In [55]:
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
In [56]:
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.
In [57]:
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 [58]:
t,W=brownian(1.0,1000)
raise NotImplementedError()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-58-9bac129aca11> in <module>()
1 t,W=brownian(1.0,1000)
----> 2 raise NotImplementedError()
NotImplementedError:
In [42]:
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 [59]:
plt.plot(t,W)
plt.xlabel('t')
plt.ylabel('W')
raise NotImplementedError()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-59-bcf349eafa8b> in <module>()
2 plt.xlabel('t')
3 plt.ylabel('W')
----> 4 raise NotImplementedError()
NotImplementedError:
In [ ]:
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 [96]:
dW=np.diff(W)
mean=dW.mean()
StDv=dW.std()
print('dW=',dW)
print('mean=',mean)
print('Deviation=',StDv)
raise NotImplementedError()
('dW=', array([ 1.33295265e-02, 6.67255678e-04, 2.92559285e-04,
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('mean=', 0.00032194680549929409)
('Deviation=', 0.031138707152554235)
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-96-457ac25a3a56> in <module>()
5 print('mean=',mean)
6 print('Deviation=',StDv)
----> 7 raise NotImplementedError()
NotImplementedError:
In [48]:
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 [97]:
def geo_brownian(t, W, X0, mu, sigma):
"Return X(t) for geometric brownian motion with drift mu, volatility sigma."""
X=np.zeros((np.size(t),1))
for i in range(np.size(t)):
X[i]=X0*2.718**(((mu-sigma**2)/2)*(t[i]+sigma*W[i]))
return X
raise NotImplementedError()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-97-55a3e140fa86> in <module>()
6 return X
7
----> 8 raise NotImplementedError()
NotImplementedError:
In [65]:
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 [100]:
X=geo_brownian(t,W,1.0,.5,.3)
plt.xlabel('t')
plt.ylabel('X')
plt.plot(t,X)
raise NotImplementedError()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-100-1b75fc1c8cec> in <module>()
3 plt.ylabel('X')
4 plt.plot(t,X)
----> 5 raise NotImplementedError()
NotImplementedError:
In [ ]:
assert True # leave this for grading
Content source: rvperry/phys202-2015-work
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