Interest Rate Swaps
===================

*Very nascent*.

Interest rate swaps are a first step towards including **rate-sensitive
instruments** in the modeling and valuation spectrum of DX Analytics.
The model used in the following is the **square-root diffusion process**
by Cox-Ingersoll-Ross (1985). Data used are UK/London OIS and Libor
rates.

.. code:: python

    import dx
    import datetime as dt
    import time
    import pandas as pd
    import numpy as np
    import matplotlib.pyplot as plt
    import seaborn as sns; sns.set()
    %matplotlib inline

OIS Data & Discounting
----------------------

We start by importing **OIS term structure data** (source:
http://www.bankofengland.co.uk) for risk-free discounting. We also
adjust the data structure somewhat for our purposes.

.. code:: python

    # UK OIS Spot Rates Yield Curve
    oiss = pd.read_excel('data/ukois09.xls', 'oiss')
    # use years as index
    oiss = oiss.set_index('years')
    # del oiss['years']

.. code:: python

    # only date information for columns, no time
    oiss.columns = [d.date() for d in oiss.columns]

.. code:: python

    oiss.tail()




.. raw:: html

    <div>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>2014-10-01</th>
          <th>2014-10-02</th>
          <th>2014-10-03</th>
          <th>2014-10-06</th>
          <th>2014-10-07</th>
          <th>2014-10-08</th>
          <th>2014-10-09</th>
          <th>2014-10-10</th>
          <th>2014-10-13</th>
          <th>2014-10-14</th>
          <th>2014-10-15</th>
          <th>2014-10-16</th>
        </tr>
        <tr>
          <th>years</th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>4.666667</th>
          <td>1.550947</td>
          <td>1.525838</td>
          <td>1.570203</td>
          <td>1.533901</td>
          <td>1.479044</td>
          <td>1.445234</td>
          <td>1.436023</td>
          <td>1.409962</td>
          <td>1.337567</td>
          <td>1.283932</td>
          <td>1.063205</td>
          <td>1.234125</td>
        </tr>
        <tr>
          <th>4.750000</th>
          <td>1.563661</td>
          <td>1.538616</td>
          <td>1.583198</td>
          <td>1.546459</td>
          <td>1.491152</td>
          <td>1.457593</td>
          <td>1.448627</td>
          <td>1.422015</td>
          <td>1.349782</td>
          <td>1.296052</td>
          <td>1.074773</td>
          <td>1.246277</td>
        </tr>
        <tr>
          <th>4.833333</th>
          <td>1.576142</td>
          <td>1.551158</td>
          <td>1.595952</td>
          <td>1.558789</td>
          <td>1.503050</td>
          <td>1.469768</td>
          <td>1.461040</td>
          <td>1.433887</td>
          <td>1.361832</td>
          <td>1.308025</td>
          <td>1.086254</td>
          <td>1.258307</td>
        </tr>
        <tr>
          <th>4.916667</th>
          <td>1.588400</td>
          <td>1.563474</td>
          <td>1.608472</td>
          <td>1.570898</td>
          <td>1.514746</td>
          <td>1.481764</td>
          <td>1.473269</td>
          <td>1.445585</td>
          <td>1.373724</td>
          <td>1.319859</td>
          <td>1.097650</td>
          <td>1.270219</td>
        </tr>
        <tr>
          <th>5.000000</th>
          <td>1.600442</td>
          <td>1.575572</td>
          <td>1.620768</td>
          <td>1.582794</td>
          <td>1.526247</td>
          <td>1.493588</td>
          <td>1.485320</td>
          <td>1.457116</td>
          <td>1.385461</td>
          <td>1.331557</td>
          <td>1.108962</td>
          <td>1.282014</td>
        </tr>
      </tbody>
    </table>
    </div>



Next we replace the **year fraction index** by a **DatetimeIndex**.

.. code:: python

    # generate time index given input data
    # starting date + 59 months
    date = oiss.columns[-1]
    index = pd.date_range(date, periods=60, freq='M') # , tz='GMT')
    index = [d.replace(day=date.day) for d in index]
    index = pd.DatetimeIndex(index)
    oiss.index = index

Let us have a look at the **most current data**, i.e. the term
structure, of the data set.

.. code:: python

    oiss.iloc[:, -1].plot(figsize=(10, 6))




.. parsed-literal::

    <matplotlib.axes._subplots.AxesSubplot at 0x7fadaeb12910>




.. image:: 10_dx_interest_rate_swaps_files/10_dx_interest_rate_swaps_13_1.png


This data is used to instantiate a ``deterministic_short_rate`` **model
for risk-neutral discounting** purposes.

.. code:: python

    # generate deterministic short rate model based on UK OIS curve
    ois = dx.deterministic_short_rate('ois', zip(oiss.index, oiss.iloc[:, -1].values / 100))

.. code:: python

    # example dates and corresponding discount factors
    dr = pd.date_range('2015-1', periods=4, freq='6m').to_pydatetime()
    ois.get_discount_factors(dr)[::-1]




.. parsed-literal::

    ([0.98982514869237104, 0.99227549016270356, 0.99562377741583252, 1.0],
     array([datetime.datetime(2015, 1, 31, 0, 0),
            datetime.datetime(2015, 7, 31, 0, 0),
            datetime.datetime(2016, 1, 31, 0, 0),
            datetime.datetime(2016, 7, 31, 0, 0)], dtype=object))



Libor Market Data
-----------------

We want to model a **3 month Libor-based interest rate swap**. To this
end, we need Libor term structure data, i.e. forward rates in this case
(source: http://www.bankofengland.co.uk), to calibrate the valuation to.
The data importing and adjustments are the same as before.

.. code:: python

    # UK Libor foward rates
    libf = pd.read_excel('data/ukblc05.xls', 'fwds')
    # use years as index
    libf = libf.set_index('years')

.. code:: python

    # only date information for columns, no time
    libf.columns = [d.date() for d in libf.columns]

.. code:: python

    libf.tail()




.. raw:: html

    <div>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>2014-10-01</th>
          <th>2014-10-02</th>
          <th>2014-10-03</th>
          <th>2014-10-06</th>
          <th>2014-10-07</th>
          <th>2014-10-08</th>
          <th>2014-10-09</th>
          <th>2014-10-10</th>
          <th>2014-10-13</th>
          <th>2014-10-14</th>
          <th>2014-10-15</th>
          <th>2014-10-16</th>
        </tr>
        <tr>
          <th>years</th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>4.666667</th>
          <td>2.722915</td>
          <td>2.686731</td>
          <td>2.678433</td>
          <td>2.681385</td>
          <td>2.639166</td>
          <td>2.542012</td>
          <td>2.528768</td>
          <td>2.489063</td>
          <td>2.470154</td>
          <td>2.349732</td>
          <td>2.372435</td>
          <td>2.157431</td>
        </tr>
        <tr>
          <th>4.750000</th>
          <td>2.733588</td>
          <td>2.697887</td>
          <td>2.690366</td>
          <td>2.692824</td>
          <td>2.650841</td>
          <td>2.554583</td>
          <td>2.543066</td>
          <td>2.502357</td>
          <td>2.483842</td>
          <td>2.366052</td>
          <td>2.387671</td>
          <td>2.176186</td>
        </tr>
        <tr>
          <th>4.833333</th>
          <td>2.744126</td>
          <td>2.708848</td>
          <td>2.702083</td>
          <td>2.704147</td>
          <td>2.662380</td>
          <td>2.567071</td>
          <td>2.557216</td>
          <td>2.515531</td>
          <td>2.497446</td>
          <td>2.382298</td>
          <td>2.402829</td>
          <td>2.194784</td>
        </tr>
        <tr>
          <th>4.916667</th>
          <td>2.754540</td>
          <td>2.719634</td>
          <td>2.713604</td>
          <td>2.715368</td>
          <td>2.673794</td>
          <td>2.579486</td>
          <td>2.571229</td>
          <td>2.528592</td>
          <td>2.510973</td>
          <td>2.398468</td>
          <td>2.417915</td>
          <td>2.213221</td>
        </tr>
        <tr>
          <th>5.000000</th>
          <td>2.764842</td>
          <td>2.730264</td>
          <td>2.724949</td>
          <td>2.726501</td>
          <td>2.685095</td>
          <td>2.591835</td>
          <td>2.585117</td>
          <td>2.541548</td>
          <td>2.524432</td>
          <td>2.414562</td>
          <td>2.432939</td>
          <td>2.231493</td>
        </tr>
      </tbody>
    </table>
    </div>



.. code:: python

    # generate time index given input data
    # starting date + 59 months
    date = libf.columns[-1]
    index = pd.date_range(date, periods=60, freq='M')  # , tz='GMT')
    index = [d.replace(day=date.day) for d in index]
    index = pd.DatetimeIndex(index)
    libf.index = index

And the short end of the **Libor term sturcture** visualized.

.. code:: python

    libf.iloc[:, -1].plot(figsize=(10, 6))




.. parsed-literal::

    <matplotlib.axes._subplots.AxesSubplot at 0x7fadae9c4c90>




.. image:: 10_dx_interest_rate_swaps_files/10_dx_interest_rate_swaps_24_1.png


Model Calibration
-----------------

Next, equipped with the Libor data, we calibrate the **square-root
diffusion short rate model**. A bit of data preparation:

.. code:: python

    t = libf.index.to_pydatetime()
    f = libf.iloc[:, -1].values / 100
    initial_value = 0.005

A **mean-squared error (MSE)** function to be minimized during
calibration.

.. code:: python

    def srd_forward_error(p0):
        global initial_value, f, t
        if p0[0] < 0 or p0[1] < 0 or p0[2] < 0:
            return 100
        f_model = dx.srd_forwards(initial_value, p0, t)
        MSE = np.sum((f - f_model) ** 2) / len(f)
        return MSE

And the **calibration** itself.

.. code:: python

    from scipy.optimize import fmin

.. code:: python

    opt = fmin(srd_forward_error, (1.0, 0.7, 0.2),
               maxiter=1000, maxfun=1000)


.. parsed-literal::

    Optimization terminated successfully.
             Current function value: 0.000000
             Iterations: 371
             Function evaluations: 649


The **optimal parameters** (kappa, theta, sigma) are:

.. code:: python

    opt




.. parsed-literal::

    array([ 0.00544441,  1.80697228,  0.23689443])



The **model fit** is not too bad in this case.

.. code:: python

    plt.figure(figsize=(10, 6))
    plt.plot(t, f, label='market forward rates')
    plt.plot(t, dx.srd_forwards(initial_value, opt, t), 'r.', label='model forward rates')
    plt.gcf().autofmt_xdate(); plt.legend(loc=0)




.. parsed-literal::

    <matplotlib.legend.Legend at 0x7fadae717fd0>




.. image:: 10_dx_interest_rate_swaps_files/10_dx_interest_rate_swaps_36_1.png


Floating Rate Modeling
----------------------

The optimal parameters from the calibration are used to model the
**floating rate (3m Libor rate)**.

.. code:: python

    # market environment
    me_srd = dx.market_environment('me_srd', dt.datetime(2014, 10, 16))

.. code:: python

    # square-root diffusion
    me_srd.add_constant('initial_value', 0.02)
    me_srd.add_constant('kappa', opt[0]) 
    me_srd.add_constant('theta', opt[1])
    me_srd.add_constant('volatility', opt[2])
    me_srd.add_curve('discount_curve', ois)
      # OIS discounting object
    me_srd.add_constant('currency', 'EUR')
    me_srd.add_constant('paths', 10000)
    me_srd.add_constant('frequency', 'w')
    me_srd.add_constant('starting_date', me_srd.pricing_date)
    me_srd.add_constant('final_date', dt.datetime(2020, 12, 31))

.. code:: python

    srd = dx.square_root_diffusion('srd', me_srd)

Let us have a look at some **simulated rate paths**.

.. code:: python

    paths = srd.get_instrument_values()

.. code:: python

    plt.figure(figsize=(10, 6))
    plt.plot(srd.time_grid, paths[:, :6])




.. parsed-literal::

    [<matplotlib.lines.Line2D at 0x7fadae6b56d0>,
     <matplotlib.lines.Line2D at 0x7fadacd926d0>,
     <matplotlib.lines.Line2D at 0x7fadacd92910>,
     <matplotlib.lines.Line2D at 0x7fadacd92ad0>,
     <matplotlib.lines.Line2D at 0x7fadacd92c90>,
     <matplotlib.lines.Line2D at 0x7fadacd92e50>]




.. image:: 10_dx_interest_rate_swaps_files/10_dx_interest_rate_swaps_44_1.png


Interest Rate Swap
------------------

Finally, we can model the **interest rate swap** itself.

Modeling
~~~~~~~~

First, the market environment with all the **parameters** needed.

.. code:: python

    # market environment for the IRS
    me_irs = dx.market_environment('irs', me_srd.pricing_date)
    me_irs.add_constant('fixed_rate', 0.01)
    me_irs.add_constant('trade_date', me_srd.pricing_date)
    me_irs.add_constant('effective_date', me_srd.pricing_date)
    me_irs.add_constant('payment_date', dt.datetime(2014, 12, 27))
    me_irs.add_constant('payment_day', 27)
    me_irs.add_constant('termination_date', me_srd.get_constant('final_date'))
    me_irs.add_constant('currency', 'EUR')
    me_irs.add_constant('notional', 1000000)
    me_irs.add_constant('tenor', '6m')
    me_irs.add_constant('counting', 'ACT/360')
    # discount curve from mar_env of floating rate

The instantiation of the **valuation object**.

.. code:: python

    irs = dx.interest_rate_swap('irs', srd, me_irs)

Valuation
~~~~~~~~~

The **present value** of the interest rate swap given the assumption, in
particular, of the fixed rate.

.. code:: python

    %time irs.present_value(fixed_seed=True)


.. parsed-literal::

    CPU times: user 185 ms, sys: 14 ms, total: 199 ms
    Wall time: 196 ms




.. parsed-literal::

    482804.62804830389



You can also generate a **full output of all present values** per
simulation path.

.. code:: python

    irs.present_value(full=True).iloc[:, :6]




.. raw:: html

    <div>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>0</th>
          <th>1</th>
          <th>2</th>
          <th>3</th>
          <th>4</th>
          <th>5</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>2014-12-27</th>
          <td>20919.908406</td>
          <td>3765.545880</td>
          <td>-10000.000000</td>
          <td>20447.662984</td>
          <td>15671.171542</td>
          <td>-461.905549</td>
        </tr>
        <tr>
          <th>2015-06-27</th>
          <td>27796.391566</td>
          <td>67649.371268</td>
          <td>-7952.865109</td>
          <td>21069.238749</td>
          <td>6547.642190</td>
          <td>56254.997722</td>
        </tr>
        <tr>
          <th>2015-12-27</th>
          <td>72797.267752</td>
          <td>65085.851729</td>
          <td>-9495.534310</td>
          <td>7184.517383</td>
          <td>-6249.109421</td>
          <td>72603.409748</td>
        </tr>
        <tr>
          <th>2016-06-27</th>
          <td>146587.773739</td>
          <td>60725.900678</td>
          <td>-9450.695252</td>
          <td>10630.874710</td>
          <td>1457.279233</td>
          <td>96390.826927</td>
        </tr>
        <tr>
          <th>2016-12-27</th>
          <td>112006.856512</td>
          <td>11685.574401</td>
          <td>-9365.011855</td>
          <td>36777.060898</td>
          <td>40752.233786</td>
          <td>42437.807055</td>
        </tr>
        <tr>
          <th>2017-06-27</th>
          <td>89091.008350</td>
          <td>19143.973791</td>
          <td>-8652.442452</td>
          <td>10394.933700</td>
          <td>81112.588234</td>
          <td>49316.809090</td>
        </tr>
        <tr>
          <th>2017-12-27</th>
          <td>153802.099096</td>
          <td>37433.554707</td>
          <td>-2231.031239</td>
          <td>3046.024441</td>
          <td>81218.496071</td>
          <td>89235.982850</td>
        </tr>
        <tr>
          <th>2018-06-27</th>
          <td>114033.046751</td>
          <td>-4200.273599</td>
          <td>8204.181410</td>
          <td>86020.232903</td>
          <td>92146.494366</td>
          <td>44010.774119</td>
        </tr>
        <tr>
          <th>2018-12-27</th>
          <td>165327.788708</td>
          <td>-7186.102024</td>
          <td>-8829.883386</td>
          <td>46785.300398</td>
          <td>171037.865434</td>
          <td>14615.688086</td>
        </tr>
        <tr>
          <th>2019-06-27</th>
          <td>101110.008932</td>
          <td>-9038.246291</td>
          <td>-6883.724709</td>
          <td>29794.380064</td>
          <td>183168.704902</td>
          <td>-4599.638746</td>
        </tr>
        <tr>
          <th>2019-12-27</th>
          <td>57890.028467</td>
          <td>-8998.424023</td>
          <td>-8573.907447</td>
          <td>12491.846129</td>
          <td>151080.678798</td>
          <td>-8960.333669</td>
        </tr>
        <tr>
          <th>2020-06-27</th>
          <td>95516.235053</td>
          <td>-8968.295497</td>
          <td>-8963.127240</td>
          <td>15234.994497</td>
          <td>67282.503413</td>
          <td>-7763.697572</td>
        </tr>
        <tr>
          <th>2020-12-27</th>
          <td>4651.947620</td>
          <td>-4500.468014</td>
          <td>-6316.280222</td>
          <td>39899.010233</td>
          <td>107971.761933</td>
          <td>-8568.679338</td>
        </tr>
      </tbody>
    </table>
    </div>



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