.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "generated_examples\savings\plot_ex2_comp_option_values.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_generated_examples_savings_plot_ex2_comp_option_values.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_generated_examples_savings_plot_ex2_comp_option_values.py:


Black-Scholes-Merton on dividend paying stock
=============================================

As the :doc:`/libraries/notebooks/savings/savings_example2` shows,
time values of options and guarantees on a GMAB policy
can be calculated using the Black-Scholes-Merton formula
on a dividend paying stock, when maintenance fees are deducted
from account value at a constant rate, by regarding the fees as dividends.

The Black-Scholes-Merton pricing formula for European put options
on a dividend paying stock
can be expressed as below, where
:math:`X`, :math:`S_{0}`, :math:`q` correspond to the sum assured,
the initial account value and the maintenence fee rate(1%) in this example.

.. math::

    p=Xe^{-rT}N\left(-d_{2}\right)-S_{0}e^{-qT}N\left(-d_{1}\right)

    d_{1}=d_{1}=\frac{\ln\left(\frac{S_{0}}{X}\right)+\left(r-q+\frac{\sigma^{2}}{2}\right)T}{\sigma\sqrt{T}}

    d_{2}=d_{1}-\sigma\sqrt{T}

The graph below compares the option values with the maintenance fee deduction
against the corresponding values without fee deduction
for various in-the-moneyness.

Reference: *Options, Futures, and Other Derivatives* by John C.Hull

.. seealso::

    * :doc:`/libraries/notebooks/savings/savings_example1` notebook in the :mod:`~savings` library
    * :doc:`/libraries/notebooks/savings/savings_example2` notebook in the :mod:`~savings` library

.. GENERATED FROM PYTHON SOURCE LINES 38-64



.. image-sg:: /generated_examples/savings/images/sphx_glr_plot_ex2_comp_option_values_001.png
   :alt: TVOG by ITM
   :srcset: /generated_examples/savings/images/sphx_glr_plot_ex2_comp_option_values_001.png
   :class: sphx-glr-single-img





.. code-block:: default

    import modelx as mx
    import pandas as pd
    import matplotlib.pyplot as plt
    from scipy.stats import norm, lognorm
    import numpy as np

    ex1 = mx.read_model("CashValue_ME_EX1").Projection
    ex2 = mx.read_model("CashValue_ME_EX2").Projection

    ex1.model_point_table = ex1.model_point_moneyness
    ex2.model_point_table = ex2.model_point_moneyness
    S0 = ex1.model_point_table['premium_pp'] * ex1.model_point_table['policy_count']

    fig, ax = plt.subplots()
    ax.scatter(S0, ex1.formula_option_put(120), s= 10, alpha=1, label='No dividends')
    ax.scatter(S0, ex2.formula_option_put(120), alpha=0.5, label='With dividends')
    ax.legend()
    ax.grid(True)
    fig.suptitle('TVOG by ITM')

    ex1.model.close()
    ex2.model.close()






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

   **Total running time of the script:** ( 0 minutes  4.844 seconds)


.. _sphx_glr_download_generated_examples_savings_plot_ex2_comp_option_values.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


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

      :download:`Download Python source code: plot_ex2_comp_option_values.py <plot_ex2_comp_option_values.py>`

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

      :download:`Download Jupyter notebook: plot_ex2_comp_option_values.ipynb <plot_ex2_comp_option_values.ipynb>`


.. only:: html

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

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