In [2]:
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
In [3]:
import antipackage
import github.ellisonbg.misc.vizarray as va
Here is a function that produces standard Brownian motion using NumPy. This is also known as a Wiener Process.
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def brownian(maxt, n):
"""Return one realization of a Brownian (Wiener) process with n steps and a max time of t."""
t = np.linspace(0.0,maxt,n)
h = t[1]-t[0]
Z = np.random.normal(0.0,1.0,n-1)
dW = np.sqrt(h)*Z
W = np.zeros(n)
W[1:] = dW.cumsum()
return t, W
Call the brownian
function to simulate a Wiener process with 1000
steps and max time of 1.0
. Save the results as two arrays t
and W
.
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brownian(1.0, 1000)
t=brownian(1.0, 1000)[0]
W=brownian(1.0, 1000)[1]
np.savez('brownian.npz', t, W)
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assert isinstance(t, np.ndarray)
assert isinstance(W, np.ndarray)
assert t.dtype==np.dtype(float)
assert W.dtype==np.dtype(float)
assert len(t)==len(W)==1000
Visualize the process using plt.plot
with t
on the x-axis and W(t)
on the y-axis. Label your x and y axes.
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plt.plot(t, W)
plt.ylabel("Wiener Process")
plt.xlabel("Time")
Out[43]:
In [44]:
assert True # this is for grading
Use np.diff
to compute the changes at each step of the motion, dW
, and then compute the mean and standard deviation of those differences.
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dW = np.diff(W)
print dW.mean()
print dW.std()
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assert len(dW)==len(W)-1
assert dW.dtype==np.dtype(float)
Write a function that takes $W(t)$ and converts it to geometric Brownian motion using the equation:
$$ X(t) = X_0 e^{((\mu - \sigma^2/2)t + \sigma W(t))} $$Use Numpy ufuncs and no loops in your function.
In [47]:
def geo_brownian(t, W, X0, mu, sigma):
"Return X(t) for geometric brownian motion with drift mu, volatility sigma."""
t = np.linspace(0.0, 1.0, 1000)
raised = ((mu - sigma**2/2)*t + sigma*W)
expfunc = np.exp(raised)
X = X0*expfunc
return t, X
In [48]:
assert True # leave this for grading
Use your function to simulate geometric brownian motion, $X(t)$ for $X_0=1.0$, $\mu=0.5$ and $\sigma=0.3$ with the Wiener process you computed above.
Visualize the process using plt.plot
with t
on the x-axis and X(t)
on the y-axis. Label your x and y axes.
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t = geo_brownian(1.0, W, 1.0, 0.5, 0.3)[0]
X = geo_brownian(1.0, W, 1.0, 0.5, 0.3)[1]
plt.plot(t,X)
plt.xlabel("Time")
plt.ylabel("Geometric Brownian Motion")
Out[49]:
In [37]:
assert True # leave this for grading