.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/smile/plot_01_sabr.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_smile_plot_01_sabr.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_smile_plot_01_sabr.py:


Hagan's SABR model
==================

Hagan, Kumar, Lesniewski and Woodward (2002) observed that local
volatility predicts the wrong dynamics: when the forward rises, a local
volatility model moves the smile the wrong way. Their stochastic
:math:`\alpha\beta\rho` model, :math:`dF = \alpha F^\beta\,dW_1`,
:math:`d\alpha = \nu\alpha\,dW_2`, has an asymptotic implied-volatility
formula that became the market standard for swaption smiles. Each
parameter has one job: :math:`\beta` the backbone, :math:`\rho` the skew,
:math:`\nu` the curvature. Here a smile is calibrated and each parameter
varied in turn.

.. GENERATED FROM PYTHON SOURCE LINES 17-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import sabr_calibrate, sabr_implied_vol
    from financialkit.stochastic_volatility.visualizers import plot_smile

    F, T = 0.03, 5.0
    K = np.linspace(0.01, 0.06, 26)
    alpha, beta = 0.035, 0.5  # at-the-money Black vol alpha / F^(1 - beta), about 20%








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Calibration to a noisy smile
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-41

.. code-block:: Python


    rng = np.random.default_rng(0)
    market = sabr_implied_vol(F, K, T, alpha, beta, -0.3, 0.4) + rng.normal(0, 5e-4, K.size)
    fit = sabr_calibrate(F, K, T, market, beta=beta)
    print(f"true alpha = {alpha}, rho = -0.3, nu = 0.4")
    print(f"fit  alpha = {fit.alpha:.4f}, rho = {fit.rho:.3f}, nu = {fit.nu:.3f}, rms error = {fit.rmse * 1e4:.1f} bp of vol")

    fig, axes = plt.subplots(1, 3, figsize=(12, 3.8))
    axes[0].plot(K, market, "o", ms=3, label="market")
    plot_smile(K, fit.implied_vol(K), ax=axes[0], label="SABR fit")




.. image-sg:: /api/gallery/stochastic_volatility/smile/images/sphx_glr_plot_01_sabr_001.png
   :alt: plot 01 sabr
   :srcset: /api/gallery/stochastic_volatility/smile/images/sphx_glr_plot_01_sabr_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    true alpha = 0.035, rho = -0.3, nu = 0.4
    fit  alpha = 0.0349, rho = -0.299, nu = 0.402, rms error = 3.8 bp of vol

    <Axes: xlabel='strike K', ylabel='implied volatility'>



.. GENERATED FROM PYTHON SOURCE LINES 42-44

Correlation and vol of vol
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-52

.. code-block:: Python


    for rho in (-0.6, 0.0, 0.6):
        plot_smile(K, sabr_implied_vol(F, K, T, alpha, beta, rho, 0.4), ax=axes[1], label=f"rho = {rho}")
    for nu in (0.1, 0.4, 0.8):
        plot_smile(K, sabr_implied_vol(F, K, T, alpha, beta, 0.0, nu), ax=axes[2], label=f"nu = {nu}")
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.071 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_smile_plot_01_sabr.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/smile/plot_01_sabr.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_sabr.ipynb <plot_01_sabr.ipynb>`

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

      :download:`Download Python source code: plot_01_sabr.py <plot_01_sabr.py>`

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

      :download:`Download zipped: plot_01_sabr.zip <plot_01_sabr.zip>`


.. only:: html

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

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