Update app.py

app.py

@@ -110,6 +110,7 @@ def plot_price_with_cones(data, bootstrap_percentiles, days, thresholds, bootstr
     return fig
 
 # Streamlit app
+st.set_page_config(layout="wide")
 st.title('Stock Price Simulation')
 
 st.sidebar.header('Input Parameters')
@@ -120,6 +121,28 @@ This application simulates future stock prices using bootstrapping simulation me
 You can specify the stock ticker, the date range, the number of simulation days, the number of simulations, and price thresholds.
 The simulation results will show the probability of the stock price falling below, between, or above the specified thresholds.
 
+**Background and Concept**
+
+The concept of bootstrapping was introduced by Bradley Efron in 1979. The primary goal of bootstrapping is to understand the variability of a statistic by generating multiple samples from the observed data. This approach assumes that the sample data represents the population, allowing us to draw inferences about the population from the sample.
+
+**Steps in Bootstrapping:**
+
+Given a dataset \( X = \{x_1, x_2, ..., x_n\} \), we aim to estimate the statistic \( \theta \) (e.g., the mean return of a stock).
+
+1. **Resampling**: Create a resample \( X^* \) by drawing \( n \) observations from \( X \) with replacement. This means that each data point can be selected multiple times in a single resample:
+""")
+st.latex(r'X^* = \{x_1^*, x_2^*, ..., x_n^*\}')
+st.write("""
+2. **Statistic Calculation**: Calculate the statistic \( \theta^* \) for the resample \( X^* \):
+""")
+st.latex(r'\theta^* = f(X^*)')
+st.write("""
+3. **Repeat**: Repeat the above steps \( B \) times to generate \( B \) bootstrap statistics:
+""")
+st.latex(r'\{\theta_1^*, \theta_2^*, ..., \theta_B^*\}')
+st.write("""
+4. **Estimate**: Use the bootstrap statistics to estimate the mean, standard error, and confidence intervals of \( \theta \).
+
 **How to use:**
 1. Enter the stock ticker, start date, and end date.
 2. Set the number of days for the simulation and the number of iterations.
@@ -138,7 +161,7 @@ end_date = st.sidebar.date_input('End Date', pd.to_datetime('2025-01-01'))
 days = st.sidebar.number_input('Number of Days for Simulation', min_value=1, max_value=365, value=30)
 n_iterations = st.sidebar.number_input('Number of Simulations', min_value=100, max_value=100000, value=10000)
 threshold1 = st.sidebar.number_input('Threshold 1', min_value=0, value=850)
-threshold2 = st.sidebar.number_input('Threshold 2', min_value=0, value=1000)
+threshold2 = st.sidebar.number_input('Threshold 2', min_value=0, value=1050)
 thresholds = [threshold1, threshold2]
 
 if st.sidebar.button('Run Simulation'):
@@ -175,4 +198,6 @@ if st.sidebar.button('Run Simulation'):
       - The probability of the stock price being below Threshold 1: {bootstrap_probabilities["below"]:.2%}
       - The probability of the stock price being between Threshold 1 and Threshold 2: {bootstrap_probabilities["between"]:.2%}
       - The probability of the stock price being above Threshold 2: {bootstrap_probabilities["above"]:.2%}
+
+    These results help in understanding the potential future movements of the stock price based on historical data and bootstrapping simulation.
     """)