There might also be a trade-off between compactness and readability in that this
implementation approach makes it quite difficult to grasp what exactly is going on on the
NumPy level. However, it shows how far one can go sometimes with NumPy vectorization.
Graphical Analysis
Finally, let us have a graphical look at the underlying mechanics (refer to Chapter 5 for an
explanation of the matplotlib plotting library). First, we plot the first 10 simulated paths
over all time steps. Figure 3-2 shows the output:
In [ 29 ]: import matplotlib.pyplot as plt
plt.plot(S[:, : 10 ])
plt.grid(True)
plt.xlabel(‘time step’)
plt.ylabel(‘index level’)
Figure 3-2. The first 10 simulated index level paths
Second, we want to see the frequency of the simulated index levels at the end of the
simulation period. Figure 3-3 shows the output, this time illustrating the (approximately)
log-normal distribution of the end-of-period index level values:
In [ 30 ]: plt.hist(S[- 1 ], bins= 50 )
plt.grid(True)
plt.xlabel(‘index level’)
plt.ylabel(‘frequency’)
The same type of figure looks completely different for the option’s end-of-period
(maturity) inner values, as Figure 3-4 illustrates:
In [ 31 ]: plt.hist(np.maximum(S[- 1 ] - K, 0 ), bins= 50 )
plt.grid(True)
plt.xlabel(‘option inner value’)
plt.ylabel(‘frequency’)
plt.ylim( 0 , 50000 )
