We present a simulation framework for limit order book (LOB) dynamics based on the Five-Type Queue-Reactive (FTQR) model coupled with multivariate Hawkes processes. The five event types — limit bid arrivals ($\lambda_b$), limit ask arrivals ($\lambda_a$), bid cancellations ($\theta_b$), ask cancellations ($\theta_a$), and market order arrivals ($\mu$) — are modelled with state-dependent intensities $a_i \, q^{\beta}$ that scale with queue size $q_b$ or $q_a$, with short-term temporal clustering captured by exponential Hawkes kernels. Price is emergent: the mid-price moves only when a best-queue is depleted below a depletion threshold, never by direct prescription. We demonstrate via simulation that the model reproduces key microstructural stylized facts including queue size mean-reversion, bid-ask asymmetry, and clustered price moves, providing a tractable foundation for reversal signal detection in liquid futures markets.
We develop a Wishart stochastic volatility framework for the two-sided futures order book and establish its role as the correct continuum limit of the multivariate FTQR-Hawkes microstructure model. The variance state $V(t)$ evolves as a $2 \times 2$ positive-definite matrix CIR process, capturing the full covariation $v_{12}(t)$ between bid-side and ask-side volatility — a quantity that the Double Heston model is structurally forced to set to zero. We prove that the Wishart model retains an exponential-affine characteristic function, with the matrix Riccati ODE solved via a $4 \times 4$ Hamiltonian matrix exponential, enabling Carr-Madan FFT calibration to the joint call-put option surface. The off-diagonal entry $v_{12}(0)$ recovered from this calibration initialises the Hawkes cross-excitation matrix $\alpha_{12}$, and three mean-field game systems yield rational intensity benchmarks $\lambda^*_b$, $\lambda^*_a$ whose signed divergence from empirical intensities generates high-conviction intraday trading signals.
We study optimal market making in a limit order book where the dealer faces inventory risk and adverse selection from informed traders, following the Avellaneda–Stoikov (2008) framework extended to include asymmetric information costs. The dealer controls bid and ask quote depths to maximise expected terminal wealth minus a quadratic inventory penalty, leading to a Hamilton–Jacobi–Bellman equation whose solution yields closed-form reservation prices and optimal spreads. The reservation price \(r(s,q,\tau)\) shifts linearly with inventory \(q\) and remaining horizon \(\tau = T - t\), while the optimal spread \(\delta^* = \delta^a + \delta^b\) decomposes into a risk-aversion component proportional to \(\gamma \sigma^2 \tau\) and an adverse-selection component \(\tfrac{2}{\gamma}\ln\!\left(1 + \tfrac{\gamma}{\kappa}\right)\). Simulated inventory paths under the optimal policy exhibit mean-reverting behaviour governed by the asymmetric quoting rule, with the dealer widening quotes on the side that would exacerbate a large inventory position.
We study the dynamics of a continuous-time limit order book model in which market orders arrive as Poisson processes and the mid-price evolves as a diffusion driven by order-flow imbalance. Price impact is decomposed into a temporary component and a permanent component that shifts the fundamental value. Using a Hamilton--Jacobi--Bellman framework we derive the optimal liquidation strategy for a large trader minimising expected execution cost subject to a terminal inventory constraint, obtaining a closed-form feedback control in the linear--quadratic case and a numerical solution via backward Euler for nonlinear impact functions.
We study multivariate Hawkes processes as models for high-frequency trade arrivals, capturing the self-exciting and cross-exciting structure of order flow across asset classes. The intensity of each arrival stream depends on the full history of all streams via a matrix of exponential kernels, whose parameters are estimated by maximum likelihood using an expectation-maximisation algorithm. Calibrated on simulated tick data, the model reproduces the clustering of trades, the elevated autocorrelation of inter-arrival times, and the asymmetric cross-excitation patterns characteristic of correlated liquid assets.
We study a continuous-time Kyle–Back insider trading model in which the informed agent receives the asset's true value $V$ at a random Poisson time $\tau \sim \mathrm{Exp}(\mu)$, rather than at inception. The model decomposes into two phases: before signal arrival, the price is uninformative and the insider's continuation value satisfies a linear ODE; after arrival, a standard Kyle–Back equilibrium operates on the residual horizon. Our main result is a closed-form formula for the insider's expected profit, $\mathbb{E}[\Pi](\mu) = \tfrac{\sigma_z \sqrt{\Sigma_0}}{2}\!\left[\sqrt{T} - \tfrac{\sqrt{\pi}}{2\sqrt{\mu}} e^{-\mu T} \,\mathrm{erfi}(\sqrt{\mu T})\right]$, involving the imaginary error function. The formula interpolates between zero profit ($\mu \to 0$) and the classical Kyle profit $\tfrac{\sigma_z\sqrt{\Sigma_0 T}}{2}$ ($\mu \to \infty$), and is strictly increasing in $\mu$, $\Sigma_0$, $T$, and $\sigma_z$.