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Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The symmetric Hawkes model, estimated from every tick-level mid-price move, gives volatility within about 15–25 percent of realized measures and a diffusion analogue with close distributional properties.

desk verdict Diffusion analogue is a useful idea with honest caveats, but the claimed closeness to Hawkes is not quantitatively established; deserves a serious referee and a revision. read the letter →

arxiv 1908.05089 v1 pith:7Y4KRDUY submitted 2019-08-14 q-fin.ST q-fin.TR

Kyungsub Lee , Byoung Ki Seo This is my paper · ORCID
classification q-fin.STq-fin.TR
keywords Hawkesprocesstickdynamicsultra-high-frequencydatavolatilityestimationsimulatedmaximumlikelihooddiffusionapproximationmarketmicrostructurerealized
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that a symmetric Hawkes process—the difference of two mutually exciting point processes describing up and down movements of the mid-price—is a practical model for ultra-high-frequency tick dynamics. Combined with maximum likelihood estimation on all price-change arrival times, it yields volatility estimates that track the two-scale realized volatility (TSRV) and the uncertainty-zone realized volatility (RRV) within roughly 15 to 25 percent for ten S&P 500 stocks over 2007–2011. The paper further claims that a diffusion analogue, with parameters explicitly tied to the Hawkes parameters, has a very close distributional property to the Hawkes model, making Itô-calculus tools available for variance, skewness, and leverage. A sympathetic reader would care because the model turns tick-by-tick data directly into volatility estimates, usable on short intraday windows, without first removing market microstructure noise.

What carries the argument

The load-bearing object is the symmetric Hawkes process: a bivariate point process $(N_1,N_2)$ with exponential-decay intensities sharing baseline $\mu$, decay $\beta$, self-excitation $\alpha_s$, and mutual excitation $\alpha_c$, whose difference scaled by the tick size $\delta$ is the mid-price. Its tractability comes from two facts: fixing $\beta$ makes the log-likelihood concave in $(\mu,\alpha_s,\alpha_c)$, so the maximum can be located reliably; and the stationarity assumption $\lambda_1(0)=\lambda_2(0)=\mu\beta/(\beta-\alpha_s-\alpha_c)$ reduces the return variance to a closed formula. The diffusion analogue $dS_t=n_tdt+\sqrt{V_t}dW^s_t$, $dn_t=-\kappa_1 n_tdt+\phi\sqrt{V_t}dW^s_t$, $dV_t=\kappa_2(\theta-V_t)dt+\gamma\sqrt{V_t}dW^v_t$ with the parameter identities above carries the second half of the paper: it imports Heston-type square-root variance machinery, a leverage parameter $\rho$, and simulated maximum likelihood estimation into the tick-level setting.

What would settle it

Take a low-activity stock or day, simulate the fitted symmetric Hawkes process, and compare one-minute mid-price return densities with the diffusion analogue's Kolmogorov-forward density; a significant distributional distance would show the normal approximation fails at that time scale.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the price process $S_t = S_0 + \delta(N_1(t)-N_2(t))$, with $N_1,N_2$ counting up and down moves of the mid-price and each intensity following a Hawkes process with common baseline $\mu$, decay $\beta$, self-excitation $\alpha_s$, and mutual excitation $\alpha_c$, can be estimated by maximum likelihood from ultra-high-frequency data. Under the stationarity condition $\lambda_1(0)=\lambda_2(0)=\mu\beta/(\beta-\alpha_s-\alpha_c)$, the return variance has the closed form $$\operatorname{Var}\left(\frac{S_t-S_0}{S_0}\right)=\frac{2\$delta^{2}$\lambda_1(0)}{$S_0^{2}$\$xi_1^{2}$}\left\{\$beta^{2}$t-\frac{2(\alpha_s-\alpha_c)\$\beta$}{\xi_1}($e^{{\xi_1 t}}$-1)+\frac{(\alpha_s-\alpha_c)^2}{2\xi_1}($e^{{2\xi_1 t}}$-1)\right\},$$ with $\xi_1=-\beta-\alpha_c+\alpha_s$, and this Hawkes volatility tracks TSRV and RRV on ten S&P 500 stocks with mean percentage errors typically in the 15–25 percent range. The companion claim is that replacing Poisson increments by Brownian increments over one-minute intervals produces the diffusion system $dS_t=n_tdt+\sqrt{V_t}dW^s_t$, $dn_t=-\kappa_1 n_tdt+\phi\sqrt{V_t}dW^s_t$, $dV_t=\kappa_2(\theta-V_t)dt+\gamma\sqrt{V_t}dW^v_t$, with $\kappa_1=b-a_s+a_c$, $\kappa_2=b-a_s-a_c$, $\theta=2bm\delta^2/(b-a_s-a_c)$, $\gamma=\delta(a_s+a_c)$, $\phi=a_s-a_c$, and $d[W^s,W^v]_t=\rho dt$; the paper reports that this diffusion has a very close distributional property to the Hawkes model, giving simpler variance and skewness formulas and a leverage parameter $\rho$.

Load-bearing premise

The load-bearing premise is that over a one-minute interval enough mid-price changes occur for the Poisson increments in the Hawkes process to be well approximated by normal (Brownian) increments; the paper gives no error bound for this replacement, and the diffusion model's very close distributional claim depends on it.

Editorial extensions

If this is right

  • Volatility can be estimated from as little as ten minutes of mid-price tick data, so the model produces an intraday volatility curve updated every ten minutes.
  • The closed-form Hawkes variance formula lets practitioners reparametrize the likelihood directly in terms of annualized volatility and estimate it without first removing microstructure noise.
  • If the diffusion analogue is accepted, variance, skewness, and leverage formulas follow from Itô calculus, making the tick-level model compatible with standard continuous-time tools.
  • The empirical comparison shows that the Hawkes volatility tracks TSRV and RRV daily, with mean percentage errors between the two measures staying around 15–25 percent for ten S&P 500 stocks.
  • The fully characterized Hawkes estimates show the symmetry assumptions are often reasonable, while cases like XOM in 2008 indicate persistence differences between self- and mutual excitation that a symmetric model cannot capture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to estimate the diffusion model on one-minute bins for less liquid names, where the normal approximation is most doubtful; if distributional closeness breaks down there, that would identify the shortest safe sampling scale.
  • The mean signature plot analysis implies that the sign of the parameter $\phi=a_s-a_c$ controls how realized variance responds to sampling frequency, and this directional prediction is directly checkable on the same high-frequency data.
  • If the diffusion analogy holds, Heston-style option pricing formulas could be applied with parameters estimated from tick arrivals, connecting ultra-high-frequency estimation to derivatives pricing.
  • The paper's 15–25 percent gap between Hawkes volatility and TSRV is left unexplained; allowing the baseline intensity to vary intraday in the likelihood would test whether parameter drift accounts for the gap.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies a symmetric bivariate Hawkes process for tick-level mid-price dynamics, estimates its parameters by maximum likelihood from ultra-high-frequency NYSE data for ten S&P 500 stocks, and compares the resulting Hawkes-based volatility with two-scale realized volatility (TSRV) and the Robert-Rosenbaum volatility (RRV). The empirical part reports daily and intraday parameter dynamics and finds that Hawkes volatility is generally within 15–25% of TSRV. The paper then proposes a continuous-time diffusion analogue of the Hawkes model, with state variables for price, drift, and variance, and derives closed-form variance and third-moment formulas. Parameter estimates for the diffusion model are obtained by simulated maximum likelihood. The central claim is that the diffusion analogue has 'very close distributional property' to the Hawkes model while offering analytical simplicity.

Significance. If the claims were fully established, the Hawkes-based volatility estimator would be a useful tool for intraday risk measurement using all tick arrivals, and the diffusion analogue would provide a tractable continuous-time model for high-frequency volatility and skewness. The paper has concrete strengths: the Hawkes likelihood estimation is standard and is checked in a simulation study; the variance derivations in the appendices are detailed and checkable; the empirical comparison with TSRV and RRV is extensive and includes an interesting intraday volatility application. The main weakness is that the diffusion analogue—a major advertised contribution—is supported only by a heuristic normal approximation with no error control, and the empirical diffusion estimates do not match the Hawkes volatilities well. The paper is therefore promising but requires substantial additional work before the central diffusion-analogy claim can be accepted.

major comments (3)
  1. [Section 3.1 and Section 3.3] The normal approximation underlying the diffusion analogue is not justified for the data actually used. Section 3.1 replaces Poisson increments by normal increments on the grounds that 'a sufficient number of price changes were observed during, e.g., one minute.' The paper's own estimates contradict this: for GE on 2011-01-03, Table 4 gives mu=0.0067, alpha_s=0.4661, alpha_c=1.3576, beta=2.2596, so the stationary intensity per direction is about 0.0347 per second, or about 4 mid-price changes per minute total; Table 5 gives roughly 5.6 per minute for T. A Poisson sum with mean 4–13 is not close to Gaussian, and no quantitative error bound is provided. Section 3.3 concedes the derivation is 'not a rigorous mathematical transform' and only 'to provide an intuition,' yet Section 3.4 uses the diffusion model as the working model for estimation. The 'very close distributional property' claim is therefore not supported for the low-intensity stocks in the empirical study; Figure 2 uses a high-intensity parameter setting (mu=0.09, 30-second horizon) and reports no error metric. Please provide a quantitative justification (e.g., a Berry-Esseen bound or a limit theorem with stationarity conditions) or explicitly restrict the claim to regimes where the normal approximation is verified.
  2. [Section 3.1, parameter mapping after Eq. (9)] Part of the agreement between the diffusion model and the Hawkes model is by construction. The parameter relation theta = 2bm*delta^2/(b-as-ac) is chosen so that the long-run variance of the diffusion model, b^2 theta t/(S0^2 kappa1^2), exactly equals the long-run variance of the Hawkes model in Remark 2 when m=mu, as=alpha_s, ac=alpha_c, b=beta. Thus the asymptotic variance comparison after Proposition 4 is not independent evidence of distributional closeness; it is baked into the parameter mapping. The finite-time variance in Proposition 4 is a genuine derivation, but its agreement with Proposition 3 is shown in only one figure (Figure 3) and one parameter set. Please clarify in the text that the long-run variance match is by construction and present finite-time comparisons over a range of empirically calibrated parameters.
  3. [Section 4.5, Tables 8 and 4] The claimed empirical similarity between the diffusion-model estimates and the Hawkes estimates is not supported by the reported numbers. For GE in January 2011, Table 8 gives diffusion volatilities that are frequently much larger than the Hawkes H.vol in Table 4: on 0105 the diffusion volatility is 0.3661 versus 0.1339 for Hawkes, on 0104 it is 0.1546 versus 0.1139, and on 0110 it is 0.1739 versus 0.1520. Several diffusion estimates also have parameter values very different from the Hawkes estimates (e.g., ac=2.9568 on 0104 vs. alpha_c=1.3941 in Table 4). The text states that the results are 'similar' to the Hawkes model without providing a quantitative comparison, confidence intervals, or an explanation of the systematic discrepancy. Please either provide a proper comparison (e.g., ratio statistics, error bars, or a scatter plot) or temper the claim about the empirical similarity of the two models.
minor comments (6)
  1. [Abstract] The sentence 'This study examine the theoretical and empirical perspectives' should read 'This study examines'; the same grammatical issue appears in the introduction.
  2. [Section 2.3] There is a typo 'Haweks' in the paragraph introducing the symmetric model; it should be 'Hawkes'.
  3. [Section 2.4] The reported estimates for 'simulation set 1' in the text appear to actually be from simulation set 2 (alpha_s=0.6590 is close to the true alpha_s=0.65 of set 2, not the true alpha_s=0.4 of set 1). Also, 'alpha_c = 0.0.4864' contains a typo and should be '0.4864'.
  4. [Figure 2 and Section 3.2] The caption of Figure 2 says 'histogram of the Hawkes model price by the simulation with 30 seconds (right)' but the panel labels are not visible; please clarify which panel shows the density and which shows the histogram, and add a legend or labels.
  5. [Section 4.5] Tables 8 and 9 report diffusion parameter estimates without standard errors or any measure of simulation variability; because the simulated likelihood depends on random numbers, at least the seed or a small Monte Carlo standard error should be reported.
  6. [Appendix C] The likelihood form in Eq. (C.1) should define the log-likelihood function with the parameter vector explicitly; the notation L(θ,T) appears only later, and the text introduces 'the conditional cumulative distributions' with an unnumbered equation that is not referenced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Hawkes volatility is derived from the model and benchmarked externally, while the diffusion model is a transparent analogy rather than a circular derivation.

full rationale

The paper's central volatility claim is not circular. Proposition 3 is derived in Appendix D from the symmetric Hawkes intensity dynamics and the stationarity condition, and it is cross-checked against Da Fonseca and Zaatour (2014a) with a noted difference in the exponential term. The empirical H.vol is a maximum-likelihood model output, not a fit to TSRV or RRV, which are independent realized-volatility benchmarks; the reported 15–25% differences are therefore informative. The diffusion section is explicitly an analogy: Section 3.1 constructs the parameter mapping (κ1, κ2, θ, γ, φ) so that the diffusion has the same eigenvalues and stationary variance as the Hawkes model, and Section 3.3 states the derivation is 'not a rigorous mathematical transform' and is meant only 'to provide an intuition not a mathematical proof.' The long-run variance agreement between Proposition 4 and Remark 2 is a transparent consequence of the θ calibration, not a hidden reuse of the target result, and the full variance formulas are not algebraically identical. Self-citations (Choe and Lee 2014a,b; Lee 2016) are used for defining third-moment variation and a moment-estimation method; Proposition 5 is proved in the appendix, so the central derivation does not rest on an unverified self-citation. The unquantified normal approximation for low-intensity stocks (e.g., GE with roughly 4–6 mid-price changes per minute) is a correctness and robustness limitation, not circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The central empirical claims rest on five fitted Hawkes parameters per stock-day plus five diffusion parameters estimated by simulated MLE. The theoretical claims rest on stationarity and diffusion-approximation assumptions that the paper itself labels as heuristic. The diffusion process is a new modeling object rather than an entity with independent physical evidence.

free parameters (6)
  • mu (baseline intensity) = e.g., 0.0067 to 0.0316 for GE January 2011 (Table 4)
    Estimated daily by MLE from mid-price tick data; drives the level of the Hawkes volatility.
  • alpha_s (self-excitation coefficient) = e.g., 0.42 to 0.70 for GE January 2011
    Estimated by MLE; measures clustering of same-direction price moves.
  • alpha_c (cross-excitation coefficient) = e.g., 0.35 to 1.39 for GE January 2011
    Estimated by MLE; measures clustering of opposite-direction price moves.
  • beta (decay rate) = e.g., 1.42 to 2.53 for GE January 2011
    Estimated by MLE; controls the memory of the intensity process.
  • Diffusion parameters m, as, ac, b, rho = GE January 2011 values in Tables 8 and 9
    Estimated by simulated maximum likelihood; no standard errors are reported.
  • Hand-chosen model parameters for comparisons = Figure 2: m=0.09, as=0.6, ac=0.3, b=2.5, delta=0.2; Figure 3: as=1.2, ac=0.3, b=2.2, m=0.01, delta/S0=0.002
    Chosen by hand to illustrate distributional closeness; no sensitivity analysis is provided.
assumptions (6)
  • standard math Spectral radius of the branching matrix Q is less than 1 (qs+qc<1) for the Hawkes process to be well defined.
    Invoked in Section 2.2 following Hawkes and Oakes (1974) and Bremaud (1981).
  • domain assumption The intensity processes are in stationarity at time 0: lambda1(0)=lambda2(0)=mu*beta/(beta-alpha_s-alpha_c), and later V0=theta, n0=0.
    Used to derive the variance formulas in Propositions 3 and 4 and in the appendix proofs; the paper argues the effect is small when expectations converge quickly but does not prove it for the data.
  • ad hoc to paper Poisson increments are approximated by normal increments at the one-minute scale.
    Stated in Section 3.1 as the basis for the diffusion analogy; no quantitative error bound is provided.
  • ad hoc to paper The diffusion model SDE is postulated as an analogy rather than derived as a limit.
    The paper explicitly says the derivation is intuitive and not an exact mathematical justification (Sections 3.1 and 3.3).
  • domain assumption Data preprocessing: price changes are multiples of half the minimal spread, larger changes are decomposed into successive minimal moves, and 1-second timestamps are redistributed uniformly.
    Section 4.1; if the tick structure is not as assumed, the arrival times used for MLE are distorted.
  • domain assumption The variance process remains nonnegative and the correlation rho is treated as constant.
    Section 3.1 assumes the square-root process and constant leverage parameter for tractability; the Feller condition is not discussed.
invented entities (1)
  • Diffusion analogue process (S_t, n_t, V_t)
    purpose: A tractable continuous-time stand-in for the symmetric Hawkes tick model that allows closed-form variance, PDE density calculation, and a leverage parameter.
    No external falsifiable handle is provided; the model is calibrated and tested only within the paper via simulation and in-sample fits.

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Cite this review

Pith. "Pith review of Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data." pith.science (2026). https://pith.science/paper/7Y4KRDUY

@misc{pith2026190805089,
  author       = {Pith},
  title        = {Pith review of: Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Y4KRDUY}},
  note         = {Machine review of arXiv:1908.05089}
}
read the original abstract

This study examine the theoretical and empirical perspectives of the symmetric Hawkes model of the price tick structure. Combined with the maximum likelihood estimation, the model provides a proper method of volatility estimation specialized in ultra-high-frequency analysis. Empirical studies based on the model using the ultra-high-frequency data of stocks in the S\&P 500 are performed. The performance of the volatility measure, intraday estimation, and the dynamics of the parameters are discussed. A new approach of diffusion analogy to the symmetric Hawkes model is proposed with the distributional properties very close to the Hawkes model. As a diffusion process, the model provides more analytical simplicity when computing the variance formula, incorporating skewness and examining the probabilistic property. An estimation of the diffusion model is performed using the simulated maximum likelihood method and shows similar patterns to the Hawkes model.

Figures

Figures reproduced from arXiv: 1908.05089 by the authors.

Figure 1
Figure 1. Maximum log-likelihood function when β is fixed for simulation set 1 (left) and 2 (right) 3 Diffusion analogy 3.1 Diffusion model This subsection proposes a new diffusion approach for the tick structure. The diffusion model is analogous to the symmetric Hawkes model and has a similar probabilistic property. When the price process is represented by the difference of the two Hawkes process, the increment of the price … view at source ↗
Figure 2
Figure 2. Numerically computed probability density function of the price driven by the diffusion model [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The comparison between the volatility computed by the diffusion model and the symmetric [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Annualized volatility surface as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Mean signature plot with fixed κ1 = 0.5 with various φ (left) and fixed φ = −0.3 and various κ1 (right) where η is a newly introduced parameter. It is believed that there are many possible ways to incorporate asymmetry into the Hawkes model. In general, estimating ρ in…
Figure 6
Figure 6. Figure 6: Convergence of the estimates of ρ [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Hawkes model and diffusion analogy 3.3 Comparison Both the Hawkes and the diffusion models well describe the microstructure of price dynamics such as trade clustering or microstructure noise. The Hawkes model directly describes the tick-by-tick structure of the asset p…
Figure 8
Figure 8. Figure 8: Symmetric Hawkes estimation result, GE, 2011 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Symmetric Hawkes estimation result, GE, 2010 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Volatility comparisons with symmetric Hawkes estimation results, T (left) and MCD (right), [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Symmetric Hawkes estimation result, GE, 2008 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Estimated cumulative intraday volatility (annualized) with every ten minutes update [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Estimation result with the fully characterized Hawkes, GE, 2011 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    A \" t-Sahalia, Y., Cacho-Diaz, J., and Laeven, R. J. (2010). Modeling financial contagion using mutually exciting jump processes. Technical report, National Bureau of Economic Research

    work page 2010

  2. [2]

    A \" t-Sahalia, Y. et al. (2008). Closed-form likelihood expansions for multivariate diffusions. The Annals of Statistics , 36(2):906--937

    work page 2008

  3. [3]

    Ait-Sahalia, Y., Fan, J., and Li, Y. (2013). The leverage effect puzzle: Disentangling sources of bias at high frequency. Journal of Financial Economics , 109:224--249

    work page 2013

  4. [4]

    A., and Zhang, L

    A \" t-Sahalia, Y., Mykland, P. A., and Zhang, L. (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial studies , 18:351--416

    work page 2005

  5. [5]

    A., and Zhang, L

    A \" t-Sahalia, Y., Mykland, P. A., and Zhang, L. (2011). Ultra high frequency volatility estimation with dependent microstructure noise. Journal of Econometrics , 160:160--175

    work page 2011

  6. [6]

    G., Bollerslev, T., Diebold, F

    Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica , 71:579--625

    work page 2003

  7. [7]

    Bacry, E., Dayri, K., and Muzy, J.-F. (2012). Non-parametric kernel estimation for symmetric Hawkes processes. application to high frequency financial data. The European Physical Journal B-Condensed Matter and Complex Systems , 85:1--12

    work page 2012

  8. [8]

    Bacry, E., Delattre, S., Hoffmann, M., and Muzy, J.-F. (2013). Modelling microstructure noise with mutually exciting point processes. Quantitative Finance , 13:65--77

    work page 2013

Show all 43 references
  1. [9]

    and Muzy, J.-F

    Bacry, E. and Muzy, J.-F. (2014). Hawkes model for price and trades high-frequency dynamics. Quantitative Finance , 14:1147--1166

    2014

  2. [10]

    Barndorff-Nielsen, O. E. and Shephard, N. (2002a). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 64:253--280

    2002

  3. [11]

    Barndorff-Nielsen, O. E. and Shephard, N. (2002b). Estimating quadratic variation using realized variance. Journal of Applied Econometrics , 17:457--477

    2002

  4. [12]

    and Hautsch, N

    Bauwens, L. and Hautsch, N. (2009). Modelling financial high frequency data using point processes. In Handbook of Financial Time Series , pages 953--979. Springer Berlin Heidelberg

    2009

  5. [13]

    Bollerslev, T., Gibson, M., and Zhou, H. (2011). Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities. Journal of econometrics , 160(1):235--245

    2011

  6. [14]

    Bowsher, C. G. (2007). Modelling security market events in continuous time: Intensity based, multivariate point process models. Journal of Econometrics , 141:876--912

    2007

  7. [15]

    Brandt, M. W. and Santa-Clara, P. (2002). Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. Journal of financial economics , 63:161--210

    2002

  8. [16]

    Br \'e maud, P. (1981). Point Processes and Queues : Martingale Dynamics . Springer

    1981

  9. [17]

    Broyden, C. G. (1970). The convergence of a class of double-rank minimization algorithms 1. general considerations. IMA Journal of Applied Mathematics , 6:76--90

    1970

  10. [18]

    Choe, G. H. and Lee, K. (2014a). Conditional correlation in asset return and garch intensity model. AStA Advances in Statistical Analysis , 98:197--224

    2014

  11. [19]

    Choe, G. H. and Lee, K. (2014b). High moment variations and their application. Journal of Futures Markets , 34:1040--1061

    2014

  12. [20]

    and Zaatour, R

    Da Fonseca, J. and Zaatour, R. (2014a). Clustering and mean reversion in a Hawkes microstructure model. Journal of Futures Markets , 35:813--838

    2014

  13. [21]

    and Zaatour, R

    Da Fonseca, J. and Zaatour, R. (2014b). Hawkes process: Fast calibration, application to trade clustering, and diffusive limit. Journal of Futures Markets , 34:548--579

    2014

  14. [22]

    Daley, D. J. and Vere-Jones, D. (2003). An introduction to the theory of point processes , volume 1. Springer

    2003

  15. [23]

    and Zhao, H

    Dassios, A. and Zhao, H. (2012). Ruin by dynamic contagion claims. Insurance: Mathematics and Economics , 51:93--106

    2012

  16. [24]

    El Euch, O., Masaaki, F., and Mathieu, R. (2016). The microstructural foundations of leverage effect and rough volatility. arXiv preprint arXiv:1609.05177

    2016 arXiv

  17. [25]

    Embrechts, P., Liniger, T., Lin, L., et al. (2011). Multivariate hawkes processes: an application to financial data. Journal of Applied Probability , 48:367--378

    2011

  18. [26]

    Errais, E., Giesecke, K., and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM Journal on Financial Mathematics , 1:642--665

    2010

  19. [27]

    Garcia, R., Lewis, M.-A., Pastorello, S., and Renault, \'E . (2011). Estimation of objective and risk-neutral distributions based on moments of integrated volatility. Journal of Econometrics , 160(1):22--32

    2011

  20. [28]

    Hansen, P. R. and Lunde, A. (2006). Realized variance and market microstructure noise. Journal of Business and Economic Statistics , 24:127--161

    2006

  21. [29]

    Hawkes, A. G. (1971a). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society. Series B (Methodological) , pages 438--443

    1971

  22. [30]

    Hawkes, A. G. (1971b). Spectra of some self-exciting and mutually exciting point processes. Biometrika , 58:83--90

    1971

  23. [31]

    Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. Journal of Applied Probability , pages 493--503

    1974

  24. [32]

    and Toomet, O

    Henningsen, A. and Toomet, O. (2011). maxlik: A package for maximum likelihood estimation in R . Computational Statistics , 26:443--458

    2011

  25. [33]

    Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies , 6:327--343

    1993

  26. [34]

    Hewlett, P. (2006). Clustering of order arrivals, price impact and trade path optimisation. In Workshop on Financial Modeling with Jump processes, Ecole Polytechnique

    2006

  27. [35]

    Jaisson, T., Rosenbaum, M., et al. (2015). Limit theorems for nearly unstable Hawkes processes. The Annals of Applied Probability , 25:600--631

    2015

  28. [36]

    Large, J. (2007). Measuring the resiliency of an electronic limit order book. Journal of Financial Markets , 10:1--25

    2007

  29. [37]

    Lee, K. (2016). Probabilistic and statistical properties of moment variations and their use in inference and estimation based on high frequency return data. Studies in Nonlinear Dynamics & Econometrics , 20:19--36

    2016

  30. [38]

    and Fukushima, M

    Li, D.-H. and Fukushima, M. (2001). On the global convergence of the bfgs method for nonconvex unconstrained optimization problems. SIAM Journal on Optimization , 11:1054--1064

    2001

  31. [39]

    Nash, J. C. et al. (2014). On best practice optimization methods in R . Journal of Statistical Software , 60

    2014

  32. [40]

    Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics , 30:243--261

    1978

  33. [41]

    Robert, C. Y. and Rosenbaum, M. (2011). A new approach for the dynamics of ultra-high-frequency data: The model with uncertainty zones. Journal of Financial Econometrics , 9:344--366

    2011

  34. [42]

    A., and A \" t-Sahalia, Y

    Zhang, L., Mykland, P. A., and A \" t-Sahalia, Y. (2005). A tale of two time scales. Journal of the American Statistical Association , 100:1394--1411

    2005

  35. [43]

    Zheng, B., Roueff, F., and Abergel, F. (2014). Modelling bid and ask prices using constrained Hawkes processes: Ergodicity and scaling limit. SIAM Journal on Financial Mathematics , 5:99--136

    2014

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Reviewed August 14, 2026 · model on record in the stance chip above.