.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/sdf/plot_01_harrison_kreps.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_asset_pricing_sdf_plot_01_harrison_kreps.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_asset_pricing_sdf_plot_01_harrison_kreps.py:


Harrison-Kreps: state prices and the stochastic discount factor
===============================================================

Michael Harrison and David Kreps (1979) proved the fundamental theorem
of asset pricing: a market has no arbitrage if and only if some strictly
positive linear rule prices every traded payoff. With finitely many
states it is a vector of state prices :math:`q_s > 0` with
:math:`p = X'q`; divided by the physical probabilities it is the
stochastic discount factor :math:`m_s = q_s / \pi_s`, so that
:math:`p = E[m x]`; normalised, it is the risk-neutral measure. In a
complete market the state prices are unique. In an incomplete one, a new
claim has an interval of arbitrage-free prices.

.. GENERATED FROM PYTHON SOURCE LINES 17-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import FiniteStateMarket

    # A one-period binomial market: bond and stock.
    S0, R = 100.0, 1.03
    up_down = np.array([125.0, 85.0])
    mkt = FiniteStateMarket(np.column_stack([[R, R], up_down]), [1.0, S0], probabilities=[0.6, 0.4])
    print("complete:", mkt.is_complete, " arbitrage free:", mkt.is_arbitrage_free())
    print("state prices:", mkt.state_prices().round(4), " risk-neutral:", mkt.risk_neutral_probabilities().round(4))
    print("SDF:", mkt.sdf().round(4))





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

 .. code-block:: none

    complete: True  arbitrage free: True
    state prices: [0.4369 0.534 ]  risk-neutral: [0.45 0.55]
    SDF: [0.7282 1.335 ]




.. GENERATED FROM PYTHON SOURCE LINES 31-33

An arbitrage when the stock is mispriced
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-40

.. code-block:: Python


    bad = FiniteStateMarket(np.column_stack([[R, R], up_down]), [1.0, 80.0])
    theta = bad.arbitrage_portfolio()
    theta = theta / theta[1]  # per share bought
    print(f"stock at 80: arbitrage free? {bad.is_arbitrage_free()}")
    print(f"buy 1 share, hold {theta[0]:+.1f} bonds: cost {bad.prices @ theta:+.2f}, payoffs {(bad.payoffs @ theta).round(2)}")





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

 .. code-block:: none

    stock at 80: arbitrage free? False
    buy 1 share, hold -80.0 bonds: cost +0.00, payoffs [42.6  2.6]




.. GENERATED FROM PYTHON SOURCE LINES 41-47

Incompleteness: a trinomial market
----------------------------------

Three states and two assets: the state prices are no longer unique, and
each call has a range of prices consistent with no arbitrage. The
binomial price lies inside it.

.. GENERATED FROM PYTHON SOURCE LINES 47-63

.. code-block:: Python


    states = np.array([125.0, 100.0, 85.0])
    tri = FiniteStateMarket(np.column_stack([np.full(3, R), states]), [1.0, S0])
    strikes = np.linspace(85, 125, 41)
    bounds = np.array([tri.price_bounds(np.maximum(states - K, 0)) for K in strikes])
    binom = np.array([mkt.state_prices() @ np.maximum(up_down - K, 0) for K in strikes])

    fig, ax = plt.subplots(figsize=(7, 4))
    ax.fill_between(strikes, bounds[:, 0], bounds[:, 1], color="C0", alpha=0.3, label="trinomial: arbitrage-free range")
    ax.plot(strikes, binom, "C3", label="binomial: unique price")
    ax.set_xlabel("strike")
    ax.set_ylabel("call price")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/sdf/images/sphx_glr_plot_01_harrison_kreps_001.png
   :alt: plot 01 harrison kreps
   :srcset: /api/gallery/asset_pricing/sdf/images/sphx_glr_plot_01_harrison_kreps_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.083 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_sdf_plot_01_harrison_kreps.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/asset_pricing/sdf/plot_01_harrison_kreps.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_harrison_kreps.ipynb <plot_01_harrison_kreps.ipynb>`

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

      :download:`Download Python source code: plot_01_harrison_kreps.py <plot_01_harrison_kreps.py>`

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

      :download:`Download zipped: plot_01_harrison_kreps.zip <plot_01_harrison_kreps.zip>`


.. only:: html

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

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