.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/var/plot_04_monte_carlo_var.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_risk_var_plot_04_monte_carlo_var.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_risk_var_plot_04_monte_carlo_var.py:


Pritsker: Monte Carlo VaR for option books
==========================================

Delta-normal VaR treats every position as linear in its risk factors.
For options that is badly wrong: a short straddle has zero delta at the
money, so its delta-normal VaR is zero, yet it loses heavily on a large
move in either direction. Pritsker (1997) compared delta, delta-gamma
and full-revaluation Monte Carlo VaR and found that only full
revaluation is reliable for books with large gamma. The cost is
repricing every position in every scenario.

.. GENERATED FROM PYTHON SOURCE LINES 15-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_greeks, black_scholes_price
    from financialkit.risk import delta_normal_var, monte_carlo_var
    from financialkit.risk.visualizers import plot_var

    S0, T, r, sigma = 100.0, 0.25, 0.03, 0.25
    K = S0 * np.exp((r + 0.5 * sigma**2) * T)  # d1 = 0: the straddle is delta-neutral
    h = 10 / 252
    straddle = lambda S, t: black_scholes_price(S, K, t, r, sigma) + black_scholes_price(S, K, t, r, sigma, option_type="put")
    value0 = straddle(S0, T)








.. GENERATED FROM PYTHON SOURCE LINES 29-34

Full revaluation against the delta approximation
------------------------------------------------

The risk factor is the log return of the stock over ten days. The
book is short 100 delta-neutral straddles.

.. GENERATED FROM PYTHON SOURCE LINES 34-50

.. code-block:: Python



    def pnl(dX):
        return -100.0 * (straddle(S0 * np.exp(dX[:, 0]), T - h) - value0)


    cov = np.array([[sigma**2]])
    res = monte_carlo_var(pnl, cov, 0.99, horizon=h, n_paths=20_000, rng=0)
    g = black_scholes_greeks(S0, K, T, r, sigma)
    gp = black_scholes_greeks(S0, K, T, r, sigma, option_type="put")
    delta = -100.0 * (g.delta + gp.delta) * S0
    print(f"position delta (in $ per unit log return): {delta:,.1f}")
    print(f"delta-normal 99% ten-day VaR: {delta_normal_var([delta], cov, 0.99, h):,.1f}")
    print(f"Monte Carlo VaR {res.var:,.1f}  (95% CI {res.ci[0]:,.1f} to {res.ci[1]:,.1f}),  ES {res.es:,.1f}")
    plot_var(res)




.. image-sg:: /api/gallery/risk/var/images/sphx_glr_plot_04_monte_carlo_var_001.png
   :alt: monte carlo
   :srcset: /api/gallery/risk/var/images/sphx_glr_plot_04_monte_carlo_var_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    position delta (in $ per unit log return): -0.0
    delta-normal 99% ten-day VaR: 0.0
    Monte Carlo VaR 440.2  (95% CI 423.5 to 461.3),  ES 562.2

    <Axes: title={'center': 'monte carlo'}, xlabel='profit and loss', ylabel='probability'>



.. GENERATED FROM PYTHON SOURCE LINES 51-53

The P&L profile behind it
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 53-64

.. code-block:: Python


    x = np.linspace(-0.3, 0.3, 200)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(x, pnl(x[:, None]), label="full revaluation")
    ax.plot(x, delta * x, "--", label="delta approximation")
    ax.set_xlabel("ten-day log return")
    ax.set_ylabel("P&L of the short straddles")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/var/images/sphx_glr_plot_04_monte_carlo_var_002.png
   :alt: plot 04 monte carlo var
   :srcset: /api/gallery/risk/var/images/sphx_glr_plot_04_monte_carlo_var_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.072 seconds)


.. _sphx_glr_download_api_gallery_risk_var_plot_04_monte_carlo_var.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/risk/var/plot_04_monte_carlo_var.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_monte_carlo_var.ipynb <plot_04_monte_carlo_var.ipynb>`

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

      :download:`Download Python source code: plot_04_monte_carlo_var.py <plot_04_monte_carlo_var.py>`

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

      :download:`Download zipped: plot_04_monte_carlo_var.zip <plot_04_monte_carlo_var.zip>`


.. only:: html

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

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