.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "generated_examples\savings\plot_ex1_option_value.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_ex1_option_value.py>`
        to download the full example code

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

.. _sphx_glr_generated_examples_savings_plot_ex1_option_value.py:


Monte Carlo vs Black-Scholes-Merton
===========================================

Time values of options and guarantees for various in-the-moneyness
are calculated using Monte Carlo simulations and the Black-Scholes-Merton
pricing formula for European put options.

The Black-Scholes-Merton pricing formula for European put options
can be expressed as below, where
:math:`X` and :math:`S_{0}` correspond to the sum assured
and the initial account value in this example.

.. math::

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

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

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


The graph below shows the results obtained from
the Monte Carlo simulations with 10,000 risk neutral scenarios,
and from the Black-Scholes-Merton formula.

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

.. seealso::

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

.. GENERATED FROM PYTHON SOURCE LINES 35-59



.. image-sg:: /generated_examples/savings/images/sphx_glr_plot_ex1_option_value_001.png
   :alt: TVOG by ITM
   :srcset: /generated_examples/savings/images/sphx_glr_plot_ex1_option_value_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

    model = mx.read_model("CashValue_ME_EX1")
    proj = model.Projection
    proj.model_point_table = proj.model_point_moneyness
    monte_carlo = pd.Series(proj.pv_claims_over_av('MATURITY'), index=proj.model_point().index)
    monte_carlo = list(np.average(monte_carlo[i]) for i in range(1, 10))
    S0 = proj.model_point_table['premium_pp'] * proj.model_point_table['policy_count']

    
    fig, ax = plt.subplots()
    ax.scatter(S0, monte_carlo, s= 10, alpha=1, label='Monte Carlo')
    ax.scatter(S0, proj.formula_option_put(120), alpha=0.5, label='Black-Scholes-Merton')
    ax.legend()
    ax.grid(True)
    fig.suptitle('TVOG by ITM')






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

   **Total running time of the script:** ( 0 minutes  2.643 seconds)


.. _sphx_glr_download_generated_examples_savings_plot_ex1_option_value.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_ex1_option_value.py <plot_ex1_option_value.py>`

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

      :download:`Download Jupyter notebook: plot_ex1_option_value.ipynb <plot_ex1_option_value.ipynb>`


.. only:: html

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

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