# Numpy Exercise 3

## Imports



In [11]:

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




In [12]:

import antipackage
import github.ellisonbg.misc.vizarray as va




Using existing version:  github.ellisonbg.misc.vizarray



## Geometric Brownian motion

Here is a function that produces standard Brownian motion using NumPy. This is also known as a Wiener Process.



In [13]:

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 [17]:

brownian(1.0,1000)

t,W = np.array(brownian(1.0,1000))




In [18]:

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 [19]:

plt.plot(t,W)




Out[19]:

[<matplotlib.lines.Line2D at 0x7fbf614017d0>]




In [20]:

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 [25]:

dW = np.diff(W)
print (dW.mean())
print (dW.std())




0.000480726110351
0.0326604764321




In [26]:

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 [45]:

def geo_brownian(t, W, X0, mu, sigma):
"Return X(t) for geometric brownian motion with drift mu, volatility sigma."""
a = (((mu - (sigma ** 2))/2)*t) + (sigma * W(t))
y = (X0)*(np.exp(a))
return(t,y)




In [46]:

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 [ ]:

raise NotImplementedError()




In [ ]:

assert True # leave this for grading