects path traversal frequency.

The gap-filled approach yields a more complete and informative market
profile by eliminating empty bins, capturing traversal information,
and providing robustness to variations in feed update frequency.  The
primary cost of this approach is computational: the gap-filling
process multiplies the dataset size by a factor of approximately
$11.7\times$, which may result in slower calculation times compared to
typical market profile construction from raw data.

%% ============================================================================
%% AUTHOR INFORMATION
%% ============================================================================
\newpage
\vspace{2em}
\section*{Author Information}
\label{sec:author_info}

\begin{center}
\textbf{Rembrant Oyangoren Albeos}~\href{https://orcid.org/0009-0006-8743-4419}{\includegraphics[height=10pt]{ORCID_icon.png}}
\end{center}

\noindent\textbf{ORCID:} \url{https://orcid.org/0009-0006-8743-4419}

\noindent\textbf{Email:} algorembrant@gmail.com

\noindent\textbf{Affiliation:} Developer \& Researcher at ConQ

\noindent\textbf{Organization:} Continual Quasars~\includegraphics[height=7pt]{ContinualQuasars_icon.png}

\noindent\textbf{Organization GitHub:} \url{https://github.com/ContinualQuasars}

\noindent\textbf{This Version:} Febuary 14, 2026

\noindent\textbf{GitHub:} \url{https://github.com/ContinualQuasars/mBA-Profile}


% ════════════════════════════════════════════════════════
\newpage
\vspace{20}
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