Options Futures and Other Derivatives 8th eDITION
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chapter 13.Wiener Process and Ito Lemma 13.1 the markov property
a type of stochastic process,only time
13.2 CONTINUOUS-TIME STOCHASTIC PROCESSES
Wiener Process(Brownian Motion)
import pandas as pd import numpy as np import statsmodels .api as sm import scipy .stats as sci import matplotlib .pyplot as plt # dz = e * (dt**0.5) T = 1000 dT = 1 k = dT ** 0.5 N = T / dT port_er = []port_dT = []cer = 0 cdT = 0 for p in range (T ): er = np .random .randn (1 )# sigma*np.random.randn()+mu产生sigma&mu的分布 cer = cer + er * k cdT = cdT + dT port_er .append (cer )port_dT .append (cdT )port_er = np .array (port_er )port_dT = np .array (port_dT )plt .figure (figsize = (8 ,4 ))# 整体区域大小 plt .scatter (port_dT ,port_er ,c = port_er ,marker = 'o' )# 按c分成不同的颜色，五颜六色；marker是点的类型，o是小圆圈 plt .grid (True )# 网格 plt .xlabel ('dT' )#累积时间,dT plt .ylabel ('Z' )#累积Z,port_e plt .colorbar (label = 'Z' )Generalized Wiener Process
dx = adt + b dz
T = 1000 dT = 1 k = dT ** 0.5 N = T / dT port_er = []port_dT = []port_x = []port_y = []cer = 0 cdT = 0 a = 0.03 b = 1.5 cx = 0 cy = 0 for p in range (T ): er = np .random .randn (1 )# sigma*np.random.randn()+mu产生sigma&mu的分布 cer = cer + er * k cdT = cdT + dT cx = cx + a * dT + b * er * k cy = cy + a * dT port_er .append (cer )port_dT .append (cdT )port_x .append (cx )port_y .append (cy )port_er = np .array (port_er )port_dT = np .array (port_dT )port_x = np .array (port_x )port_y = np .array (port_y )plt .figure (figsize = (8 ,4 ))# 整体区域大小 plt .scatter (port_dT ,port_er ,c = port_er ,marker = 'o' )# 按c分成不同的颜色，五颜六色；marker是点的类型，o是小圆圈 plt .scatter (port_dT ,port_x ,c = port_er ,marker = 'o' )plt .scatter (port_dT ,port_y ,c = port_er ,marker = 'o' )plt .grid (True )# 网格 plt .xlabel ('dT' )#累积时间,dT plt .ylabel ('Z' )#累积Z,port_e plt .colorbar (label = 'Z' )ps.1.2.q world vs. p world