# 1. Quickstart¶

This brief first part illustrates—without much explanation—the usage of the DX Analytics library. It models two risk factors, two derivatives instruments and values these in a portfolio context.

[1]:

import dx
import datetime as dt
import pandas as pd
from pylab import plt
plt.style.use('seaborn')


## 1.1. Risk Factor Models¶

The first step is to define a model for the risk-neutral discounting.

[2]:

r = dx.constant_short_rate('r', 0.01)


We then define a market environment containing the major parameter specifications needed,

[3]:

me_1 = dx.market_environment('me', dt.datetime(2016, 1, 1))

[4]:

me_1.add_constant('initial_value', 100.)
# starting value of simulated processes
# volatiltiy factor
# horizon for simulation
# currency of instrument
# frequency for discretization
# number of paths
# number of paths


Next, the model object for the first risk factor, based on the geometric Brownian motion (Black-Scholes-Merton (1973) model).

[5]:

gbm_1 = dx.geometric_brownian_motion('gbm_1', me_1)


Some paths visualized.

[6]:

pdf = pd.DataFrame(gbm_1.get_instrument_values(), index=gbm_1.time_grid)

[7]:

%matplotlib inline
pdf.iloc[:, :10].plot(legend=False, figsize=(10, 6));


Second risk factor with higher volatility. We overwrite the respective value in the market environment.

[8]:

me_2 = dx.market_environment('me_2', me_1.pricing_date)

[9]:

gbm_2 = dx.geometric_brownian_motion('gbm_2', me_2)

[10]:

pdf = pd.DataFrame(gbm_2.get_instrument_values(), index=gbm_2.time_grid)

[11]:

pdf.iloc[:, :10].plot(legend=False, figsize=(10, 6));


## 1.2. Valuation Models¶

Based on the risk factors, we can then define derivatives models for valuation. To this end, we need to add at least one (the maturity), in general two (maturity and strike), parameters to the market environments.

[12]:

me_opt = dx.market_environment('me_opt', me_1.pricing_date)


The first derivative is an American put option on the first risk factor gbm_1.

[13]:

am_put = dx.valuation_mcs_american_single(
name='am_put',
underlying=gbm_1,
mar_env=me_opt,
payoff_func='np.maximum(strike - instrument_values, 0)')


Let us calculate a Monte Carlo present value estimate and estimates for the option Greeks.

[14]:

am_put.present_value()

[14]:

15.013

[15]:

am_put.delta()

[15]:

-0.5411

[16]:

am_put.gamma()

[16]:

-0.0082

[17]:

am_put.vega()

[17]:

51.1584

[18]:

am_put.theta()

[18]:

-3.7421

[19]:

am_put.rho()

[19]:

-71.1355


The second derivative is a European call option on the second risk factor gbm_2.

[20]:

eur_call = dx.valuation_mcs_european_single(
name='eur_call',
underlying=gbm_2,
mar_env=me_opt,
payoff_func='np.maximum(maturity_value - strike, 0)')


Valuation and Greek estimation for this option.

[21]:

eur_call.present_value()

[21]:

20.641978

[22]:

eur_call.delta()

[22]:

0.8496

[23]:

eur_call.gamma()

[23]:

0.0057

[24]:

eur_call.vega()

[24]:

48.2888

[25]:

eur_call.theta()

[25]:

-8.7493

[26]:

eur_call.rho()

[26]:

54.3995


## 1.3. Excursion: SABR Model¶

To illustrate how general the approach of DX Analytics is, let us quickly analyze an option based on a SABR stochastic volatility process. In what follows herafter, the SABR model does not play a role.

We need to define different parameters obviously.

[27]:

me_3 = dx.market_environment('me_3', me_1.pricing_date)

[28]:

# interest rate like parmeters
# initial value
# initial variance
# exponent
# correlation factor
# volatility of volatility/variance


The model object instantiation.

[29]:

sabr = dx.sabr_stochastic_volatility('sabr', me_3)


The valuation object instantiation.

[30]:

me_opt.add_constant('strike', me_3.get_constant('initial_value'))

[31]:

sabr_call = dx.valuation_mcs_european_single(
name='sabr_call',
underlying=sabr,
mar_env=me_opt,
payoff_func='np.maximum(maturity_value - strike, 0)')


Some statistics — same syntax/API even if the model is more complex.

[32]:

sabr_call.present_value(fixed_seed=True)

[32]:

0.025849

[33]:

sabr_call.delta()

[33]:

0.8151

[34]:

sabr_call.rho()

[34]:

-0.0385

[35]:

# resetting the option strike


## 1.4. Options Portfolio¶

### 1.4.1. Modeling¶

In a portfolio context, we need to add information about the model class(es) to be used to the market environments of the risk factors.

[36]:

me_1.add_constant('model', 'gbm')


To compose a portfolio consisting of our just defined options, we need to define derivatives positions. Note that this step is independent from the risk factor model and option model definitions. We only use the market environment data and some additional information needed (e.g. payoff functions).

[37]:

put = dx.derivatives_position(
name='put',
quantity=2,
underlyings=['gbm_1'],
mar_env=me_opt,
otype='American single',
payoff_func='np.maximum(strike - instrument_values, 0)')

[38]:

call = dx.derivatives_position(
name='call',
quantity=3,
underlyings=['gbm_2'],
mar_env=me_opt,
otype='European single',
payoff_func='np.maximum(maturity_value - strike, 0)')


Let us define the relevant market by 2 Python dictionaries, the correlation between the two risk factors and a valuation environment.

[39]:

risk_factors = {'gbm_1': me_1, 'gbm_2' : me_2}
correlations = [['gbm_1', 'gbm_2', -0.4]]
positions = {'put' : put, 'call' : call}

[40]:

val_env = dx.market_environment('general', dt.datetime(2016, 1, 1))


These are used to define the derivatives portfolio.

[41]:

port = dx.derivatives_portfolio(
name='portfolio',  # name
positions=positions,  # derivatives positions
val_env=val_env,  # valuation environment
risk_factors=risk_factors, # relevant risk factors
correlations=correlations, # correlation between risk factors
parallel=False)  # parallel valuation


### 1.4.2. Simulation and Valuation¶

Now, we can get the position values for the portfolio via the get_values method.

[42]:

port.get_values()

Total
pos_value    91.269955
dtype: float64

[42]:

position name quantity otype risk_facts value currency pos_value
0 put put 2 American single [gbm_1] 15.029000 EUR 30.058000
1 call call 3 European single [gbm_2] 20.403985 EUR 61.211955

Via the get_statistics methods delta and vega values are provided as well.

[43]:

port.get_statistics()

Totals
pos_value     91.2700
pos_delta      0.5310
pos_vega     225.1372
dtype: float64

[43]:

position name quantity otype risk_facts value currency pos_value pos_delta pos_vega
0 put put 2 American single [gbm_1] 15.029 EUR 30.058 -1.1802 87.3484
1 call call 3 European single [gbm_2] 20.404 EUR 61.212 1.7112 137.7888

Much more complex scenarios are possible with DX Analytics

### 1.4.3. Risk Reports¶

Having modeled the derivatives portfolio, risk reports are only two method calls away.

[44]:

deltas, benchvalue = port.get_port_risk(Greek='Delta')


gbm_2
0.8
0.9
1.0
1.1
1.2

gbm_1
0.8
0.9
1.0
1.1
1.2


[45]:

deltas

[45]:

gbm_2_Delta gbm_1_Delta
major minor
0.8 factor 80.000000 80.000000
value 61.767112 122.001955
0.9 factor 90.000000 90.000000
value 75.421234 104.903955
1.0 factor 100.000000 100.000000
value 91.269955 91.269955
1.1 factor 110.000000 110.000000
value 109.161973 80.751955
1.2 factor 120.000000 120.000000
value 128.739235 73.195955
[46]:

deltas.loc(axis=0)[:, 'value'] - benchvalue

[46]:

gbm_2_Delta gbm_1_Delta
major minor
0.8 value -29.502843 30.732
0.9 value -15.848721 13.634
1.0 value 0.000000 0.000
1.1 value 17.892018 -10.518
1.2 value 37.469280 -18.074
[47]:

vegas, benchvalue = port.get_port_risk(Greek='Vega', step=0.05)


gbm_2
0.8
0.8500000000000001
0.9000000000000001
0.9500000000000002
1.0000000000000002
1.0500000000000003
1.1000000000000003
1.1500000000000004
1.2000000000000004

gbm_1
0.8
0.8500000000000001
0.9000000000000001
0.9500000000000002
1.0000000000000002
1.0500000000000003
1.1000000000000003
1.1500000000000004
1.2000000000000004


[48]:

vegas

[48]:

gbm_2_Vega gbm_1_Vega
major minor
0.80 factor 0.400000 0.160000
value 77.334907 87.597955
0.85 factor 0.430000 0.170000
value 80.843542 88.473955
0.90 factor 0.450000 0.180000
value 84.336475 89.377955
0.95 factor 0.480000 0.190000
value 87.812380 90.351955
1.00 factor 0.500000 0.200000
value 91.269955 91.269955
1.05 factor 0.530000 0.210000
value 94.708426 92.143955
1.10 factor 0.550000 0.220000
value 98.125972 93.137955
1.15 factor 0.580000 0.230000
value 101.519533 94.045955
1.20 factor 0.600000 0.240000
value 104.887072 95.013955
[49]:

vegas.loc(axis=0)[:, 'value'] - benchvalue

[49]:

gbm_2_Vega gbm_1_Vega
major minor
0.80 value -13.935048 -3.672
0.85 value -10.426413 -2.796
0.90 value -6.933480 -1.892
0.95 value -3.457575 -0.918
1.00 value 0.000000 0.000
1.05 value 3.438471 0.874
1.10 value 6.856017 1.868
1.15 value 10.249578 2.776
1.20 value 13.617117 3.744

© Dr. Yves J. Hilpisch | The Python Quants GmbH