{ "cells": [ { "cell_type": "markdown", "id": "39981e04", "metadata": {}, "source": [ "
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📚 Prerequisites — Multi-asset OU optimal execution

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Free sample · Companion course €34.99 / Education tier €19/mo

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Mathematical scaffolding required to fully understand Cross-Asset Microstructure & Portfolio-Level Short-Term PnL: multi-dim Itô calculus, Ornstein–Uhlenbeck dynamics, HJB on $\\mathbb{R}^d$, the quadratic ansatz, matrix Riccati ODEs, Hayashi–Yoshida non-synchronous covariance, optimal trading speed and a fully-solved 2D toy example with rich visuals.

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\n", "\n", "> **License**: Free sample notebook. CC BY-NC 4.0 — non-commercial use only. Citation: HFThot Research Lab, *Microstructure OU Prerequisites*, 2026." ] }, { "cell_type": "markdown", "id": "0287a764", "metadata": {}, "source": [ "## 📑 Table of contents\n", "\n", "1. [Multi-dimensional Itô calculus](#1) · *visuals: 1D & 2D Brownian paths*\n", "2. [Ornstein–Uhlenbeck (OU) dynamics](#2) · *visuals: paths & stationary distribution*\n", "3. [Hamilton–Jacobi–Bellman equation](#3)\n", "4. [The quadratic ansatz](#4)\n", "5. [Matrix Riccati system — solved 2D example](#5) · *visuals: norms, components, heatmap*\n", "6. [Optimal trading speed under OU](#6) · *visuals: inventory & speed trajectories*\n", "7. [Hayashi–Yoshida non-synchronous covariance](#7) · *visuals: Epps effect*\n", "8. [What the full course covers (paid)](#8)\n", "\n", "---\n", "\n", "**Reproducibility:** all cells use `numpy` + `matplotlib` only. Run top-to-bottom on Python ≥ 3.10. Set the random seed in the next cell for identical figures." ] }, { "cell_type": "code", "execution_count": null, "id": "425c5152", "metadata": {}, "outputs": [], "source": [ "# ─── Setup ────────────────────────────────────────────────\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# Reproducibility\n", "RNG_SEED = 42\n", "rng = np.random.default_rng(RNG_SEED)\n", "\n", "# HFThot dark theme palette\n", "HF_BG = '#0b132b'\n", "HF_PANEL = '#1c2541'\n", "HF_GOLD = '#d4af37'\n", "HF_TEAL = '#5bc0be'\n", "HF_AMBER = '#fbbf24'\n", "HF_RED = '#ef4444'\n", "HF_GREEN = '#10b981'\n", "HF_TEXT = '#e0fbfc'\n", "\n", "def hf_fig(figsize=(10, 4), nrows=1, ncols=1):\n", " fig, ax = plt.subplots(nrows, ncols, figsize=figsize, facecolor=HF_BG)\n", " axes = np.atleast_1d(ax).ravel() if (nrows*ncols > 1) else [ax]\n", " for a in axes:\n", " a.set_facecolor(HF_BG)\n", " a.tick_params(colors=HF_TEXT)\n", " for spine in a.spines.values():\n", " spine.set_color(HF_TEXT); spine.set_alpha(0.4)\n", " a.grid(True, ls=':', alpha=0.3, color=HF_TEXT)\n", " return fig, ax\n", "\n", "print(f'NumPy {np.__version__} · seed = {RNG_SEED}')" ] }, { "cell_type": "markdown", "id": "dea81848", "metadata": {}, "source": [ "\n", "## 1 · Multi-dimensional Itô calculus" ] }, { "cell_type": "markdown", "id": "c4f84777", "metadata": {}, "source": [ "### 1.1 The setting\n", "\n", "Let $X_t \\in \\mathbb{R}^d$ be an Itô process driven by an $m$-dimensional Brownian motion $W_t$:\n", "\n", "$$\n", "dX_t \\;=\\; \\mu(X_t, t)\\,dt \\;+\\; \\sigma(X_t, t)\\,dW_t,\n", "\\qquad \\mu \\in \\mathbb{R}^d, \\quad \\sigma \\in \\mathbb{R}^{d\\times m}.\n", "$$\n", "\n", "The **drift** $\\mu$ describes the deterministic trend, the **diffusion** $\\sigma$ the local volatility/correlation structure." ] }, { "cell_type": "markdown", "id": "4d3a1465", "metadata": {}, "source": [ "### 1.2 The Itô formula\n", "\n", "For any sufficiently smooth $f: \\mathbb{R}^d \\times [0,T] \\to \\mathbb{R}$ ($C^{1,2}$):\n", "\n", "$$\n", "df(X_t, t) \\;=\\; \\Bigl[\\partial_t f \\;+\\; (\\nabla_x f)^{\\top} \\mu \\;+\\; \\tfrac12\\,\\mathrm{tr}\\bigl(\\sigma\\sigma^{\\top}\\,\\nabla_x^2 f\\bigr)\\Bigr]\\,dt \\;+\\; (\\nabla_x f)^{\\top}\\sigma\\,dW_t.\n", "$$\n", "\n", "The last (martingale) term has **zero expectation**; the bracketed term — the *Itô drift* — is what dynamic programming will extremize." ] }, { "cell_type": "markdown", "id": "8ca7a7f6", "metadata": {}, "source": [ "### 1.3 Why it matters here\n", "\n", "When $\\mu = \\Phi(\\bar\\mu - X_t)$ (mean-reverting) and $\\sigma\\sigma^\\top = \\Sigma$ (constant), and $f$ is the **quadratic value function**\n", "\n", "$$J(t,q,s) = q^\\top A(t)\\,q + q^\\top B(t)(s-\\bar\\mu) + q^\\top C(t) + D(t),$$\n", "\n", "the Itô drift collapses into a *finite-dimensional system of matrix ODEs* — the **Riccati system** of section 5. This is the same trick that powers the LQR controller in classical engineering." ] }, { "cell_type": "markdown", "id": "cea7bde9", "metadata": {}, "source": [ "### 1.4 Visual — single Brownian path with quadratic-variation envelope" ] }, { "cell_type": "code", "execution_count": null, "id": "ceedecb8", "metadata": {}, "outputs": [], "source": [ "# Single 1D Brownian motion + ±√t envelope\n", "T = 1.0; N = 1000\n", "t = np.linspace(0, T, N+1)\n", "dW = rng.normal(scale=np.sqrt(T/N), size=N)\n", "W = np.concatenate([[0.0], np.cumsum(dW)])\n", "\n", "fig, ax = hf_fig((10, 4))\n", "ax.plot(t, W, color=HF_TEAL, lw=1.5, label='$W_t$')\n", "ax.fill_between(t, -np.sqrt(t), np.sqrt(t), color=HF_GOLD, alpha=0.15, label=r'$\\pm\\sqrt{t}$ envelope')\n", "ax.axhline(0, color=HF_TEXT, lw=0.5, alpha=0.4)\n", "ax.set_xlabel('time $t$', color=HF_TEXT)\n", "ax.set_ylabel('$W_t$', color=HF_TEXT)\n", "ax.set_title('1D Brownian motion sample path', color=HF_GOLD)\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT, loc='upper left')\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "67b11c1b", "metadata": {}, "source": [ "### 1.5 Visual — 2D correlated Brownian motion" ] }, { "cell_type": "code", "execution_count": null, "id": "d9bf1a2c", "metadata": {}, "outputs": [], "source": [ "# 2D correlated Brownian motion — Cholesky factorisation\n", "rho = 0.6\n", "L = np.linalg.cholesky(np.array([[1.0, rho], [rho, 1.0]]))\n", "\n", "dW = rng.normal(scale=np.sqrt(T/N), size=(N, 2)) @ L.T\n", "W2 = np.vstack([np.zeros((1, 2)), np.cumsum(dW, axis=0)])\n", "\n", "fig, ax = hf_fig((6, 6))\n", "ax.plot(W2[:, 0], W2[:, 1], color=HF_AMBER, lw=1.2, alpha=0.85)\n", "ax.scatter(W2[0, 0], W2[0, 1], color=HF_GREEN, zorder=5, s=60, label='start')\n", "ax.scatter(W2[-1, 0], W2[-1, 1], color=HF_RED, zorder=5, s=60, label='end')\n", "ax.set_xlabel('$W^{(1)}_t$', color=HF_TEXT)\n", "ax.set_ylabel('$W^{(2)}_t$', color=HF_TEXT)\n", "ax.set_title(f'2D Brownian path · correlation $\\\\rho={rho}$', color=HF_GOLD)\n", "ax.set_aspect('equal')\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT)\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "12557032", "metadata": {}, "source": [ "\n", "## 2 · Ornstein–Uhlenbeck (OU) dynamics" ] }, { "cell_type": "markdown", "id": "02a8f7b7", "metadata": {}, "source": [ "### 2.1 SDE\n", "\n", "The OU process is the canonical mean-reverting Itô process:\n", "\n", "$$\n", "dS_t \\;=\\; \\Phi\\,(\\bar\\mu - S_t)\\,dt \\;+\\; \\Sigma^{1/2}\\,dW_t.\n", "$$\n", "\n", "Here $\\Phi \\in \\mathbb{R}^{d\\times d}$ is the **mean-reversion matrix** (eigenvalues control the speed of return to $\\bar\\mu$), $\\bar\\mu$ is the long-run mean, and $\\Sigma$ is the (constant) covariance." ] }, { "cell_type": "markdown", "id": "ceec4308", "metadata": {}, "source": [ "### 2.2 Half-life\n", "\n", "For a diagonal $\\Phi = \\mathrm{diag}(\\phi_1, \\ldots, \\phi_d)$, each component has the half-life\n", "\n", "$$\n", "\\tau^{1/2}_i \\;=\\; \\frac{\\ln 2}{\\phi_i}.\n", "$$\n", "\n", "In HFT/microstructure, $\\tau^{1/2}$ ranges from **seconds** (tick noise) to **hours** (basket convergence trades)." ] }, { "cell_type": "markdown", "id": "3037efc3", "metadata": {}, "source": [ "### 2.3 Visual — OU paths with three different half-lives" ] }, { "cell_type": "code", "execution_count": null, "id": "5f18a0c7", "metadata": {}, "outputs": [], "source": [ "# Compare three OU half-lives on the same horizon\n", "def simulate_ou(phi, sigma, T, N, x0=0.0, mu_bar=0.0, seed=0):\n", " rng_ = np.random.default_rng(seed)\n", " dt = T/N\n", " x = np.empty(N+1); x[0] = x0\n", " sqdt = np.sqrt(dt)\n", " for k in range(N):\n", " x[k+1] = x[k] + phi*(mu_bar - x[k])*dt + sigma*sqdt*rng_.normal()\n", " return x\n", "\n", "T, N = 600.0, 6000 # 10 min @ 100 ms\n", "half_lives = [10.0, 60.0, 300.0] # 10s, 1min, 5min\n", "colors_ou = [HF_TEAL, HF_AMBER, HF_RED]\n", "\n", "fig, ax = hf_fig((11, 4.5))\n", "t = np.linspace(0, T, N+1)\n", "for hl, col in zip(half_lives, colors_ou):\n", " phi = np.log(2)/hl\n", " x = simulate_ou(phi, sigma=0.05, T=T, N=N, seed=int(hl))\n", " ax.plot(t, x, color=col, lw=1.2, alpha=0.9, label=fr'$\\tau_{{1/2}} = {hl:.0f}$ s')\n", "\n", "ax.axhline(0, color=HF_GOLD, lw=0.8, ls='--', alpha=0.6, label=r'$\\bar\\mu$')\n", "ax.set_xlabel('time $t$ (s)', color=HF_TEXT)\n", "ax.set_ylabel('$S_t$', color=HF_TEXT)\n", "ax.set_title('OU paths · varying half-lives (same noise scale)', color=HF_GOLD)\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT, loc='upper right')\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "ef6cbb79", "metadata": {}, "source": [ "### 2.4 Visual — empirical stationary distribution" ] }, { "cell_type": "code", "execution_count": null, "id": "5295acae", "metadata": {}, "outputs": [], "source": [ "# Long simulation → stationary histogram vs theoretical Gaussian N(0, sigma^2/(2*phi))\n", "phi = np.log(2)/30.0 # half-life 30s\n", "sigma = 0.05\n", "T_long, N_long = 50_000.0, 500_000\n", "x = simulate_ou(phi, sigma, T_long, N_long, seed=7)\n", "x_stationary = x[N_long//5:] # discard burn-in\n", "\n", "# theoretical std of the stationary distribution\n", "sd_theory = sigma / np.sqrt(2*phi)\n", "xs = np.linspace(x_stationary.min(), x_stationary.max(), 400)\n", "pdf = np.exp(-xs**2 / (2*sd_theory**2)) / (sd_theory*np.sqrt(2*np.pi))\n", "\n", "fig, ax = hf_fig((10, 4))\n", "ax.hist(x_stationary, bins=80, density=True, color=HF_TEAL, alpha=0.55,\n", " edgecolor=HF_GOLD, lw=0.4, label='empirical')\n", "ax.plot(xs, pdf, color=HF_AMBER, lw=2.5, label=fr'$\\mathcal{{N}}(0,\\,\\sigma^2/2\\phi)$, sd={sd_theory:.4f}')\n", "ax.set_xlabel('$S_\\\\infty$', color=HF_TEXT)\n", "ax.set_ylabel('density', color=HF_TEXT)\n", "ax.set_title('OU stationary distribution — empirical vs. theory', color=HF_GOLD)\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT)\n", "plt.tight_layout(); plt.show()\n", "print(f'empirical std = {x_stationary.std():.5f} · theoretical = {sd_theory:.5f}')" ] }, { "cell_type": "markdown", "id": "6f41d920", "metadata": {}, "source": [ "\n", "## 3 · Hamilton–Jacobi–Bellman equation" ] }, { "cell_type": "markdown", "id": "bd40e7df", "metadata": {}, "source": [ "### 3.1 The control problem\n", "\n", "The trader chooses a **trading speed** $v_t \\in \\mathbb{R}^d$ (positive = buy, negative = sell). The state evolves as\n", "\n", "$$\n", "\\begin{aligned}\n", "dq_t &= v_t\\,dt && \\text{(inventory)} \\\\\n", "dS_t &= \\Phi(\\bar\\mu - S_t)\\,dt + \\Sigma^{1/2} dW_t && \\text{(asset prices, OU)} \\\\\n", "dX_t &= -\\,v_t \\!\\cdot\\! S_t\\,dt \\;-\\; \\tfrac12\\,v_t^\\top \\eta\\, v_t\\,dt && \\text{(cash, with quadratic impact)}\n", "\\end{aligned}\n", "$$\n", "\n", "where $\\eta \\in \\mathbb{R}^{d\\times d}$ (s.p.d.) is the temporary market-impact matrix." ] }, { "cell_type": "markdown", "id": "1005cf2f", "metadata": {}, "source": [ "### 3.2 The objective\n", "\n", "Maximise the **mean–variance functional**\n", "\n", "$$\n", "J(t, q, s) \\;=\\; \\sup_{v} \\mathbb{E}\\Bigl[\\; X_T \\;+\\; q_T^\\top S_T \\;-\\; \\tfrac{\\gamma}{2}\\!\\int_t^T\\! q_u^\\top \\Sigma\\, q_u\\,du \\;\\Big|\\; \\mathcal{F}_t\\Bigr],\n", "$$\n", "\n", "with the running risk-aversion penalty $\\gamma > 0$ on inventory exposure to volatility." ] }, { "cell_type": "markdown", "id": "cd6457a8", "metadata": {}, "source": [ "### 3.3 The HJB equation\n", "\n", "By dynamic programming + Itô on $J$:\n", "\n", "$$\n", "\\partial_t J \\;+\\; \\sup_{v\\in\\mathbb{R}^d}\\Bigl\\{(\\nabla_q J)^\\top v - v^\\top s - \\tfrac12 v^\\top\\eta v\\Bigr\\} \\;+\\; (\\nabla_s J)^\\top \\Phi(\\bar\\mu-s) \\;+\\; \\tfrac12\\,\\mathrm{tr}(\\Sigma\\,\\nabla_s^2 J) \\;-\\; \\tfrac\\gamma2\\,q^\\top\\Sigma q \\;=\\; 0.\n", "$$" ] }, { "cell_type": "markdown", "id": "2831d857", "metadata": {}, "source": [ "### 3.4 Hamiltonian — closed-form maximiser\n", "\n", "The bracketed term is concave-quadratic in $v$, so the first-order condition gives the **optimal trading speed**:\n", "\n", "$$\n", "\\boxed{\\;v^\\star \\;=\\; \\eta^{-1}\\bigl(\\nabla_q J - s\\bigr)\\;}\n", "$$\n", "\n", "Substituting $v^\\star$ back yields a *parabolic, semilinear PDE* for $J$ — solved next by ansatz." ] }, { "cell_type": "markdown", "id": "a2199b0c", "metadata": {}, "source": [ "\n", "## 4 · The quadratic ansatz" ] }, { "cell_type": "markdown", "id": "d54d84aa", "metadata": {}, "source": [ "### 4.1 Postulate (Bergault–Drissi–Guéant 2022)\n", "\n", "$$\n", "J(t,q,s) \\;=\\; q^\\top A(t)\\,q \\;+\\; q^\\top B(t)\\,(s-\\bar\\mu) \\;+\\; q^\\top C(t) \\;+\\; D(t).\n", "$$\n", "\n", "Here $A(t) \\in \\mathbb{S}^d_+$, $B(t) \\in \\mathbb{R}^{d\\times d}$, $C(t) \\in \\mathbb{R}^d$, $D(t) \\in \\mathbb{R}$." ] }, { "cell_type": "markdown", "id": "03cf921f", "metadata": {}, "source": [ "### 4.2 The Riccati system\n", "\n", "Plugging the ansatz into the HJB equation and matching by powers of $(q, s)$ produces:\n", "\n", "$$\n", "\\begin{aligned}\n", "\\dot A(t) &= -\\,2\\,A(t)\\,\\eta^{-1}\\,A(t) \\;+\\; \\tfrac{\\gamma}{2}\\,\\Sigma \\\\\n", "\\dot B(t) &= -\\,2\\,A(t)\\,\\eta^{-1}\\,B(t) \\;+\\; \\Phi^\\top\\,B(t) \\\\\n", "\\dot C(t) &= -\\,2\\,A(t)\\,\\eta^{-1}\\,C(t)\n", "\\end{aligned}\n", "$$\n", "\n", "with **terminal conditions** (mark-to-market at horizon $T$):\n", "\n", "$$\n", "A(T) = 0, \\qquad B(T) = I, \\qquad C(T) = 0.\n", "$$" ] }, { "cell_type": "markdown", "id": "01fc3f1e", "metadata": {}, "source": [ "### 4.3 Why this works\n", "\n", "The HJB operator on linear-Gaussian dynamics **preserves quadratic forms**: a quadratic guess produces only quadratic, linear and constant terms after differentiation and the trace operation. Matching powers of $(q,s)$ therefore *closes* the equations into an ODE system — exactly the same trick as in the **Linear-Quadratic Regulator** of classical control." ] }, { "cell_type": "markdown", "id": "22eb15f5", "metadata": {}, "source": [ "\n", "## 5 · Matrix Riccati system — solved 2D example" ] }, { "cell_type": "markdown", "id": "d39910e0", "metadata": {}, "source": [ "### 5.1 Toy parameters\n", "\n", "Two crypto-style assets with **30-min** and **1-h** half-lives, 60 % correlation, identical impact and a moderate risk aversion. Horizon $T = 30$ min." ] }, { "cell_type": "code", "execution_count": null, "id": "ea2935d5", "metadata": {}, "outputs": [], "source": [ "# 2-asset toy parameters\n", "d = 2\n", "Phi = np.diag([np.log(2)/3600.0, np.log(2)/1800.0]) # half-lives 1 h, 30 min\n", "Sigma = np.array([[1.0, 0.6], [0.6, 1.0]]) # 60 % correlation\n", "eta = np.diag([1e-4, 1e-4]) # impact\n", "gamma = 1e-3 # risk aversion\n", "T = 1800.0 # horizon: 30 min\n", "N = 600 # outer steps\n", "n_sub = 20 # inner RK4 sub-steps\n", "\n", "print('Φ (1/s) =', np.diag(Phi))\n", "print('half-lives (s) =', np.log(2)/np.diag(Phi))\n", "print('Σ =\\n', Sigma)\n", "print('η =\\n', eta)\n", "print(f'γ = {gamma}, T = {T:.0f}s, N = {N}')" ] }, { "cell_type": "markdown", "id": "258135ee", "metadata": {}, "source": [ "### 5.2 Backward RK4 integrator" ] }, { "cell_type": "code", "execution_count": null, "id": "cba01c04", "metadata": {}, "outputs": [], "source": [ "# Allocate trajectories\n", "t_grid = np.linspace(0, T, N+1)\n", "A = np.zeros((N+1, d, d))\n", "B = np.zeros((N+1, d, d))\n", "C = np.zeros((N+1, d))\n", "A[-1] = np.zeros((d, d))\n", "B[-1] = np.eye(d)\n", "C[-1] = np.zeros(d)\n", "\n", "eta_inv = np.linalg.inv(eta)\n", "\n", "def rhs(A_, B_, C_):\n", " dA = -2*A_ @ eta_inv @ A_ + 0.5*gamma*Sigma\n", " dB = -2*A_ @ eta_inv @ B_ + Phi.T @ B_\n", " dC = -2*A_ @ eta_inv @ C_\n", " return dA, dB, dC\n", "\n", "dt_outer = T / N\n", "for k in range(N, 0, -1):\n", " Ak, Bk, Ck = A[k], B[k], C[k]\n", " h = dt_outer / n_sub\n", " for _ in range(n_sub):\n", " k1 = rhs(Ak, Bk, Ck)\n", " k2 = rhs(Ak - 0.5*h*k1[0], Bk - 0.5*h*k1[1], Ck - 0.5*h*k1[2])\n", " k3 = rhs(Ak - 0.5*h*k2[0], Bk - 0.5*h*k2[1], Ck - 0.5*h*k2[2])\n", " k4 = rhs(Ak - h*k3[0], Bk - h*k3[1], Ck - h*k3[2])\n", " Ak = Ak - (h/6)*(k1[0] + 2*k2[0] + 2*k3[0] + k4[0])\n", " Bk = Bk - (h/6)*(k1[1] + 2*k2[1] + 2*k3[1] + k4[1])\n", " Ck = Ck - (h/6)*(k1[2] + 2*k2[2] + 2*k3[2] + k4[2])\n", " A[k-1], B[k-1], C[k-1] = Ak, Bk, Ck\n", "\n", "print('A(0) =\\n', A[0])\n", "print('B(0) =\\n', B[0])\n", "print('C(0) =', C[0])" ] }, { "cell_type": "markdown", "id": "2687a502", "metadata": {}, "source": [ "### 5.3 Visual — Frobenius norms grow backward in time" ] }, { "cell_type": "code", "execution_count": null, "id": "7849b05b", "metadata": {}, "outputs": [], "source": [ "# Plot ||A(t)||_F and ||B(t)||_F\n", "fig, ax = hf_fig((10, 4.5))\n", "ax.plot(t_grid/60, np.linalg.norm(A, axis=(1,2)), color=HF_TEAL, lw=2.2, label='$\\\\|A(t)\\\\|_F$')\n", "ax.plot(t_grid/60, np.linalg.norm(B, axis=(1,2)), color=HF_AMBER, lw=2.2, label='$\\\\|B(t)\\\\|_F$')\n", "ax.axvline(T/60, color=HF_GOLD, ls='--', alpha=0.6, label='terminal $T$')\n", "ax.set_xlabel('time $t$ (min)', color=HF_TEXT)\n", "ax.set_ylabel('Frobenius norm', color=HF_TEXT)\n", "ax.set_title('2D Riccati — Frobenius norms of $A(t)$ and $B(t)$', color=HF_GOLD)\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT)\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "05ea1b4a", "metadata": {}, "source": [ "### 5.4 Visual — individual entries of $A(t)$" ] }, { "cell_type": "code", "execution_count": null, "id": "3f2b9b2d", "metadata": {}, "outputs": [], "source": [ "# Decompose A(t) into its 3 unique entries (symmetric)\n", "fig, ax = hf_fig((10, 4.5))\n", "ax.plot(t_grid/60, A[:, 0, 0], color=HF_TEAL, lw=2, label='$A_{11}(t)$')\n", "ax.plot(t_grid/60, A[:, 1, 1], color=HF_AMBER, lw=2, label='$A_{22}(t)$')\n", "ax.plot(t_grid/60, A[:, 0, 1], color=HF_RED, lw=2, label='$A_{12}(t)$')\n", "ax.set_xlabel('time $t$ (min)', color=HF_TEXT)\n", "ax.set_ylabel('entry value', color=HF_TEXT)\n", "ax.set_title('Individual entries of $A(t)$ — symmetric matrix has 3 d.o.f.', color=HF_GOLD)\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT)\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "a686d6d9", "metadata": {}, "source": [ "### 5.5 Visual — heatmap of $A(0)$" ] }, { "cell_type": "code", "execution_count": null, "id": "9dc3b79f", "metadata": {}, "outputs": [], "source": [ "# Visualise A(0) as a heatmap\n", "fig, ax = hf_fig((5, 4))\n", "im = ax.imshow(A[0], cmap='magma', aspect='equal')\n", "for i in range(d):\n", " for j in range(d):\n", " ax.text(j, i, f'{A[0,i,j]:.2e}', ha='center', va='center',\n", " color='white' if abs(A[0,i,j]) < A[0].max()/2 else 'black',\n", " fontsize=11, fontweight='bold')\n", "ax.set_xticks([0,1]); ax.set_yticks([0,1])\n", "ax.set_xticklabels(['asset 1', 'asset 2'], color=HF_TEXT)\n", "ax.set_yticklabels(['asset 1', 'asset 2'], color=HF_TEXT)\n", "ax.set_title('$A(0)$ — initial value-function quadratic form', color=HF_GOLD)\n", "cbar = fig.colorbar(im, ax=ax)\n", "cbar.ax.tick_params(colors=HF_TEXT)\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "cf60fe4a", "metadata": {}, "source": [ "\n", "## 6 · Optimal trading speed under OU" ] }, { "cell_type": "markdown", "id": "a343cacb", "metadata": {}, "source": [ "### 6.1 Closed-form\n", "\n", "Plugging the quadratic ansatz back into $v^\\star = \\eta^{-1}(\\nabla_q J - s)$ gives the **time-varying linear feedback law**:\n", "\n", "$$\n", "v^\\star(t,q,s) \\;=\\; \\eta^{-1}\\bigl(\\,2\\,A(t)\\,q \\;+\\; B(t)(s-\\bar\\mu) \\;+\\; C(t) \\;-\\; s\\,\\bigr).\n", "$$\n", "\n", "The trader liquidates inventory ($A(t)\\,q$ term) while exploiting the OU drift ($B(t)(s-\\bar\\mu)$ term)." ] }, { "cell_type": "markdown", "id": "76f656f1", "metadata": {}, "source": [ "### 6.2 Closed-loop simulation" ] }, { "cell_type": "code", "execution_count": null, "id": "0ca3bb7b", "metadata": {}, "outputs": [], "source": [ "# Forward simulation of the closed-loop optimal policy\n", "n_paths = 200\n", "S0 = np.array([100.0, 100.0])\n", "q0 = np.array([5.0, -3.0]) # long asset 1, short asset 2\n", "mu_bar = S0.copy()\n", "dt = T / N\n", "sqdt = np.sqrt(dt)\n", "L_Sigma = np.linalg.cholesky(Sigma)\n", "\n", "q_paths = np.zeros((n_paths, N+1, d))\n", "S_paths = np.zeros((n_paths, N+1, d))\n", "v_paths = np.zeros((n_paths, N, d))\n", "q_paths[:, 0] = q0\n", "S_paths[:, 0] = S0\n", "\n", "for p in range(n_paths):\n", " rng_p = np.random.default_rng(1000 + p)\n", " for k in range(N):\n", " s = S_paths[p, k]\n", " q = q_paths[p, k]\n", " v = eta_inv @ (2*A[k] @ q + B[k] @ (s - mu_bar) + C[k] - s)\n", " v_paths[p, k] = v\n", " q_paths[p, k+1] = q + v*dt\n", " dW_ = sqdt * rng_p.normal(size=d)\n", " S_paths[p, k+1] = s + Phi @ (mu_bar - s)*dt + L_Sigma @ dW_\n", "\n", "# Median + IQR for inventory\n", "median_q = np.median(q_paths, axis=0)\n", "q25 = np.quantile(q_paths, 0.25, axis=0)\n", "q75 = np.quantile(q_paths, 0.75, axis=0)\n", "print(f'Simulated {n_paths} closed-loop paths, {N} steps each.')" ] }, { "cell_type": "markdown", "id": "6d5e5222", "metadata": {}, "source": [ "### 6.3 Visual — inventory trajectories with confidence band" ] }, { "cell_type": "code", "execution_count": null, "id": "42d1b731", "metadata": {}, "outputs": [], "source": [ "fig, axes = hf_fig((11, 4.5), ncols=2)\n", "for j in range(d):\n", " ax = axes[j]\n", " ax.plot(t_grid/60, q_paths[:, :, j].T, color=HF_TEAL, lw=0.4, alpha=0.15)\n", " ax.plot(t_grid/60, median_q[:, j], color=HF_GOLD, lw=2.5, label='median')\n", " ax.fill_between(t_grid/60, q25[:, j], q75[:, j], color=HF_AMBER, alpha=0.35, label='IQR')\n", " ax.axhline(0, color=HF_TEXT, ls='--', alpha=0.4)\n", " ax.set_xlabel('time (min)', color=HF_TEXT)\n", " ax.set_ylabel(f'$q^{{({j+1})}}_t$', color=HF_TEXT)\n", " ax.set_title(f'Inventory · asset {j+1}', color=HF_GOLD)\n", " ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT, loc='upper right')\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "d53cd7a0", "metadata": {}, "source": [ "### 6.4 Visual — distribution of optimal trading speed at $t=0$" ] }, { "cell_type": "code", "execution_count": null, "id": "783c08e1", "metadata": {}, "outputs": [], "source": [ "# Histogram of v*(0) across the 200 paths\n", "fig, axes = hf_fig((11, 4), ncols=2)\n", "for j in range(d):\n", " ax = axes[j]\n", " ax.hist(v_paths[:, 0, j], bins=30, color=(HF_TEAL if j==0 else HF_AMBER),\n", " edgecolor=HF_GOLD, alpha=0.75)\n", " ax.axvline(np.median(v_paths[:, 0, j]), color=HF_RED, lw=2, ls='--', label='median')\n", " ax.set_xlabel(f'$v^\\\\star_0$ — asset {j+1}', color=HF_TEXT)\n", " ax.set_ylabel('count', color=HF_TEXT)\n", " ax.set_title(f'Initial optimal speed — asset {j+1}', color=HF_GOLD)\n", " ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT)\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "7f8c9e7b", "metadata": {}, "source": [ "\n", "## 7 · Hayashi–Yoshida non-synchronous covariance" ] }, { "cell_type": "markdown", "id": "d22b3f79", "metadata": {}, "source": [ "### 7.1 The non-synchronous problem\n", "\n", "Crypto venues emit price updates at **different cadences** (BTC: ~10 ms ticks, SOL: ~50 ms, ETH options: ~1 s). Aggregating to a common grid of width $\\Delta t$ introduces the **Epps effect**: observed correlation decays toward zero as $\\Delta t \\to 0$." ] }, { "cell_type": "markdown", "id": "00b641d7", "metadata": {}, "source": [ "### 7.2 The Hayashi–Yoshida estimator\n", "\n", "Hayashi & Yoshida (2005) avoid grid aggregation by summing return products over **all overlapping native intervals**:\n", "\n", "$$\n", "\\widehat\\Sigma^{HY}_{ij} \\;=\\; \\sum_{k}\\sum_{\\ell}\\, \\Delta r_i^{(k)}\\,\\Delta r_j^{(\\ell)}\\;\\mathbb{1}_{\\bigl\\{I_i^{(k)} \\cap I_j^{(\\ell)} \\neq \\emptyset\\bigr\\}}\n", "$$\n", "\n", "where $I_i^{(k)} = (t_{i,k-1}, t_{i,k}]$ is the $k$-th native interval of asset $i$." ] }, { "cell_type": "markdown", "id": "29d251f4", "metadata": {}, "source": [ "### 7.3 Key properties\n", "\n", "| Property | Statement |\n", "|---|---|\n", "| **Consistency** | $\\widehat\\Sigma^{HY}_{ij} \\xrightarrow{p} \\langle r_i, r_j\\rangle_T$ as observation density → ∞ |\n", "| **Unbiasedness** | unbiased under non-synchronous Poisson sampling |\n", "| **Asymptotic normality** | central limit theorem → direct confidence intervals |\n", "| **Robustness** | no bandwidth, no bias-variance trade-off (unlike kernel methods) |" ] }, { "cell_type": "markdown", "id": "60a054dc", "metadata": {}, "source": [ "### 7.4 Visual — the Epps effect, simulated" ] }, { "cell_type": "code", "execution_count": null, "id": "6489721a", "metadata": {}, "outputs": [], "source": [ "# Simulate two correlated assets sampled at different rates,\n", "# then compute naive grid-aggregated correlation as a function of grid width.\n", "T_sim = 60.0 # 1 minute\n", "N_fine = 60_000\n", "true_rho = 0.6\n", "L_corr = np.linalg.cholesky(np.array([[1.0, true_rho], [true_rho, 1.0]]))\n", "\n", "dt_fine = T_sim / N_fine\n", "dW = rng.normal(scale=np.sqrt(dt_fine), size=(N_fine, 2)) @ L_corr.T\n", "prices = np.exp(np.cumsum(0.001 * dW, axis=0)) # log-normal\n", "times = np.linspace(0, T_sim, N_fine, endpoint=False)\n", "\n", "# Naive grid aggregation at multiple Δt\n", "grid_widths_ms = [10, 25, 50, 100, 250, 500, 1000, 2500, 5000]\n", "empirical_rho = []\n", "for ms in grid_widths_ms:\n", " dt_grid = ms / 1000.0\n", " n_grid = int(T_sim / dt_grid)\n", " edges = np.linspace(0, T_sim, n_grid + 1)\n", " bin_idx = np.searchsorted(edges, times, side='right') - 1\n", " bin_idx = np.clip(bin_idx, 0, n_grid-1)\n", " last_in_bin = np.full((n_grid, 2), np.nan)\n", " for b, p in zip(bin_idx, prices):\n", " last_in_bin[b] = p\n", " # forward-fill\n", " for b in range(1, n_grid):\n", " if np.isnan(last_in_bin[b, 0]): last_in_bin[b] = last_in_bin[b-1]\n", " rets = np.diff(np.log(last_in_bin), axis=0)\n", " rets = rets[~np.isnan(rets).any(axis=1)]\n", " empirical_rho.append(np.corrcoef(rets.T)[0, 1])\n", "\n", "fig, ax = hf_fig((10, 4.5))\n", "ax.plot(grid_widths_ms, empirical_rho, 'o-', color=HF_TEAL, lw=2, ms=8, label='naive grid correlation')\n", "ax.axhline(true_rho, color=HF_GOLD, ls='--', lw=2, label=f'true $\\\\rho$ = {true_rho}')\n", "ax.set_xscale('log')\n", "ax.set_xlabel('grid width $\\\\Delta t$ (ms, log)', color=HF_TEXT)\n", "ax.set_ylabel('measured correlation', color=HF_TEXT)\n", "ax.set_title('Epps effect — naive correlation decays as $\\\\Delta t \\\\to 0$', color=HF_GOLD)\n", "ax.legend(facecolor=HF_PANEL, edgecolor=HF_GOLD, labelcolor=HF_TEXT)\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "markdown", "id": "286e5ecf", "metadata": {}, "source": [ "\n", "## 8 · What the full course covers (paid)" ] }, { "cell_type": "markdown", "id": "f0b9fcca", "metadata": {}, "source": [ "### 8.1 Topic ladder\n", "\n", "The full **80-page LaTeX course** (€34.99) and **Education tier** (€19/mo, 35+ notebooks) cover in rigorous detail:\n", "\n", "1. **Stochastic calculus prerequisites** — measure theory, filtered probability spaces, Brownian motion, Itô integral, Stratonovich, Girsanov change of measure.\n", "2. **HJB derivation from dynamic programming**, viscosity solutions, comparison theorem.\n", "3. **Quadratic ansatz proof** — why it solves the HJB exactly under linear-Gaussian dynamics, and what fails when leaving this class.\n", "4. **Matrix Riccati existence/uniqueness** — Bergault–Drissi–Guéant a-priori estimates, monotonicity, blow-up criteria.\n", "5. **Numerical integration** — backward RK4 stability analysis, sub-stepping, condition-number diagnostics.\n", "6. **Hayashi–Yoshida full derivation** — Doob–Meyer decomposition, CLT, finite-sample bias correction.\n", "7. **Multi-frequency $\\Phi$ calibration** — spectral decomposition, weighted least squares.\n", "8. **Option overlay theory** — Itô on $\\Pi_t = X_t + q_t S_t + \\Theta_t$, gamma harvest, theta decay, break-even constraint.\n", "9. **Adaptive Bayesian extension** (Baldacci–Manziuk 2020) — finite differences, deep RL, fill intensity models.\n", "10. **40+ solved exercises** with companion notebook reproducing every figure of the blog." ] }, { "cell_type": "markdown", "id": "f344af3a", "metadata": {}, "source": [ "### 8.2 References\n", "\n", "1. Bergault, P., Drissi, F. & Guéant, O. (2022). *Multi-asset optimal execution and statistical arbitrage strategies under Ornstein–Uhlenbeck dynamics.* SIAM J. Fin. Math. **13**(1), 353–390.\n", "2. Baldacci, B. & Manziuk, I. (2020). *Adaptive trading strategies across liquidity pools.* arXiv:2008.07807.\n", "3. Almgren, R. & Chriss, N. (2001). *Optimal execution of portfolio transactions.* J. Risk **3**(2), 5–40.\n", "4. Avellaneda, M. & Stoikov, S. (2008). *High-frequency trading in a limit order book.* Quant. Finance **8**(3), 217–224.\n", "5. Hayashi, T. & Yoshida, N. (2005). *On covariance estimation of non-synchronously observed diffusion processes.* Bernoulli **11**(2), 359–379.\n", "6. Cartea, Á., Jaimungal, S. & Penalva, J. (2015). *Algorithmic and High-Frequency Trading.* Cambridge University Press." ] }, { "cell_type": "markdown", "id": "8d884e19", "metadata": {}, "source": [ "---\n", "\n", "
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" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.10" } }, "nbformat": 4, "nbformat_minor": 5 }