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

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


"""
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()