Python for Finance

Volatility, Implied Volatility, ARCH, and GARCH
Here, is the variance at time t, is the ith coefficient, is the squared error
term for the period of t-i, and q is the order of error terms. When q is 1, we have the
simplest ARCH (1) process as follows:
Simulating an ARCH (1) process
It is a good idea that we simulate an ARCH (1) process and have a better
understanding of the volatility clustering, which means that high volatility is usually
followed by a high-volatility period while low volatility is usually followed by a lowvolatility
period. The following code reflects this phenomenon:
import scipy as sp
import matplotlib.pyplot as plt
#
sp.random.seed(12345)
n=1000 # n is the number of observations
n1=100 # we need to drop the first several observations
n2=n+n1 # sum of two numbers
#
a=(0.1,0.3) # ARCH (1) coefficients alpha0 and alpha1, see Equation
(3)
errors=sp.random.normal(0,1,n2)
t=sp.zeros(n2)
t[0]=sp.random.normal(0,sp.sqrt(a[0]/(1-a[1])),1)
for i in range(1,n2-1):
t[i]=errors[i]*sp.sqrt(a[0]+a[1]*t[i-1]**2)
y=t[n1-1:-1] # drop the first n1 observations
#
plt.title('ARCH (1) process')
x=range(n)
plt.plot(x,y)
plt.show()
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