2.2.3 The Almgren–Chriss Model

Basic Prerequisites: 02 - The Execution Problem and Calculus & Optimization (Euler-Lagrange / convex minimization).

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1. Intuition & Practical Objective

The execution problem has now been reduced to two scalars: an expected cost $E[x]$ and a variance $V[x]$, both functions of the trading trajectory $x=(x_0,\dots ,x_N)$. The Almgren–Chriss model closes the problem by choosing the trajectory that minimizes

$$U[x]=E[x]+\lambda V[x],$$

a quadratic objective in $x$. For linear impact it is exactly a convex quadratic program, so the optimum is unique, static, and - in continuous time - has a closed form: a hyperbolic (exponential-decay) trajectory. This page does the derivation twice: the Euler-Lagrange route AC themselves used, and the dynamic-programming / Hamilton–Jacobi–Bellman route that is the modern control-theoretic statement, and shows they agree.

The practical objective: understand where the shape comes from, so that when the trajectory later disappoints (page 05) you know exactly which assumption to blame.

The one-sentence essence. “Minimize expected temporary impact plus risk aversion times variance, and the Euler-Lagrange equation reduces to $\eta \ddot{x}=\lambda {\sigma }^{2}x$ - whose solutions are exponentials with decay rate $\kappa =\sqrt{\lambda {\sigma }^2/\eta }$, so the optimal schedule decays exponentially and its half-life is $1/\kappa$.“

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2. Mathematical Ground Truth & Derivations

2.1 The discrete quadratic problem (AC §2.2)

With linear impact, from page 02:

$$E[x]=\frac{1}{2}\gamma X^2+\epsilon \sum _{k=1}^N|n_k|+\frac{\tilde{\eta }}{\tau }\sum _{k=1}^N{n}_k^2,V[x]={\sigma }^2\sum _{k=1}^N\tau x_k^2,\tilde{\eta }=\eta -\frac{1}{2}\gamma \tau .$$

Dropping the schedule-independent constant $\frac{1}{2}\gamma X^2$ and the fixed $\epsilon X$ (assume all $n_k\ge 0$), $U$ is a strictly convex quadratic in the free variables $x_1,\dots ,x_{N-1}$ (since $\tilde{\eta }>0$). Setting $\partial U/\partial x_j=0$ for $1\le j\le N-1$:

$$\frac{\tilde{\eta }}{\tau }\cdot 2(-x_{j-1}+2x_j-x_{j+1})+2\lambda {\sigma }^2\tau x_j=0⟹\frac{1}{{\tau }^2}(x_{j-1}-2x_j+x_{j+1})={\tilde{\kappa }}^2{x}_j,{\tilde{\kappa }}^2=\frac{\lambda {\sigma }^2}{\tilde{\eta }}.$$

2.2 Euler-Lagrange and the closed form

This is a second-order linear difference equation; its solutions are $\exp (\pm \kappa t_j)$ where $\kappa$ solves $\frac{2}{{\tau }^2}(\cosh (\kappa \tau )-1)={\tilde{\kappa }}^2$. Imposing $x_0=X$ and $x_N=0$ gives (AC eq 17-18)

$$\boxed{x_j=X\frac{\sinh (\kappa (T-t_j))}{\sinh (\kappa T)}},n_j=\frac{2\sinh \left(\frac{1}{2}\kappa \tau \right)}{\sinh (\kappa T)}\cosh (\kappa (T-t_{j-\frac{1}{2}}))X.$$

In continuous time ($\tau \to 0$) the same argument on $U={\int }_0^T[\eta {\dot{x}}^2+\lambda {\sigma }^2{x}^2]dt$ gives the Euler-Lagrange equation

$$\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=0⟹2\lambda {\sigma }^{2}x-2\eta \ddot{x}=0⟹\boxed{\ddot{x}={\kappa }^{2}x},\kappa =\sqrt{\frac{\lambda {\sigma }^2}{\eta }},$$

whose solution with $x(0)=X,x(T)=0$ is the same $\sinh$ trajectory. Limits: $\lambda \to 0$ ($\kappa \to 0$) gives $x_t=X(1-t/T)$ (TWAP); $\lambda \to \infty$ ($\kappa \to \infty$) gives instant liquidation at $t=0$ (a block). $\theta =1/\kappa$ is the trade half-life. The regime is read off $\kappa T=T/\theta$: if $T\gg \theta$ (large $\kappa T$) the trade is risk-constrained - the bulk of liquidation happens well before $T$ and the curve looks like a fast block; if $T\ll \theta$ (small $\kappa T$) it is impact-constrained and approaches the straight line (TWAP) - see AC §2.3 (here $\kappa T=3.0$ is squarely in the interior).

2.3 Time-homogeneity and the HJB

AC’s Theorem (time-homogeneity). The static optimum is also the dynamic optimum: re-solving at any interior time $t_k$ merely reproduces the continuation of the original trajectory. AC prove this two ways - algebraically (the continuation $X\sinh (\kappa (T-t_j))/\sinh (\kappa (T-t_k))$ matches) and by optimal control (no new information arrives, since the price has no serial correlation). This is why a static curve is legitimate.

The HJB statement. Cast the risk-neutral version as a dynamic program: with value function $V(t,x)$ (minimal expected cost to liquidate $x$ by $T$), trading rate $\nu =-\dot{x}$, and temporary cost rate $\nu h(\nu )=\epsilon \nu +\eta {\nu }^2$,

$$-V_t=\underset{\nu \ge 0}{\min }[\epsilon \nu +\eta {\nu }^2-\nu V_x],V(T,x)=0.$$

The first-order condition $\epsilon +2\eta \nu -V_x=0$ gives ${\nu }^{\star }=(V_x-\epsilon )/(2\eta )$, and substituting yields the HJB equation (note the sign: the minimised bracket is $-(V_x-\epsilon {)}^2/(4\eta )$)

$$\boxed{-V_t=-\frac{(V_x-\epsilon {)}^2}{4\eta }⟺V_t=\frac{(V_x-\epsilon {)}^2}{4\eta }}.$$

Its solution $V(t,x)=\epsilon x+\eta x^2/(T-t)$ (readily checked) gives ${\nu }^{\star }=x/(T-t)$ - the constant-rate TWAP optimum - and $V(0,X)=\epsilon X+\eta X^2/T$, the risk-neutral cost. Adding the risk penalty $\lambda {\sigma }^2{x}^2$ to the running cost turns the same HJB into the one whose characteristics solve $\ddot{x}={\kappa }^{2}x$: the risk-aversion term is precisely what bends TWAP into the hyperbolic curve.

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3. Computational Implementation - the closed form is the discrete optimum

The strongest possible check: build the exact quadratic $U$ and minimize it by linear algebra; compare against the closed-form trajectory. numpy + stdlib.

The closed form and the exact discrete minimizer agree to 2 shares in 1,000,000 (the residual is the $O(\tau )$ difference between $\kappa$ and the discrete $\tilde{\kappa }$), and the objective values match to 10 significant figures. This is the statement “the AC formula solves the AC problem,” verified numerically.

HJB check. With $\eta =2.5\times 10^{-6}$, $\epsilon =0.02$, the ansatz $V(t,x)=\epsilon x+\eta x^2/(T-t)$ satisfies $-V_t=(V_x-\epsilon {)}^2/(4\eta )$ with maximum relative residual $8.94\times 10^{-10}$ and gives $V(0,X)=\epsilon X+\eta X^2/T=$ $520,000 - the risk-neutral TWAP cost.

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4. Failure Modes & First-Principles Breakdowns

1. The model is a quadratic; reality is not. The entire closed form rests on $U=E+\lambda V$ being a convex quadratic. Nonlinear impact ($\eta v^{\alpha }$) destroys it - the Euler-Lagrange equation becomes nonlinear and the $\sinh$ solution is only an approximation (06 - Advanced Extensions).
2. Mean-variance is not expected utility. $E+\lambda V$ equals quadratic utility only up to second order; for non-Gaussian (fat-tailed, jump) shortfall the objective mis-specifies risk. AC handle this with the L-VaR construction (§3.2), which replaces variance with a quantile.
3. Static ≠ adaptive. Time-homogeneity holds only because the price has no serial correlation or drift. With drift, momentum, or mean reversion, the optimal strategy is genuinely dynamic and the static curve is stale (AC §4; Hasbrouck Ch 15 gives the drift-corrected $s_t^{\star }$).
4. $\kappa$ inherits every parameter error. $\kappa =\sqrt{\lambda {\sigma }^2/\eta }$: urgency scales as $\sqrt{\lambda }$, $\sigma$, and $1/\sqrt{\eta }$. A 4x error in the risk aversion mis-sets the half-life by 2x. See 05 - Failure Modes.
5. The objective is dimensioned loosely. $E$ is in dollars, $V$ in dollars${}^2$, so $\lambda$ carries units of 1/$. Quoting a $\lambda$ without its units (or in the wrong currency scale) silently rescales the entire schedule.

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5. References

• Almgren, Robert; Chriss, Neil - “Optimal execution of portfolio transactions,” Journal of Risk 3(2), 5-40 (2000)
• Bertsimas, Dimitris; Lo, Andrew W. - “Optimal control of execution costs,” Journal of Financial Markets 1(1), 1-50 (1998). The dynamic-programming / HJB formulation ($s_t^{\star }=\bar{s}/T$ in the zero-drift case).
• Cartea, A.; Jaimungal, S.; Penalva, J. - Algorithmic and High-Frequency Trading (2015)
• Gueant, Olivier - The Financial Mathematics of Market Liquidity (2016)
• Hasbrouck, Joel - Empirical Market Microstructure (2007)

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6. Connected Graph Bridges

• Back: 02 - The Execution Problem · Index Hub
• Forward: 04 - Efficient Frontier & Trajectory
• Theory: Calculus & Optimization (Euler-Lagrange, convex QP) · Stochastic Calculus (martingale conditioning under the dynamic program)
• Control-theory twin: Avellaneda–Stoikov & Optimal Quoting (same HJB machinery, inventory instead of execution)