.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/greeks/plot_02_pathwise.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_greeks_plot_02_pathwise.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_greeks_plot_02_pathwise.py:


Broadie-Glasserman pathwise Greeks
==================================

Mark Broadie and Paul Glasserman (1996) differentiated each simulated
payoff with respect to the parameter and averaged:

.. math:: \Delta = E\left[e^{-rT}\mathbf 1\{S_T > K\}\frac{S_T}{S}\right], \qquad
          \mathcal V = E\left[e^{-rT}\mathbf 1\{S_T > K\} S_T(\sqrt T Z - \sigma T)\right].

Interchanging derivative and expectation is valid when the payoff is
Lipschitz, as a call's is. The estimator is then far less noisy than the
likelihood ratio on the same paths, but it has nothing to say about a
digital, or about the gamma of a call, whose derivatives jump.

.. GENERATED FROM PYTHON SOURCE LINES 18-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.monte_carlo import likelihood_ratio_greeks, pathwise_greeks
    from financialkit.options import black_scholes_greeks

    S, K, T, r, sigma = 100.0, 100.0, 0.5, 0.03, 0.25
    exact = black_scholes_greeks(S, K, T, r, sigma)








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Pathwise against likelihood ratio, on common random numbers
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-39

.. code-block:: Python


    z = np.random.default_rng(0).standard_normal(100_000)
    pw, lr = pathwise_greeks(S, K, T, r, sigma, z=z), likelihood_ratio_greeks(S, K, T, r, sigma, z=z)
    for name in ("delta", "vega"):
        print(
            f"{name}: exact {getattr(exact, name):.4f}; pathwise {getattr(pw, name):.4f} +/- {getattr(pw, name + '_se'):.4f}; "
            f"likelihood ratio {getattr(lr, name):.4f} +/- {getattr(lr, name + '_se'):.4f}"
        )





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

 .. code-block:: none

    delta: exact 0.5688; pathwise 0.5670 +/- 0.0018; likelihood ratio 0.5674 +/- 0.0043
    vega: exact 27.7893; pathwise 27.7650 +/- 0.1619; likelihood ratio 27.7868 +/- 0.5900




.. GENERATED FROM PYTHON SOURCE LINES 40-42

The variance advantage across strikes
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-56

.. code-block:: Python


    strikes = np.linspace(70, 140, 15)
    ratio = {"delta": [], "vega": []}
    for k in strikes:
        p, l = pathwise_greeks(S, k, T, r, sigma, z=z), likelihood_ratio_greeks(S, k, T, r, sigma, z=z)
        for name in ratio:
            ratio[name].append((getattr(l, name + "_se") / getattr(p, name + "_se")) ** 2)
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    for name, v in ratio.items():
        axes[0].semilogy(strikes, v, "o-", label=name)
    axes[0].set_xlabel("strike K")
    axes[0].set_ylabel("variance(LR) / variance(pathwise)")
    axes[0].legend()




.. image-sg:: /api/gallery/monte_carlo/greeks/images/sphx_glr_plot_02_pathwise_001.png
   :alt: plot 02 pathwise
   :srcset: /api/gallery/monte_carlo/greeks/images/sphx_glr_plot_02_pathwise_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1125201a0>



.. GENERATED FROM PYTHON SOURCE LINES 57-59

Pathwise deltas converge with the paths
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 59-69

.. code-block:: Python


    ns = 1000 * 4 ** np.arange(5)
    g = [pathwise_greeks(S, K, T, r, sigma, n_paths=n, rng=n) for n in ns]
    axes[1].errorbar(ns, [x.delta for x in g], yerr=[2 * x.delta_se for x in g], fmt="o", capsize=3, label="pathwise delta")
    axes[1].axhline(exact.delta, color="k", label="Black-Scholes")
    axes[1].set_xscale("log")
    axes[1].set_xlabel("paths")
    axes[1].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.142 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_greeks_plot_02_pathwise.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/greeks/plot_02_pathwise.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_pathwise.ipynb <plot_02_pathwise.ipynb>`

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

      :download:`Download Python source code: plot_02_pathwise.py <plot_02_pathwise.py>`

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

      :download:`Download zipped: plot_02_pathwise.zip <plot_02_pathwise.zip>`


.. only:: html

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

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