.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/simulation/plot_01_andersen_qe.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_stochastic_volatility_simulation_plot_01_andersen_qe.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_stochastic_volatility_simulation_plot_01_andersen_qe.py:


Andersen's quadratic-exponential scheme
=======================================

Leif Andersen (2008) simulated the Heston variance by matching the mean
and variance of its exact noncentral chi-square transition with
something cheap to sample: a scaled squared Gaussian when the variance
is well away from zero, and a point mass at zero plus an exponential
tail when it is close, switching at :math:`\psi = s^2/m^2 = 1.5`. The
spot then takes a step that integrates the variance with the
trapezoidal rule. On Andersen's hardest test case (Feller condition
badly violated, ten years) QE is nearly unbiased with a few steps per
year, where Euler schemes are far off.

.. GENERATED FROM PYTHON SOURCE LINES 17-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import HestonModel, heston_mc_price, heston_paths

    m = HestonModel(v0=0.04, kappa=0.5, theta=0.04, xi=1.0, rho=-0.9)
    S, K, T, r = 100.0, 100.0, 10.0, 0.0
    exact = m.price(S, K, T, r)
    print(f"Andersen Case I, Feller satisfied: {m.feller_satisfied}; exact price {exact:.4f} (Andersen: 13.0847)")





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

 .. code-block:: none

    Andersen Case I, Feller satisfied: False; exact price 13.0847 (Andersen: 13.0847)




.. GENERATED FROM PYTHON SOURCE LINES 28-30

Bias against step size
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-44

.. code-block:: Python


    steps = np.array([5, 10, 20, 40, 80])
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    for scheme in ("qe", "full_truncation"):
        res = [heston_mc_price(m, S, K, T, r, n_steps=n, n_paths=20_000, scheme=scheme, rng=1) for n in steps]
        bias = np.array([x.estimate - exact for x in res])
        se = np.array([x.std_error for x in res])
        ax1.errorbar(T / steps, bias, yerr=2 * se, marker="o", ms=3, capsize=2, label=scheme)
    ax1.axhline(0.0, color="k", lw=0.8)
    ax1.set_xscale("log")
    ax1.set_xlabel("time step")
    ax1.set_ylabel("price bias")
    ax1.legend()




.. image-sg:: /api/gallery/stochastic_volatility/simulation/images/sphx_glr_plot_01_andersen_qe_001.png
   :alt: plot 01 andersen qe
   :srcset: /api/gallery/stochastic_volatility/simulation/images/sphx_glr_plot_01_andersen_qe_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x117db0d70>



.. GENERATED FROM PYTHON SOURCE LINES 45-50

The variance hits zero
----------------------

The exponential branch puts an atom at zero, as the exact transition
has when the Feller condition fails.

.. GENERATED FROM PYTHON SOURCE LINES 50-60

.. code-block:: Python


    v = heston_paths(m, S, 1.0, r, n_steps=10, n_paths=50_000, scheme="qe", rng=2).variance[:, -1]
    print(f"fraction of paths with v(1) = 0: {np.mean(v == 0):.3f}")
    ax2.hist(v[v > 0], bins=80, range=(0, 0.2))
    ax2.set_xlabel("v(1)")
    ax2.set_ylabel("paths")
    ax2.set_title(f"QE variance after one year ({np.mean(v == 0):.0%} of paths at 0)")
    fig.tight_layout()

    plt.show()




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

 .. code-block:: none

    fraction of paths with v(1) = 0: 0.793





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

   **Total running time of the script:** (0 minutes 0.956 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_simulation_plot_01_andersen_qe.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/stochastic_volatility/simulation/plot_01_andersen_qe.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_andersen_qe.ipynb <plot_01_andersen_qe.ipynb>`

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

      :download:`Download Python source code: plot_01_andersen_qe.py <plot_01_andersen_qe.py>`

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

      :download:`Download zipped: plot_01_andersen_qe.zip <plot_01_andersen_qe.zip>`


.. only:: html

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

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