Update app.py

app.py

@@ -18,6 +18,8 @@ It simulates various market scenarios to analyze the dependencies between differ
 Users can assess potential outcomes in terms of best, worst, and mean case scenarios.
 """)
 
+st.sidebar.title("Input Parameters")
+
 # Sidebar: How to Use (closed by default)
 with st.sidebar.expander("How to Use", expanded=False):
     st.write("""
@@ -64,23 +66,53 @@ if run_button and len(portfolio_tickers) == len(portfolio_weights):
     st.markdown("This section explores how the portfolio performs under simulated market drop scenarios using copula models.")
 
     with st.expander("Methodology", expanded=False):
+        st.markdown("## Simulating Market Drop Scenarios: Best, Worst, and Mean Cases")
+        #st.markdown("## Transforming Returns")
         st.markdown("""
-        **Step 1: Transforming Returns**
-        - We transform the returns to a uniform distribution using quantile transformation. 
-        - This ensures that the data fits within the range required by the copula model.
-
-        **Step 2: Fitting the Copula Model**
-        - A multivariate Student-t copula is fitted to the transformed data.
-        - The copula model captures the dependencies between the asset returns.
-
-        **Step 3: Simulating Scenarios**
-        - We simulate numerous return scenarios using the fitted copula model to assess the portfolio's risk under various market conditions.
-
-        **Step 4: Portfolio Returns Calculation**
-        - For each identified scenario, we calculate the portfolio returns using a weighted sum of the individual asset returns.
-
-        **Step 5: Visualization**
-        - We visualize the simulated portfolio returns using histograms, CDFs, and KDEs to understand potential outcomes.
+        To prepare the returns for copula modeling, we transform them to a uniform distribution using quantile transformation. 
+        This is to ensure that the data fits within the range required by the copula model.
+        """)
+        st.latex(r"""
+        u_i = \frac{\text{rank}(R_i)}{n + 1}
+        """)
+    
+        #st.markdown("## Fitting the Copula Model")
+        st.markdown("""
+        Next, we fit a multivariate Student-t copula to the transformed data. The copula model captures the dependencies between the stock returns to understand the joint behavior of the assets.
+        """)
+        st.latex(r"""
+        C_{\nu}(u_1, u_2, \ldots, u_d) = t_{\nu, \Sigma}(t_{\nu}^{-1}(u_1), t_{\nu}^{-1}(u_2), \ldots, t_{\nu}^{-1}(u_d))
+        """)
+    
+        #st.markdown("## Simulating Scenarios")
+        st.markdown("""
+        Using the fitted copula model, we simulate a large number of return scenarios. This simulation helps us explore potential future outcomes 
+        and assess the portfolio's risk under various market conditions.
+        """)
+    
+        #st.markdown("## Market Drop Scenarios")
+        st.markdown("""
+        We identify scenarios where the specified ticker drops by the given percentage. By analyzing these scenarios, we can evaluate how the portfolio performs under stress.
+        """)
+    
+        #st.markdown("## Portfolio Returns")
+        st.markdown("""
+        For each identified scenario, we calculate the portfolio returns. The portfolio return is a weighted sum of the individual stock returns.
+        """)
+        st.latex(r"""
+        R_p = \sum_{i=1}^{n} w_i R_i
+        """)
+    
+        #st.markdown("## Visualizing Results")
+        st.markdown("""
+        We visualize the simulated portfolio returns using histograms, cumulative distribution functions, and kernel density estimates. 
+        These visualizations help us understand the distribution and characteristics of potential returns.
+        """)
+    
+        #st.markdown("## Portfolio Price Trajectory")
+        st.markdown("""
+        Finally, we visualize the portfolio's price trajectory under the worst-case, best-case, and mean scenarios. 
+        This helps in understanding the potential impact of market drops on the portfolio value.
         """)
 
     # Define the portfolio and allocation