.. _sphx_glr_api_gallery_interest_rate_models_short_rate:

Short-rate models
-----------------

Models of the instantaneous short rate that produce their own yield
curve: Merton, Vasicek and Cox-Ingersoll-Ross, the affine family that
contains them, and the two-factor G2++, all checked against Feynman-Kac
Monte Carlo.


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Richard Feynman&#x27;s path integrals (1948) and Mark Kac&#x27;s proof (1949) tie a parabolic PDE to an expectation over Brownian paths. For a short-rate model the bond-pricing PDE \partial_t P + \mu\,\partial_r P + \tfrac12\sigma^2\partial_{rr} P = rP is solved by">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_01_feynman_kac_bond_price_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_01_feynman_kac_bond_price`

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      <div class="sphx-glr-thumbnail-title">Feynman-Kac: bond prices by Monte Carlo</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="William Feller (1951) classified the boundaries of diffusions such as the square-root process">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_02_feller_condition_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_02_feller_condition`

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      <div class="sphx-glr-thumbnail-title">Feller's boundary condition</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In Section 6 of &quot;Theory of Rational Option Pricing&quot; (1973), Robert Merton let the short rate follow an arithmetic Brownian motion, dr = \mu\,dt + \sigma\,dW, the first stochastic interest rate in continuous time. Bonds price in closed form,">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_03_merton_short_rate_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_03_merton_short_rate`

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      <div class="sphx-glr-thumbnail-title">Merton's Gaussian short rate</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Oldrich Vasicek (1977) derived the term structure from a short rate pulled back towards a long-run mean,">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_04_vasicek_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_04_vasicek`

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      <div class="sphx-glr-thumbnail-title">Vasicek's mean-reverting short rate</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="John Cox, Jonathan Ingersoll and Stephen Ross (1985) derived the short rate from a general-equilibrium economy and found">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_05_cir_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_05_cir`

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      <div class="sphx-glr-thumbnail-title">Cox-Ingersoll-Ross: the square-root short rate</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Darrell Duffie and Rui Kan (1996) found the common structure behind Vasicek, CIR and their multi-factor extensions. If the state&#x27;s drift and variance are affine in the state,">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_06_duffie_kan_affine_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_06_duffie_kan_affine`

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      <div class="sphx-glr-thumbnail-title">Duffie-Kan: affine term structures</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In a one-factor model a single shock moves every yield, so the 1-year and 30-year rates are perfectly correlated, which the data flatly reject. Damiano Brigo and Fabio Mercurio&#x27;s G2++ (2001) adds a second Gaussian factor,">

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  .. image:: /api/gallery/interest_rate_models/short_rate/images/thumb/sphx_glr_plot_07_g2pp_thumb.png
    :alt:

  :doc:`/api/gallery/interest_rate_models/short_rate/plot_07_g2pp`

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      <div class="sphx-glr-thumbnail-title">Brigo-Mercurio G2++: two Gaussian factors</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/interest_rate_models/short_rate/plot_01_feynman_kac_bond_price
   /api/gallery/interest_rate_models/short_rate/plot_02_feller_condition
   /api/gallery/interest_rate_models/short_rate/plot_03_merton_short_rate
   /api/gallery/interest_rate_models/short_rate/plot_04_vasicek
   /api/gallery/interest_rate_models/short_rate/plot_05_cir
   /api/gallery/interest_rate_models/short_rate/plot_06_duffie_kan_affine
   /api/gallery/interest_rate_models/short_rate/plot_07_g2pp