.. _sphx_glr_api_gallery_asset_pricing_sdf:

The stochastic discount factor
------------------------------

No arbitrage as the existence of positive state prices, and the
volatility a discount factor needs to price the assets.


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Michael Harrison and David Kreps (1979) proved the fundamental theorem of asset pricing: a market has no arbitrage if and only if some strictly positive linear rule prices every traded payoff. With finitely many states it is a vector of state prices q_s &gt; 0 with p = X&#x27;q; divided by the physical probabilities it is the stochastic discount factor m_s = q_s / \pi_s, so that p = E[m x]; normalised, it is the risk-neutral measure. In a complete market the state prices are unique. In an incomplete one, a new claim has an interval of arbitrage-free prices.">

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  .. image:: /api/gallery/asset_pricing/sdf/images/thumb/sphx_glr_plot_01_harrison_kreps_thumb.png
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  :doc:`/api/gallery/asset_pricing/sdf/plot_01_harrison_kreps`

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      <div class="sphx-glr-thumbnail-title">Harrison-Kreps: state prices and the stochastic discount factor</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Lars Peter Hansen and Ravi Jagannathan (1991) asked what asset returns alone reveal about the stochastic discount factor. Any m that prices the returns, E[m R] = 1, must satisfy">

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  .. image:: /api/gallery/asset_pricing/sdf/images/thumb/sphx_glr_plot_02_hansen_jagannathan_thumb.png
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  :doc:`/api/gallery/asset_pricing/sdf/plot_02_hansen_jagannathan`

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      <div class="sphx-glr-thumbnail-title">The Hansen-Jagannathan bound</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/asset_pricing/sdf/plot_01_harrison_kreps
   /api/gallery/asset_pricing/sdf/plot_02_hansen_jagannathan