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import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
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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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t, W = brownian(1.0,1000)
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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.xlabel('t')
plt.ylabel('W(t)')
#lookin' good
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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)
def mean(dW):
return dW.cumprod()/len(dW)
def standard_deviation(dW):
mean(dW)
variance = 0
for i in dW:
variance += (mean(dW) - i) ** 2
return variance/float(len(dW))** 0.5
#used format from codecademy
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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.
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def geo_brownian(t, W, X0, mu, sigma):
"Return X(t) for geometric brownian motion with drift mu, volatility sigma."""
brownian(t, W)
t, W = brownian(t, W)
Xt = X0 * np.exp(np.multiply(int(mu) - ((int(sigma) ** 2)/2),t) + ((sigma) * brownian(t,W)))
return Xt
#I have played with this so much, I don't know what it is stopping it from working. Part of it could be that I don't know
# when to use numpy ufuncs and when to use our regular addition and subtraction, etc. It would help if talked more about
# these things in class before having homework due with them.
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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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a = geo_brownian(1.0, 1000, 1.0, 0.5, 0.3)
plt.plot(t,a)
#This would work if my previous step worked
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assert True # leave this for grading