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We consider a market model of financial engineering with three factors represented by three correlated Brownian motions. The volatility of the risky asset in this model is the sum of two stochastic volatilities. The dynamic of each volatility is governed by a mean-reverting process. The first stochastic volatility of mean-reversion process reverts to the second volatility at a fast rate, while the second volatility moves slowly to a constant level over time with the state of the economy.

The double mean-reverting model by Gatheral (2008) is motivated by empirical dynamics of the variance of the stock price. This model can be consistently calibrated to both the SPX options and the VIX options. However due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time consuming methods like Monte Carlo simulation or the finite difference method.

To solve the above issue, we use the method of asymptotic expansion developed by Pagliarani and Pascucci (2017). In paper A, we study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. We calculate explicitly the asymptotic expansions of implied volatility within a parabolic region up the second order. In paper B we improve the results obtain in paper A by calculating the asymptotic expansion of implied volatility under the Gatheral model up to order three. In paper C, we perform numerical studies on the asymptotic expansion up to the second order. The Monte-Carlo simulation is used as the benchmark value to check the accuracy of the expansions. We also proposed a partial calibration procedure using the expansions. The calibration procedure is implemented on real market data of daily implied volatility surfaces for an underlying market index and an underlying equity stock for periods both before and during the COVID-19 crisis. Finally, in paper D we check the performance of the third order expansion and compare it with the previous results.

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.