https://github.com/luphord/longstaff_schwartz

A Python implementation of the Longstaff-Schwartz linear regression algorithm for the evaluation of call rights an American options.
https://github.com/luphord/longstaff_schwartz

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A Python implementation of the Longstaff-Schwartz linear regression algorithm for the evaluation of call rights an American options.

• Host: GitHub
• URL: https://github.com/luphord/longstaff_schwartz
• Owner: luphord
• License: mit
• Created: 2019-08-05T04:40:39.000Z (about 5 years ago)
• Default Branch: master
• Last Pushed: 2023-11-02T21:42:25.000Z (11 months ago)
• Last Synced: 2024-07-17T07:26:19.920Z (2 months ago)
• Language: Jupyter Notebook
• Size: 10.1 MB
• Stars: 37
• Watchers: 2
• Forks: 13
• Open Issues: 2
• Metadata Files:
  • Readme: README.rst
  • Changelog: HISTORY.rst
  • Contributing: CONTRIBUTING.rst
  • License: LICENSE
  • Authors: AUTHORS.rst

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README

        
============================

Longstaff-Schwartz Algorithm

============================

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        :alt: Documentation Status

A Python implementation of the Longstaff-Schwartz linear regression algorithm for the evaluation of call rights and American options.

* Seminal paper: **Francis A. Longstaff, Eduardo S. Schwartz**, *Valuing American Options by Simulation: A Simple Least-Squares Approach* (The Review of Financial Studies) (2001) Vol 14, No 1, pp. 113-147

* Documentation: https://longstaff-schwartz.readthedocs.io

* Free software: MIT license

Talks

-----

* PyConDE 2019-10-10: `Slides `__, `Jupyter Notebook `__

* PyData Meetup 2019-09-18: Slides_, `Jupyter Notebook`_

.. _Slides: https://raw.githack.com/luphord/longstaff_schwartz/master/talks/talk_meetup_2019-09-18/index.html

.. _`Jupyter Notebook`: https://github.com/luphord/longstaff_schwartz/blob/master/talks/talk_meetup_2019-09-18/Notebook_Meetup_2019-09-18.ipynb

Usage

-----

.. code-block:: python

        from longstaff_schwartz.algorithm import longstaff_schwartz

        from longstaff_schwartz.stochastic_process import GeometricBrownianMotion

        import numpy as np

        # Model parameters

        t = np.linspace(0, 5, 100)  # timegrid for simulation

        r = 0.01  # riskless rate

        sigma = 0.15  # annual volatility of underlying

        n = 100  # number of simulated paths

        # Simulate the underlying

        gbm = GeometricBrownianMotion(mu=r, sigma=sigma)

        rnd = np.random.RandomState(1234)

        x = gbm.simulate(t, n, rnd)  # x.shape == (t.size, n)

        # Payoff (exercise) function

        strike = 0.95

        def put_payoff(spot):

            return np.maximum(strike - spot, 0.0)

        # Discount factor function

        def constant_rate_df(t_from, t_to):

            return np.exp(-r * (t_to - t_from))

        # Approximation of continuation value

        def fit_quadratic(x, y):

            return np.polynomial.Polynomial.fit(x, y, 2, rcond=None)

        # Selection of paths to consider for exercise

        # (and continuation value approxmation)

        def itm(payoff, spot):

            return payoff > 0

        # Run valuation of American put option

        npv_american = longstaff_schwartz(x, t, constant_rate_df,

                                          fit_quadratic, put_payoff, itm)

        # European put option for comparison

        npv_european = constant_rate_df(t[0], t[-1]) * put_payoff(x[-1]).mean()

        # Check results

        assert np.round(npv_american, 4) == 0.0734

        assert np.round(npv_european, 4) == 0.0626

        assert npv_american > npv_european

Plots

-----

For details see `PyData Meetup Jupyter Notebook`_.

.. _`PyData Meetup Jupyter Notebook`: https://github.com/luphord/longstaff_schwartz/blob/master/talks/talk_meetup_2019-09-18/Notebook_Meetup_2019-09-18.ipynb

Approximation of continuation value

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. image:: docs/_static/approximated-continuation-and-exercise-value-1.svg

Favourable exercise

~~~~~~~~~~~~~~~~~~~

.. image:: docs/_static/exercise-or-hold.svg

.. image:: docs/_static/first-exercises.svg

Credits

-------

Main developer is luphord_.

.. _luphord: https://github.com/luphord

Primary source for the algorithm is **Francis A. Longstaff, Eduardo S. Schwartz**, *Valuing American Options by Simulation: A Simple Least-Squares Approach* (The Review of Financial Studies) (2001) Vol 14, No 1, pp. 113-147.

There is no affiliation between the authors of the paper and this code.

This package was prepared with Cookiecutter_ and the `audreyr/cookiecutter-pypackage`_ project template.

.. _Cookiecutter: https://github.com/audreyr/cookiecutter

.. _`audreyr/cookiecutter-pypackage`: https://github.com/audreyr/cookiecutter-pypackage