Jump-diffusion models of parametric volume-price distributions

TL;DR

This work introduces a data-driven, KM-based framework to model intraday volume–price distributions on the NYSE by fitting four distribution families ($\Gamma$, $IG$, Weibull, Log-Normal) with shape $\phi$ and scale $\theta$ and detrending them. Using adaptive zone-adaptive binning and KM coefficients up to order 6, the study classifies dynamics as diffusion or jump–diffusion and estimates global jump parameters, revealing a robust split: for $\,f_G$, $f_{IG}$, and $f_W$, $\phi$ is diffusive while $\theta$ is jump-diffusion; for $f_{LN}$ the pattern reverses due to log-space parameterization. Global inversions show that jumps can account for a large portion of variance in $\theta$ (approx. 40–63%), underscoring the importance of rare discontinuities in volatility. The methodology combines Markov verification, tail-stabilizing binning, and finite-lag corrections to provide a rigorous characterization of intraday microstructure dynamics with implications for risk management and intraday liquidity modeling.

Abstract

We present a data-driven framework to model the stochastic evolution of volume-price distribution from the New York Stock Exchange (NYSE) equities. The empirical distributions are sampled every 10 minutes over 976 trading days, and fitted to different models, namely Gamma, Inverse Gamma, Weibull, and Log-Normal distributions. Each of these models is parameterized by a shape parameter, $φ$, and a scale parameter, $θ$, which are detrended from their daily average behavior. The time series of the detrended parameters is analyzed using adaptive binning and regression-based extraction of the Kramers-Moyal (KM) coefficients, up to their sixth order, enabling to classification of its intrinsic dynamics. We show that (i) $φ$ is well described as a pure diffusion with a linear mean regression for the Gamma, Inverse Gamma, and Weibull models, while $θ$ shows dominant jump-diffusion dynamics, with an elevated fourth- and sixth-order moment contributions; (ii) the log-normal model shows however the opposite: $θ$ is predominantly diffusive, with $φ$ showing weak jump signatures; (iii) global moment inversion yields jump rates and amplitudes that account for a large share of total variance for $θ$, confirming that rare discontinuities dominate volatility.

Figures (15)

• Figure 1: Daily variation and trends in $\theta$ and $\phi$ for the Gamma, Inverse Gamma, Weibull, and Log-Normal distributions. Each subplot shows original values (blue), 21-day moving average (red), and fluctuations (green, inset). Vertical lines mark daily cycles every 38 points.
• Figure 1: Summary of KM coefficient analysis for all distribution–parameter combinations.
• Figure 2: Left: average intraday profiles of $\theta$ (first row) and $\phi$ (second row) for $f_G$, $f_{IG}$, $f_{LN}$ and $f_W$, computed over 976 trading days at 10-minute resolution. Markers show sample means at each intraday time index; red curves are cubic polynomial fits summarizing the deterministic daily pattern. Right: empirical distributions of the detrended fluctuations $\theta'(t)$ and $\phi'(t)$ for the same four families.
• Figure 3: Raw conditional moments $K^{(n)}(x,\tau)$ for $f_G(\phi)$ (left) and $f_G(\theta)$ (right). Rows 1--2: 3D surfaces with contour projections for $n=1$ (signed) and $n=2$ (positive). Rows 3--4: $\log_{10}|K^{(4)}|$ and $\log_{10}|K^{(6)}|$. $f_G(\phi)$ shows flat higher-order structure consistent with diffusion; $f_G(\theta)$ shows state-localized peaks consistent with jump-diffusion. Analogous figures for $f_{IG}$, $f_{LN}$, and $f_W$ appear in Appendix \ref{['app:conditional_moments']}.
• Figure 4: KM coefficients $D^{(n)}(x)$ for $f_G(\phi)$ (left) and $f_G(\theta)$ (right). Each panel shows $D^{(1)}(x)$, $D^{(2)}(x)$, $D^{(4)}(x)$, $D^{(6)}(x)$, and the diagnostic ratio $R(x)=D^{(4)}(x)/D^{(2)}(x)$ with a reference line at $0.1$. $f_G(\phi)$ exhibits linear mean-reverting drift, stable diffusion ($\sim 10^{-4}$), and negligible higher-order terms ($D^{(4)}\sim 10^{-7}$, $D^{(6)}\sim 10^{-9}$), yielding $R(x)<0.10$ (diffusion). $f_G(\theta)$ shows nonlinear drift, varying diffusion ($\sim 10^{10}$), and large higher-order terms ($D^{(4)}\sim 10^{22}$, $D^{(6)}\sim 10^{33}$), with $R(x)\gg 1$ (jump–diffusion).