Wiener Processes¶

One of the important parameters when evaluating forecast economically is the Oil Price. Several authors have stated that the Oil Price follows a Wiener process in which the future value does not depend on the past but follows Gaussian increments each step.

Dcapy.wiener module provides somes classes to implement simple Random Walks, Brownian Geometric Motion and Mean Reversion. This sections review how to initialize those proccesesss and in laters sections are going to be implemented directly on a cashflow modeling

In [1]:

import os
from dcapy.wiener import Brownian, GeometricBrownian,MeanReversion
from dcapy.dca import ProbVar
import numpy as np 
import pandas as pd
from datetime import date
import matplotlib.pyplot as plt
import seaborn as sns 
from scipy import stats

import os from dcapy.wiener import Brownian, GeometricBrownian,MeanReversion from dcapy.dca import ProbVar import numpy as np import pandas as pd from datetime import date import matplotlib.pyplot as plt import seaborn as sns from scipy import stats

In [3]:

rw = Brownian()
rw

rw = Brownian() rw

Out[3]:

Brownian(initial_condition=0, ti=0, steps=1, processes=1, generator=ProbVar(dist='norm', kw={'loc': 0, 'scale': 1}, factor=1.0, seed=None), freq_input='D', freq_output='D', drift=0)

By default, a Brownian class is initialized with random generator defined by a ProbVar class (Seen on Montecarlo analysis when forecasting) with a standard normal distribution.

By calling the method generate and pass the number of steps and processes it returns a dataframe with the data.

In [9]:

steps = 500 
processes = 20

df = rw.generate(steps, processes,seed=21)
df.plot(legend=False)

steps = 500 processes = 20 df = rw.generate(steps, processes,seed=21) df.plot(legend=False)

Out[9]:

<AxesSubplot:>

Add custom parameters¶

Set initial conditions and drift

In [18]:

rw = Brownian(
    initial_condition=100,
    ti=0,
    drift = 0.2
)
df = rw.generate(steps, processes, seed=21)
df.plot(legend=False)

rw = Brownian( initial_condition=100, ti=0, drift = 0.2 ) df = rw.generate(steps, processes, seed=21) df.plot(legend=False)

Out[18]:

<AxesSubplot:>

Try different generator parameters

In [20]:

rw = Brownian(
    initial_condition=100,
    ti=0,
    drift = 0.2,
    generator=ProbVar(dist='norm', kw={'loc': 0, 'scale': 3})
)
df = rw.generate(steps, processes, seed=21)
df.plot(legend=False)

rw = Brownian( initial_condition=100, ti=0, drift = 0.2, generator=ProbVar(dist='norm', kw={'loc': 0, 'scale': 3}) ) df = rw.generate(steps, processes, seed=21) df.plot(legend=False)

Out[20]:

<AxesSubplot:>

In [21]:

df.iloc[-5:,2:4].to_dict()

df.iloc[-5:,2:4].to_dict()

Out[21]:

{2: {495: 232.45683887750448,
  496: 232.13758213986907,
  497: 226.4127500997084,
  498: 230.31191895112065,
  499: 231.53885761432832},
 3: {495: 188.09406097045363,
  496: 187.18454346136707,
  497: 184.99190959455308,
  498: 187.43233147587208,
  499: 186.18314679682655}}

Add dates¶

In [6]:

rw = Brownian(
    initial_condition=100,
    ti=date(2021,1,1,),
    freq_output='D',
    drift = 0.2,
    generator=ProbVar(dist='norm', kw={'loc': 0, 'scale': 3})
)
df = rw.generate(steps, processes)
df.plot(legend=False)

rw = Brownian( initial_condition=100, ti=date(2021,1,1,), freq_output='D', drift = 0.2, generator=ProbVar(dist='norm', kw={'loc': 0, 'scale': 3}) ) df = rw.generate(steps, processes) df.plot(legend=False)

Out[6]:

<AxesSubplot:>

In [7]:

x1 = Brownian(initial_condition=100,drift=0.5, freq_input='A')

fig,ax = plt.subplots(figsize=(12,7))
df1 = x1.generate(50*12,50, freq_output='M')
df1.plot(ax=ax,legend=False)
df1a = x1.generate(50*12,3, freq_output='M',interval=0.07)
df1a.plot(ax=ax)

x1 = Brownian(initial_condition=100,drift=0.5, freq_input='A') fig,ax = plt.subplots(figsize=(12,7)) df1 = x1.generate(50*12,50, freq_output='M') df1.plot(ax=ax,legend=False) df1a = x1.generate(50*12,3, freq_output='M',interval=0.07) df1a.plot(ax=ax)

Out[7]:

<AxesSubplot:>

In [8]:

x2 = Brownian(
    initial_condition=0,
    generator = ProbVar(dist='norm',kw={'loc':0,'scale':0.2887}, seed=910821),
    drift=0.2/12,
    freq_input='M')

fig,ax = plt.subplots(figsize=(12,7))
df2 = x2.generate(50*12,10, freq_output='M')
df2.plot(ax=ax,legend=False)
df2a = x2.generate(50*12,3, freq_output='M',interval=0.07)
df2a.plot(ax=ax)

x2 = Brownian( initial_condition=0, generator = ProbVar(dist='norm',kw={'loc':0,'scale':0.2887}, seed=910821), drift=0.2/12, freq_input='M') fig,ax = plt.subplots(figsize=(12,7)) df2 = x2.generate(50*12,10, freq_output='M') df2.plot(ax=ax,legend=False) df2a = x2.generate(50*12,3, freq_output='M',interval=0.07) df2a.plot(ax=ax)

Out[8]:

<AxesSubplot:>

Geometric Brownian Motion¶

You can create a Geometric Brownian Motion in the same way

In [23]:

x3 = GeometricBrownian(
    initial_condition=80,
    generator = ProbVar(dist='norm',kw={'loc':0,'scale':0.26}, seed=9113),
    drift=0.01,
    freq_input='A')

fig,ax = plt.subplots(figsize=(12,8))
df3 = x3.generate(12,20, freq_output='A', seed=21)
df3.plot(ax=ax,legend=False)

x3 = GeometricBrownian( initial_condition=80, generator = ProbVar(dist='norm',kw={'loc':0,'scale':0.26}, seed=9113), drift=0.01, freq_input='A') fig,ax = plt.subplots(figsize=(12,8)) df3 = x3.generate(12,20, freq_output='A', seed=21) df3.plot(ax=ax,legend=False)

Out[23]:

<AxesSubplot:>

In [10]:

#x4 = Weiner(initial_condition=10,generator=stats.norm, mu=0.03/12, kw_generator={'scale':np.sqrt(np.power(0.26,2)/12)},freq_mu='A', seed=9)

x4 = GeometricBrownian(
    initial_condition=10,
    generator = ProbVar(dist='norm',kw={'loc':0,'scale':np.sqrt(np.power(0.26,2)/12)}, seed=9),
    drift=0.03/12,
    freq_input='A')

fig,ax = plt.subplots(figsize=(12,8))
df4 = x4.generate(20*12,20, freq_output='A')
df4.plot(ax=ax,legend=False)
df4a = x4.generate(20*12,3, freq_output='A',interval=0.1)
df4a.plot(ax=ax, linewidth=4)

#x4 = Weiner(initial_condition=10,generator=stats.norm, mu=0.03/12, kw_generator={'scale':np.sqrt(np.power(0.26,2)/12)},freq_mu='A', seed=9) x4 = GeometricBrownian( initial_condition=10, generator = ProbVar(dist='norm',kw={'loc':0,'scale':np.sqrt(np.power(0.26,2)/12)}, seed=9), drift=0.03/12, freq_input='A') fig,ax = plt.subplots(figsize=(12,8)) df4 = x4.generate(20*12,20, freq_output='A') df4.plot(ax=ax,legend=False) df4a = x4.generate(20*12,3, freq_output='A',interval=0.1) df4a.plot(ax=ax, linewidth=4)

Out[10]:

<AxesSubplot:>

In [11]:

df4 = x4.generate(20*12,3, freq_output='A',interval=0.1)
df4.plot()

df4 = x4.generate(20*12,3, freq_output='A',interval=0.1) df4.plot()

Out[11]:

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Mean Reversion¶

To create the mean reversion instance privide the mean m and the velocity of reversion eta eta.

In [30]:

oil_mr = MeanReversion(
    initial_condition = 66,
    ti = 0,
    generator = {'dist':'norm','kw':{'loc':0,'scale':5.13}},
    m=46.77,
    eta=0.112652,
    freq_input = 'A'
)

price_mr = oil_mr.generate(12,50, freq_output='A', seed=21)

price_mr.plot(legend=False)

oil_mr = MeanReversion( initial_condition = 66, ti = 0, generator = {'dist':'norm','kw':{'loc':0,'scale':5.13}}, m=46.77, eta=0.112652, freq_input = 'A' ) price_mr = oil_mr.generate(12,50, freq_output='A', seed=21) price_mr.plot(legend=False)

Out[30]:

<AxesSubplot:>

In [31]:

price_mr.iloc[-5:,2:4].to_dict()

price_mr.iloc[-5:,2:4].to_dict()

Out[31]:

{2: {7: 75.17152947067616,
  8: 65.04597525741526,
  9: 54.13505845726961,
  10: 55.41636945148142,
  11: 60.879395238090595},
 3: {7: 68.96120878051282,
  8: 71.31717910163093,
  9: 65.83984995324853,
  10: 62.34497287513045,
  11: 55.34394126548259}}

In [ ]: