Symbols - Advanced Analytics and Algorithmic Trading

This page collects the mathematical notation used throughout the book. Symbols that carry chapter-specific meaning are listed under the relevant section below; general conventions apply everywhere.

General conventions¶

Symbol	Meaning
$\log(x)$	Natural logarithm
$\mathbf{x}, \mathbf{y}$	Vectors (bold lowercase); column vectors written $\mathbf{x}=[x_1,\dots,x_n]^T$
$\mathbf{X}, \mathbf{Y}$	Matrices (bold uppercase)
$X, Y$	Random variables (uppercase Roman)
$\boldsymbol{\theta}$	Model parameters (Greek lowercase)
$\hat\theta$	Point estimate of $\theta$
$\mathbb{E}[X \mid Y]$	Expectation of $X$ given $Y$
$\operatorname{Var}[X \mid Y]$	Variance of $X$ given $Y$
$\operatorname{Cov}[X, Y]$	Covariance of $X$ and $Y$
$\boldsymbol{\Sigma}$	Covariance matrix
$X \sim p$	Random variable $X$ is distributed as $p$
$p(\cdot)$	Probability density or probability mass function
$p(y \mid \mathbf{x})$	Probability (density) of $y$ given $\mathbf{x}$
$\mathbb{KL}(p \parallel q)$	Kullback-Leibler divergence from $p$ to $q$
$\mathcal{N}(\mu, \sigma^2)$	Gaussian distribution with mean $\mu$ and variance $\sigma^2$
$\mathbf{1}[\cdot]$	Indicator function
$\mathbb{1}_A$	Indicator of event $A$
$\lVert \mathbf{x} \rVert$	Euclidean norm of $\mathbf{x}$
$\lVert \mathbf{x} \rVert_1$	$L^1$ norm ($\sum_i
$(\cdot)^+$	Positive part: $\max(\cdot, 0)$

Probability and statistics¶

Symbol	Meaning
$p(\theta)$	Prior distribution over parameters
$p(\theta \mid D)$	Posterior distribution given data $D$
$p(D \mid \theta)$	Likelihood function
$p(D)$	Marginal likelihood (model evidence)
$\hat\theta_\text{MLE}$	Maximum likelihood estimate
$\hat\theta_\text{MAP}$	Maximum a posteriori estimate
$\alpha, \beta$	Beta/Dirichlet distribution shape parameters
$BF_{1,2}$	Bayes factor $p(D\mid H_1)/p(D\mid H_2)$
$\mathcal{GP}(\mu, k)$	Gaussian process with mean $\mu$ and kernel $k$
$k(x,x')$	Kernel (covariance) function
$\lambda_\text{reg}$	Regularisation hyperparameter (ridge/Lasso)
$\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$	Multivariate Gaussian distribution

Stochastic calculus¶

Symbol	Meaning
$W_t$	Standard Brownian motion (Wiener process)
$\mathcal{F}_t$	Filtration (information available at time $t$ )
$dW_t$	Brownian increment
$\mu$	Drift coefficient (GBM, OU)
$\sigma$	Diffusion coefficient / volatility
$\theta$	Mean-reversion speed (OU: $dS_t = \theta(\mu-S_t)dt + \sigma\,dW_t$ )
$\lambda_\text{Poi}$	Poisson process arrival rate
$\phi$	Hawkes process self-excitation amplitude ( $\lambda(t) = \mu_0 + \sum_{t_i<t}\phi\,e^{-\beta(t-t_i)}$ )
$\beta$	Hawkes process decay rate
$\mu_0$	Hawkes process baseline intensity
$N_t$	Counting process (number of events by time $t$ )
$\tau$	Stopping time / time remaining

Financial instruments and markets¶

Symbol	Meaning
$S_t$	Asset price at time $t$
$r$	Risk-free interest rate
$y$	Yield to maturity
$\mathcal{D}$	Modified duration
$C$	Option / instrument price; coupon payment (context-dependent)
$K$	Strike price
$T$	Maturity / terminal time
$F$	Forward price
$\Delta$	Option delta (sensitivity to underlying price)
$\Gamma$	Option gamma (second derivative in price)
$\Theta$	Option theta (time decay)
$\mathcal{V}$	Option vega (sensitivity to volatility)
$R_i$	Return of asset $i$
$\beta_i$	CAPM / factor beta of asset $i$
$R_f$	Risk-free rate
$\alpha_i$	Jensen’s alpha (excess return over CAPM benchmark)

Market microstructure and LOB¶

Symbol	Meaning
$P^a_{best}$	Best ask price
$P^b_{best}$	Best bid price
$M_t$	Mid-price: $M_t = \tfrac{1}{2}(P^a_{best}+P^b_{best})$
$S$	Bid-ask spread: $S = P^a_{best} - P^b_{best}$
$I$	Order imbalance: $I = (V^b_{best}-V^a_{best})/(V^b_{best}+V^a_{best})$
$\text{TFI}_{t,\tau}$	Trade flow imbalance over window $\tau$
$V^b_{best}, V^a_{best}$	Volume at best bid / best ask
$\lambda(t)$	Point process intensity at time $t$
$\lambda(\delta)$	Fill intensity at depth $\delta$ : $\lambda(\delta)=Ae^{-k\delta}$
$A$	Base fill rate (fill intensity at zero depth)
$k$	Depth-sensitivity / demand-elasticity parameter
$P(\text{fill}\mid\delta,\tau)$	Fill probability: $1-e^{-\lambda(\delta)\tau}$
$\text{PIN}$	Probability of Informed Trading: $\alpha\mu_{\text{PIN}}/(\alpha\mu_{\text{PIN}}+2\varepsilon)$
$\text{VPIN}_t$	Volume-synchronised PIN: $\frac{1}{n}\sum
$\text{ILLIQ}_t$	Amihud illiquidity: $\frac{1}{D}\sum
$\lambda_K$	Kyle’s lambda: $\lambda_K=\sigma_v/(2\sigma_u)$
$Y$	Square-root impact coefficient: $\text{MI}\approx Y\sigma\sqrt{Q/V_{ADV}}$
$V_{ADV}$	Average daily volume
$\alpha_\text{inf}$	Fraction of informed traders (Glosten-Milgrom; also written $\alpha$ )
$V_H, V_L$	High and low asset values (Glosten-Milgrom)
$p$	Prior probability that asset value is $V_H$

Execution algorithms¶

Symbol	Meaning
$Q$	Parent order size (initial inventory)
$q_t$	Remaining inventory at time $t$
$v_t$	Trading rate: $v_t = -\dot{q}_t$
$v_i$	Quantity traded in discrete interval $i$
$p_0$	Arrival / decision price
$p_\text{avg}$	Average execution price
$\text{IS}$	Implementation shortfall: $Q(p_\text{avg}-p_0)$
$\text{VWAP}$	Volume-weighted average price: $\sum v_i P_i/\sum v_i$
$\text{TWAP}$	Time-weighted average price
$\eta$	Temporary price-impact coefficient ( $h(v)=\eta v$ )
$\gamma_\text{perm}$	Permanent price-impact coefficient ( $g(v)=\gamma_\text{perm} v$ )
$\rho$	Risk-aversion coefficient in execution mean-variance objective
$\kappa$	Almgren-Chriss trajectory decay rate: $\kappa=\sqrt{\rho\sigma^2/\eta}$
$x_t$	Remaining inventory in Almgren-Chriss (also written $q_t$ )

Market making and optimal quoting¶

Symbol	Meaning
$\delta^a, \delta^b$	Ask-side and bid-side half-spreads (both non-negative)
$\delta^*(t,q)$	Optimal half-spread (HJB / AS solution)
$r(t,q)$	Reservation price: mid-point of optimal quotes
$\gamma$	CARA risk-aversion coefficient ( $U(W)=-e^{-\gamma W}$ )
$\mathbf{q}_t$	Inventory vector (multi-asset)
$H(t,q)$	Value function in execution-tactic / AS HJB
$u(t,\mathbf{q})$	Scalar indifference function (AS multi-asset)
$\boldsymbol{\Gamma}$	Asset covariance matrix (market-making context)
$\text{spread}_\text{GL}$	Glosten-Milgrom equilibrium spread: $\alpha(V_H-V_L)$
$n$	Number of competing market makers (Grossman-Miller)
$\Lambda^a, \Lambda^b$	Ask-side and bid-side order arrival processes (Poisson rate $Ae^{-k\delta}$ )
$\delta_\text{res}$	Reservation half-spread (RfQ / dealer context)
$s$	Side variable ( $s=+1$ ask, $s=-1$ bid) in RfQ context
$P_m$	Mid-price in RfQ context (equivalent to $M_t$ )

Optimal control and dynamic programming¶

Symbol	Meaning
$V(t,x)$	Value function (cost-to-go or reward-to-go)
$J$	Objective functional
$u_t$	Control variable at time $t$
$\mathcal{H}$	Hamiltonian
$\mathbf{P}_t$	Riccati matrix (LQSC solution)
$\lambda_\text{DP}$	Risk-aversion / Lagrange multiplier in DP objectives

Machine learning and data-driven methods¶

Symbol	Meaning
$\ell(y, \hat y)$	Loss function
$R[f]$	Expected (generalisation) risk
$\hat R[f]$	Empirical risk (training loss)
$\mathbf{w}$	Weight vector (neural network / linear model)
$f^*$	Bayes-optimal decision function
$K$	Number of classes; number of cross-validation folds (context-dependent)
$N$	Number of training observations
$H$	Hurst exponent ( $H=0.5$ random walk, $H<0.5$ mean-reverting, $H>0.5$ trending)
$z_t$	Z-score: $(x_t-\hat\mu)/\hat\sigma$
$Q(s,a)$	Action-value function (Q-learning)
$\pi$	Policy (reinforcement learning)

Portfolio and investment¶

Symbol	Meaning
$\mathbf{w}$	Portfolio weight vector
$\boldsymbol{\mu}$	Vector of expected returns
$\boldsymbol{\Sigma}$	Asset covariance matrix
$S_a$	Sharpe ratio: $\mathbb{E}[R_a-R_b]/\sigma_a$
$\mathbf{h}^*$	Minimum-variance hedge vector: $\boldsymbol{\Sigma}_{HH}^{-1}\boldsymbol{\sigma}_{XH}$
$\beta_\text{hedge}$	Hedge ratio (scalar, single hedge instrument)
$t_{1/2}$	Mean-reversion half-life: $\ln 2/\theta$