.. slideconf::
   :slide_classes: appear

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Implied Volatility and the Volatility Term Structure
==============================================================================

Implied Volatility
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Let :math:`\smash{C_{mkt}}` and :math:`\smash{P_{mkt}}` denote the
market prices of call and put options. Recall that the implied
volatility for strike :math:`\smash{X}` is defined as the value
:math:`\smash{\sigma_{imp}}` such that

.. math::

   \begin{align}
   C_{mkt} & = S_0 \Phi(d_1) - X e^{-rT} \Phi(d_2) \\
   P_{mkt} & = X e^{-rT} \Phi(-d_2) - S_0 \Phi(-d_1)\\
   d_1 & = \frac{\log(S_0/X) + (r+\sigma_{imp}^2/2)T}{\sigma_{imp} \sqrt{T}} \\
   d_2 & = \frac{\log(S_0/X) + (r-\sigma_{imp}^2/2)T}{\sigma_{imp} \sqrt{T}} = d_1
   - \sigma_{imp} \sqrt{T}.
   \end{align}

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- That is, :math:`\smash{\sigma_{imp}}` is the value such that
  :math:`\smash{C_{mkt} = C_{bs}}` and :math:`\smash{P_{mkt} =
  P_{bs}}`.

Put-Call Parity
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Both the BSM and market prices satisfy put-call parity:

.. math::

   \begin{align}
   C_{mkt} + X e^{r_f} & = S_0 + P_{mkt} \\
   C_{bs} + X e^{r_f} & = S_0 + P_{bs}.
   \end{align}

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- As a result :math:`\smash{C_{mkt} - C_{bs} = P_{mkt} - P_{bs}}`.

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- Thus, the implied vol for a particular strike that sets
  :math:`\smash{C_{mkt} = C_{bs}}` is also the implied vol that sets
  :math:`\smash{P_{mkt} = P_{bs}}` (for the same strike).


Foreign Exchange Vol Smile
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.. image:: ImpliedVol/currencySmile.png
   :width: 6in
   :align: center

Foreign Exchange Implied Distribution
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.. image:: ImpliedVol/currencyImpliedDist.png
   :width: 6in
   :align: center

Evidence of Heavy Tails
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The first column of the following table shows the fraction of times
over a 10-year period that 12 currencies experienced a daily move that
exceeded various thresholds. The second column compares the
theoretical probabilities of a lognormal distribution.

.. image:: ImpliedVol/heavyTailTable.png
   :width: 6in
   :align: center
	   
Equities Exchange Vol Smile
==============================================================================
.. image:: ImpliedVol/equitySmile.png
   :width: 6in
   :align: center

Equities Implied Distribution
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.. image:: ImpliedVol/equityImpliedDist.png
   :width: 6in
   :align: center

Volatility Surface
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.. image:: ImpliedVol/volSurface.png
   :width: 6in
   :align: center