.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/market_microstructure/execution/plot_03_almgren_chriss.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_market_microstructure_execution_plot_03_almgren_chriss.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_market_microstructure_execution_plot_03_almgren_chriss.py:


Almgren and Chriss: the efficient frontier of execution
=======================================================

Robert Almgren and Neil Chriss (2000) split impact into a permanent part
:math:`\gamma` and a temporary part :math:`\eta` that punishes trading
fast. Selling slowly is cheap in expectation but leaves the position
exposed to volatility for longer. Minimising :math:`E + \lambda V` gives

.. math::

   x(t) = X \frac{\sinh(\kappa(T - t))}{\sinh(\kappa T)}, \qquad \kappa \approx \sqrt{\lambda\sigma^2/\eta},

and sweeping the risk aversion :math:`\lambda` traces an efficient
frontier of expected cost against variance. The parameters are those of
their worked example: one million shares at $50, 30% annual volatility,
sold over five days.

.. GENERATED FROM PYTHON SOURCE LINES 21-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.market_microstructure import almgren_chriss
    from financialkit.market_microstructure.visualizers import plot_execution

    X, T, N = 1e6, 5.0, 5
    sigma, eta, gamma, eps = 0.95, 2.5e-6, 2.5e-7, 0.0625








.. GENERATED FROM PYTHON SOURCE LINES 31-33

Trajectories
------------

.. GENERATED FROM PYTHON SOURCE LINES 33-41

.. code-block:: Python


    lams = [0.0, 2e-7, 2e-6, 2e-5]
    results = [almgren_chriss(X, T, N, sigma, eta, lam, gamma, eps) for lam in lams]
    ax = plot_execution(results, labels=[f"$\\lambda$ = {lam:g}" for lam in lams])
    ax.set_xlabel("days")
    for lam, r in zip(lams, results, strict=True):
        print(f"lambda {lam:7.0e}: E = ${r.expected_cost / 1e6:.3f}M, sd = ${np.sqrt(r.variance) / 1e6:.3f}M")




.. image-sg:: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_03_almgren_chriss_001.png
   :alt: plot 03 almgren chriss
   :srcset: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_03_almgren_chriss_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    lambda   0e+00: E = $0.662M, sd = $1.041M
    lambda   2e-07: E = $0.688M, sd = $0.896M
    lambda   2e-06: E = $1.141M, sd = $0.449M
    lambda   2e-05: E = $2.110M, sd = $0.101M




.. GENERATED FROM PYTHON SOURCE LINES 42-44

The efficient frontier
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-56

.. code-block:: Python


    frontier = [almgren_chriss(X, T, N, sigma, eta, lam, gamma, eps) for lam in np.logspace(-8, -4, 60)]
    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot([np.sqrt(r.variance) / 1e6 for r in frontier], [r.expected_cost / 1e6 for r in frontier], "C0-")
    for lam, r in zip(lams, results, strict=True):
        ax.plot(np.sqrt(r.variance) / 1e6, r.expected_cost / 1e6, "o", label=f"$\\lambda$ = {lam:g}")
    ax.set_xlabel("standard deviation of cost ($M)")
    ax.set_ylabel("expected cost ($M)")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_03_almgren_chriss_002.png
   :alt: plot 03 almgren chriss
   :srcset: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_03_almgren_chriss_002.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_market_microstructure_execution_plot_03_almgren_chriss.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/market_microstructure/execution/plot_03_almgren_chriss.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_03_almgren_chriss.ipynb <plot_03_almgren_chriss.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_03_almgren_chriss.py <plot_03_almgren_chriss.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_almgren_chriss.zip <plot_03_almgren_chriss.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_