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Signal-Aware Optimal Execution

Algorithm description

Signal-aware optimal execution extends classical optimal-execution theory (Almgren-Chriss, Obizhaeva-Wang, Gatheral-Schied-Slynko, Cartea-Jaimungal) by adding a short-horizon predictive signal \(I_t\) to the underlying price dynamics. The trader solves a stochastic control problem in which three objectives compete:

  1. Impact cost — the unwanted price move caused by their own trading.
  2. Inventory risk — exposure to price moves while holding non-zero inventory.
  3. Signal alpha — the expected return from holding inventory aligned with the signal direction.

Formally, for inventory \(X_t\) decreasing from \(x\) to 0 over \([0, T]\), the cost functional is:

\[\mathcal{C}(X) \;=\; \mathbb{E}\left[\underbrace{\int_0^T I_s X_s ds}_{\text{signal term}} \;+\; \underbrace{\tfrac{1}{2}\int_0^T\!\!\int_0^T G(|t-s|)\, dX_t\, dX_s}_{\text{impact cost}} \;+\; \underbrace{\phi \int_0^T X_t^2 dt}_{\text{risk penalty}} \right]\]

where \(G\) is the transient impact kernel and \(\phi \geq 0\) is the risk aversion. The optimal \(X^*\) minimises \(\mathcal{C}\) over admissible (deterministic, finite-variation, fuel-constrained) strategies.

Two flavours

Framework Impact kernel \(G\) Terminal condition Optimal form
GSS (Gatheral-Schied-Slynko) transient, e.g. \(G(t) = \kappa\rho e^{-\rho t}\) hard fuel constraint \(X_T = 0\) singular (jumps at \(t = 0, T\))
CJ (Cartea-Jaimungal) instantaneous, \(G(dt) = \kappa \delta_0\) soft penalty \(\varrho X_T^2\) smooth (absolutely continuous)

As the decay rate \(\rho \to \infty\), the GSS framework's singular jumps vanish and the schedule converges to the CJ smooth schedule (Lehalle-Neuman 2019).

Closed-form for OU signal + exponential impact

Under the assumptions:

  • Signal \(I_t\) is Ornstein-Uhlenbeck: \(dI_t = -\gamma I_t\, dt + \sigma dW_t\)
  • Impact kernel: \(G(t) = \kappa \rho e^{-\rho t}\)
  • Zero risk aversion: \(\phi = 0\)

The optimal schedule decomposes into three components (Lehalle-Neuman Corollary 2.7):

\[X^*_t \;=\; b_0(t) \cdot x \;+\; b_1(t) \cdot I_0 \;+\; b_2(t) \cdot \int_0^t e^{-\gamma(t-s)} dW_s\]

where: - \(b_0(t)\) is the no-signal schedule (Obizhaeva-Wang style liquidation curve). - \(b_1(t)\) scales the initial-signal correction — weighted linearly in \(I_0\). - \(b_2(t)\) is the OU residual correction — depends on the signal path after \(t = 0\) (only non-zero for signal-adaptive strategies).

Jumps at \(t = 0\) (initial burst) and \(t = T\) (final burst) are part of the singular control; the continuous middle portion rides the signal.

Key parameters

Parameter Interpretation
\(T\) Execution horizon (fuel window)
\(x\) Initial inventory to liquidate
\(\gamma\) Signal mean-reversion rate (1/minute typically)
\(\sigma\) Signal volatility
\(\kappa\) Impact scale
\(\rho\) Impact decay rate
\(\phi\) Inventory risk aversion
\(\varrho\) Terminal penalty (CJ only)

Interaction effect: if \(\rho \ll \gamma\) (impact persists longer than signal auto-correlation), the signal tilt in the schedule is muted. If \(\rho \gg \gamma\), the framework collapses to the instantaneous-impact CJ limit.

Empirical calibration recipe

For deploying signal-aware execution on a real universe:

  1. Choose the signal. Typical choice: order-book imbalance (OBI) at best, multi-level OFI, or integrated OFI. OBI is empirically OU on NASDAQ OMX data.
  2. Calibrate \(\gamma, \sigma\) of the OU signal from a recent window (30 min – 1 day of minute bars).
  3. Estimate \(\kappa, \rho\) from your own trading history. \(\rho\) is notoriously unstable — see propagator-model literature for alternative transient-impact calibrations.
  4. Choose \(\phi\) from your risk budget — trades off execution time vs holding risk.
  5. Plug into closed form. For OU signal + exponential impact, Corollary 2.7's formula gives the schedule in closed form. For other kernels, solve the integral equation (2.7) numerically.

Non-monotonicity and price manipulation

A striking consequence of signal-aware execution is that the optimal inventory path is not always monotone. If the signal opposes the trade direction early, the optimal strategy may increase inventory (buy while liquidating) before reversing.

This creates a theoretical possibility of transaction-triggered price manipulation: a trader could in principle engineer a sequence of trades whose total expected cost is lower than unwinding the position, by exploiting the interaction between their own impact and the signal. Open problem: what restrictions on \(G\) and \(I\) rule this out? (Remark 2.10 of the paper.)

Practical implication: pre-trade risk and compliance checks may flag non-monotone schedules. A monotonicity constraint imposes extra cost but may be required.

Time inconsistency

Under transient impact, the problem is time-inconsistent: the optimum on \([0, T]\) computed at \(t = 0\) is not the concatenation of optima on \([t, T]\) computed at each \(t\). Three practical resolutions:

  1. Commit: use the \(t = 0\) optimal schedule for the full horizon. Gives the best total cost but can't react to new information.
  2. Re-plan: solve a new \(t_0 = t\) problem periodically with remaining inventory as initial condition. Approximate — no longer globally optimal.
  3. Use CJ limit: drop transient impact (\(\rho \to \infty\)). Time-consistent, fully adaptive, but loses the structural impact-decay information.

In practice, re-planning at a moderate frequency (e.g. every 1 minute) is the pragmatic choice.

When to use / when not to

Use when: - You have a predictive short-horizon signal with documented mean-reversion (OBI, OFI, microprice deviation). - You execute large orders and need to balance signal alpha vs market impact. - Your execution horizon is minutes to hours — where signal persistence is comparable to impact decay.

Don't use when: - Signal horizon \(\ll\) execution horizon: signal has decayed before you can act on it. Use the no-signal Obizhaeva-Wang schedule. - Signal horizon \(\gg\) execution horizon: the signal is essentially constant drift. Almgren-Chriss with a simple drift term is simpler and sufficient. - Non-linear impact matters (large order / thin market): the quadratic-impact assumption underpredicts cost for big trades.

Relationship to other execution methods

Method Signal-aware? Impact kernel Strategy type
TWAP / VWAP No Implicit linear Static schedule
Almgren-Chriss No Linear, no decay Static mean-variance
Obizhaeva-Wang No Exponential decay Singular
Cartea-Jaimungal Bounded Markov signal Instantaneous Absolutely continuous
GSS (Gatheral-Schied-Slynko) No Arbitrary positive-definite Singular
Lehalle-Neuman (signal in GSS) Markov signal Transient Singular
MPC execution Data-driven signal Learned Absolutely continuous

Connections