Multi-Risk Derivatives Portfolios
=================================

The step from multi-risk derivatives instruments to multi-risk
derivatives instrument portfolios is not a too large one. This part of
the tutorial shows how to model an economy with three risk factors

.. code:: python

    from dx import *
    import seaborn as sns; sns.set()

Risk Factors
------------

This sub-section models the single risk factors. We start with
definition of the risk-neutral discounting object.

.. code:: python

    # constant short rate
    r = constant_short_rate('r', 0.02)

**Three risk factors** ares modeled:

-  geometric Brownian motion
-  jump diffusion
-  stochastic volatility process

.. code:: python

    # market environments
    me_gbm = market_environment('gbm', dt.datetime(2015, 1, 1))
    me_jd = market_environment('jd', dt.datetime(2015, 1, 1))
    me_sv = market_environment('sv', dt.datetime(2015, 1, 1))

Assumptions for the ``geometric_brownian_motion`` object.

.. code:: python

    # geometric Brownian motion
    me_gbm.add_constant('initial_value', 36.)
    me_gbm.add_constant('volatility', 0.2) 
    me_gbm.add_constant('currency', 'EUR')
    me_gbm.add_constant('model', 'gbm')

Assumptions for the ``jump_diffusion`` object.

.. code:: python

    # jump diffusion
    me_jd.add_constant('initial_value', 36.)
    me_jd.add_constant('volatility', 0.2)
    me_jd.add_constant('lambda', 0.5)
        # probability for jump p.a.
    me_jd.add_constant('mu', -0.75)
        # expected jump size [%]
    me_jd.add_constant('delta', 0.1)
        # volatility of jump
    me_jd.add_constant('currency', 'EUR')
    me_jd.add_constant('model', 'jd')

Assumptions for the ``stochastic_volatility`` object.

.. code:: python

    # stochastic volatility model
    me_sv.add_constant('initial_value', 36.)
    me_sv.add_constant('volatility', 0.2)
    me_sv.add_constant('vol_vol', 0.1)
    me_sv.add_constant('kappa', 2.5)
    me_sv.add_constant('theta', 0.4)
    me_sv.add_constant('rho', -0.5)
    me_sv.add_constant('currency', 'EUR')
    me_sv.add_constant('model', 'sv')

Finally, the unifying valuation assumption for the **valuation
environment**.

.. code:: python

    # valuation environment
    val_env = market_environment('val_env', dt.datetime(2015, 1, 1))
    val_env.add_constant('paths', 10000)
    val_env.add_constant('frequency', 'W')
    val_env.add_curve('discount_curve', r)
    val_env.add_constant('starting_date', dt.datetime(2015, 1, 1))
    val_env.add_constant('final_date', dt.datetime(2015, 12, 31))

These are added to the single ``market_environment`` objects of the risk
factors.

.. code:: python

    # add valuation environment to market environments
    me_gbm.add_environment(val_env)
    me_jd.add_environment(val_env)
    me_sv.add_environment(val_env)

Finally, the **market model** with the risk factors and the correlations
between them.

.. code:: python

    risk_factors = {'gbm' : me_gbm, 'jd' : me_jd, 'sv' : me_sv}
    correlations = [['gbm', 'jd', 0.66], ['jd', 'sv', -0.75]]

Derivatives
-----------

In this sub-section, we model the single derivatives instruments.

American Put Option
~~~~~~~~~~~~~~~~~~~

The first derivative instrument is an **American put option**.

.. code:: python

    gbm = geometric_brownian_motion('gbm_obj', me_gbm)

.. code:: python

    me_put = market_environment('put', dt.datetime(2015, 1, 1))
    me_put.add_constant('maturity', dt.datetime(2015, 12, 31))
    me_put.add_constant('strike', 40.)
    me_put.add_constant('currency', 'EUR')
    me_put.add_environment(val_env)

.. code:: python

    am_put = valuation_mcs_american_single('am_put', mar_env=me_put, underlying=gbm,
                           payoff_func='np.maximum(strike - instrument_values, 0)')

.. code:: python

    am_put.present_value(fixed_seed=True, bf=5)




.. parsed-literal::

    5.012



European Maximum Call on 2 Assets
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The second derivative instrument is a **European maximum call option on
two risk factors**.

.. code:: python

    jd = jump_diffusion('jd_obj', me_jd)

.. code:: python

    me_max_call = market_environment('put', dt.datetime(2015, 1, 1))
    me_max_call.add_constant('maturity', dt.datetime(2015, 9, 15))
    me_max_call.add_constant('currency', 'EUR')
    me_max_call.add_environment(val_env)

.. code:: python

    payoff_call = "np.maximum(np.maximum(maturity_value['gbm'], maturity_value['jd']) - 34., 0)"

.. code:: python

    assets = {'gbm' : me_gbm, 'jd' : me_jd}
    asset_corr = [correlations[0]]

.. code:: python

    asset_corr




.. parsed-literal::

    [['gbm', 'jd', 0.66]]



.. code:: python

    max_call = valuation_mcs_european_multi('max_call', me_max_call, assets, asset_corr,
            payoff_func=payoff_call)

.. code:: python

    max_call.present_value(fixed_seed=False)




.. parsed-literal::

    8.334



.. code:: python

    max_call.delta('jd')




.. parsed-literal::

    0.7597222222222231



.. code:: python

    max_call.delta('gbm')




.. parsed-literal::

    0.2819444444444461



American Minimum Put on 2 Assets
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The third derivative instrument is an **American minimum put on two risk
factors**.

.. code:: python

    sv = stochastic_volatility('sv_obj', me_sv)

.. code:: python

    me_min_put = market_environment('min_put', dt.datetime(2015, 1, 1))
    me_min_put.add_constant('maturity', dt.datetime(2015, 6, 17))
    me_min_put.add_constant('currency', 'EUR')
    me_min_put.add_environment(val_env)

.. code:: python

    payoff_put = "np.maximum(32. - np.minimum(instrument_values['jd'], instrument_values['sv']), 0)"

.. code:: python

    assets = {'jd' : me_jd, 'sv' : me_sv}
    asset_corr = [correlations[1]]
    asset_corr




.. parsed-literal::

    [['jd', 'sv', -0.75]]



.. code:: python

    min_put = valuation_mcs_american_multi(
                    'min_put', val_env=me_min_put, risk_factors=assets,
                    correlations=asset_corr, payoff_func=payoff_put)

.. code:: python

    min_put.present_value(fixed_seed=True)




.. parsed-literal::

    4.302



.. code:: python

    min_put.delta('jd')




.. parsed-literal::

    -0.1083333333333325



.. code:: python

    min_put.delta('sv')




.. parsed-literal::

    -0.21944444444444372



Portfolio
---------

To compose a derivatives portfolio, ``derivatives_position`` objects are
needed.

.. code:: python

    am_put_pos = derivatives_position(
                        name='am_put_pos',
                        quantity=2,
                        underlyings=['gbm'],
                        mar_env=me_put,
                        otype='American single',
                        payoff_func='np.maximum(instrument_values - 36., 0)')

.. code:: python

    max_call_pos = derivatives_position(
                        'max_call_pos', 3, ['gbm', 'jd'],
                        me_max_call, 'European multi',
                        payoff_call)

.. code:: python

    min_put_pos = derivatives_position(
                        'min_put_pos', 5, ['sv', 'jd'],
                        me_min_put, 'American multi',
                        payoff_put)

These objects are to be collected in ``dictionary`` objects.

.. code:: python

    positions = {'am_put_pos' : am_put_pos, 'max_call_pos' : max_call_pos,
                 'min_put_pos' : min_put_pos}

All is together to instantiate the ``derivatives_portfolio`` class.

.. code:: python

    port = derivatives_portfolio(name='portfolio',
                                 positions=positions,
                                 val_env=val_env,
                                 risk_factors=risk_factors,
                                 correlations=correlations)

Let us have a look at the major **portfolio statistics**.

.. code:: python

    %time stats = port.get_statistics()
    stats


.. parsed-literal::

    Totals
    pos_value    51.764
    dtype: float64
    CPU times: user 1.72 s, sys: 3 ms, total: 1.72 s
    Wall time: 1.72 s




.. raw:: html

    <div>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>position</th>
          <th>name</th>
          <th>quantity</th>
          <th>otype</th>
          <th>risk_facts</th>
          <th>value</th>
          <th>currency</th>
          <th>pos_value</th>
          <th>pos_delta</th>
          <th>pos_vega</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>max_call_pos</td>
          <td>max_call_pos</td>
          <td>3</td>
          <td>European multi</td>
          <td>[gbm, jd]</td>
          <td>8.165</td>
          <td>EUR</td>
          <td>24.495</td>
          <td>{'jd': 2.266667, 'gbm': 0.866667}</td>
          <td>{'jd': 10.5, 'gbm': 15.0}</td>
        </tr>
        <tr>
          <th>1</th>
          <td>am_put_pos</td>
          <td>am_put_pos</td>
          <td>2</td>
          <td>American single</td>
          <td>[gbm]</td>
          <td>3.182</td>
          <td>EUR</td>
          <td>6.364</td>
          <td>1.2166</td>
          <td>30.4</td>
        </tr>
        <tr>
          <th>2</th>
          <td>min_put_pos</td>
          <td>min_put_pos</td>
          <td>5</td>
          <td>American multi</td>
          <td>[sv, jd]</td>
          <td>4.181</td>
          <td>EUR</td>
          <td>20.905</td>
          <td>{'jd': -0.618056, 'sv': -1.180556}</td>
          <td>{'jd': 10.0, 'sv': 10.0}</td>
        </tr>
      </tbody>
    </table>
    </div>



.. code:: python

    stats['pos_value'].sum()




.. parsed-literal::

    51.763999999999996



Finally, a graphical look at **two selected, simulated paths** of the
stochastic volatility risk factor and the jump diffusion risk factor,
respectively.

.. code:: python

    path_no = 1
    paths1 = port.underlying_objects['sv'].get_instrument_values()[:, path_no]
    paths2 = port.underlying_objects['jd'].get_instrument_values()[:, path_no]

.. code:: python

    paths1




.. parsed-literal::

    array([ 36.        ,  36.34884124,  34.9130166 ,  36.31001777,
            34.85894675,  35.63213975,  39.0707393 ,  44.35899961,
            44.47413556,  43.13168786,  47.48647475,  46.27393881,
            43.49655379,  47.23009583,  49.53908109,  43.71731937,
            51.76008829,  52.58524036,  54.2573447 ,  51.6460015 ,
            44.28088435,  39.01540236,  32.91109921,  30.24790867,
            29.61667412,  31.76473582,  27.30258542,  28.39127869,
            28.23333544,  27.08972766,  27.32993764,  28.65926992,
            26.9785575 ,  31.8938528 ,  34.58971037,  32.7699367 ,
            29.35365969,  28.66856698,  31.14196767,  31.37051847,
            30.10832392,  33.11928276,  34.48108709,  32.97416103,
            28.08648082,  26.52705819,  21.57771857,  20.42912064,
            20.99231801,  19.23163105,  21.37234395,  20.80944619,
            20.73301676,  20.79953889,  20.34822548,  18.69820953])



.. code:: python

    paths2




.. parsed-literal::

    array([ 36.        ,  36.09036184,  37.30618631,  36.53472814,
            36.79572901,  36.66944957,  35.56227649,  33.3644349 ,
            33.42674479,  34.22659123,  32.22562864,  33.05209641,
            33.82798732,  33.20431052,  33.78753542,  36.17164509,
            33.76072474,  33.00278459,  32.79744011,  33.50195594,
            34.40874205,  34.57037634,  37.00098086,  37.7309694 ,
            38.32223737,  38.2477447 ,  39.92233666,  39.6591103 ,
            39.43998269,  39.88306801,  40.35338195,  40.23418088,
            41.48106905,  39.66194269,  39.32622142,  39.48978055,
            40.67396363,  40.76122376,  40.00499204,  39.725126  ,
            40.05448393,  39.06546207,  37.674332  ,  38.48264394,
            40.86222422,  41.3749628 ,  44.02458974,  45.32751049,
            44.704871  ,  45.06224883,  43.38741804,  43.68855748,
            44.75023229,  45.4285521 ,  45.27511078,  45.68070094])



The resulting plot illustrates the strong **negative correlation**.

.. code:: python

    import matplotlib.pyplot as plt
    %matplotlib inline
    plt.figure(figsize=(10, 6))
    plt.plot(port.time_grid, paths1, 'r', label='sv')
    plt.plot(port.time_grid, paths2, 'b', label='jd')
    plt.gcf().autofmt_xdate()
    plt.legend(loc=0); plt.grid(True)
    # negatively correlated underlyings



.. image:: 05_dx_portfolio_multi_risk_files/05_dx_portfolio_multi_risk_67_0.png


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