.. _sphx_glr_api_gallery_pde_methods_accuracy:

Accuracy
--------

Getting more digits from the same work: Richardson extrapolation across
grids, and grids that concentrate their nodes where the payoff bends.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Lewis Fry Richardson (1911), computing stresses in masonry dams by finite differences, noticed that when the error behaves like C h^p, two solutions on grids h and h/2 determine both the constant and the limit: V \approx V_{h/2} + (V_{h/2} - V_h)/(2^p - 1). richardson_extrapolation applies this to Crank-Nicolson prices; three grids also confirm that the observed order is 2, which needs the strike on a node and Rannacher&#x27;s start so that the error is a clean power of h.">

.. only:: html

  .. image:: /api/gallery/pde_methods/accuracy/images/thumb/sphx_glr_plot_01_richardson_extrapolation_thumb.png
    :alt:

  :doc:`/api/gallery/pde_methods/accuracy/plot_01_richardson_extrapolation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Richardson extrapolation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Away from the strike a vanilla option is almost linear, so a uniform grid spends most of its nodes where nothing happens. Tavella and Randall (2000) map a uniform coordinate through a hyperbolic sine to cluster the nodes around the strike, keeping the grid smooth so the three-point differences stay second order. tavella_randall_grid builds the grid, and grid=&quot;tavella_randall&quot; prices on it: the same accuracy with a fraction of the nodes.">

.. only:: html

  .. image:: /api/gallery/pde_methods/accuracy/images/thumb/sphx_glr_plot_02_tavella_randall_grid_thumb.png
    :alt:

  :doc:`/api/gallery/pde_methods/accuracy/plot_02_tavella_randall_grid`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Tavella-Randall non-uniform grids</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/pde_methods/accuracy/plot_01_richardson_extrapolation
   /api/gallery/pde_methods/accuracy/plot_02_tavella_randall_grid