.. _sphx_glr_api_gallery_stochastic_volatility_simulation:

Simulation
----------

Monte Carlo for Heston when the variance can hit zero: Andersen's
quadratic-exponential scheme and the full-truncation Euler scheme.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="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 \psi = s^2/m^2 = 1.5. The spot then takes a step that integrates the variance with the trapezoidal rule. On Andersen&#x27;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.">

.. only:: html

  .. image:: /api/gallery/stochastic_volatility/simulation/images/thumb/sphx_glr_plot_01_andersen_qe_thumb.png
    :alt:

  :doc:`/api/gallery/stochastic_volatility/simulation/plot_01_andersen_qe`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Andersen's quadratic-exponential scheme</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="An Euler step for the square-root variance can go negative, and the repairs proposed in the literature differ in where they take the positive part: in the diffusion only (partial truncation, Deelstra and Delbaen 1998), as an absolute value in the diffusion (Higham and Mao 2005), or in both the drift and the diffusion while letting the variance itself go negative (full truncation). Lord, Koekkoek and van Dijk (2010) put them in one framework and found full truncation the least biased. All three schemes here run through financialkit.integrators.euler_maruyama with correlated increments.">

.. only:: html

  .. image:: /api/gallery/stochastic_volatility/simulation/images/thumb/sphx_glr_plot_02_full_truncation_thumb.png
    :alt:

  :doc:`/api/gallery/stochastic_volatility/simulation/plot_02_full_truncation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lord, Koekkoek and van Dijk's full truncation</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/stochastic_volatility/simulation/plot_01_andersen_qe
   /api/gallery/stochastic_volatility/simulation/plot_02_full_truncation