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

import time
t0 = time.time()

"""
Explanation: <img src="http://hilpisch.com/tpq_logo.png" alt="The Python Quants" width="45%" align="right" border="4">
Multi-Risk Derivatives Valuation
A specialty of DX Analytics is the valuation of derivatives instruments defined on multiple risk factors and portfolios composed of such derivatives. This section of the documentation illustrates the usage of the dedicated multi-risk valuation classes.
End of explanation
"""


r = constant_short_rate('r', 0.06)

me1 = market_environment('me1', dt.datetime(2015, 1, 1))
me2 = market_environment('me2', dt.datetime(2015, 1, 1))

me1.add_constant('initial_value', 36.)
me1.add_constant('volatility', 0.1)  # low volatility
me1.add_constant('currency', 'EUR')
me1.add_constant('model', 'gbm')

me2.add_environment(me1)
me2.add_constant('initial_value', 36.)
me2.add_constant('volatility', 0.5)  # high volatility

"""
Explanation: There are the following multiple risk factor valuation classes available:

valuation_mcs_european_multi for the valuation of multi-risk derivatives with European exercise
valuation_mcs_american_multi for the valuation of multi-risk derivatives with American exercise

The handling of these classes is similar to building a portfolio of single-risk derivatives positions.
Market Environments
Market environments for the risk factors are the starting point.
End of explanation
"""


risk_factors = {'gbm1' : me1, 'gbm2' : me2}
correlations = [['gbm1', 'gbm2', 0.5]]

"""
Explanation: We assum a positive correlation between the two risk factors.
End of explanation
"""


val_env = market_environment('val_env', dt.datetime(2015, 1, 1))

val_env.add_constant('starting_date', val_env.pricing_date)
val_env.add_constant('final_date', dt.datetime(2015, 12, 31))
val_env.add_constant('frequency', 'W')
val_env.add_constant('paths', 5000)
val_env.add_curve('discount_curve', r)
val_env.add_constant('maturity', dt.datetime(2015, 12, 31))
val_env.add_constant('currency', 'EUR')

"""
Explanation: Valuation Environment
Similar to the instantiation of a derivatives_portfolio object, a valuation environment is needed (unifying certain parameters/assumptions for all relevant risk factors of a derivative).
End of explanation
"""


# European maximum call option
payoff_func = "np.maximum(np.maximum(maturity_value['gbm1'], maturity_value['gbm2']) - 38, 0)"
vc = valuation_mcs_european_multi(
            name='European maximum call',  # name
            val_env=val_env,  # valuation environment
            risk_factors=risk_factors,  # the relevant risk factors
            correlations=correlations,  # correlations between risk factors
            payoff_func=payoff_func)  # payoff function

vc.risk_factors

"""
Explanation: valuation_mcs_european_multi
As an example for a multi-risk derivative with European exercise consider a maximum call option. With multiple risk factors, payoff functions are defined by adding key (the name strings) to the maturity_value array object. As with the portfolio valuation class, the multi-risk factor valuation classes get passed market_environment objects only and not the risk factor model objects themsemselves.
End of explanation
"""


vc.underlying_objects

"""
Explanation: At instantiation, the respective risk factor model objects are instantiated as well.
End of explanation
"""


vc.correlations

vc.correlation_matrix

vc.val_env.get_list('cholesky_matrix')

"""
Explanation: Correlations are stored as well and the resulting corrleation and Cholesky matrices are generated.
End of explanation
"""


np.shape(vc.generate_payoff())

"""
Explanation: The payoff of a European option is a one-dimensional ndarray object.
End of explanation
"""


vc.present_value()

"""
Explanation: Present value estimations are generated by a call of the present_value method.
End of explanation
"""


vc.update('gbm1', initial_value=50.)

vc.present_value()

vc.update('gbm2', volatility=0.6)

vc.present_value()

"""
Explanation: The update method allows updating of certain parameters.
End of explanation
"""


vc.update('gbm1', initial_value=36., volatility=0.1)
vc.update('gbm2', initial_value=36., volatility=0.5)

"""
Explanation: Let us reset the values to the original parameters.
End of explanation
"""


vc.delta('gbm2', interval=0.5)

vc.vega('gbm1')

"""
Explanation: When calculating Greeks the risk factor now has to be specified by providing its name.
End of explanation
"""


%%time
s_list = np.arange(28., 46.1, 2.)
pv = []; de = []; ve = []
for s in s_list:
    vc.update('gbm1', initial_value=s)
    pv.append(vc.present_value())
    de.append(vc.delta('gbm1', .5))
    ve.append(vc.vega('gbm1', 0.2))
vc.update('gbm1', initial_value=36.)

%matplotlib inline

plot_option_stats(s_list, pv, de, ve)

"""
Explanation: Sensitivities Positive Correlation
Almos in complete analogy to the single-risk valuation classes, sensitivities can be estimated for the multi-risk valuation classes. 
Sensitivities Risk Factor 1
Consider first the case from before with positive correlation between the two risk factors. The following estimates and plots the sensitivities for the first risk factor gbm1.
End of explanation
"""


%%time
s_list = np.arange(28., 46.1, 2.)
pv = []; de = []; ve = []
for s in s_list:
    vc.update('gbm2', initial_value=s)
    pv.append(vc.present_value())
    de.append(vc.delta('gbm2', .5))
    ve.append(vc.vega('gbm2', 0.2))

plot_option_stats(s_list, pv, de, ve)

"""
Explanation: Sensitivities Risk Factor 2
Now the sensitivities for the second risk factor.
End of explanation
"""


correlations = [['gbm1', 'gbm2', -0.9]]

# European maximum call option
payoff_func = "np.maximum(np.maximum(maturity_value['gbm1'], maturity_value['gbm2']) - 38, 0)"
vc = valuation_mcs_european_multi(
            name='European maximum call',
            val_env=val_env,
            risk_factors=risk_factors,
            correlations=correlations,
            payoff_func=payoff_func)

"""
Explanation: Sensitivities with Negative Correlation
The second case is for highly negatively correlated risk factors.
End of explanation
"""


%%time
s_list = np.arange(28., 46.1, 2.)
pv = []; de = []; ve = []
for s in s_list:
    vc.update('gbm1', initial_value=s)
    pv.append(vc.present_value())
    de.append(vc.delta('gbm1', .5))
    ve.append(vc.vega('gbm1', 0.2))
vc.update('gbm1', initial_value=36.)

plot_option_stats(s_list, pv, de, ve)

"""
Explanation: Sensitivities Risk Factor 1
Again, sensitivities for the first risk factor first.
End of explanation
"""


%%time
s_list = np.arange(28., 46.1, 2.)
pv = []; de = []; ve = []
for s in s_list:
    vc.update('gbm2', initial_value=s)
    pv.append(vc.present_value())
    de.append(vc.delta('gbm2', .5))
    ve.append(vc.vega('gbm2', 0.2))

plot_option_stats(s_list, pv, de, ve)

"""
Explanation: Sensitivities Risk Factor 2
Finally, the sensitivities for the second risk factor for this second scenario.
End of explanation
"""


correlations = [['gbm1', 'gbm2', 0.5]]

# European maximum call option
payoff_func = "np.maximum(np.maximum(maturity_value['gbm1'], maturity_value['gbm2']) - 38, 0)"
vc = valuation_mcs_european_multi(
            name='European maximum call',
            val_env=val_env,
            risk_factors=risk_factors,
            correlations=correlations,
            payoff_func=payoff_func)

"""
Explanation: Surfaces for Positive Correlation Case
Let us return to the case of positive correlation between the two relevant risk factors.
End of explanation
"""


asset_1 = np.arange(28., 46.1, 4.)  # range of initial values
asset_2 = asset_1
a_1, a_2 = np.meshgrid(asset_1, asset_2)
  # two-dimensional grids out of the value vectors
value = np.zeros_like(a_1)

"""
Explanation: Value Surface
We are now interested in the value surface of the derivative instrument for both different initial values of the first and second risk factor.
End of explanation
"""


%%time
for i in range(np.shape(value)[0]):
    for j in range(np.shape(value)[1]):
        vc.update('gbm1', initial_value=a_1[i, j])
        vc.update('gbm2', initial_value=a_2[i, j])
        value[i, j] = vc.present_value()

"""
Explanation: The following estimates for all possible combinations of the initial values---given the assumptions from above---the present value of the European maximum call option.
End of explanation
"""


plot_greeks_3d([a_1, a_2, value], ['gbm1', 'gbm2', 'present value'])

"""
Explanation: The resulting plot then looks as follows. Here, a helper plot function of DX Analytics is used.
End of explanation
"""


delta_1 = np.zeros_like(a_1)
delta_2 = np.zeros_like(a_1)

%%time
for i in range(np.shape(delta_1)[0]):
    for j in range(np.shape(delta_1)[1]):
        vc.update('gbm1', initial_value=a_1[i, j])
        vc.update('gbm2', initial_value=a_2[i, j])
        delta_1[i, j] = vc.delta('gbm1')
        delta_2[i, j] = vc.delta('gbm2')

"""
Explanation: Delta Surfaces
Applying a very similar approach, a delta surface for all possible combinations of the intial values is as easily generated.
End of explanation
"""


plot_greeks_3d([a_1, a_2, delta_1], ['gbm1', 'gbm2', 'delta gbm1'])

"""
Explanation: The plot for the delta surface of the first risk factor.
End of explanation
"""


plot_greeks_3d([a_1, a_2, delta_2], ['gbm1', 'gbm2', 'delta gbm2'])

"""
Explanation: And the plot for the delta of the second risk factor.
End of explanation
"""


vega_1 = np.zeros_like(a_1)
vega_2 = np.zeros_like(a_1)

for i in range(np.shape(vega_1)[0]):
    for j in range(np.shape(vega_1)[1]):
        vc.update('gbm1', initial_value=a_1[i, j])
        vc.update('gbm2', initial_value=a_2[i, j])
        vega_1[i, j] = vc.vega('gbm1')
        vega_2[i, j] = vc.vega('gbm2')

"""
Explanation: Vega Surfaces
The same approach can of course be applied to generate vega surfaces.
End of explanation
"""


plot_greeks_3d([a_1, a_2, vega_1], ['gbm1', 'gbm2', 'vega gbm1'])

"""
Explanation: The surface for the first risk factor.
End of explanation
"""


plot_greeks_3d([a_1, a_2, vega_2], ['gbm1', 'gbm2', 'vega gbm2'])

"""
Explanation: And the one for the second risk factor.
End of explanation
"""


# restore initial values
vc.update('gbm1', initial_value=36., volatility=0.1)
vc.update('gbm2', initial_value=36., volatility=0.5)

"""
Explanation: Finally, we reset the intial values and the volatilities for the two risk factors.
End of explanation
"""


# American put payoff
payoff_am = "np.maximum(34 - np.minimum(instrument_values['gbm1'], instrument_values['gbm2']), 0)"
# finer time grid and more paths
val_env.add_constant('frequency', 'B')
val_env.add_curve('time_grid', None)
  # delete existing time grid information
val_env.add_constant('paths', 5000)

# American put option on minimum of two assets
vca = valuation_mcs_american_multi(
            name='American minimum put',
            val_env=val_env,
            risk_factors=risk_factors,
            correlations=correlations,
            payoff_func=payoff_am)

vca.present_value()

for key, obj in vca.instrument_values.items():
    print np.shape(vca.instrument_values[key])

"""
Explanation: valuation_mcs_american_multi
In general, the modeling and handling of the valuation classes for American exercise is not too different from those for European exercise. The major difference is in the definition of payoff function.
Present Values
This example models an American minimum put on the two risk factors from before.
End of explanation
"""


asset_1 = np.arange(28., 44.1, 4.)
asset_2 = asset_1
a_1, a_2 = np.meshgrid(asset_1, asset_2)
value = np.zeros_like(a_1)

%%time
for i in range(np.shape(value)[0]):
    for j in range(np.shape(value)[1]):
        vca.update('gbm1', initial_value=a_1[i, j])
        vca.update('gbm2', initial_value=a_2[i, j])
        value[i, j] = vca.present_value()

plot_greeks_3d([a_1, a_2, value], ['gbm1', 'gbm2', 'present value'])

"""
Explanation: The present value surface is generated in the same way as before for the European option on the two risk factors. The computational burden is of course much higher for the American option, which are valued by the use of the Least-Squares Monte Carlo approach (LSM) according to Longstaff-Schwartz (2001).
End of explanation
"""


delta_1 = np.zeros_like(a_1)
delta_2 = np.zeros_like(a_1)

%%time
for i in range(np.shape(delta_1)[0]):
    for j in range(np.shape(delta_1)[1]):
        vca.update('gbm1', initial_value=a_1[i, j])
        vca.update('gbm2', initial_value=a_2[i, j])
        delta_1[i, j] = vca.delta('gbm1')
        delta_2[i, j] = vca.delta('gbm2')

plot_greeks_3d([a_1, a_2, delta_1], ['gbm1', 'gbm2', 'delta gbm1'])

plot_greeks_3d([a_1, a_2, delta_2], ['gbm1', 'gbm2', 'delta gbm2'])

"""
Explanation: Delta Surfaces
The same exercise as before for the two delta surfaces.
End of explanation
"""


vega_1 = np.zeros_like(a_1)
vega_2 = np.zeros_like(a_1)

%%time
for i in range(np.shape(vega_1)[0]):
    for j in range(np.shape(vega_1)[1]):
        vca.update('gbm1', initial_value=a_1[i, j])
        vca.update('gbm2', initial_value=a_2[i, j])
        vega_1[i, j] = vca.vega('gbm1')
        vega_2[i, j] = vca.vega('gbm2')

plot_greeks_3d([a_1, a_2, vega_1], ['gbm1', 'gbm2', 'vega gbm1'])

plot_greeks_3d([a_1, a_2, vega_2], ['gbm1', 'gbm2', 'vega gbm2'])

"""
Explanation: Vega Surfaces
And finally for the vega surfaces.
End of explanation
"""


me3 = market_environment('me3', dt.datetime(2015, 1, 1))
me4 = market_environment('me4', dt.datetime(2015, 1, 1))

me3.add_environment(me1)
me4.add_environment(me1)

# for jump-diffusion
me3.add_constant('lambda', 0.5)
me3.add_constant('mu', -0.6)
me3.add_constant('delta', 0.1)
me3.add_constant('model', 'jd')

# for stoch volatility model
me4.add_constant('kappa', 2.0)
me4.add_constant('theta', 0.3)
me4.add_constant('vol_vol', 0.2)
me4.add_constant('rho', -0.75)
me4.add_constant('model', 'sv')

val_env.add_constant('paths', 10000)
val_env.add_constant('frequency', 'W')
val_env.add_curve('time_grid', None)

"""
Explanation: More than Two Risk Factors
The principles of working with multi-risk valuation classes can be illustrated quite well in the two risk factor case. However, there is---in theory---no limitation on the number of risk factors used for derivatives modeling.
Four Asset Basket Option
Consider a maximum basket option on four different risk factors. We add a jump diffusion as well as a stochastic volatility model to the mix
End of explanation
"""


risk_factors = {'gbm1' : me1, 'gbm2' : me2, 'jd' : me3, 'sv' : me4}
correlations = [['gbm1', 'gbm2', 0.5], ['gbm2', 'jd', -0.5], ['gbm1', 'sv', 0.7]]

"""
Explanation: In this case, we need to specify three correlation values.
End of explanation
"""


# European maximum call payoff
payoff_1 = "np.maximum(np.maximum(np.maximum(maturity_value['gbm1'], maturity_value['gbm2']),"
payoff_2 = " np.maximum(maturity_value['jd'], maturity_value['sv'])) - 40, 0)"
payoff = payoff_1 + payoff_2

payoff

"""
Explanation: The payoff function in this case gets a bit more complex.
End of explanation
"""


vc = valuation_mcs_european_multi(
            name='European maximum call',
            val_env=val_env,
            risk_factors=risk_factors,
            correlations=correlations,
            payoff_func=payoff)

"""
Explanation: However, the instantiation of the valuation classe remains the same.
End of explanation
"""


vc.risk_factors

vc.underlying_objects

vc.present_value()

"""
Explanation: Example Output and Calculations
The following just displays some example output and the results from certain calculations.
End of explanation
"""


vc.correlation_matrix

vc.val_env.get_list('cholesky_matrix')

"""
Explanation: The correlation and Cholesky matrices now are of shape 4x4.
End of explanation
"""


vc.delta('jd', interval=0.1)

vc.delta('sv')

vc.vega('jd')

vc.vega('sv')

"""
Explanation: Delta and vega estimates are generated in exactly the same fashion as in the two risk factor case.
End of explanation
"""


delta_1 = np.zeros_like(a_1)
delta_2 = np.zeros_like(a_1)

%%time
for i in range(np.shape(delta_1)[0]):
    for j in range(np.shape(delta_1)[1]):
        vc.update('jd', initial_value=a_1[i, j])
        vc.update('sv', initial_value=a_2[i, j])
        delta_1[i, j] = vc.delta('jd')
        delta_2[i, j] = vc.delta('sv')

plot_greeks_3d([a_1, a_2, delta_1], ['jump diffusion', 'stochastic vol', 'delta jd'])

plot_greeks_3d([a_1, a_2, delta_2], ['jump diffusion', 'stochastic vol', 'delta sv'])

"""
Explanation: Delta for Jump Diffusion and Stochastic Vol Process
Of course, we cannot visualize Greek surfaces dependent on initial values for all four risk factors but still for two. In what follows we generate the delta surfaces with respect to the jump diffusion- and stochastic volatility-based risk factors.
End of explanation
"""


vega_1 = np.zeros_like(a_1)
vega_2 = np.zeros_like(a_1)

%%time
for i in range(np.shape(vega_1)[0]):
    for j in range(np.shape(vega_1)[1]):
        vc.update('jd', initial_value=a_1[i, j])
        vc.update('sv', initial_value=a_2[i, j])
        vega_1[i, j] = vc.vega('jd')
        vega_2[i, j] = vc.vega('sv')

plot_greeks_3d([a_1, a_2, vega_1], ['jump diffusion', 'stochastic vol', 'vega jd'])

plot_greeks_3d([a_1, a_2, vega_2], ['jump diffusion', 'stochastic vol', 'vega sv'])

"""
Explanation: Vega for Jump Diffusion and Stochastic Vol Process
Now the same exercise for the vega surfaces for the same two risk factors.
End of explanation
"""


# payoff of American minimum put option
payoff_am_1 = "np.maximum(40 - np.minimum(np.minimum(instrument_values['gbm1'], instrument_values['gbm2']),"
payoff_am_2 = "np.minimum(instrument_values['jd'], instrument_values['sv'])), 0)"
payoff_am = payoff_am_1 + payoff_am_2

vca = valuation_mcs_american_multi(
            name='American minimum put',
            val_env=val_env,
            risk_factors=risk_factors,
            correlations=correlations,
            payoff_func=payoff_am)

"""
Explanation: American Exercise
As a final illustration consider the case of an American minimum put option on the four risk factors. This again is a step that leads to a much increased computational burden due to the necessity to apply the least-squares regression approach.
End of explanation
"""


# restore initial values
vc.update('jd', initial_value=36., volatility=0.1)
vc.update('sv', initial_value=36., volatility=0.1)
%time vc.present_value()

%time vca.present_value()

%time vca.delta('gbm1')

%time vca.delta('gbm2')

%time vca.vega('jd')

%time vca.vega('sv')

print "Duration for whole notebook %.2f in min" % ((time.time() - t0) / 60)

"""
Explanation: However, another illustration that even such a complex instrument can be handled as elegantly as the most simple one (i.e. European option on single risk factor). Let us compare the present value estimates for both the European and American maximum basket options.
End of explanation
"""


