.. 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_03_antithetic_variates.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_03_antithetic_variates.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_03_antithetic_variates.py:


Hammersley-Morton antithetic variates
=====================================

John Hammersley and Keith Morton (1956) proposed pairing each random
input with its mirror image. If :math:`f` is monotone, :math:`f(Z)` and
:math:`f(-Z)` are negatively correlated and their average is less
variable than two independent draws:

.. math:: \operatorname{Var}\left[\tfrac12(f(Z) + f(-Z))\right] = \tfrac12\operatorname{Var} f(Z)\,(1 + \rho).

The same pairing applied to whole Brownian paths is
:func:`financialkit.integrators.antithetic_increments`.

.. GENERATED FROM PYTHON SOURCE LINES 17-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.integrators import antithetic_increments, gbm_paths
    from financialkit.monte_carlo import antithetic_estimate, sample_mean, terminal_prices

    S, T, r, sigma = 100.0, 1.0, 0.05, 0.2








.. GENERATED FROM PYTHON SOURCE LINES 26-30

The gain depends on the payoff's monotonicity
---------------------------------------------
A call is monotone in :math:`Z` and gains; a straddle is nearly
symmetric, so the pair is positively correlated and antithetics hurt.

.. GENERATED FROM PYTHON SOURCE LINES 30-48

.. code-block:: Python


    strikes = np.linspace(70, 130, 13)
    gain = {"call": [], "straddle": []}
    for K in strikes:
        for name, payoff in [("call", lambda s, K=K: np.maximum(s - K, 0)), ("straddle", lambda s, K=K: np.abs(s - K))]:
            f = lambda z, p=payoff: np.exp(-r * T) * p(terminal_prices(S, T, r, sigma, z))  # noqa: E731
            crude = sample_mean(f(np.random.default_rng(0).standard_normal(20_000)))
            anti = antithetic_estimate(f, 10_000, rng=1)
            gain[name].append(anti.variance_ratio(crude))

    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    for name, g in gain.items():
        axes[0].plot(strikes, g, "o-", label=name)
    axes[0].axhline(1.0, color="k", lw=1)
    axes[0].set_xlabel("strike K")
    axes[0].set_ylabel("variance reduction factor")
    axes[0].legend()




.. image-sg:: /api/gallery/monte_carlo/variance_reduction/images/sphx_glr_plot_03_antithetic_variates_001.png
   :alt: plot 03 antithetic variates
   :srcset: /api/gallery/monte_carlo/variance_reduction/images/sphx_glr_plot_03_antithetic_variates_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x112237cb0>



.. GENERATED FROM PYTHON SOURCE LINES 49-52

Antithetic Brownian paths
-------------------------
An Asian call on 12 monthly averages, priced from mirrored paths.

.. GENERATED FROM PYTHON SOURCE LINES 52-68

.. code-block:: Python


    dW = antithetic_increments(T, 12, 20_000, rng=2)
    _, paths = gbm_paths(S, r, sigma, T, 12, 20_000, dW=dW)
    pay = np.exp(-r * T) * np.maximum(paths[:, 1:].mean(axis=1) - 100.0, 0)
    pairs = sample_mean(0.5 * (pay[:10_000] + pay[10_000:]), n_evaluations=20_000)
    _, ind = gbm_paths(S, r, sigma, T, 12, 20_000, rng=3)
    crude = sample_mean(np.exp(-r * T) * np.maximum(ind[:, 1:].mean(axis=1) - 100.0, 0))
    print(f"Asian call: crude {crude.estimate:.4f} +/- {crude.std_error:.4f}, antithetic {pairs.estimate:.4f} +/- {pairs.std_error:.4f}")
    axes[1].plot(paths[0], label="path")
    axes[1].plot(paths[10_000], label="its antithetic mirror")
    axes[1].set_xlabel("month")
    axes[1].set_ylabel("S")
    axes[1].legend()
    axes[1].set_title(f"Asian call: variance reduction x{pairs.variance_ratio(crude):.1f}")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    Asian call: crude 6.1235 +/- 0.0599, antithetic 6.1736 +/- 0.0419





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

   **Total running time of the script:** (0 minutes 0.074 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_variance_reduction_plot_03_antithetic_variates.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_03_antithetic_variates.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_antithetic_variates.ipynb <plot_03_antithetic_variates.ipynb>`

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

      :download:`Download Python source code: plot_03_antithetic_variates.py <plot_03_antithetic_variates.py>`

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

      :download:`Download zipped: plot_03_antithetic_variates.zip <plot_03_antithetic_variates.zip>`


.. only:: html

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

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