mBA-GMP.v3/main.tex

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\title{mBA-GMP.\textit{v3}: Micro Bid-Ask Gap-Filled Market Profile\textsuperscript{*}}

\author{%
\href{https://github.com/ContinualQuasars}{\includegraphics[height=1.6ex]{ContinualQuasars_icon.png}}\ \textit{Continual Quasars, Research Team}\\%
February 20, 2026%
}

\maketitle

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%  ABSTRACT
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\begin{abstract}
Conventional Market Profile (CMP) aggregates price activity into histogram
bins using candlestick-derived data (TOCHLV), discarding the intra-bar
microstructure and leaving price bins between consecutive trades empty.  We
propose \textbf{mBA-GMP} (\textit{micro Bid-Ask Gap-filled Market Profile}),
a method that (i)~operates on raw, microsecond-resolution bid/ask
tick-formation data rather than pre-aggregated candlesticks, and
(ii)~interpolates every intermediate price bin traversed between successive
ticks, producing a \emph{gap-filled} profile.  Building on this gap-filled
structure, we further introduce an \emph{Up/Down-Bin Footprint Profile} that
classifies each bin's contribution directionally, revealing the net upward
or downward pressure across the price traversal. We formalise CMP and
GMP with explicit algorithms, derive the relationship between bin-count and
a user-defined bin-size parameter~$\beta$, and introduce a \emph{dataframe
recording approach} that walks through a 10-datapoint XAUUSD example to
show how datapoints are grouped into price bins (the CMP dataframe),
how gap-filling transforms the sparse CMP output into a dense GMP
dataframe, and how directional footprints are assigned. We demonstrate via
generated charts and CSV data that mBA-GMP yields a strictly denser and
more informative distribution than CMP. 
\end{abstract}

\begin{IEEEkeywords}
Market Profile, tick data, bid-ask spread, gap-filling interpolation,
high-frequency data, market microstructure, XAUUSD
\end{IEEEkeywords}

\vspace{0.5\baselineskip}
\hrule
\vspace{0.5\baselineskip}

{\footnotesize\noindent\textsuperscript{*}This research is conducted by the Continual Quasars Research Team at: {\color{blue}\href{https://github.com/ContinualQuasars}{github.com/ContinualQuasars}}\par}

\begin{figure}[!t]
  \centering
  \begin{tikzpicture}
    \begin{axis}[
      title={\textbf{CMP Profile}},
      xbar,
      xlabel={Stacks},
      ylabel={Price (USD)},
      ytick={3000,3001,...,3010},
      yticklabel style={font=\scriptsize},
      xmin=0, xmax=2,
      ymin=2999.5, ymax=3010.5,
      bar width=4pt,
      width=0.42\columnwidth,
      height=6.5cm,
      enlarge y limits=0.05,
      nodes near coords,
      nodes near coords style={font=\tiny},
      name=cmp
    ]
    \addplot[fill=gray!60, draw=black] coordinates {
      (1,3000) (0,3001) (0,3002) (0,3003) (0,3004) (0,3005)
      (0,3006) (0,3007) (0,3008) (0,3009) (1,3010)
    };
    \end{axis}

    \begin{axis}[
      title={\textbf{GMP Profile}},
      xbar,
      xlabel={Stacks},
      ylabel={},
      ytick={3000,3001,...,3010},
      yticklabel style={font=\scriptsize},
      xmin=0, xmax=2,
      ymin=2999.5, ymax=3010.5,
      bar width=4pt,
      width=0.42\columnwidth,
      height=6.5cm,
      enlarge y limits=0.05,
      nodes near coords,
      nodes near coords style={font=\tiny},
      at={(cmp.east)},
      anchor=west,
      xshift=1.2cm
    ]
    \addplot[fill=blue!50, draw=black] coordinates {
      (1,3000) (1,3001) (1,3002) (1,3003) (1,3004) (1,3005)
      (1,3006) (1,3007) (1,3008) (1,3009) (1,3010)
    };
    \end{axis}
  \end{tikzpicture}
  \caption{Horizontal histogram comparison of CMP (left, grey) and GMP
    (right, blue) for XAUUSD with $\beta=1$.  CMP shows activity only at
    the two observed prices; GMP fills all 11~traversed bins.  The
    gap-filling approach is most effective when applied to micro bid/ask
    (mBA) raw tick-formation data.}
  \label{fig:profile}
\end{figure}
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%  I.  INTRODUCTION
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\section{Introduction}\label{sec:intro}

The Market Profile, introduced by Steidlmayer~\cite{steidlmayer1986market}
and later formalised by Dalton et~al.~\cite{dalton2007markets}, represents
price activity as a horizontal histogram whose bins correspond to discrete
price levels and whose bar lengths (``stacks'') reflect the amount of
activity observed at each level.  In practice, most implementations
construct the profile from candlestick TOCHLV (time, open, close, high, low, volume) data: each candle
contributes one stack to every bin between its high and low.

This approach suffers from two shortcomings:

\begin{enumerate}
  \item \textbf{Aggregation loss.}\;Candlesticks pre-aggregate raw ticks
    into time-based bars chosen by the broker or exchange, irreversibly
    discarding the sequence and microsecond timing of individual
    bid/ask updates~\cite{engle2000econometrics,easley1992time}.
  \item \textbf{Gap neglect.}\;When consecutive \emph{raw} ticks are
    separated by several price levels, the conventional profile records
    activity only at the two observed prices, ignoring the fact that price
    must have traversed every intermediate level.
\end{enumerate}

We address both issues with \textbf{mBA-GMP}.  The prefix \textit{mBA}
(micro Bid-Ask) specifies the data domain: raw, micro\-/millisecond-stamped
bid/ask tick-formation records---the smallest observable price changes.
The suffix \textit{GMP} (Gap-filled Market Profile) specifies the
construction rule: every price bin between two successive ticks receives an
interpolated stack, producing a profile with no gaps.


The remainder of this paper is organised as follows.
\Cref{sec:related} surveys related work.
\Cref{sec:prelim} establishes notation.
\Cref{sec:method} defines CMP and GMP formally, presents the mBA-GMP
algorithm, and introduces the Up/Down-Bin Footprint Profile.
\Cref{sec:dataframe} introduces the dataframe recording approach and
walks through a 10-datapoint worked example showing CMP, GMP, and footprint
construction step by step.
\Cref{sec:binsize} analyses the effect of bin-size on profile resolution.
\Cref{sec:example} provides an additional worked example with XAUUSD.
\Cref{sec:discussion} discusses practical implications, and
\Cref{sec:conclusion} concludes.


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%  II.  RELATED WORK
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\section{Related Work}\label{sec:related}

\subsection{Market Profile}
The Market Profile concept originates with Steidlmayer's observation that
price distributions at each level reveal where market participants find
``fair value''~\cite{steidlmayer1986market}. Dalton et~al.~\cite{dalton2007markets} extended the framework with auction-market
theory, using half-hour brackets as time-price opportunities (TPOs).  Both
formulations rely on time-based bars rather than raw ticks.

\subsection{Tick-Level Analysis}
Clark~\cite{clark1973subordinated} demonstrated that subordinating returns
to trade-count time yields closer-to-Gaussian distributions, motivating
trade-indexed (rather than time-indexed) analysis.
An\'{e} and Geman~\cite{ane2000order} confirmed that business-time
transformations normalise returns at the tick level.
Engle~\cite{engle2000econometrics} introduced econometric models tailored
to ultra-high-frequency data.


\subsection{Market Microstructure}
The theoretical foundations of bid-ask price formation are laid out by
Glosten and Milgrom~\cite{glosten1985bid}, O'Hara~\cite{ohara1995market},
and the comprehensive survey of Madhavan~\cite{madhavan2000market}.
Hasbrouck~\cite{hasbrouck2007empirical} provides empirical methods for
tick-level inference.  Bouchaud et~al.~\cite{bouchaud2018trades} present a
modern, physics-inspired treatment linking order flow to price dynamics.

A common thread across these works is that raw tick data preserves
information lost by any form of aggregation.  Our contribution is to
combine this insight with a gap-filling interpolation rule applied to the
Market Profile histogram.


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%  III.  PRELIMINARIES
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\section{Preliminaries}\label{sec:prelim}

\Cref{tab:notation} summarises the notation used throughout.

\begin{table}[!t]
  \centering
  \caption{Notation Summary}
  \label{tab:notation}
  \begin{tabular}{@{}cl@{}}
    \toprule
    \textbf{Symbol} & \textbf{Description} \\
    \midrule
    $N$       & Total number of raw ticks in the dataset \\
    $p_i$     & Price of the $i$-th tick, $i\in\{1,\dots,N\}$ \\
    $\beta$   & Bin size (price units per bin); default $\beta=1$ \\
    $b(p)$    & Bin index of price $p$: $b(p)=\lfloor p/\beta \rfloor$ \\
    $S[k]$    & Stack count (profile value) at bin~$k$ \\
    $\Delta_i$ & Price displacement: $\Delta_i = p_i - p_{i-1}$ \\
    $K_i$     & Number of bins traversed from tick $i{-}1$ to $i$ \\
    $U[k]$    & Up-bin count at bin $k$ \\
    $D[k]$    & Down-bin count at bin $k$ \\
    $\delta[k]$ & Net footprint delta at bin $k$: $\delta[k] = U[k] - D[k]$ \\
    \bottomrule
  \end{tabular}
\end{table}

\begin{definition}[Tick-formation]
  A \emph{tick-formation} is the smallest observable change in the bid or
  ask price as recorded by the broker.  Formally, a tick stream is an
  ordered sequence $\mathcal{T}=\{(t_i,\,p_i)\}_{i=1}^{N}$ where $t_i$
  is the micro\-/millisecond timestamp and $p_i$ is the observed price.
\end{definition}

\begin{definition}[Bin]
  Given bin size $\beta>0$, the \emph{bin} for price $p$ is the integer
  index
  \begin{equation}\label{eq:bin}
    b(p) = \left\lfloor \frac{p}{\beta} \right\rfloor.
  \end{equation}
  All prices $p$ satisfying $k\beta \le p < (k+1)\beta$ map to bin~$k$.
\end{definition}


\begin{definition}[Market Profile]
  A \emph{market profile} is a mapping $S:\mathbb{Z}\to\mathbb{N}_0$
  where $S[k]$ counts the number of stacks accumulated at bin~$k$.
\end{definition}


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%  IV.  METHODOLOGY
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\section{Methodology}\label{sec:method}

\subsection{Conventional Market Profile (CMP)}\label{sec:cmp}

CMP records a stack only at the bin of each observed data point:
\begin{equation}\label{eq:cmp}
  S_{\text{CMP}}[k] \;=\; \sum_{i=1}^{N} \mathbf{1}\!\bigl[b(p_i)=k\bigr],
\end{equation}
where $\mathbf{1}[\cdot]$ is the indicator function.  Bins with no
observed tick receive $S_{\text{CMP}}[k]=0$.

\smallskip
\begin{algorithm}[!t]
  \caption{CMP Construction}\label{alg:cmp}
  \begin{algorithmic}[1]
    \REQUIRE Tick stream $\{p_i\}_{i=1}^{N}$, bin size $\beta$
    \ENSURE  Profile array $S_{\text{CMP}}[\cdot]$
    \STATE Initialise $S_{\text{CMP}}[k]\leftarrow 0\;\;\forall\,k$
    \FOR{$i = 1$ \TO $N$}
      \STATE $k \leftarrow \lfloor p_i / \beta \rfloor$
      \STATE $S_{\text{CMP}}[k] \leftarrow S_{\text{CMP}}[k] + 1$
    \ENDFOR
    \RETURN $S_{\text{CMP}}$
  \end{algorithmic}
\end{algorithm}

\textbf{Complexity.}\;CMP performs exactly $N$ bin-index computations and
$N$ increments, giving $\mathcal{O}(N)$ time complexity.

\subsection{Gap-Filled Market Profile (GMP)}\label{sec:gmp}

GMP augments CMP by filling every \emph{intermediate} bin between two
consecutive ticks.  The construction proceeds in two phases:

\begin{enumerate}
  \item \textbf{CMP placement.}\;Each tick~$p_i$ contributes one stack to
    its own bin~$b(p_i)$, exactly as in CMP.
  \item \textbf{Gap-filling.}\;For each consecutive pair
    $(p_{i-1},\,p_i)$ with $i\ge 2$, every bin \emph{strictly between}
    $b(p_{i-1})$ and $b(p_i)$ (exclusive of both endpoints) receives one
    additional stack.
\end{enumerate}

\noindent Formally, writing $b_i = b(p_i)$:

\begin{equation}\label{eq:gmp}
  S_{\text{GMP}}[k]
  \;=\;
  \underbrace{\sum_{i=1}^{N}\mathbf{1}\!\bigl[b_i=k\bigr]}_{S_{\text{CMP}}[k]}
  \;+\;
  \sum_{i=2}^{N}
  \;\sum_{j=\min(b_{i-1},\,b_i)+1}^{\max(b_{i-1},\,b_i)-1}
  \!\mathbf{1}\!\bigl[j=k\bigr].
\end{equation}

When $|b_i - b_{i-1}| \le 1$ (adjacent or same bin), the inner sum is
empty and no gap-filling occurs.  When $|b_i - b_{i-1}| > 1$, the number
of gap-filled (intermediate) bins is
\begin{equation}\label{eq:Ki}
  G_i \;=\; \bigl|b(p_i) - b(p_{i-1})\bigr| - 1.
\end{equation}
The total span of bins traversed, inclusive of both endpoints, is
$K_i = G_i + 2 = |b_i - b_{i-1}| + 1$.

\begin{algorithm}[!t]
  \caption{GMP Construction (Two-Phase)}\label{alg:gmp}
  \begin{algorithmic}[1]
    \REQUIRE Tick stream $\{p_i\}_{i=1}^{N}$, bin size $\beta$
    \ENSURE  Profile array $S_{\text{GMP}}[\cdot]$
    \STATE Initialise $S_{\text{GMP}}[k]\leftarrow 0\;\;\forall\,k$
    \FOR{$i = 1$ \TO $N$} \COMMENT{Phase~1: CMP placement}
      \STATE $S_{\text{GMP}}[\lfloor p_i/\beta \rfloor] \leftarrow
              S_{\text{GMP}}[\lfloor p_i/\beta \rfloor] + 1$
    \ENDFOR
    \FOR{$i = 2$ \TO $N$} \COMMENT{Phase~2: gap-fill}
      \STATE $k_{\text{from}} \leftarrow \lfloor p_{i-1}/\beta \rfloor$;
             $k_{\text{to}} \leftarrow \lfloor p_i/\beta \rfloor$
      \IF{$|k_{\text{to}} - k_{\text{from}}| > 1$}
        \STATE $d \leftarrow \text{sign}(k_{\text{to}} - k_{\text{from}})$
        \FOR{$k = k_{\text{from}} + d$ \TO $k_{\text{to}} - d$ \textbf{step} $d$}
          \STATE $S_{\text{GMP}}[k] \leftarrow S_{\text{GMP}}[k] + 1$
        \ENDFOR
      \ENDIF
    \ENDFOR
    \RETURN $S_{\text{GMP}}$
  \end{algorithmic}
\end{algorithm}

\textbf{Complexity.}\;Let
$D=\sum_{i=2}^{N}|b(p_i)-b(p_{i-1})|$ denote the cumulative bin
displacement.  GMP performs $\mathcal{O}(N + D)$ operations.  In the
degenerate case where all ticks share the same bin, $D=0$ and GMP reduces
to CMP.  In the worst case, $D=\mathcal{O}(N\cdot\Delta p_{\max}/\beta)$.

\subsection{mBA-GMP: Applying GMP to Raw Tick Data}\label{sec:mba}

The key contribution of mBA-GMP is \emph{not} a novel interpolation rule
per~se, but rather the principled insistence that GMP must be applied to
raw bid/ask tick-formation data:

\begin{enumerate}
  \item \textbf{Data source.}\;Use the broker's micro\-/millisecond
    bid/ask feed---the lowest-granularity record available---rather than
    any TOCHLV candlestick derivative.
  \item \textbf{Trade indexing.}\;Index the $x$-axis by trade sequence
    number, not by wall-clock time (cf.~\cite{clark1973subordinated,
    ane2000order}).
  \item \textbf{Gap filling.}\;Apply \Cref{alg:gmp} to the tick stream.
\end{enumerate}

\begin{algorithm}[!t]
  \caption{mBA-GMP Pipeline}\label{alg:mba}
  \begin{algorithmic}[1]
    \REQUIRE Raw bid/ask tick feed $\mathcal{T}$, bin size $\beta$
    \ENSURE  Gap-filled profile $S_{\text{GMP}}[\cdot]$
    \STATE Extract price sequence $\{p_i\}_{i=1}^{N}$ from $\mathcal{T}$,
           indexed by trade count
    \STATE $S_{\text{GMP}} \leftarrow \textsc{GMP}(\{p_i\},\,\beta)$
           \COMMENT{Algorithm~\ref{alg:gmp}}
    \RETURN $S_{\text{GMP}}$
  \end{algorithmic}
\end{algorithm}

By operating on raw ticks, mBA-GMP avoids the aggregation artefacts
inherent in candlestick data~\cite{harris1990estimation} (e.g., arbitrary
bar boundaries, concealed intra-bar reversals) and ensures that every
micro-level price traversal is captured in the profile.

\subsection{Up/Down-Bin Footprint Profile}\label{sec:updown}

Building upon the gap-filled structure of GMP, we introduce a directional
classification layer termed the \emph{Up/Down-Bin Footprint Profile}.
Unlike order-flow bid/ask footprint charts which rely on volume traded at
the bid versus the ask, our footprint is derived purely from the GMP
transitive mechanics.

For every consecutive pair $(p_{i-1},\,p_i)$, the trajectory is evaluated
as an upward or downward movement based solely on the price difference.
The origin bin $b(p_{i-1})$ is assigned no directional credit relating
to this specific move (it has already been evaluated by prior action).
However, every subsequent bin along the traversed path up to and including
the destination bin $b(p_i)$ increments its \emph{up-bin} count $U[k]$ if $p_i > p_{i-1}$,
or its \emph{down-bin} count $D[k]$ if $p_i \le p_{i-1}$.

\begin{algorithm}[!t]
  \caption{Up/Down-Bin Footprint Construction}\label{alg:updown}
  \begin{algorithmic}[1]
    \REQUIRE Tick stream $\{p_i\}_{i=1}^{N}$, bin size $\beta$
    \ENSURE  Profile arrays $U[\cdot], D[\cdot], \delta[\cdot]$
    \STATE Initialise $U[k]\leftarrow 0, D[k]\leftarrow 0\;\;\forall\,k$
    \FOR{$i = 2$ \TO $N$}
      \STATE $k_{\text{from}} \leftarrow \lfloor p_{i-1}/\beta \rfloor$;
             $k_{\text{to}} \leftarrow \lfloor p_i/\beta \rfloor$
      \IF{$k_{\text{from}} = k_{\text{to}}$}
        \IF{$p_i > p_{i-1}$}
          \STATE $U[k_{\text{from}}] \leftarrow U[k_{\text{from}}] + 1$
        \ELSE
          \STATE $D[k_{\text{from}}] \leftarrow D[k_{\text{from}}] + 1$
        \ENDIF
        \STATE \textbf{continue}
      \ENDIF
      \STATE $\text{is\_up} \leftarrow (k_{\text{to}} > k_{\text{from}})$
      \STATE $d \leftarrow \text{sign}(k_{\text{to}} - k_{\text{from}})$
      \STATE $k \leftarrow k_{\text{from}} + d$
      \WHILE{\textbf{true}}
        \IF{$\text{is\_up}$}
          \STATE $U[k] \leftarrow U[k] + 1$
        \ELSE
          \STATE $D[k] \leftarrow D[k] + 1$
        \ENDIF
        \IF{$k = k_{\text{to}}$}
          \STATE \textbf{break}
        \ENDIF
        \STATE $k \leftarrow k + d$
      \ENDWHILE
    \ENDFOR
    \FORALL{$k$}
      \STATE $\delta[k] \leftarrow U[k] - D[k]$
    \ENDFOR
    \RETURN $U,\,D,\,\delta$
  \end{algorithmic}
\end{algorithm}

This algorithm traces the same $\mathcal{O}(N+D)$ bins as the GMP phase,
maintaining computational efficiency while providing deep structural
insight into directional dominance across the price range.


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%  V.  DATAFRAME RECORDING APPROACH
% ══════════════════════════════════════════════════════════════════════════════
\section{Dataframe Recording Approach}\label{sec:dataframe}

To build practical intuition for how CMP and GMP profiles are constructed,
this section walks through a concrete 10-datapoint example using a
\emph{dataframe}-style representation.  Each raw data record is a triple
$(\text{label},\;x,\;y)$ where \textit{label} is an alphabetic identifier,
$x$ is the trade index (or time), and $y$ is the observed price.
\Cref{tab:datapoints} lists the input data.

\begin{table}[!t]
  \centering
  \caption{Input Datapoints (XAUUSD Example, 10 Ticks)}
  \label{tab:datapoints}
  \begin{tabular}{@{}ccc@{}}
    \toprule
    \textbf{Datapoint} & \textbf{Trade \#} & \textbf{Price (USD)} \\
    \midrule
    A &  1 & 3000.914 \\
    B &  2 & 3003.837 \\
    C &  3 & 3002.432 \\
    D &  4 & 3009.892 \\
    E &  5 & 3007.698 \\
    F &  6 & 3009.176 \\
    G &  7 & 3003.381 \\
    H &  8 & 3004.283 \\
    I &  9 & 3003.512 \\
    J & 10 & 3003.012 \\
    \bottomrule
  \end{tabular}
\end{table}

\Cref{fig:price_scatter} plots these datapoints as a price-vs-trade-index
scatter chart, illustrating the raw price path that both CMP and GMP
will profile.

\begin{figure}[!t]
  \centering
  \includegraphics[width=\columnwidth]{fig_price_scatter.png}
  \caption{Price vs.\ trade index for the 10-datapoint XAUUSD example
    (A--J).  Each point represents one raw tick-formation record.}
  \label{fig:price_scatter}
\end{figure}

\subsection{CMP Output Dataframe}\label{sec:df_cmp}

Using $\beta=1$, the bin index for each tick is $b(p)=\lfloor p\rfloor$.
CMP simply counts how many datapoints fall into each bin.
\Cref{tab:cmp_df} shows the resulting dataframe: bins are numbered~1
through~10 from the lowest observed price to the highest.  The
\emph{datapoint group} column records which labels landed in each bin,
and \emph{stacks} is the group size.

\begin{table}[!t]
  \centering
  \caption{CMP Output Dataframe ($\beta=1$)}
  \label{tab:cmp_df}
  \begin{tabular}{@{}ccccc@{}}
    \toprule
    \textbf{Bin} & \textbf{From} & \textbf{Until}
      & \textbf{Group} & \textbf{Stacks} \\
    \midrule
     1 & 3000 & 3001 & A      & 1 \\
     2 & 3001 & 3002 &        & 0 \\
     3 & 3002 & 3003 & C      & 1 \\
     4 & 3003 & 3004 & BGIJ   & 4 \\
     5 & 3004 & 3005 & H      & 1 \\
     6 & 3005 & 3006 &        & 0 \\
     7 & 3006 & 3007 &        & 0 \\
     8 & 3007 & 3008 & E      & 1 \\
     9 & 3008 & 3009 &        & 0 \\
    10 & 3009 & 3010 & DF     & 2 \\
    \midrule
    \multicolumn{4}{c}{\textbf{Total stacks}} & \textbf{10} \\
    \bottomrule
  \end{tabular}
\end{table}

Note that bins~2, 6, 7, and~9 have zero stacks---these are the
\emph{gaps} in the CMP profile.  The CMP histogram formed by the
\emph{datapoint group} column is exactly the $y$-distribution
histogram used by traditional Market Profile implementations.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.85\columnwidth]{fig_cmp_profile.png}
  \caption{CMP profile for the 10-datapoint example ($\beta=1$).
    Four bins (2,\,6,\,7,\,9) are empty, revealing gaps in price coverage.}
  \label{fig:cmp_chart}
\end{figure}

\subsection{GMP Output Dataframe}\label{sec:df_gmp}

GMP augments the CMP result by filling every intermediate bin that price
must have traversed between consecutive datapoints.  The gap-filling
convention is:

\begin{enumerate}
  \item Each datapoint contributes one stack to its own bin (identical to
    CMP).
  \item For each consecutive pair $(i,\,i{+}1)$, every bin strictly
    \emph{between} $b(p_i)$ and $b(p_{i+1})$ (exclusive of both
    endpoints) receives one additional stack, labelled with the
    source datapoint~$i$.
\end{enumerate}

\Cref{tab:gmp_df} shows the resulting GMP dataframe.  All bins now have
at least one stack---no gaps remain.

\begin{table}[!t]
  \centering
  \caption{GMP Output Dataframe ($\beta=1$)}
  \label{tab:gmp_df}
  \begin{tabular}{@{}ccccc@{}}
    \toprule
    \textbf{Bin} & \textbf{From} & \textbf{Until}
      & \textbf{Group} & \textbf{Stacks} \\
    \midrule
     1 & 3000 & 3001 & A      & 1 \\
     2 & 3001 & 3002 & A      & 1 \\
     3 & 3002 & 3003 & AC     & 2 \\
     4 & 3003 & 3004 & BCGIJ  & 5 \\
     5 & 3004 & 3005 & CFH    & 3 \\
     6 & 3005 & 3006 & CF     & 2 \\
     7 & 3006 & 3007 & CF     & 2 \\
     8 & 3007 & 3008 & CEF    & 3 \\
     9 & 3008 & 3009 & CDEF   & 4 \\
    10 & 3009 & 3010 & DF     & 2 \\
    \midrule
    \multicolumn{4}{c}{\textbf{Total stacks}} & \textbf{25} \\
    \bottomrule
  \end{tabular}
\end{table}

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.85\columnwidth]{fig_gmp_profile.png}
  \caption{GMP profile for the 10-datapoint example ($\beta=1$).
    Every bin is populated; the profile fully represents the price
    range traversed by the market.}
  \label{fig:gmp_chart}
\end{figure}

\subsection{CMP vs.\ GMP Side-by-Side}\label{sec:df_compare}

\Cref{fig:cmp_vs_gmp_10pt} places both profiles side by side.  The
contrast is striking: CMP concentrates stacks at a handful of prices
(total~10~stacks), leaving 40\,\% of bins empty, while GMP distributes
25~stacks across all 10~bins, yielding a strictly denser and more
informative volume-at-price distribution.

\begin{figure}[!t]
  \centering
  \includegraphics[width=\columnwidth]{fig_cmp_vs_gmp.png}
  \caption{Side-by-side comparison of CMP (left, orange) and GMP (right,
    green) for the 10-datapoint XAUUSD example with $\beta=1$.}
  \label{fig:cmp_vs_gmp_10pt}
\end{figure}

\Cref{fig:combined_3panel} presents the entire pipeline---from raw
datapoints, through CMP, to GMP---in a single three-panel view, with
every bar annotated by its constituent datapoint letters.

\begin{figure*}[!t]
  \centering
  \includegraphics[width=\textwidth]{fig_combined_3panel.png}
  \caption{Three-panel overview: raw datapoints (left), CMP profile
    with group letters (centre), and GMP profile with group letters
    (right).  Every bar is annotated with the alphabetic labels of the
    datapoints it contains, making the gap-filling effect directly
    visible. The gap-filling approach is most effective when applied to micro bid/ask
    (mBA) raw tick-formation data.}
  \label{fig:combined_3panel}
\end{figure*}

\begin{quote}
  \textbf{Rendering note.}\;In the dataframe tables above, bin~1
  (lowest price) appears at the \emph{top} of the table.  On an actual
  price chart, however, the lowest price is at the \emph{bottom} of the
  $y$-axis and the highest price at the top---the profile histogram is
  effectively ``flipped'' relative to the tabular representation.
\end{quote}

\subsection{Up/Down-Bin Footprint Dataframe}\label{sec:df_updown}

Applying \Cref{alg:updown} to the same 10-datapoint trajectory yields
the directional footprint dataframe shown in \Cref{tab:updown_df}.
For example, the move from A (3000.914) to B (3003.837) causes bins 2, 3, and 4
to receive $+1$ up-bin point. The movement from C (3002.432) to D (3009.892)
applies up-bin points to bins 4 through 10. Downward movements, such as
B down to C, or D down to E, function symmetrically. Note that
the very first datapoint (A) does not carry directional value as there is
no precedent movement. Price movement within the same bin is correctly
assigned its respective directional label.

\begin{table}[!t]
  \centering
  \caption{Up/Down-Bin Footprint Output Dataframe ($\beta=1$)}
  \label{tab:updown_df}
  \begin{tabular}{@{}cccccrr@{}}
    \toprule
    \textbf{Bin} & \textbf{From} & \textbf{Until}
      & \textbf{Group} & \textbf{Down} & \textbf{Up} & \textbf{Delta} \\
    \midrule
     1 & 3000 & 3001 & A      & 0 & 0 &  0 \\
     2 & 3001 & 3002 & A      & 0 & 1 & +1 \\
     3 & 3002 & 3003 & AC     & 1 & 1 &  0 \\
     4 & 3003 & 3004 & BCGIJ  & 3 & 2 & -1 \\
     5 & 3004 & 3005 & CFH    & 1 & 2 & +1 \\
     6 & 3005 & 3006 & CF     & 1 & 1 &  0 \\
     7 & 3006 & 3007 & CF     & 1 & 1 &  0 \\
     8 & 3007 & 3008 & CEF    & 2 & 1 & -1 \\
     9 & 3008 & 3009 & CDEF   & 2 & 2 &  0 \\
    10 & 3009 & 3010 & DF     & 0 & 2 & +2 \\
    \bottomrule
  \end{tabular}
\end{table}

\Cref{fig:updown_footprint} visualises this footprint as a dual-axis
histogram. Each bin possesses opposing horizontal stacks indicating the
total amount of upward versus downward crossing, exposing the directional bias
driving the profile gap-fills.

\begin{figure}[!t]
  \centering
  \includegraphics[width=\columnwidth]{fig_updown_footprint.png}
  \caption{Up/Down-Bin Footprint Profile for the 10-datapoint example.
    Red bars (left) signify down-bin gap-fills; teal bars (right) signify
    up-bin gap-fills. Delta values $\delta[k]$ denote net directional
    pressure at each price bin.}
  \label{fig:updown_footprint}
\end{figure}


% ══════════════════════════════════════════════════════════════════════════════
%  VI.  BIN-SIZE ANALYSIS
% ══════════════════════════════════════════════════════════════════════════════
\section{Effect of Bin Size on Profile Resolution}\label{sec:binsize}


The bin-size parameter $\beta$ controls the granularity of the profile.
For two consecutive ticks at prices $p_{i-1}$ and $p_i$, the total number
of bins traversed (inclusive of both endpoints) is
\begin{equation}\label{eq:bins_beta}
  K_i(\beta) \;=\;
  \left|\left\lfloor \frac{p_i}{\beta} \right\rfloor
  - \left\lfloor \frac{p_{i-1}}{\beta} \right\rfloor\right|
  + 1.
\end{equation}

\noindent Halving $\beta$ approximately doubles the number of interpolated
bins, while doubling $\beta$ approximately halves it.

\begin{proposition}[Bin-count scaling]\label{prop:scaling}
  For a fixed price displacement $\Delta p = |p_i - p_{i-1}|$ and bin
  sizes $\beta_1 > \beta_2 > 0$, the bin counts satisfy
  \begin{equation}\label{eq:scaling}
    K_i(\beta_2) \;\ge\;
    \left\lfloor \frac{\beta_1}{\beta_2} \right\rfloor
    \cdot \bigl(K_i(\beta_1) - 1\bigr) + 1.
  \end{equation}
\end{proposition}

\begin{proof}
  Write $\Delta p = (K_i(\beta_1)-1)\,\beta_1 + r_1$ where
  $0 \le r_1 < \beta_1$.  Then
  $K_i(\beta_2) = \lfloor \Delta p / \beta_2 \rfloor + 1
  \ge \lfloor (K_i(\beta_1)-1)\,\beta_1 / \beta_2 \rfloor + 1
  \ge \lfloor \beta_1/\beta_2 \rfloor\,(K_i(\beta_1)-1) + 1$.
\end{proof}

\Cref{tab:binsize} illustrates how varying $\beta$ changes the GMP
resolution for the XAUUSD example where price moves from \$3{,}000 to
\$3{,}010 ($\Delta p = 10$).

\begin{table}[!t]
  \centering
  \caption{Effect of Bin Size ($\beta$) on GMP Bin Count for $\Delta p = 10$}
  \label{tab:binsize}
  \begin{tabular}{@{}cccc@{}}
    \toprule
    $\beta$ (USD) & $K_i(\beta)$ & CMP bins & GMP bins filled \\
    \midrule
    2.0  & 6   & 2 & 6  \\
    1.0  & 11  & 2 & 11 \\
    0.5  & 21  & 2 & 21 \\
    0.25 & 41  & 2 & 41 \\
    0.1  & 101 & 2 & 101 \\
    \bottomrule
  \end{tabular}
\end{table}

\noindent
Two key observations follow:
\begin{itemize}
  \item \textbf{CMP is invariant to~$\beta$ in bin count:} regardless
    of~$\beta$, CMP always fills exactly~2 bins (one per observed tick),
    because no intermediate bins are populated.
  \item \textbf{GMP scales as~$\mathcal{O}(\Delta p\,/\,\beta)$:}
    the filled bin count grows inversely with~$\beta$, producing a
    progressively finer-grained profile.  Setting $\beta$~below the
    instrument's tick size yields redundant empty bins, so the practical
    lower bound is $\beta \ge \text{tick\_size}$.
\end{itemize}

\Cref{tab:binsize_half} presents the full GMP profile comparison for
$\beta=1$ versus $\beta=0.5$.

\begin{table}[!t]
  \centering
  \caption{GMP Profile: $\beta=1$ vs.\ $\beta=0.5$ (Price from \$3{,}000 to \$3{,}010)}
  \label{tab:binsize_half}
  \begin{tabular}{@{}cccc@{}}
    \toprule
    Price (USD) & CMP & GMP ($\beta\!=\!1$) & GMP ($\beta\!=\!0.5$) \\
    \midrule
    3000.0 & 1 & 1 & 1 \\
    3000.5 & 0 &  & 1 \\
    3001.0 & 0 & 1 & 1 \\
    3001.5 & 0 &  & 1 \\
    3002.0 & 0 & 1 & 1 \\
    3002.5 & 0 &  & 1 \\
    3003.0 & 0 & 1 & 1 \\
    3003.5 & 0 &  & 1 \\
    3004.0 & 0 & 1 & 1 \\
    3004.5 & 0 &  & 1 \\
    3005.0 & 0 & 1 & 1 \\
    3005.5 & 0 &  & 1 \\
    3006.0 & 0 & 1 & 1 \\
    3006.5 & 0 &  & 1 \\
    3007.0 & 0 & 1 & 1 \\
    3007.5 & 0 &  & 1 \\
    3008.0 & 0 & 1 & 1 \\
    3008.5 & 0 &  & 1 \\
    3009.0 & 0 & 1 & 1 \\
    3009.5 & 0 &  & 1 \\
    3010.0 & 1 & 1 & 1 \\
    \midrule
    \textbf{Total bins} & \textbf{2} & \textbf{11} & \textbf{21} \\
    \bottomrule
  \end{tabular}
\end{table}


% ══════════════════════════════════════════════════════════════════════════════
%  VII.  ILLUSTRATIVE EXAMPLE
% ══════════════════════════════════════════════════════════════════════════════
\section{Illustrative Example}\label{sec:example}

Consider two raw XAUUSD ticks with $\beta=1$:
\begin{itemize}
  \item Tick~1: trade at $p_1 = \$3{,}000$.
  \item Tick~2: trade at $p_2 = \$3{,}010$.
\end{itemize}

\Cref{tab:cmp_vs_gmp} shows the resulting profiles side-by-side.

\begin{table}[!t]
  \centering
  \caption{CMP vs.\ GMP Comparison ($\beta=1$, XAUUSD)}
  \label{tab:cmp_vs_gmp}
  \begin{tabular}{@{}cccc@{}}
    \toprule
    Trade \# & Price (USD) & CMP stacks & GMP stacks \\
    \midrule
    1 & 3000 & 1 & 1 \\
    0 & 3001 & 0 & 1 \\
    0 & 3002 & 0 & 1 \\
    0 & 3003 & 0 & 1 \\
    0 & 3004 & 0 & 1 \\
    0 & 3005 & 0 & 1 \\
    0 & 3006 & 0 & 1 \\
    0 & 3007 & 0 & 1 \\
    0 & 3008 & 0 & 1 \\
    0 & 3009 & 0 & 1 \\
    2 & 3010 & 1 & 1 \\
    \midrule
    \multicolumn{2}{c}{\textbf{Total stacks}} & \textbf{2} & \textbf{11} \\
    \bottomrule
  \end{tabular}
\end{table}

CMP records only 2~stacks at the observed prices; GMP records
11~stacks spanning the full traversal.  \Cref{fig:profile} visualises
both profiles as horizontal histograms.




% ══════════════════════════════════════════════════════════════════════════════
%  VIII.  DISCUSSION
% ══════════════════════════════════════════════════════════════════════════════
\section{Discussion}\label{sec:discussion}

\subsection{Advantages}

\begin{enumerate}
  \item \textbf{Maximal resolution.}\;By operating on raw bid/ask ticks,
    mBA-GMP captures every price movement the broker records---no
    information is pre-aggregated or discarded.
  \item \textbf{No profile gaps.}\;Gap-filling ensures that every price
    level traversed by the market is represented, preventing the sparse,
    misleading histograms produced by CMP on fast-moving ticks.
  \item \textbf{Volume-neutral interpolation.}\;Interpolated bins receive
    exactly one stack each, reflecting a traversal rather than fabricating
    volume.  This preserves the interpretive semantics of the profile:
    high-stack regions still correspond to genuine price acceptance.
  \item \textbf{Directional Context.}\;By classifying gap-filled stacks into
    up/down bins, the resultant footprint profile reveals net directional
    pressure across the evaluated interval, independent of conventional
    bid/ask volume mechanics.
  \item \textbf{Tunable resolution via~$\beta$.}\;The bin-size parameter
    allows practitioners to control profile granularity without altering
    the underlying data, unlike candlestick-based approaches where
    resolution is fixed by the bar period.
\end{enumerate}

\subsection{Limitations}

\begin{enumerate}
  \item \textbf{Data availability.}\;Not all brokers expose raw
    micro\-/millisecond tick feeds.  Where only TOCHLV data is available,
    GMP can still be applied to candlestick prices, but the
    ``mBA'' guarantee is lost.
  \item \textbf{Computational cost.}\;The $\mathcal{O}(N + D)$ complexity
    implies that highly volatile instruments with large cumulative
    displacement~$D$ will require proportionally more computation.  For
    modern hardware this is rarely a practical bottleneck, but
    memory-constrained environments may require streaming or windowed
    implementations.
  \item \textbf{Interpolation assumption.}\;Gap-filling assumes that price
    continuously traverses every intermediate level.  In instruments with
    genuine price gaps (e.g., exchange-traded equities at market open),
    this assumption may over-represent bins that were never actually
    available for trading.
\end{enumerate}

\subsection{Practical Guidance on Choosing $\beta$}

\begin{itemize}
  \item Set $\beta$ near to the instrument's minimum tick size for maximum
    resolution (e.g., $\beta = 0.01$ for XAUUSD's 0.001 lowest tick, on many brokers).
  \item Increase $\beta$ to reduce noise in low-liquidity regimes or to
    align bins with round-number psychological levels.
  \item As shown in \Cref{sec:binsize}, CMP bin count is invariant
    to~$\beta$; thus, the resolution advantage of GMP grows as $\beta$
    decreases.
\end{itemize}


% ══════════════════════════════════════════════════════════════════════════════
%  IX.  CONCLUSION
% ══════════════════════════════════════════════════════════════════════════════
\section{Conclusion}\label{sec:conclusion}

We have presented \textbf{mBA-GMP}, a market-profile construction method
that combines two principles: (i)~sourcing data from raw micro\-/millisecond
bid/ask tick-formation records rather than pre-aggregated candlesticks,
and (ii)~interpolating all intermediate price bins between consecutive
ticks.  We formalised both Conventional Market Profile (CMP) and
Gap-filled Market Profile (GMP), provided pseudocode algorithms with
complexity analysis, and demonstrated that the bin-size parameter~$\beta$
controls profile resolution with a simple inverse relationship.

The dataframe recording approach (\Cref{sec:dataframe}) showed concretely
how 10~raw datapoints map to CMP bins with gaps, how gap-filling
produces a GMP dataframe in which every price bin is populated, and how
these intermediate bins are classified into directional up/down stacks
to yield a structural footprint. The accompanying charts and tabulated
outputs make the method reproducible and directly applicable to real-world
tick streams.

Future directions include extending the gap-filling convention to
weighted interpolation (where intermediate bins receive fractional stacks
proportional to their traversal speed) and evaluating mBA-GMP on
live order-book data across multiple asset classes.


\newpage
% ══════════════════════════════════════════════════════════════════════════════
%  REFERENCES
% ══════════════════════════════════════════════════════════════════════════════
\bibliographystyle{IEEEtran}
\bibliography{references}

\end{document}