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In this paper, a numerical forward-backward stochastic differential equations (FBSDEs) approach is used for pricing American options under the Heston model. The motivation behind this approach lies in Heston model’s ability to capture significant market features, including stochastic volatility and the correlation between the underlying asset and the volatility. Yet, to the best of our knowledge, there is a lack of dedicated investigation on American option pricing using FBSDE approach under this model. In the present study, the FBSDE representation is obtained and utilized to develop a numerical scheme, which enables the approximation of option prices and hedge ratios. The primary focus lies in approximating American option price as the solution to the backward stochastic differential equation (BSDE) within the FBSDE representation using a θ discretization, which we name as the BSDE-θ Scheme. This BSDE-θ scheme offers computational advantages and flexibility for handling both lower and higher dimensions. Finally, numerical experimental studies are conducted to assess the accuracy and stability of the scheme across various choices of θ. Additionally, the impact of the choice of basis functions in the form of global polynomials on the estimation of conditional expectation functions is investigated. The findings indicate that the choice of θ influences the accuracy of the numerical scheme in particular cases. Moreover, the choice of basis functions under consideration does not exhibit any observable impact on the 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.

In this chapter, we consider the problem of American option pricing when the asset price dynamics follow a binomial model dominated by a varying economic situation. We assume that the economic situation transits based on a semi-Markov chain. The pricing procedure is formulated using a semi-Markov decision process, as the decision-maker decides whether to exercise early or hold the option. Some properties of the optimal exercise regions and the monotonicity of option prices are discussed based on simulation results when each parameter is changed, and the optimal strategies are investigated.

The Gatheral model is a three factor model with mean-reverting stochastic volatility that reverts to a stochastic long run mean. This chapter reviews previous analytical results on the first and second order implied volatility expansions under this model. Using the Monte Carlo simulation as the benchmark method, numerical studies are conducted to investigate the accuracy and properties of these analytical expansions. The classical Black–Scholes option pricing model assumes that the underlying asset follows a geometric Brownian motion with constant volatility. The chapter discusses partial calibration procedure is proposed and synthetic and real data calibration. If a full calibration is desired, we can use the results from the partial calibration as inputs for the final local optimization over all model parameters. In implementing the calibration procedure, the effect of the Covid-19 pandemic on the model calibration is high.

The Gatheral double stochastic volatility model is a three-factor model with mean-reverting stochastic volatility that reverts to a stochastic long-run mean. Our previous paper investigated the performance of the first and second-order implied volatilities expansions under this model. Moreover, a simple partial calibration method has been proposed. This paper reviews and extends previous results to the third-order implied volatility expansions under the same model. Using Monte-Carlo simulation as the benchmark method, extensive numerical studies are conducted to investigate the accuracy and properties of the third-order expansion. 

In this research, we consider the pricing of American options when the underlying asset is governed by the Markovian regime-switching process. We assume that the price dynamics depend on the economy, the state of which transits based on a discrete-time Markov chain. The underlying economy cannot be known directly but can be partially observed by receiving a signal stochastically related to the real state of the economy. The pricing procedure and optimal stopping problem are formulated using a partially observable Markov decision process. Some structural properties of the American option prices are derived under certain assumptions. These properties establish the existence of a monotonic optimal exercise policy with respect to the holding time, asset price, and economic conditions. 

In this article, a model under which the underlying asset follows a Markov regime-switching process is considered. The underlying economy is partially observable in a form of a signal stochastically related to the actual state of the economy. The American option pricing problem is formulated using a partially observable Markov decision process (POMDP). Through the article, a three-state economy is assumed with a focus on the threshold for the early exercise, hold regions, and its monotonicity. An extensive numerical experimental study is conducted in order to clarify the relationship between the monotonicity of the exercising strategy and the sufficient conditions which are obtained in Jin, Dimitrov, and Ni. In this article, the effect of sufficient conditions is confirmed. It was shown that sufficient conditions are not necessary for the monotonicity of the exercising strategy, and a discussion including milder conditions is presented based on the numerical studies.

Recently, an Almost-Exact Simulation (AES) scheme was introduced for the Heston stochastic volatility model and tested for European option pricing. This paper extends this scheme for pricing Bermudan and American options under both Heston and double Heston models. The AES improves Monte Carlo simulation efficiency by using the non-central chi-square distribution for the variance process. We derive the AES scheme for the double Heston model and compare the performance of the AES schemes under both models with the Euler scheme. Our numerical experiments validate the effectiveness of the AES scheme in providing accurate option prices with reduced computational time, highlighting its robustness for both models. In particular, the AES achieves higher accuracy and computational efficiency when the number of simulation steps matches the exercise dates for Bermudan options.