.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/market_microstructure/dealers/plot_02_ho_stoll.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_dealers_plot_02_ho_stoll.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_dealers_plot_02_ho_stoll.py:


Ho and Stoll: the dealer's inventory problem
============================================

Thomas Ho and Hans Stoll (1981) gave the dealer risk aversion. Holding
inventory exposes it to price moves, so a dealer who is long lowers
*both* quotes, making a sale to it less attractive and a purchase from
it more so, until the inventory returns to zero. With order arrivals
:math:`\lambda = \alpha - \beta\delta` on each side, the spread itself,
:math:`\alpha/\beta`, does not depend on the inventory: only its position
does.

.. GENERATED FROM PYTHON SOURCE LINES 15-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.market_microstructure import ho_stoll_quotes

    alpha, beta, sigma = 10.0, 40.0, 0.5
    inventory = np.linspace(-30, 30, 61)








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Quotes shade against inventory
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-38

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 4))
    for g, ls in [(0.0, ":"), (0.05, "--"), (0.2, "-")]:
        b, a = ho_stoll_quotes(100.0, inventory, alpha, beta, g, sigma)
        ax.plot(inventory, a, "C1" + ls, label=f"ask, $\\gamma$ = {g}")
        ax.plot(inventory, b, "C0" + ls, label=f"bid, $\\gamma$ = {g}")
    ax.axhline(100.0, color="0.5", lw=0.8)
    ax.set_xlabel("dealer inventory")
    ax.set_ylabel("quote")
    ax.legend(ncol=3, fontsize="small")
    fig.tight_layout()




.. image-sg:: /api/gallery/market_microstructure/dealers/images/sphx_glr_plot_02_ho_stoll_001.png
   :alt: plot 02 ho stoll
   :srcset: /api/gallery/market_microstructure/dealers/images/sphx_glr_plot_02_ho_stoll_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 39-45

Inventory control in action
---------------------------

Each period, buy and sell orders arrive as Poisson counts at the rates
the quotes imply. A risk-neutral dealer's inventory wanders like a
random walk; the risk-averse dealer's is mean-reverting.

.. GENERATED FROM PYTHON SOURCE LINES 45-69

.. code-block:: Python


    rng = np.random.default_rng(4)
    n_paths, n_periods = 400, 300
    paths = {}
    for g in (0.0, 0.2):
        I = np.zeros((n_paths, n_periods + 1))
        for t in range(n_periods):
            b, a = ho_stoll_quotes(100.0, I[:, t], alpha, beta, g, sigma)
            lam_a = np.clip(alpha - beta * (a - 100.0), 0, None)
            lam_b = np.clip(alpha - beta * (100.0 - b), 0, None)
            I[:, t + 1] = I[:, t] - rng.poisson(lam_a) + rng.poisson(lam_b)
        paths[g] = I
        print(f"gamma = {g}: inventory sd after {n_periods} periods {I[:, -1].std():.1f}")

    fig, ax = plt.subplots(figsize=(7, 3.5))
    for g, color in [(0.0, "C3"), (0.2, "C0")]:
        ax.plot(paths[g][:10].T, color=color, lw=0.7, alpha=0.7)
        ax.plot([], [], color=color, label=f"$\\gamma$ = {g}")
    ax.set_xlabel("period")
    ax.set_ylabel("inventory")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/market_microstructure/dealers/images/sphx_glr_plot_02_ho_stoll_002.png
   :alt: plot 02 ho stoll
   :srcset: /api/gallery/market_microstructure/dealers/images/sphx_glr_plot_02_ho_stoll_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    gamma = 0.0: inventory sd after 300 periods 53.6
    gamma = 0.2: inventory sd after 300 periods 3.5





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

   **Total running time of the script:** (0 minutes 0.117 seconds)


.. _sphx_glr_download_api_gallery_market_microstructure_dealers_plot_02_ho_stoll.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/dealers/plot_02_ho_stoll.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_ho_stoll.ipynb <plot_02_ho_stoll.ipynb>`

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

      :download:`Download Python source code: plot_02_ho_stoll.py <plot_02_ho_stoll.py>`

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

      :download:`Download zipped: plot_02_ho_stoll.zip <plot_02_ho_stoll.zip>`


.. only:: html

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

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