This submission features projects that applied quantitative tools to address problems in financial modeling and risk management. The first project focused on constructing high- and low-risk portfolios using quadratic optimization. Results demonstrated strong in-sample performance with expected degradation during backtesting highlighting the importance of robust validation. The second project investigated the assumption of normality in financial data. Through rolling Shapiro-Wilk tests on log returns, transient periods of normality were found. We demonstrated how these periods can be exploited to build portfolios with some evidence of normality in small time windows. The third project analyzed the sensitivity of Black-Scholes price of option on time to expiration and spot price. Finally, the fourth project simulated delta hedging in stochastic volatility environments showing that while hedging reduces variance it cannot eliminate risk exposure entirely unlike the idealized Black-Scholes