:orphan:

Examples
========

This gallery walks through every public feature of
``financialkit.monte_carlo``: crude Monte Carlo with error bars, variance
reduction, quasi-Monte Carlo, Monte Carlo Greeks and multilevel Monte
Carlo. Each script illustrates one dated breakthrough from the
:doc:`history page </history/monte_carlo_breakthroughs>`.

See also the narrative tutorial:

- :doc:`/tutorials/one_option_five_pricers`

Each script in this gallery is self-contained and can be run directly with
``python examples/monte_carlo/<section>/<script>.py``.

Sections
--------

- **basics** -- the Monte Carlo method of Metropolis and Ulam, Box-Muller
  normal variates and Boyle's option prices.
- **variance_reduction** -- Neyman stratification, importance sampling,
  antithetic variates, Latin hypercube sampling, control variates and
  moment matching.
- **quasi_monte_carlo** -- Halton and Sobol sequences, the Koksma-Hlawka
  inequality, Owen scrambling, Paskov-Traub's high-dimensional
  integrals and the Brownian bridge.
- **greeks** -- likelihood-ratio and pathwise Greeks.
- **multilevel** -- Giles's multilevel Monte Carlo.


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


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Crude Monte Carlo
-----------------

Expectations as sample means with standard errors and confidence
intervals, normal variates from uniforms, and option prices by
simulation.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Stanislaw Ulam, convalescing in 1946 and playing solitaire, wondered whether it was quicker to estimate his chance of winning by playing many games than by combinatorics. With John von Neumann and Nicholas Metropolis he turned the idea into a method for the neutron-diffusion problems at Los Alamos, and Metropolis and Ulam named it in 1949. An integral is an expectation, so">

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  .. image:: /api/gallery/monte_carlo/basics/images/thumb/sphx_glr_plot_01_crude_monte_carlo_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/basics/plot_01_crude_monte_carlo`

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      <div class="sphx-glr-thumbnail-title">Metropolis-Ulam: the Monte Carlo method</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every Black-Scholes simulation needs normal random numbers, and a computer produces uniforms. George Box and Mervin Muller (1958) gave an exact transform: if u_1, u_2 are independent uniforms,">

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  .. image:: /api/gallery/monte_carlo/basics/images/thumb/sphx_glr_plot_02_box_muller_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/basics/plot_02_box_muller`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Box-Muller normal variates</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Phelim Boyle (1977) observed that risk-neutral valuation turns an option price into an expectation, and so into a simulation:">

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  .. image:: /api/gallery/monte_carlo/basics/images/thumb/sphx_glr_plot_03_boyle_option_pricing_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/basics/plot_03_boyle_option_pricing`

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      <div class="sphx-glr-thumbnail-title">Boyle: options by Monte Carlo</div>
    </div>


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    </div>

Monte Carlo Greeks
------------------

Sensitivities without bump-and-reprice: the likelihood-ratio and
pathwise estimators, checked against the Black-Scholes Greeks.


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    <div class="sphx-glr-thumbcontainer" tooltip="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:">

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  .. image:: /api/gallery/monte_carlo/greeks/images/thumb/sphx_glr_plot_01_likelihood_ratio_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/greeks/plot_01_likelihood_ratio`

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      <div class="sphx-glr-thumbnail-title">Glynn: likelihood-ratio Greeks</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Mark Broadie and Paul Glasserman (1996) differentiated each simulated payoff with respect to the parameter and averaged:">

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  .. image:: /api/gallery/monte_carlo/greeks/images/thumb/sphx_glr_plot_02_pathwise_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/greeks/plot_02_pathwise`

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      <div class="sphx-glr-thumbnail-title">Broadie-Glasserman pathwise Greeks</div>
    </div>


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    </div>

Multilevel Monte Carlo
----------------------

Giles's telescoping sum over Euler discretisations of increasing
resolution, and its cost against standard Monte Carlo.


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    <div class="sphx-glr-thumbcontainer" tooltip="Pricing with a discretised SDE has two errors: the bias of the time step and the statistical error. Balancing them with Euler&#x27;s scheme costs O(\varepsilon^{-3}). Michael Giles (2008), borrowing the idea of multigrid (and of Heinrich&#x27;s multilevel integration), wrote the finest approximation as a telescoping sum,">

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  .. image:: /api/gallery/monte_carlo/multilevel/images/thumb/sphx_glr_plot_01_giles_mlmc_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/multilevel/plot_01_giles_mlmc`

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      <div class="sphx-glr-thumbnail-title">Giles multilevel Monte Carlo</div>
    </div>


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    </div>

Quasi-Monte Carlo
-----------------

Low-discrepancy points instead of random ones: Halton and Sobol
sequences, the Koksma-Hlawka error bound, randomized QMC with error bars,
and high-dimensional path integrals with and without the Brownian bridge.


.. raw:: html

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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Johannes van der Corput (1935) built a one-dimensional sequence that fills the unit interval evenly by reflecting the binary digits of k about the radix point: 1/2, 1/4, 3/4, 1/8, 5/8, ... John Halton (1960) took one such radical-inverse sequence per coordinate, in distinct prime bases, to fill the d-dimensional cube. The points are deterministic, yet integrate far better than random ones in low dimension, with an error close to (\log n)^d / n.">

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  .. image:: /api/gallery/monte_carlo/quasi_monte_carlo/images/thumb/sphx_glr_plot_01_halton_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/quasi_monte_carlo/plot_01_halton`

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      <div class="sphx-glr-thumbnail-title">Halton sequences</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Jurjen Koksma (1942) in one dimension and Edmund Hlawka (1961) in many bounded the error of an equal-weight rule by a product of two numbers, one about the integrand and one about the points:">

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  .. image:: /api/gallery/monte_carlo/quasi_monte_carlo/images/thumb/sphx_glr_plot_02_koksma_hlawka_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/quasi_monte_carlo/plot_02_koksma_hlawka`

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      <div class="sphx-glr-thumbnail-title">Koksma-Hlawka and discrepancy</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Ilya Sobol (1967) constructed sequences in base 2 whose leading points form (t, m, s)-nets: every dyadic box of volume 2^{t - m} holds exactly 2^t of the first 2^m points. With well-chosen direction numbers (Joe and Kuo&#x27;s, which scipy.stats.qmc.Sobol uses) they stay balanced in thousands of dimensions and are the workhorse of quasi-Monte Carlo in finance.">

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  .. image:: /api/gallery/monte_carlo/quasi_monte_carlo/images/thumb/sphx_glr_plot_03_sobol_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/quasi_monte_carlo/plot_03_sobol`

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      <div class="sphx-glr-thumbnail-title">Sobol sequences</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A deterministic point set gives one number and no error bar. Art Owen (1995) randomly permuted the digits of a net, nested so that the result is still a net: every scrambled point is uniform on the cube, so the estimate is unbiased, independent scramblings give an honest standard error, and for smooth integrands the root-mean-square error falls like N^{-3/2}, faster than the unscrambled sequence&#x27;s N^{-1}.">

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  .. image:: /api/gallery/monte_carlo/quasi_monte_carlo/images/thumb/sphx_glr_plot_04_owen_scrambling_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/quasi_monte_carlo/plot_04_owen_scrambling`

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      <div class="sphx-glr-thumbnail-title">Owen scrambling: randomized QMC</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Theory said quasi-Monte Carlo should lose its edge beyond a dozen dimensions, since (\log N)^d / N only beats N^{-1/2} for astronomically large N. Spassimir Paskov and Joseph Traub (1995) priced a Goldman Sachs collateralized mortgage obligation, a 360-dimensional integral, and found Sobol points beat Monte Carlo anyway. The explanation, later formalized as effective dimension: financial integrands depend mostly on a few coordinates and their low-order interactions.">

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  .. image:: /api/gallery/monte_carlo/quasi_monte_carlo/images/thumb/sphx_glr_plot_05_paskov_traub_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/quasi_monte_carlo/plot_05_paskov_traub`

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      <div class="sphx-glr-thumbnail-title">Paskov-Traub: QMC in high dimension</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Quasi-random points are best distributed in their first coordinates. If those coordinates drive the first time steps, the coarse shape of each path is left to the poorly distributed later ones. Russel Caflisch and Bradley Moskowitz (1995), and Caflisch, Morokoff and Owen (1997) for mortgage-backed securities, reordered the construction: the first coordinate sets W(T), the second W(T/2) given the ends, and so on by bisection,">

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  .. image:: /api/gallery/monte_carlo/quasi_monte_carlo/images/thumb/sphx_glr_plot_06_brownian_bridge_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/quasi_monte_carlo/plot_06_brownian_bridge`

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      <div class="sphx-glr-thumbnail-title">Caflisch-Moskowitz Brownian bridge</div>
    </div>


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    </div>

Variance reduction
------------------

The same expectation from fewer samples: stratification, importance
sampling, antithetic variates, Latin hypercubes, control variates and
moment matching, each compared with crude Monte Carlo.


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    <div class="sphx-glr-thumbcontainer" tooltip="Jerzy Neyman (1934) showed survey statisticians how to sample a stratified population: draw from each stratum separately and allocate the sample in proportion to the stratum&#x27;s weight times its standard deviation,">

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  .. image:: /api/gallery/monte_carlo/variance_reduction/images/thumb/sphx_glr_plot_01_neyman_stratification_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/variance_reduction/plot_01_neyman_stratification`

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      <div class="sphx-glr-thumbnail-title">Neyman stratification</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Herman Kahn and Andy Marshall (1953), at RAND, wrote down the trick for shielding calculations in which almost no simulated neutron gets through: sample from a density that makes the rare event common, and reweight by the likelihood ratio. For normal drivers a mean shift \theta gives">

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  .. image:: /api/gallery/monte_carlo/variance_reduction/images/thumb/sphx_glr_plot_02_importance_sampling_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/variance_reduction/plot_02_importance_sampling`

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      <div class="sphx-glr-thumbnail-title">Kahn-Marshall importance sampling</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="John Hammersley and Keith Morton (1956) proposed pairing each random input with its mirror image. If f is monotone, f(Z) and f(-Z) are negatively correlated and their average is less variable than two independent draws:">

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  .. image:: /api/gallery/monte_carlo/variance_reduction/images/thumb/sphx_glr_plot_03_antithetic_variates_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/variance_reduction/plot_03_antithetic_variates`

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      <div class="sphx-glr-thumbnail-title">Hammersley-Morton antithetic variates</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Michael McKay, Richard Beckman and William Conover (1979), at Los Alamos, needed to run expensive computer codes on few input combinations. A Latin hypercube design of n points stratifies every coordinate into n equal intervals at once: each interval of each axis holds exactly one point, the axes shuffled independently. The variance of an additive integrand falls by an order of n; interactions are left alone.">

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  .. image:: /api/gallery/monte_carlo/variance_reduction/images/thumb/sphx_glr_plot_04_latin_hypercube_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/variance_reduction/plot_04_latin_hypercube`

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      <div class="sphx-glr-thumbnail-title">McKay-Beckman-Conover Latin hypercube</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="If a quantity X simulated alongside the target Y has a known mean, its sampling error tells us about the error in \bar Y. Stephen Lavenberg and Peter Welch (1981) framed the correction as a regression: estimate \beta by least squares and use">

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  .. image:: /api/gallery/monte_carlo/variance_reduction/images/thumb/sphx_glr_plot_05_control_variates_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/variance_reduction/plot_05_control_variates`

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      <div class="sphx-glr-thumbnail-title">Lavenberg-Welch control variates</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A finite sample of normals never has mean exactly 0 and variance exactly 1, and the payoff inherits those sampling errors. Jérôme Barraquand (1995), pricing multi-asset options, simply rescaled the sample,">

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  .. image:: /api/gallery/monte_carlo/variance_reduction/images/thumb/sphx_glr_plot_06_moment_matching_thumb.png
    :alt:

  :doc:`/api/gallery/monte_carlo/variance_reduction/plot_06_moment_matching`

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      <div class="sphx-glr-thumbnail-title">Barraquand moment matching</div>
    </div>


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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/monte_carlo/basics/index.rst
   /api/gallery/monte_carlo/greeks/index.rst
   /api/gallery/monte_carlo/multilevel/index.rst
   /api/gallery/monte_carlo/quasi_monte_carlo/index.rst
   /api/gallery/monte_carlo/variance_reduction/index.rst


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  .. container:: sphx-glr-footer sphx-glr-footer-gallery

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

      :download:`Download all examples in Python source code: monte_carlo_python.zip </api/gallery/monte_carlo/monte_carlo_python.zip>`

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

      :download:`Download all examples in Jupyter notebooks: monte_carlo_jupyter.zip </api/gallery/monte_carlo/monte_carlo_jupyter.zip>`


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

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