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This thesis presents advancements in the valuation and modeling of financial derivatives, with a focus on American and Bermudan options. Traditional models such as Black–Scholes assume constant volatility, often leading to inaccurate pricing during periods of high market turbulence. This research addresses these limitations by considering more flexible regime–switching and stochastic volatility models.

The first part of the thesis focuses on the pricing of American options under a Markovian regime–switching model, extending previous works by addressing assumptions of asset returns across economic states. By incorporating the Totally Positive of Order 2 (TP2) property for Transition Probability Matrix (TPM) and Conditional Probability Matrix (CPM), the model ensures probabilistic progression between economic states. Extensive numerical experiments confirm the importance of TPM in maintaining the monotonicity of optimal exercise boundaries.

Secondly, the thesis investigates asymptotic expansions of implied volatility under the Gatheral model. Numerical analysis reveals the accuracy of first and second–order expansions, with a partial calibration method validated using market data from the COVID–19 pandemic.

Next, the thesis introduces a Backward Stochastic Differential Equation (BSDE)–θ scheme for pricing American options under the Heston model, simplifying the computational process and requiring only one parameter for pricing while also deriving schemes for Delta and Vega hedging strategies. Extensive numerical experiments validate the scheme’s accuracy and robustness, especially for in–the–money options.

Finally, the thesis develops an Almost–Exact Simulation (AES) scheme for Bermudan and American option pricing under Heston–type models. The AES scheme ensures non–negative variance and significantly improves simulation accuracy compared to the Euler scheme when the number of steps equals the number of exercise dates. Numerical experiments reveal that the AES scheme offers improvements in accuracy, efficiency, and memory usage, particularly for in–the–money and at–the–money options with minimal time steps.