.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/interest_rate_models/options/plot_01_jamshidian_bond_option.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_interest_rate_models_options_plot_01_jamshidian_bond_option.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_interest_rate_models_options_plot_01_jamshidian_bond_option.py:


Jamshidian: an exact bond option formula
========================================

Farshid Jamshidian (1989) priced a European option on a zero-coupon bond
in Vasicek's model in closed form. Under the measure that uses the
option-expiry bond as numeraire, the forward bond price
:math:`P(0, S)/P(0, T)` is a lognormal martingale, so

.. math:: c = P(0, S) N(h) - K P(0, T) N(h - \sigma_P), \quad
          h = \frac{1}{\sigma_P}\ln\frac{P(0, S)}{K P(0, T)} + \frac{\sigma_P}{2},

which is Black-76 on the forward bond. Only the total volatility
:math:`\sigma_P` depends on the model, and the same formula holds in
Ho-Lee, Hull-White and G2++.

.. GENERATED FROM PYTHON SOURCE LINES 19-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.interest_rate_models import G2PlusPlusModel, HoLeeModel, HullWhiteModel, VasicekModel, path_discount_factors, zero_coupon_bond_option

    vasicek = VasicekModel(0.03, 0.3, 0.05, 0.015)
    T, S = 2.0, 7.0








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Closed form against Monte Carlo
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-47

.. code-block:: Python


    K = np.linspace(0.70, 0.90, 9)
    closed = zero_coupon_bond_option(vasicek, T, S, K, "call")
    times, r = vasicek.simulate(T, 100, 20_000, rng=0)
    D = path_discount_factors(times, r)[:, -1]
    bond_at_T = vasicek.zero_coupon_bond(S, t=T, state=r[:, -1])
    payoffs = D[:, None] * np.maximum(bond_at_T[:, None] - K[None, :], 0.0)
    mc, se = payoffs.mean(axis=0), payoffs.std(axis=0) / np.sqrt(D.size)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(K, closed, "k-", label="Jamshidian closed form")
    axes[0].errorbar(K, mc, yerr=1.96 * se, fmt="o", ms=3, capsize=2, label="Monte Carlo")
    axes[0].set_xlabel("strike")
    axes[0].set_ylabel(f"call on a {S:g}y bond, expiring in {T:g}y")
    axes[0].legend()
    print(f"largest |MC - closed form| / SE: {np.max(np.abs(mc - closed) / se):.2f}")




.. image-sg:: /api/gallery/interest_rate_models/options/images/sphx_glr_plot_01_jamshidian_bond_option_001.png
   :alt: plot 01 jamshidian bond option
   :srcset: /api/gallery/interest_rate_models/options/images/sphx_glr_plot_01_jamshidian_bond_option_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    largest |MC - closed form| / SE: 1.14




.. GENERATED FROM PYTHON SOURCE LINES 48-53

One formula, four Gaussian models
---------------------------------

At the forward strike, the option value is driven by the total bond
volatility each model implies.

.. GENERATED FROM PYTHON SOURCE LINES 53-71

.. code-block:: Python


    curve = vasicek.curve()
    models = {
        "Vasicek": vasicek,
        "Hull-White": HullWhiteModel(curve, 0.3, 0.015),
        "Ho-Lee": HoLeeModel(curve, 0.015),
        "G2++": G2PlusPlusModel(curve, 0.8, 0.012, 0.05, 0.008, -0.7),
    }
    expiry = np.linspace(0.25, 6.0, 40)
    for name, m in models.items():
        Kf = m.zero_coupon_bond(S) / m.zero_coupon_bond(expiry)
        axes[1].plot(expiry, zero_coupon_bond_option(m, expiry, S, Kf), label=name)
    axes[1].set_xlabel("option expiry T (years)")
    axes[1].set_ylabel(f"at-the-money call on the {S:g}y bond")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.109 seconds)


.. _sphx_glr_download_api_gallery_interest_rate_models_options_plot_01_jamshidian_bond_option.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/interest_rate_models/options/plot_01_jamshidian_bond_option.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_jamshidian_bond_option.ipynb <plot_01_jamshidian_bond_option.ipynb>`

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

      :download:`Download Python source code: plot_01_jamshidian_bond_option.py <plot_01_jamshidian_bond_option.py>`

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

      :download:`Download zipped: plot_01_jamshidian_bond_option.zip <plot_01_jamshidian_bond_option.zip>`


.. only:: html

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

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