Calibrating Temporary Market Impact in an Optimal Execution Model
Summary
The document sets out a continuous-time optimal execution model for selling a fixed stock position over a chosen horizon. It assumes an arithmetic Brownian unaffected price and a linear temporary impact cost proportional to trading rate. Under these assumptions, expected shortfall is determined by the squared trading rate, while risk is measured by the integral of the squared remaining inventory. Combining both produces a variational objective and a hyperbolic-sine inventory trajectory, with its speed governed by volatility, risk aversion, and the impact parameter eta.
The practical question is how to estimate eta for one listed stock using Refinitiv or Bloomberg data in a manner consistent with this model. The document supplies the model equations and cites a finance optimization text, but it does not propose a calibration procedure, dataset, or empirical results. Its assumptions are restrictive: impact is temporary and linear, and price uncertainty follows a simple diffusion, so any estimate would depend on how real trading data are mapped to those assumptions.
Key ideas
- Temporary impact is modeled as a linear penalty on the execution rate.
- The objective trades off expected execution shortfall against inventory risk over a fixed horizon.
- The resulting optimal inventory path has a hyperbolic-sine form controlled by eta and volatility.
- The document asks how to estimate eta from market data but does not give an estimation method.
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Full text
# Calibrating eta in model from Refinitiv data
# Calibrating eta in model from Refinitiv data
I am trying to calibrate the temporary market impact parameter $\eta$ in the following continuous-time optimal execution model for a single listed stock.
We determine a continuous trading trajectory
$$ x(t), \qquad t \in [0,T], $$
with boundary conditions
$$ x(0)=X, \qquad x(T)=0. $$
Here, $x(t)$ is the number of shares still held at time $t$. The trading rate is
$$ y(t)=-\dot{x}(t). $$
The unaffected market price is modeled as an arithmetic Brownian motion:
$$ S(t)=S(0)+\sigma B(t). $$
The actual execution price received at time $t$ is
$$ \widetilde{S}(t)=S(t)-\eta y(t), $$
where $\eta$ is the temporary market impact parameter.
The execution shortfall is
$$ C(x) = XS(0)-\int_0^T \widetilde{S}(t)y(t)\,dt. $$
Using $y(t)=-\dot{x}(t)$, this gives
$$ C(x) = \eta\int_0^T \dot{x}(t)^2\,dt - \sigma\int_0^T x(t)\,dB(t). $$
Therefore, the expected shortfall and variance are
$$ \mathbb{E}(x) = \eta\int_0^T \dot{x}(t)^2\,dt, $$
and
$$ V(x) = \sigma^2\int_0^T x(t)^2\,dt. $$
The efficient execution trajectory is the solution of
$$ \min_{x(t)} \int_0^T \left( \eta\dot{x}(t)^2 + \lambda\sigma^2x(t)^2 \right)dt $$
subject to
$$ x(0)=X, \qquad x(T)=0. $$
The Euler equation is
$$ \ddot{x}(t) = \frac{\lambda\sigma^2}{\eta}x(t), $$
and the closed-form solution is
$$ x(t) = X \frac{\sinh(\kappa(T-t))} {\sinh(\kappa T)}, $$
where
$$ \kappa = \sqrt{\frac{\lambda\sigma^2}{\eta}}. $$
I want to calibrate the parameter $\eta$ for one listed stock using Refinitiv market data.
How should $\eta$ be estimated or calibrated from Refinitiv Workspace/Bloomberg Terminal data in a way that is consistent with this continuous-time model?
The model is taken from Cornuéjols and Tütüncü, Optimization Methods in Finance, Chapter 12, Section 12.3, Execution Costs, page 205.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.