.. 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_05_paskov_traub.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_05_paskov_traub.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_05_paskov_traub.py:


Paskov-Traub: QMC in high dimension
===================================

Theory said quasi-Monte Carlo should lose its edge beyond a dozen
dimensions, since :math:`(\log N)^d / N` only beats :math:`N^{-1/2}` for
astronomically large :math:`N`. Spassimir Paskov and Joseph Traub (1995)
priced a Goldman Sachs collateralized mortgage obligation, a
360-dimensional integral, and found Sobol points beat Monte Carlo
anyway. The explanation, later formalized as *effective dimension*:
financial integrands depend mostly on a few coordinates and their
low-order interactions.

Here: a discretely monitored arithmetic Asian call on 64 dates, a
64-dimensional integral, with Sobol increments driving the steps in
time order (:func:`financialkit.integrators.sobol_increments` with
``bridge=False``).

.. GENERATED FROM PYTHON SOURCE LINES 21-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

    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)









.. GENERATED FROM PYTHON SOURCE LINES 36-40

Standard error against paths
----------------------------
Both estimates are replicated eight times with fresh seeds; the
standard errors are on the same footing.

.. GENERATED FROM PYTHON SOURCE LINES 40-51

.. code-block:: Python


    ns = 2 ** np.arange(7, 12)
    results = {s: [gbm_path_estimate(asian, S, T, r, sigma, d, n, sampler=s, n_replications=8, rng=n) for n in ns] for s in ("pseudo", "sobol")}
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    plot_error_convergence(ns, [x.std_error for x in results["pseudo"]], ax=axes[0], label="Monte Carlo", rates=(0.5,))
    plot_error_convergence(ns, [x.std_error for x in results["sobol"]], ax=axes[0], label="Sobol, 64 dimensions", rates=())
    axes[0].set_title("arithmetic Asian call, 64 dates")
    for s in ("pseudo", "sobol"):
        x = results[s][-1]
        print(f"{s:7s}: {x.estimate:.4f} +/- {x.std_error:.4f}")




.. image-sg:: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_05_paskov_traub_001.png
   :alt: arithmetic Asian call, 64 dates
   :srcset: /api/gallery/monte_carlo/quasi_monte_carlo/images/sphx_glr_plot_05_paskov_traub_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    pseudo : 5.8934 +/- 0.0410
    sobol  : 5.8454 +/- 0.0124




.. GENERATED FROM PYTHON SOURCE LINES 52-57

The integrand's effective dimension is low
------------------------------------------
Payoff sensitivity to each Sobol coordinate, by perturbing one normal
at a time: the average depends on every increment, but those of the
early dates matter most, since they move more of the average.

.. GENERATED FROM PYTHON SOURCE LINES 57-70

.. code-block:: Python


    z = np.random.default_rng(0).standard_normal((4000, d))
    base = asian(gbm_paths(S, r, sigma, T, d, 4000, dW=np.sqrt(T / d) * z)[1])
    sens = []
    for i in range(d):
        zi = z.copy()
        zi[:, i] += 0.1
        sens.append(np.mean(np.abs(asian(gbm_paths(S, r, sigma, T, d, 4000, dW=np.sqrt(T / d) * zi)[1]) - base)))
    axes[1].bar(np.arange(1, d + 1), sens)
    axes[1].set_xlabel("coordinate (time step)")
    axes[1].set_ylabel("mean |payoff change|")
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.256 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_quasi_monte_carlo_plot_05_paskov_traub.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_05_paskov_traub.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_paskov_traub.ipynb <plot_05_paskov_traub.ipynb>`

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

      :download:`Download Python source code: plot_05_paskov_traub.py <plot_05_paskov_traub.py>`

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

      :download:`Download zipped: plot_05_paskov_traub.zip <plot_05_paskov_traub.zip>`


.. only:: html

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

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