[SOLVED] Question 1.

Question 1.

Please use Numpy to generate the following ndarray.

1 2 3 4 5 6 7 8 9 10

10 9 8 7 6 5 4 3 2 1

1 3 5 7 9 11 13 15 17 19

2 4 6 8 10 12 14 16 18 20

1 1 2 3 5 8 13 21 34 55

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1 0 0

Tip: you may use the numpy.vstack function

Question 2.

Import the provided stock data into a DataFrame and use Matplotlib to plot the daily K bar with the

dataframe.

Tips: For each K bar, only the open, high, low, close price need to be considered in this question.

Question 3.

Please use Numpy to generate random numbers and use Monte Carlo method to calculate

(3.1415926)

Tip: please read the appendix to understand the Monte Carlo method to calculate .

Appendix:
We start the familiar example of finding the area of a circle. Figure 1 below shows a circle with

radiusinscribed within a square. The area of the circle is , and the area of the

square is .The ratio of the area of the circle to the area of the square is

Figure 1.

If we could compute ratio, then we could multiple it by four to obtain the value .One particularly

simple way to do this is to pick lattice points in the square and count how many of them lie inside the

circle, see Figure 2.Suppose for example that the pointsare
selected, then there are 812 points inside the circle and 212 points outside the circle and the percentage

of points inside the circle is .Then the area of the circle is
approximated with the following calculation

Figure 2.

Monte Carlo Method for

Monte Carlo methods can be thought of as statistical simulation methods that utilize a sequences of

random numbers to perform the simulation.The name Monte Carlo was coined by Nicholas

Constantine Metropolis (1915-1999) and inspired by Stanslaw Ulam (1909-1986), because of the
similarity of statistical simulation to games of chance, and because Monte Carlo is a center for gambling

and games of chance.In a typical process one compute the number of points in a set A that lies inside

box R.The ratio of the number of points that fall inside A to the total number of points tried is equal to

the ratio of the two areas (or volume in 3 dimensions).The accuracy of the ratiodepends on the

number of points used, with more points leading to a more accurate value.

A simple Monte Carlo simulation to approximate the value ofcould involve randomly selecting

pointsin the unit square and determining the ratio , whereis number of points

that satisfy .In a typical simulation of sample sizethere were points satisfying

, shown in Figure 3.Using this data, we obtain

and

http://scienceworld.wolfram.com/biography/Metropolis.html
http://scienceworld.wolfram.com/biography/Metropolis.html
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ulam.html

Figure 3.

Every time a Monte Carlo simulation is made using the same sample sizeit will come up with a

slightly different value.The values converge very slowly of the order .This property is a

consequence of the Central Limit Theorem.