.. 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_02_bertsimas_lo.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_02_bertsimas_lo.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_02_bertsimas_lo.py:


Bertsimas and Lo: optimal control of execution costs
====================================================

Dimitris Bertsimas and Andrew Lo (1998) treated buying :math:`\bar S`
shares over :math:`T` periods as a dynamic program. If each purchase
moves the price permanently by :math:`\theta` per share,
:math:`P_k = P_{k-1} + \theta S_k + \varepsilon_k`, the best a
risk-neutral trader can do is split the order evenly. If a serially
correlated signal :math:`X_k` also drives prices, the optimal order is
linear in the remaining shares and the latest signal, and trading
ahead of predicted rises pays.

.. GENERATED FROM PYTHON SOURCE LINES 16-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.market_microstructure import bertsimas_lo

    S, T, theta = 100_000.0, 20, 5e-5
    gamma, rho, sigma_eta = 0.05, 0.8, 1.0








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Without information: the even split
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-31

.. code-block:: Python


    plain = bertsimas_lo(S, T, theta)
    print(f"trades {plain.trades[:3]} ..., excess cost {plain.expected_cost:.0f} = theta S^2 (T + 1) / (2T)")





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

 .. code-block:: none

    trades [5000. 5000. 5000.] ..., excess cost 262500 = theta S^2 (T + 1) / (2T)




.. GENERATED FROM PYTHON SOURCE LINES 32-37

A signal tilts the schedule
---------------------------

With a positive signal prices are expected to rise, so the trader buys
early; with a negative one it waits.

.. GENERATED FROM PYTHON SOURCE LINES 37-47

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 3.5))
    k = np.arange(T)
    for x0, color in [(3.0, "C3"), (0.0, "0.5"), (-3.0, "C0")]:
        ax.bar(k + 0.27 * np.sign(x0), bertsimas_lo(S, T, theta, gamma, rho, sigma_eta, x0=x0).trades, width=0.27, color=color, label=f"$X_{{-1}}$ = {x0:g}")
    ax.set_xlabel("period")
    ax.set_ylabel("shares bought")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_02_bertsimas_lo_001.png
   :alt: plot 02 bertsimas lo
   :srcset: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_02_bertsimas_lo_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 48-53

The value of reacting to the signal
-----------------------------------

Simulate the signal and prices, follow the closed-loop policy along each
realised path, and compare costs with the even split.

.. GENERATED FROM PYTHON SOURCE LINES 53-76

.. code-block:: Python


    rng = np.random.default_rng(14)
    n_paths = 2000
    cost_opt, cost_twap = np.empty(n_paths), np.empty(n_paths)
    for j in range(n_paths):
        X = np.empty(T + 1)
        X[0] = rng.standard_normal() * sigma_eta / np.sqrt(1 - rho**2)
        for t in range(T):
            X[t + 1] = rho * X[t] + sigma_eta * rng.standard_normal()
        eps = 0.05 * rng.standard_normal(T)
        for trades, out in [(bertsimas_lo(S, T, theta, gamma, rho, sigma_eta, x0=X[0], signals=X[:-1]).trades, cost_opt), (np.full(T, S / T), cost_twap)]:
            P = np.cumsum(theta * trades + gamma * X[1:] + eps)
            out[j] = P @ trades
    saving = cost_twap - cost_opt
    print(f"average saving over TWAP {saving.mean():.0f} ± {saving.std() / np.sqrt(n_paths):.0f} (TWAP excess cost {plain.expected_cost:.0f})")

    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.hist(saving, bins=50, color="C0")
    ax.axvline(0, color="k", lw=0.8)
    ax.set_xlabel("cost of TWAP minus cost of the optimal policy")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_02_bertsimas_lo_002.png
   :alt: plot 02 bertsimas lo
   :srcset: /api/gallery/market_microstructure/execution/images/sphx_glr_plot_02_bertsimas_lo_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    average saving over TWAP 4573 ± 289 (TWAP excess cost 262500)





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

   **Total running time of the script:** (0 minutes 0.152 seconds)


.. _sphx_glr_download_api_gallery_market_microstructure_execution_plot_02_bertsimas_lo.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_02_bertsimas_lo.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_bertsimas_lo.ipynb <plot_02_bertsimas_lo.ipynb>`

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

      :download:`Download Python source code: plot_02_bertsimas_lo.py <plot_02_bertsimas_lo.py>`

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

      :download:`Download zipped: plot_02_bertsimas_lo.zip <plot_02_bertsimas_lo.zip>`


.. only:: html

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

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