Numpy Exercise 3

Imports



In [79]:

    
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns



In [80]:

    
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.



In [81]:

    
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.



In [82]:

    
# YOUR CODE HERE
#raise NotImplementedError()
a = brownian(1.0,1000)
t = a[0]
W = a[1]



In [83]:

    
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.



In [84]:

    
# YOUR CODE HERE
#raise NotImplementedError()
plt.plot(t,W)
plt.title('Wiener Process Simulation')
plt.xlabel('Time')
plt.ylabel('W(t)')









    Out[84]:





<matplotlib.text.Text at 0x7f50755c3f90>



In [85]:

    
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.



In [86]:

    
# YOUR CODE HERE
#raise NotImplementedError()
dW = np.diff(W)
dW.mean(), dW.std()









    Out[86]:





(0.00016663495363132113, 0.032128921029840628)



In [87]:

    
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 [88]:

    
def geo_brownian(t, W, X0, mu, sigma):
    "Return X(t) for geometric brownian motion with drift mu, volatility sigma."""
    # YOUR CODE HERE
    #raise NotImplementedError()
    Result = X0*np.exp((mu - 0.5*sigma**2)*t + (sigma*W))
    return Result



In [89]:

    
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.



In [90]:

    
# YOUR CODE HERE
plt.plot(t , geo_brownian(t, W, 1.0, 0.5, 0.3))
plt.title('Geometric Brownian Motion Simulation')
plt.xlabel('time')
plt.ylabel('X(t)')
#raise NotImplementedError()









    Out[90]:





<matplotlib.text.Text at 0x7f50754f7ed0>



In [91]:

    
assert True # leave this for grading