Numpy Exercise 3

Imports



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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

Geometric Brownian motion

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 = np.array(brownian(1,1000))



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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)









    Out[58]:





[<matplotlib.lines.Line2D at 0x7f67c8fcc650>]



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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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#Standard numpy functions
dW = np.array(np.diff(W))
dW.mean()
dW.var(), dW.std()









    Out[59]:





(0.0010415154621963086, 0.03227251868380137)



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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."""
    #Plug into the equation above
    X = X0 * np.exp((mu-sigma**2/2)**t + sigma * W)
    return X



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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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#Standard Plot
plt.plot(geo_brownian(t,W,1.0,0.5,0.3))









    Out[61]:





[<matplotlib.lines.Line2D at 0x7f67c8efe490>]



In [ ]:

    
assert True # leave this for grading