.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/trees/plot_01_crr_convergence.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_options_trees_plot_01_crr_convergence.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_options_trees_plot_01_crr_convergence.py:


Cox-Ross-Rubinstein: the binomial tree converges to Black-Scholes
=================================================================

John Cox, Stephen Ross and Mark Rubinstein (1979) showed that the
Black-Scholes price is the limit of a much simpler model: in each short
period the stock moves up by :math:`u = e^{\sigma\sqrt{\Delta t}}` or
down by :math:`d = 1/u`, and the option is replicated one step at a
time. The risk-neutral probability :math:`p = (e^{r\Delta t} - d)/(u -
d)` falls out of the replication, and backward induction prices any
payoff. The error shrinks like :math:`1/n`, oscillating as the strike
moves between nodes.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import CRRTree, black_scholes_price
    from financialkit.options.visualizers import plot_tree_convergence








.. GENERATED FROM PYTHON SOURCE LINES 23-29

One-period replication
----------------------

A call struck at 100 on a stock at 100 that moves to 110 or 90 in a
year, with r = 5%: hold :math:`\Delta` shares and borrow so the
portfolio matches the call in both states.

.. GENERATED FROM PYTHON SOURCE LINES 29-38

.. code-block:: Python


    S, K, u, d, r = 100.0, 100.0, 1.1, 0.9, 0.05
    cu, cd = max(S * u - K, 0), max(S * d - K, 0)
    delta = (cu - cd) / (S * u - S * d)
    bond = (cu - delta * S * u) * np.exp(-r)
    p = (np.exp(r) - d) / (u - d)
    print(f"replicate with {delta:.2f} shares and {bond:.3f} in bonds: price {delta * S + bond:.4f}")
    print(f"risk-neutral expectation with p = {p:.4f}: {np.exp(-r) * (p * cu + (1 - p) * cd):.4f}")





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

 .. code-block:: none

    replicate with 0.50 shares and -42.805 in bonds: price 7.1947
    risk-neutral expectation with p = 0.7564: 7.1947




.. GENERATED FROM PYTHON SOURCE LINES 39-41

Convergence to Black-Scholes
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-60

.. code-block:: Python


    S, K, T, r, sigma = 100.0, 95.0, 1.0, 0.05, 0.2
    exact = black_scholes_price(S, K, T, r, sigma)
    steps = np.arange(5, 301, 1)
    prices = np.array([CRRTree(n).price(S, K, T, r, sigma).price for n in steps])
    print(f"Black-Scholes {exact:.5f}; CRR with 300 steps {prices[-1]:.5f}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(steps, prices, lw=1, label="CRR tree")
    axes[0].axhline(exact, color="k", ls=":", label="Black-Scholes")
    axes[0].set_xlabel("time steps n")
    axes[0].set_ylabel("call price")
    axes[0].legend()
    plot_tree_convergence(steps, prices, exact, ax=axes[1], label="CRR error")
    axes[1].loglog(steps, 1.0 / steps, "k:", label="1/n")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/trees/images/sphx_glr_plot_01_crr_convergence_001.png
   :alt: plot 01 crr convergence
   :srcset: /api/gallery/options/trees/images/sphx_glr_plot_01_crr_convergence_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Black-Scholes 13.34646; CRR with 300 steps 13.34822





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

   **Total running time of the script:** (0 minutes 0.217 seconds)


.. _sphx_glr_download_api_gallery_options_trees_plot_01_crr_convergence.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/options/trees/plot_01_crr_convergence.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_crr_convergence.ipynb <plot_01_crr_convergence.ipynb>`

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

      :download:`Download Python source code: plot_01_crr_convergence.py <plot_01_crr_convergence.py>`

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

      :download:`Download zipped: plot_01_crr_convergence.zip <plot_01_crr_convergence.zip>`


.. only:: html

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

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