Asian Journal of Pure and Applied Mathematics

Accurately simulating systems driven by random fluctuations is essential in diverse scientific domains, including financial forecasting and physical modeling. This study conducts a thorough comparative analysis of two foundational numerical methods the Euler Maruyama and Milstein schemes for solving stochastic differential equations. Going beyond conventional benchmark models, we implement these techniques within the sophisticated Heston stochastic volatility framework, which better reflects real world financial market dynamics, and further extend it to multi asset environments with correlated processes. A central innovation is our application of a ”Full Truncation” technique to preserve the plausibility and positivity of volatility paths during simulation. Our methodology is structured in three phases: visual assessment of simulated trajectories, rigorous statistical evaluation of convergence properties using Monte Carlo experiments, and practical benchmarking through pricing exotic financial options with path-dependent payoffs. Results unequivocally show that the Milstein method achieves higher accuracy and faster convergence, yielding more reliable prices for exotic options and reducing inherent simulation errors. Its scalability in multi asset settings underscores its applicability in complex systems. Meanwhile, the Euler method remains effective for large scale simulations prioritizing computational speed over precision. We conclude that selecting the appropriate numerical method depends on carefully balancing the need for precision against computational demands. The Milstein scheme proves essential for critical applications where accuracy is paramount, such as in risk management and hedging strategies.