Python for Finance

Chapter 12
Using simulation to estimate the pi value
It is a good exercise to estimate π value by simulation. Let's draw a square with 2R as
its side. If putting the largest circle inside the square, its radius will be R, described
by the following equation:
On the other hand, the square is the product of its sides:
Dividing Equation (4) by Equation (5), we have the following result:
Reorganize it; we end up with the following equation:
In other words, the value of π will be 4* Scircle/Square. When running the simulation,
we generate n pairs of x and y from a uniform distribution with a range of zero and
0.5. Then we estimate a distance that is the square root of the summation of the
squared x and y, that is, .
Obviously, when d is less than 0.5 (value of R), it will fall into the circle. We can
imagine throwing a dart that falls into the circle. The value of the pi will take the
following form:
[ 429 ]