Source code:

import numpy as np

import matplotlib.pyplot as plt

import seaborn

seaborn.set_style("whitegrid")

N = 100

X = np.random.choice([-1,1],N)

X[0]=0

RW = np.cumsum(X)

plt.figure(figsize=(19.2,10.8))

plt.plot(RW,label='Random Walk')

plt.legend()

Geometric simple random walk

Source code:

import numpy as np

import matplotlib.pyplot as plt

import seaborn

seaborn.set_style("whitegrid")

N = 2 ** 7

n = np.arange(0, N)

X = np.random.choice([-1,1],N)

X[0]=0

S0=200

RW = np.cumsum(X)

GRW=S0*np.exp(RW/100)

plt.figure(figsize=(19.2,10.8))

plt.plot(n,GRW,label='Geometric Random Walk')

plt.legend()

Source code:

import numpy as np

import matplotlib.pyplot as plt

from pylab import *

import japanize_matplotlib

T = 1

N = 2 ** 20

dt = T / N

t = np.arange(0, T, dt)

dW = np.sqrt(dt) * np.random.randn(N)

dW[0] = 0

W = np.cumsum(dW)

plt.figure(figsize=(38.4,21.6))

plt.grid(True)

plt.title(r'ブラウン運動のサンプルパス$\{W_t\}_{t\in[0,1]}$',fontsize=60,y=1.025)

mpl.rcParams['axes.xmargin'] = 0.0

plt.xlabel("$t$",fontsize=60)

plt.ylabel("$W_t$",fontsize=60)

plt.yticks(fontsize=60)

plt.xticks(fontsize=60)

plt.plot(t, W, color='royalblue',lw=0.5)

plt.savefig('bm.pdf',transparent=True)

plt.show()

Source code:

import numpy as np

import matplotlib.pyplot as plt

from pylab import *

T = 1

S0 = 1.14514

sigma = 1.14514

mu = 0.810

N = 2 ** 15

dt = T / N

t = np.arange(0, T, dt)

dW = np.sqrt(dt) * np.random.randn(N)

dW[0] = 0

W = np.cumsum(dW)

GW = S0 * np.exp(mu * t -(1/2) * (sigma ** 2) * t + sigma * W)

plt.figure(figsize=(19.2,10.8))

plt.xlabel("$t$")

plt.ylabel("$S_t$")

plt.plot(t, GW, color='m')

title('A sample path of an exponential Brownian motion: $dS_t=\mu S_t dt+ \sigma S_t dW_t, \mu=0.810,\sigma=1.14514, S_0=1.14514,t\in [0,1].$')

grid(True)

plt.show()

plt.savefig('GBM.pdf')

source code:

import numpy as np

import matplotlib.pyplot as plt

from pylab import *

sigma = 0.810

mu = 0.05

S0 = 114.514

T = 1

N = 2**22

dt = T / N

gamma=0.60

t = np.arange(0.0, T, dt)

dW = np.sqrt(dt) * np.random.randn(N)

dW[0] = 0

W = np.cumsum(dW)

Sem = np.zeros(N)

Stemp = S0

Sem[0] = Stemp

for j in range(1, N):

Stemp = Stemp + dt * mu * (Stemp) + sigma * pow(Stemp,gamma) * dW[j]

Sem[j] = Stemp

plt.figure(figsize=(19.2,10.8))

title('$dS_t=\mu S_t dt+ \sigma S_t^{\gamma} dW_t, \mu=0.05,\sigma=0.810,\gamma=0.60, S_0=114.514,t\in [0,1].$')

grid(True)

plt.plot(t, Sem, color='b')

plt.xlabel("$t$")

plt.ylabel("$S_t$")

plt.show()