2.2 Optimal Execution & Almgren–Chriss

Basic Prerequisites: Calculus & Optimization and Stochastic Calculus. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Liquidating a large block is a two-sided loss. Trade fast and you pay the market its liquidity premium - market impact. Trade slowly and you hold inventory that drifts against you - volatility risk. Neither cost can be pushed to zero; every schedule trades one against the other. The Almgren–Chriss (2000) framework turns that conflict into a single, solvable optimization: minimize expected cost plus a risk-aversion times the variance of cost, and the solution is an explicit hyperbolic trading trajectory.

This folder is the optimal-execution topic-folder for Pillar 2. It is a hub: it gives you the (a) fast formula lookup below (job #1), and (b) routes you to six sub-pages that walk from zero-knowledge intuition through the execution problem, the Almgren–Chriss model, the efficient frontier and closed-form trajectory, the failure modes and desk practice, and the modern extensions (nonlinear impact, resilient books, dark pools, adaptive control).

The one-sentence essence. “Expected liquidation cost rises as $\eta X^2/T$ (impact), its standard deviation rises as $\sigma X\sqrt{T/3}$ (risk); Almgren–Chriss picks the trajectory that minimizes $E+\lambda V$ - and with linear impact it is $x_t=X\sinh (\kappa (T-t))/\sinh (\kappa T)$ with urgency $\kappa =\sqrt{\lambda {\sigma }^2/\eta }$.”

Scope note (vs the sibling folder). This folder is the scheduling view - how to trade a given order. For why the price moves and how to measure it, see Market Impact & Depth (Kyle’s $\lambda$, square-root law, transient impact). The two meet at the impact-parameter calibration $\eta ,\gamma$.

Primary verified sources: Almgren & Chriss (2000); Almgren (2003); Obizhaeva & Wang (2013); Gatheral (2010, 2013); Cont, Kukanov & Stoikov (2014); Bertsimas & Lo (1998); Hasbrouck, Empirical Market Microstructure Ch 14-15; Cartea–Jaimungal–Penalva (2015); Gueant (2016). All numbers below were re-executed and reproduced (see §3).

2. Mathematical Ground Truth & Derivations

Notation. $X$ = initial position to liquidate (units); $T$ = horizon; $N$ = number of trading intervals of length $\tau =T/N$; $t_k=k\tau$; $x_k$ = units still held at $t_k$ (so $x_0=X$, $x_N=0$); $n_k=x_{k-1}-x_k$ = units sold in interval $k$; $S_0$ = initial price; $\sigma$ = volatility ($/share/√day); $\gamma$ = permanent impact coefficient ($/share/$share); $\eta$ = temporary impact coefficient (($/share)/(share/day)); $\epsilon$ = fixed cost per share (half-spread + fees); $\lambda$ = risk-aversion (1/$).

Price dynamics (AC eqs 1-2). Arithmetic random walk with permanent impact $g$, plus a temporary impact $h$ paid only on the shares traded:

with linear $g(v)=\gamma v$ and $h(v)=\epsilon +\frac{\eta }{\tau }v$.

Cost / risk (AC eqs 4-5, 8). Expected shortfall and its variance, expressed in the trajectory $x$:

Note the constant $\frac{1}{2}\gamma X^2$: permanent impact costs the same no matter how you pace the trade - only temporary impact and risk depend on the schedule. This is the single most useful simplification in the model.

Objective and solution (AC §2). Minimize $U(x)=E[x]+\lambda V[x]$. Setting $\partial U/\partial x_j=0$ gives the linear difference equation

whose solution with $x_0=X$, $x_N=0$ is the hyperbolic (exponential-decay) trajectory:

with the continuous-time urgency $\kappa =\sqrt{\lambda {\sigma }^2/\eta }$ (AC eq 19: $\kappa =\tilde{\kappa }+O(\tau )$).

Quick-Reference Lookup - the fast facts of this folder (all reproduced in §3):

3. Computational Implementation - the formula engine

Runs on numpy (and stdlib) only. It reproduces every verified number above: the closed-form trajectory equals the exact discrete optimum to $\sim 2$ shares in $10^6$, and the Monte Carlo implementation shortfall matches the closed-form $E$ and $V$.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the full failure analysis lives in 05 - Failure Modes & Practice. In one line each:

1. Impact-model risk - the linear $h(v)=\eta v$ is empirically false (real per-share impact grows as $v^{0.5}$); the AC schedule is provably suboptimal under the true concave law, and the error is a real dollar cost.
2. Risk-aversion misestimation - $\lambda$ is not observable; the trajectory’s half-life scales as $\sqrt{\lambda }$, so a 4x error in $\lambda$ doubles or halves your urgency.
3. Non-stationarity - $\sigma$, $\eta$ and the drift are assumed constant; volatility spikes and depth evaporation mid-execution break the assumed path and make the static trajectory stale.
4. Predictability / predatory trading - a deterministic trajectory is reverse-engineerable; faster participants front-run it, adding an adverse-selection cost the model ignores.

5. References

• Almgren, Robert; Chriss, Neil - “Optimal execution of portfolio transactions,” Journal of Risk 3(2), 5-40 (2000). *The canonical paper; the source of eqs (1)-(20) above.
• Almgren, Robert - “Optimal execution with nonlinear impact functions and trading-enhanced risk,” Applied Mathematical Finance 10(1), 1-18 (2003). Extends linear impact to power laws $h(v)=\eta v^{\alpha }$; the source of the $\sqrt{\cdot }$ correction used in sub-page 05.
• Almgren, Thum, Hauptmann, Li - “Direct estimation of equity market impact,” Risk 18(7), 58-62 (2005). *How to fit $\eta$ (and the power $\alpha$) to real order data
• Obizhaeva, Anna; Wang, Jiang - “Optimal trading strategy and supply/demand dynamics,” Journal of Financial Markets 16(1), 1-32 (2013). Resilient book: optimal schedule is discrete at both ends (block-continuous-block).
• Gatheral, Jim - “No-dynamic-arbitrage and market impact,” Quantitative Finance 10(7), 749-759 (2010). The consistency constraint (Huberman–Stanzl) every impact model must satisfy; rules out arbitrary decay kernels.
• Gatheral, Jim; Schied, Alexander; Slynko, Alla - “Transient linear price impact and Fredholm integral equations,” Mathematical Finance 22(3), 445-474 (2012). Optimal execution for general (power-law) decay kernels.
• Bertsimas, Dimitris; Lo, Andrew W. - “Optimal control of execution costs,” Journal of Financial Markets 1(1), 1-50 (1998). The dynamic-programming predecessor; gives $s_t^{\star }=\bar{s}/T$ under zero drift.
• Cartea, A.; Jaimungal, S.; Penalva, J. - Algorithmic and High-Frequency Trading (2015)
• Gueant, Olivier - The Financial Mathematics of Market Liquidity (2016). Rigorous modern monograph: execution and market making as one theory.
• Hasbrouck, Joel - Empirical Market Microstructure (2007)

6. Connected Graph Bridges

• Foundational base: Calculus & Optimization (Euler-Lagrange, convex QP) · Stochastic Calculus (arithmetic Brownian motion, quadratic variation)
• Sibling in-pillar: Execution Algorithms: VWAP, TWAP, POV (the heuristic schedules AC optimizes) · Market Microstructure & Order Types · Queue Position & Fill Probability
• Impact-model view: Market Impact & Depth (Kyle $\lambda$, square-root law, transient impact)
• Natural partner: Avellaneda–Stoikov & Optimal Quoting (the inventory risk-aversion control, mirror image of AC)
• Risk & portfolio: Liquidity Risk & Funding (L-VaR) · Constraints & Transaction Costs

Beginner: start at 01 · Practitioner: start at 05