Python for Finance

Monte Carlo Simulation
Efficiency, Quasi-Monte Carlo, and Sobol
sequences
When applying the Monte Carlo simulation to solve various finance-related
problems, a set of random numbers is generated. When the accuracy is very high,
we have to draw a huge amount of such random numbers. For example, when
pricing options, we use very small intervals or a large number of steps to increase
the accuracy of our solutions. Thus, the efficiency of our Monte Carlo simulation
would be a vital issue in terms of computational time and costs. This is especially
true if several thousand options are to be priced. One way to increase the efficiency is
to apply a better algorithm, that is, optimize our codes. Another way is to use some
special types of random numbers that are more evenly distributed. This is called
Quasi-Monte Carlo Simulation. A typical example is a so-called Sobol sequence.
Sobol sequences belong to the so-called low-discrepancy sequences, which satisfy the
properties of random numbers, but are distributed more evenly:
import numpy as np
import matplotlib.pyplot as plt
np.random.seed(12345)
n=200
a = np.random.uniform(size=(n*2))
plt.scatter(a[:n], a[n:])
plt.show()
The related graph is shown on the left panel:
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