.. 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_01_likelihood_ratio.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_01_likelihood_ratio.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_01_likelihood_ratio.py:


Glynn: likelihood-ratio Greeks
==============================

Bumping a parameter and re-simulating gives a biased, noisy derivative.
Peter Glynn (1987), and Reiman and Weiss (1989), differentiated the
*density* instead of the payoff:

.. math:: \frac{\partial}{\partial\theta} E[f(S_T)] = E\left[f(S_T)\,\frac{\partial \ln p_\theta(S_T)}{\partial\theta}\right].

Under Black-Scholes the delta weight is :math:`Z / (S\sigma\sqrt T)`. The
payoff is never differentiated, so the method prices the Greeks of a
digital, whose payoff has no useful derivative at all.

.. GENERATED FROM PYTHON SOURCE LINES 17-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import norm

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

    S, T, r, sigma = 100.0, 0.5, 0.03, 0.25
    spots = np.linspace(70, 130, 13)








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Vanilla call: delta, vega and gamma
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-41

.. code-block:: Python


    lr = [likelihood_ratio_greeks(s, 100.0, T, r, sigma, n_paths=50_000, rng=i) for i, s in enumerate(spots)]
    bs = black_scholes_greeks(spots, 100.0, T, r, sigma)
    fig, axes = plt.subplots(1, 4, figsize=(18, 4))
    for ax, name in zip(axes[:3], ["delta", "vega", "gamma"], strict=True):
        ax.plot(spots, getattr(bs, name), "k", label="Black-Scholes")
        ax.errorbar(spots, [getattr(g, name) for g in lr], yerr=[2 * getattr(g, name + "_se") for g in lr], fmt="o", ms=3, capsize=2, label="likelihood ratio")
        ax.set_xlabel("spot S")
        ax.set_title(name)
    axes[0].legend()




.. image-sg:: /api/gallery/monte_carlo/greeks/images/sphx_glr_plot_01_likelihood_ratio_001.png
   :alt: delta, vega, gamma
   :srcset: /api/gallery/monte_carlo/greeks/images/sphx_glr_plot_01_likelihood_ratio_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1123d6120>



.. GENERATED FROM PYTHON SOURCE LINES 42-46

Digital call delta
------------------
Cash-or-nothing call: :math:`V = e^{-rT}N(d_2)`, delta
:math:`e^{-rT} n(d_2) / (S\sigma\sqrt T)`.

.. GENERATED FROM PYTHON SOURCE LINES 46-58

.. code-block:: Python


    d2 = (np.log(spots / 100.0) + (r - 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    dig = [likelihood_ratio_greeks(s, 100.0, T, r, sigma, digital=True, n_paths=50_000, rng=100 + i) for i, s in enumerate(spots)]
    axes[3].plot(spots, np.exp(-r * T) * norm.pdf(d2) / (spots * sigma * np.sqrt(T)), "k", label="closed form")
    axes[3].errorbar(spots, [g.delta for g in dig], yerr=[2 * g.delta_se for g in dig], fmt="o", ms=3, capsize=2, label="likelihood ratio")
    axes[3].set_xlabel("spot S")
    axes[3].set_title("digital call delta")
    axes[3].legend()
    g = lr[6]
    print(f"ATM: delta {g.delta:.4f} +/- {g.delta_se:.4f} (BS {bs.delta[6]:.4f}), gamma {g.gamma:.5f} +/- {g.gamma_se:.5f} (BS {bs.gamma[6]:.5f})")
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    ATM: delta 0.5758 +/- 0.0061 (BS 0.5688), gamma 0.02308 +/- 0.00065 (BS 0.02223)





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

   **Total running time of the script:** (0 minutes 0.127 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_greeks_plot_01_likelihood_ratio.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_01_likelihood_ratio.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_likelihood_ratio.ipynb <plot_01_likelihood_ratio.ipynb>`

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

      :download:`Download Python source code: plot_01_likelihood_ratio.py <plot_01_likelihood_ratio.py>`

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

      :download:`Download zipped: plot_01_likelihood_ratio.zip <plot_01_likelihood_ratio.zip>`


.. only:: html

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

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