.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/variance_reduction/plot_01_neyman_stratification.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_monte_carlo_variance_reduction_plot_01_neyman_stratification.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_monte_carlo_variance_reduction_plot_01_neyman_stratification.py:


Neyman stratification
=====================

Jerzy Neyman (1934) showed survey statisticians how to sample a
stratified population: draw from each stratum separately and allocate
the sample in proportion to the stratum's weight times its standard
deviation,

.. math:: n_h \propto p_h \sigma_h, \qquad
          \operatorname{Var}\hat\mu = \sum_h \frac{p_h^2 \sigma_h^2}{n_h}.

In Monte Carlo the "population" is the unit interval that drives the
terminal stock price through :math:`Z = \Phi^{-1}(U)`. Equal-probability
strata of :math:`U` remove the between-strata variance, and Neyman's
allocation puts the remaining samples where the payoff varies most: in
the money.

.. GENERATED FROM PYTHON SOURCE LINES 21-37

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import norm

    from financialkit.monte_carlo import european_option_mc, stratified_estimate, terminal_prices, vanilla_payoff
    from financialkit.monte_carlo.visualizers import plot_allocation, plot_estimator_comparison
    from financialkit.options import black_scholes_price

    S, K, T, r, sigma = 100.0, 110.0, 1.0, 0.05, 0.2
    exact = black_scholes_price(S, K, T, r, sigma)


    def call(u):
        return np.exp(-r * T) * vanilla_payoff(terminal_prices(S, T, r, sigma, norm.ppf(u)), K)









.. GENERATED FROM PYTHON SOURCE LINES 38-40

Crude, proportional and Neyman allocations
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-51

.. code-block:: Python


    n = 10_000
    crude = european_option_mc(S, K, T, r, sigma, n_paths=n, rng=0)
    prop = stratified_estimate(call, n, n_strata=20, allocation="proportional", rng=1)
    ney = stratified_estimate(call, n, n_strata=20, allocation="neyman", rng=2)
    for res in (crude, prop, ney):
        print(f"{res.method:24s} {res.estimate:.4f} +/- {res.std_error:.4f}  variance reduction x{res.variance_ratio(crude):.1f}")

    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    plot_estimator_comparison([crude, prop, ney], exact=exact, ax=axes[0])




.. image-sg:: /api/gallery/monte_carlo/variance_reduction/images/sphx_glr_plot_01_neyman_stratification_001.png
   :alt: plot 01 neyman stratification
   :srcset: /api/gallery/monte_carlo/variance_reduction/images/sphx_glr_plot_01_neyman_stratification_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    crude                    6.0910 +/- 0.1148  variance reduction x1.0
    stratified (proportional) 6.0174 +/- 0.0273  variance reduction x17.7
    stratified (neyman)      6.0396 +/- 0.0107  variance reduction x113.0

    <Axes: ylabel='estimate (95% CI); variance reduction'>



.. GENERATED FROM PYTHON SOURCE LINES 52-56

Where Neyman puts the samples
-----------------------------
Strata of low :math:`U` end out of the money with zero variance and get
only the two-sample floor.

.. GENERATED FROM PYTHON SOURCE LINES 56-60

.. code-block:: Python


    plot_allocation(ney, ax=axes[1])
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.064 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_variance_reduction_plot_01_neyman_stratification.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/monte_carlo/variance_reduction/plot_01_neyman_stratification.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_neyman_stratification.ipynb <plot_01_neyman_stratification.ipynb>`

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

      :download:`Download Python source code: plot_01_neyman_stratification.py <plot_01_neyman_stratification.py>`

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

      :download:`Download zipped: plot_01_neyman_stratification.zip <plot_01_neyman_stratification.zip>`


.. only:: html

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

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