.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/quasi_monte_carlo/plot_06_brownian_bridge.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_quasi_monte_carlo_plot_06_brownian_bridge.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_quasi_monte_carlo_plot_06_brownian_bridge.py:


Caflisch-Moskowitz Brownian bridge
==================================

Quasi-random points are best distributed in their first coordinates. If
those coordinates drive the first time steps, the coarse shape of each
path is left to the poorly distributed later ones. Russel Caflisch and
Bradley Moskowitz (1995), and Caflisch, Morokoff and Owen (1997) for
mortgage-backed securities, reordered the construction: the first
coordinate sets :math:`W(T)`, the second :math:`W(T/2)` given the ends,
and so on by bisection,

.. math:: W(t_m) \mid W(t_l), W(t_r) \sim N\left(\frac{(t_r - t_m)W(t_l) + (t_m - t_l)W(t_r)}{t_r - t_l},\;
          \frac{(t_m - t_l)(t_r - t_m)}{t_r - t_l}\right),

which concentrates the variance in the leading coordinates and lowers
the effective dimension.

.. GENERATED FROM PYTHON SOURCE LINES 21-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.integrators import brownian_bridge
    from financialkit.monte_carlo import gbm_path_estimate
    from financialkit.monte_carlo.visualizers import plot_error_convergence








.. GENERATED FROM PYTHON SOURCE LINES 29-33

Coarse shape first
------------------
The same normals, used in bridge order: after 1, 2 and 4 coordinates the
path's outline is already in place.

.. GENERATED FROM PYTHON SOURCE LINES 33-44

.. code-block:: Python


    z = np.random.default_rng(3).standard_normal(16)
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    t = np.linspace(0, 1, 17)
    for k in (1, 2, 4, 16):
        zk = np.where(np.arange(16) < k, z, 0.0)
        axes[0].plot(t, brownian_bridge(zk[None, :], 1.0)[0], "o-" if k < 16 else "k-", ms=3, label=f"first {k} coordinates")
    axes[0].set_xlabel("t")
    axes[0].set_ylabel("W(t)")
    axes[0].legend()




.. image-sg:: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_06_brownian_bridge_001.png
   :alt: plot 06 brownian bridge
   :srcset: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_06_brownian_bridge_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x111efdd30>



.. GENERATED FROM PYTHON SOURCE LINES 45-48

Bridge Sobol against time-ordered Sobol
---------------------------------------
The 64-date Asian call of the Paskov-Traub example.

.. GENERATED FROM PYTHON SOURCE LINES 48-63

.. code-block:: Python


    S, K, T, r, sigma, d = 100.0, 100.0, 1.0, 0.05, 0.2, 64


    def asian(paths):
        return np.maximum(paths[:, 1:].mean(axis=1) - K, 0.0)


    ns = 2 ** np.arange(7, 12)
    for s, label in [("pseudo", "Monte Carlo"), ("sobol", "Sobol, time order"), ("bridge", "Sobol, Brownian bridge")]:
        res = [gbm_path_estimate(asian, S, T, r, sigma, d, n, sampler=s, n_replications=8, rng=n) for n in ns]
        plot_error_convergence(ns, [x.std_error for x in res], ax=axes[1], label=label, rates=(0.5,) if s == "pseudo" else ())
        print(f"{label:24s}: {res[-1].estimate:.4f} +/- {res[-1].std_error:.5f}")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    Monte Carlo             : 5.8934 +/- 0.04095
    Sobol, time order       : 5.8454 +/- 0.01236
    Sobol, Brownian bridge  : 5.8393 +/- 0.00283





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

   **Total running time of the script:** (0 minutes 0.245 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_quasi_monte_carlo_plot_06_brownian_bridge.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/quasi_monte_carlo/plot_06_brownian_bridge.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_06_brownian_bridge.ipynb <plot_06_brownian_bridge.ipynb>`

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

      :download:`Download Python source code: plot_06_brownian_bridge.py <plot_06_brownian_bridge.py>`

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

      :download:`Download zipped: plot_06_brownian_bridge.zip <plot_06_brownian_bridge.zip>`


.. only:: html

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

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