A Mathematical Approach to Financial Risk Analysis

Financial markets are characterized by uncertainty, volatility clustering, heavy-tailed return distributions, and complex dependence structures, posing significant challenges for effective risk measurement and management. Traditional risk assessment approaches based on variance and linear correlation often fail to capture extreme losses and systemic risk, particularly during periods of market stress. This paper proposes a comprehensive mathematical framework for financial risk analysis that integrates stochastic return modeling, time-varying volatility dynamics, copula-based dependence structures, and coherent risk measures. Asset returns are modeled as stochastic processes with conditional Non-constant variance using GARCH-type models to account for volatility persistence. Dependence among financial assets is captured through copula functions, enabling flexible modeling of nonlinear and tail dependence beyond conventional correlation measures. Financial risk is quantified using Value at Risk and Conditional Value at Risk, with emphasis on CVaR due to its coherence and effectiveness in assessing extreme downside risk. Portfolio optimization is formulated as a convex mean–CVaR optimization problem, providing robust asset allocations under risk constraints. Monte Carlo simulation and stress testing are employed to evaluate portfolio performance under adverse market scenarios.The proposed framework offers a unified and mathematically rigorous approach to financial risk analysis, with practical relevance for portfolio management, risk control, and regulatory applications in complex and volatile financial environments.