.. _sphx_glr_api_gallery_options_closed_form:

Closed-form pricers
-------------------

European option prices in closed form, from Bachelier's normal model to
Black-Scholes-Merton and its forward, futures and currency variants, and
the model-free relations every price must obey.


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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="In his 1900 thesis Théorie de la spéculation Louis Bachelier modelled the price of a French government bond as a Brownian motion and priced options on it, five years before Einstein&#x27;s paper on Brownian motion. In his model price changes are normal with an absolute volatility \sigma_N, so an at-the-money option is worth \sigma_N\sqrt{T/(2\pi)}: the square-root-of-time law. Below, the formula is checked against simulated arithmetic Brownian paths and compared with Black-Scholes at a matched volatility.">

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  .. image:: /api/gallery/options/closed_form/images/thumb/sphx_glr_plot_01_bachelier_thumb.png
    :alt:

  :doc:`/api/gallery/options/closed_form/plot_01_bachelier`

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      <div class="sphx-glr-thumbnail-title">Bachelier's normal model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Hans Stoll (1969) showed that a European call, a put and the stock are tied together with no model at all: buying a call and selling a put with the same strike and maturity replicates a forward, so">

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  .. image:: /api/gallery/options/closed_form/images/thumb/sphx_glr_plot_02_stoll_put_call_parity_thumb.png
    :alt:

  :doc:`/api/gallery/options/closed_form/plot_02_stoll_put_call_parity`

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      <div class="sphx-glr-thumbnail-title">Stoll's put-call parity</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fischer Black and Myron Scholes (1973) showed that a continuously rebalanced position in the stock and a bond replicates a European option when the stock follows geometric Brownian motion, so the option price cannot depend on investors&#x27; expected return:">

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  .. image:: /api/gallery/options/closed_form/images/thumb/sphx_glr_plot_03_black_scholes_thumb.png
    :alt:

  :doc:`/api/gallery/options/closed_form/plot_03_black_scholes`

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      <div class="sphx-glr-thumbnail-title">The Black-Scholes formula</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Robert Merton&#x27;s Theory of Rational Option Pricing (1973) did two things at once. It derived, with no model, the bounds every European option price must respect,">

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  .. image:: /api/gallery/options/closed_form/images/thumb/sphx_glr_plot_04_merton_bounds_and_dividends_thumb.png
    :alt:

  :doc:`/api/gallery/options/closed_form/plot_04_merton_bounds_and_dividends`

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      <div class="sphx-glr-thumbnail-title">Merton: dividends and no-arbitrage bounds</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fischer Black (1976) priced options on commodity futures. A futures contract costs nothing to enter, so its risk-neutral drift is zero and the Black-Scholes argument gives">

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  .. image:: /api/gallery/options/closed_form/images/thumb/sphx_glr_plot_05_black76_thumb.png
    :alt:

  :doc:`/api/gallery/options/closed_form/plot_05_black76`

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      <div class="sphx-glr-thumbnail-title">Black-76: options on futures</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Mark Garman and Steven Kohlhagen (1983) noticed that a unit of foreign currency is an asset that pays the foreign interest rate r_f, just as a stock pays a dividend yield. Merton&#x27;s dividend formula then prices FX options:">

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  .. image:: /api/gallery/options/closed_form/images/thumb/sphx_glr_plot_06_garman_kohlhagen_thumb.png
    :alt:

  :doc:`/api/gallery/options/closed_form/plot_06_garman_kohlhagen`

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      <div class="sphx-glr-thumbnail-title">Garman-Kohlhagen currency options</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/options/closed_form/plot_01_bachelier
   /api/gallery/options/closed_form/plot_02_stoll_put_call_parity
   /api/gallery/options/closed_form/plot_03_black_scholes
   /api/gallery/options/closed_form/plot_04_merton_bounds_and_dividends
   /api/gallery/options/closed_form/plot_05_black76
   /api/gallery/options/closed_form/plot_06_garman_kohlhagen