## Instructions

### Objective

Write a Python assignment program to solve probability questions in Python. You are tasked with creating a program that calculates probabilities based on given inputs. Utilize Python's mathematical libraries to handle complex probability calculations efficiently. Your assignment should showcase your understanding of fundamental probability concepts and your ability to translate them into functional code. Ensure your program is well-documented, with clear explanations of the implemented logic and any assumptions made. This assignment will not only assess your Python programming skills but also your grasp of probability theory. Keep the code concise, organized, and thoroughly tested to demonstrate your proficiency.

## Requirements and Specifications

**Source Code**

```
import random
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import math
def Tpath(T, sigma, p, x0):
"""
QUESTION 1: Simulate the path of T
"""
# Create the list with the initial value
x = np.zeros(T+1)
# Now, simulate
for t in range(1,T+1):
# Generate epsilon
prob = random.uniform(0,1)
epsilon = 1 if prob <= p else -1
# Generate value
xt = x[t-1] + sigma*epsilon
# Append
x[t] = xt
# Return
return x
def stock_betas(filename):
# First, open file
data = pd.read_excel(filename, engine='openpyxl', skiprows = 1)
sp500 = data.iloc[:,1].to_numpy()
P = data.iloc[:,2:].to_numpy()
# Now for each stock calculate all RTs and beta
betas = []
for i in range(P.shape[1]):
prices = P[:,i]
x = []
y = []
xy = []
xx = []
# Calculate RTs
for t in range(1,len(prices)):
Pt = prices[t]
Ptold = prices[t-1]
if not math.isnan(Pt) and not math.isnan(Ptold) and Pt >= 0 and Ptold > 0:
Rt = (Pt-Ptold)/Ptold
y.append(Rt)
dsp500 = (sp500[t]-sp500[t-1])/sp500[t-1]
x.append(dsp500)
xy.append(Rt*dsp500)
xx.append(dsp500**2)
# Now calculate
beta = (len(x) * sum(xy) - sum(x) * sum(y)) / (len(x)*sum(xx) - sum(x) ** 2)
betas.append(beta)
return betas
if __name__ == '__main__':
# Question 2
T = 120
sigma = 0.6
p = 0.5
x0 = 10
# Run the simulation of 1,000,000 cases
prob = 0.0
N = 10000
for _ in range(N):
x = Tpath(T, sigma, p, x0)
# Pick the number of times that x is less than x0
prob += len(np.where(x <= 0.75*x0)[0])/(T+1)
prob = prob/N
print("The probability that xt drops at least 25% below its initial value at least once is {:.4f}%".format(prob*100.0))
""" Question 3 """
# generate a new path
x = Tpath(T, sigma, p, x0)
plt.figure()
plt.plot(range(T+1), x)
plt.xlabel('Time (month)')
plt.ylabel('Probability')
plt.title('Probability vs. Time')
plt.grid(True)
plt.show()
""" Question 4 """
betas = stock_betas('stockprices2020b.xlsx')
print(betas)
```

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