Jump-diffusion models of parametric volume-price distributions

1. Problem Statement

Financial markets exhibit complex dynamics characterized by both continuous fluctuations and abrupt, discontinuous shifts (jumps) in asset prices. Traditional stochastic models, such as Geometric Brownian Motion (GBM) or standard Langevin dynamics, effectively capture smooth price evolution but fail to account for extreme events like market crashes or liquidity shocks, which manifest as heavy-tailed distributions in empirical data.

While Jump-Diffusion models (combining continuous diffusion with Poisson-driven jumps) exist, applying them to high-frequency financial data is challenging due to:

• Non-stationarity: Market parameters change over time.
• Sampling issues: Finite sampling effects, outliers, and the need for dense state-space sampling.
• Model selection: Determining whether a specific financial variable is purely diffusive or requires a jump-diffusion description is often ambiguous without rigorous non-parametric testing.

This paper addresses the need for a robust, data-driven framework to classify the intrinsic stochastic dynamics of volume-price distributions in the New York Stock Exchange (NYSE), specifically distinguishing between continuous diffusion and jump-diffusion regimes.

2. Methodology

The authors propose a four-step non-parametric analysis pipeline based on Kramers–Moyal (KM) expansion to reconstruct the stochastic differential equation (SDE) governing the system.

A. Data Preparation

• Dataset: 1,750 NYSE-listed companies over 976 trading days.
• Frequency: 10-minute intervals (09:10–15:30).
• Variable: A composite volume-price variable $s(t) = p(t)v(t)$.
• Preprocessing:
  • Removal of after-hours trading and error days.
  • Outlier Removal: A $10\sigma$ rule flagged 4 specific outlier days, which were removed from all series to ensure a consistent calendar.
  • Detrending: To isolate stochastic fluctuations, a 21-day centered moving average was subtracted from the daily parameter time series.
  • Stationarity: Verified using Augmented Dickey–Fuller (ADF) and KPSS tests, confirming weak stationarity for the detrended series.

B. Parametric Fitting

The volume-price data at the intraday level was fitted to four positive-support distribution families:

1. Gamma ($f_G$)
2. Inverse Gamma ($f_{IG}$)
3. Weibull ($f_W$)
4. Log-Normal ($f_{LN}$)

Each distribution is parameterized by a shape parameter ($\phi$) and a scale parameter ($\theta$). The study analyzes the stochastic evolution of these detrended parameters, $\phi'(t)$ and $\theta'(t)$.

C. Kramers–Moyal (KM) Analysis Pipeline

The core methodology involves estimating the KM coefficients $D^{(n)}(x)$ directly from the data:

1. Markov Property Verification: Used conditional entropy and Partial Autocorrelation Function (PACF) to determine the Markov time $\tau_M$ (found to be 5–9 time steps), ensuring the process is memoryless at the chosen lag.
2. Adaptive State-Space Discretization: Instead of classical binning (which fails with heavy tails), a zone-adaptive binning strategy was used. The state space was partitioned into "core," "shoulder," and "tail" regions to ensure sufficient data points (350–400 in the core, 150–200 in tails) for stable higher-order moment estimation.
3. Conditional Moments Calculation: Computed raw conditional moments $K^{(n)}(x, \tau)$ up to the sixth order ($n=1, 2, 4, 6$) for lags $\tau \in \{1, \dots, 6\}$.
4. Infinitesimal Moment Extraction & Correction:
   • Regressed normalized raw moments against time lag to extract infinitesimal moments $M^{(n)}(x)$.
   • Applied finite-lag corrections (using Bell polynomials) to remove cross-contamination between lower-order moments (e.g., removing the squared drift contribution from the second-order moment).
   • Calculated the final KM coefficients: $D^{(n)}(x) = F^{(n)}(x)/n!$.

D. Classification Criteria

• Pure Diffusion: If the ratio $R(x) = D^{(4)}(x)/D^{(2)}(x) < 0.1$ across all state bins, the process is described by a Langevin equation (pure diffusion).
• Jump-Diffusion: If $R(x) \gg 0.1$ (specifically if $D^{(4)}$ and $D^{(6)}$ are significant), the process is dominated by jump-diffusion dynamics.
• Global Inversion: Used Eq. (5) to globally estimate jump rates ($\lambda$) and jump amplitudes ($\sigma_\xi$) to quantify the variance share contributed by jumps.

3. Key Contributions

1. Systematic Classification of Volume-Price Dynamics: The paper provides the first rigorous, non-parametric classification of the stochastic nature of volume-price distribution parameters (shape vs. scale) across multiple distribution families.
2. Methodological Robustness: It demonstrates the critical importance of finite-lag corrections and adaptive binning when analyzing high-order moments in financial data. Without these, jump signatures can be spurious or missed entirely.
3. Discovery of Regime Asymmetry: The study reveals a fundamental asymmetry in market dynamics: the shape parameter ($\phi$) generally follows pure diffusion, while the scale parameter ($\theta$) is dominated by jump-diffusion (with the notable exception of the Log-Normal model).
4. Quantification of Jump Impact: The authors quantify that rare discontinuities (jumps) account for a massive share (40%–63%) of the total variance in the scale parameter, confirming that volatility is driven more by rare events than continuous noise.

4. Key Results

The analysis yielded distinct dynamics for the shape ($\phi$) and scale ($\theta$) parameters across the four distribution models:

Distribution	Shape Parameter ($\phi$)	Scale Parameter ($\theta$)
Gamma	Pure Diffusion: Linear mean-reverting drift; $D^{(4)}/D^{(2)} < 0.1$.	Jump-Diffusion: Nonlinear drift; $D^{(4)}/D^{(2)} \approx 10^{11}$.
Inverse Gamma	Pure Diffusion: Stable diffusion; negligible higher-order terms.	Jump-Diffusion: Large higher-order moments; $D^{(4)}/D^{(2)} \approx 10^9$.
Weibull	Pure Diffusion: Smooth surfaces; $D^{(4)}/D^{(2)} < 0.1$.	Jump-Diffusion: Strongest jump signature; $D^{(4)}/D^{(2)} \approx 10^{10}$.
Log-Normal	Weak Jump-Diffusion: Elevated but low ratio ($\approx 0.05$).	Pure Diffusion: Predominantly diffusive; $D^{(4)}/D^{(2)} < 0.01$.

• Gamma, InvG, Weibull: The scale parameter ($\theta$) exhibits dominant jump-diffusion dynamics, with jump rates of 4–15 events per unit time and jump variance shares of 40%–63%. The shape parameter ($\phi$) is purely diffusive.
• Log-Normal: Shows an inverted pattern. The log-scale ($\theta$) is diffusive, while the log-mean ($\phi$) shows weak jump signatures. The authors attribute this to the logarithmic transformation, which converts multiplicative shocks in the original scale into additive variations in log space, effectively "regularizing" the scale parameter.
• Global Variance: For the scale parameters of Gamma, InvG, and Weibull, global inversion confirms that jumps contribute the majority of the total variance, validating the "rare discontinuity" hypothesis.

5. Significance and Implications

• Risk Management: The findings suggest that risk models relying solely on diffusion (e.g., standard Black-Scholes) significantly underestimate volatility for the scale parameter of volume-price distributions. Risk controls, stop-loss placement, and stress testing must explicitly account for jump risks, particularly in liquidity and volume metrics.
• Market Microstructure: The separation of dynamics implies that shape ($\phi$) reflects the slow, structural organization of trading flow (diffusive), while scale ($\theta$) tracks liquidity shocks and order-book thinning (jump-prone).
• Modeling Framework: The paper validates the use of non-parametric KM analysis over pre-specified Lévy models for financial data where the jump measure is unknown. It highlights that high-order moment analysis ($n=4, 6$) is essential for detecting jumps but requires rigorous preprocessing (detrending, adaptive binning, finite-lag correction) to avoid artifacts.
• Log-Normal Nuance: The results caution against blindly applying jump-diffusion models to Log-Normal parameters, as the transformation itself can mask jump dynamics in the scale parameter.

In conclusion, the paper establishes that intraday volume-price dynamics are best described as a diffusive core with a jump-prone shell (for Gamma, InvG, Weibull), providing a refined theoretical basis for modeling financial volatility and managing market risk.