.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/multilevel/plot_01_giles_mlmc.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_multilevel_plot_01_giles_mlmc.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_multilevel_plot_01_giles_mlmc.py:


Giles multilevel Monte Carlo
============================

Pricing with a discretised SDE has two errors: the bias of the time
step and the statistical error. Balancing them with Euler's scheme costs
:math:`O(\varepsilon^{-3})`. Michael Giles (2008), borrowing the idea of
multigrid (and of Heinrich's multilevel integration), wrote the finest
approximation as a telescoping sum,

.. math:: E[P_L] = E[P_0] + \sum_{l=1}^{L} E[P_l - P_{l-1}],

and simulated each correction on coupled fine and coarse paths that
share Brownian increments. The corrections have small variance, so most
samples go to the cheap levels and the cost falls to
:math:`O(\varepsilon^{-2}(\log\varepsilon)^2)`. The paths are Euler
solutions from :func:`financialkit.integrators.euler_maruyama`.

.. GENERATED FROM PYTHON SOURCE LINES 21-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.monte_carlo import multilevel_monte_carlo
    from financialkit.monte_carlo.visualizers import plot_mlmc
    from financialkit.options import black_scholes_price

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








.. GENERATED FROM PYTHON SOURCE LINES 32-36

Level diagnostics
-----------------
The variance of :math:`P_l - P_{l-1}` falls by about 2 per level, and
so does the mean (the weak error of Euler).

.. GENERATED FROM PYTHON SOURCE LINES 36-43

.. code-block:: Python


    res = multilevel_monte_carlo(S, K, T, r, sigma, eps=0.02, rng=0)
    print(f"MLMC {res.estimate:.4f} +/- {res.std_error:.4f} (target RMSE {res.target_rmse}), Black-Scholes {exact:.4f}")
    print("samples per level:", res.level_samples.tolist())
    fig, axes = plt.subplots(1, 3, figsize=(16, 4))
    plot_mlmc(res, axes=axes[:2])




.. image-sg:: /api/gallery/monte_carlo/multilevel/images/sphx_glr_plot_01_giles_mlmc_001.png
   :alt: eps = 0.02
   :srcset: /api/gallery/monte_carlo/multilevel/images/sphx_glr_plot_01_giles_mlmc_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    MLMC 10.4453 +/- 0.0141 (target RMSE 0.02), Black-Scholes 10.4506
    samples per level: [1697090, 118731, 60806, 31872, 16009, 8055]

    array([<Axes: xlabel='level l'>,
           <Axes: title={'center': 'eps = 0.02'}, xlabel='level l', ylabel='samples $N_l$'>],
          dtype=object)



.. GENERATED FROM PYTHON SOURCE LINES 44-48

Cost against accuracy
---------------------
:math:`\varepsilon^2 \times` cost grows only logarithmically for MLMC,
like :math:`1/\varepsilon` for standard Monte Carlo.

.. GENERATED FROM PYTHON SOURCE LINES 48-58

.. code-block:: Python


    eps = np.array([0.2, 0.1, 0.05, 0.03, 0.02])
    runs = [multilevel_monte_carlo(S, K, T, r, sigma, eps=e, rng=i) for i, e in enumerate(eps)]
    axes[2].loglog(eps, eps**2 * np.array([x.cost for x in runs]), "o-", label="multilevel")
    axes[2].loglog(eps, eps**2 * np.array([x.standard_mc_cost for x in runs]), "s-", label="standard MC")
    axes[2].set_xlabel(r"target RMSE $\varepsilon$")
    axes[2].set_ylabel(r"$\varepsilon^2 \times$ cost (time steps)")
    axes[2].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 1.294 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_multilevel_plot_01_giles_mlmc.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/multilevel/plot_01_giles_mlmc.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_giles_mlmc.ipynb <plot_01_giles_mlmc.ipynb>`

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

      :download:`Download Python source code: plot_01_giles_mlmc.py <plot_01_giles_mlmc.py>`

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

      :download:`Download zipped: plot_01_giles_mlmc.zip <plot_01_giles_mlmc.zip>`


.. only:: html

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

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