In this project, we achieved the following four objectives:

1. We constructs two portfolios—defensive (low‐risk) and aggressive (high‐risk)—from nine US equities/ETFs using mean–variance optimization, and compares their annualized return, volatility, and maximum drawdown.

2. We tests whether the daily log‐returns of individual stocks and the constructed portfolios follow a normal distribution via D’Agostino–Pearson tests on rolling windows and after excluding outliers.

3. We analyses the behavior of the rate of change of the Black-Scholes call and put options prices with respect to time, when time processes or spot prices change.

4. We explore how a non-constant volatility (more specifically, Heston model) will affect the result of the usual delta-hedging in the setting of Black-Scholes model by simulating the underlying asset paths and option prices.