This project investigates topics in quantitative finance through four integrated studies. High- and low-risk equity portfolios are optimized for maximum Sharpe ratio under strict constraints on beta, volatility, and drawdown. The normality of log returns across 20 major equities is empirically tested, revealing that global normality rarely holds, but may emerge locally or after outlier removal. Option sensitivities—theta and delta—are analyzed using the Black-Scholes model, highlighting key regions of time decay and price risk. Hedging strategies are further evaluated under both constant and stochastic volatility models (Heston, GARCH), demonstrating that hedging effectiveness is highly sensitive to volatility dynamics. Overall, the findings emphasize the importance of empirical validation, robust modeling, and adaptive strategies in modern financial practice.