Modeling of American-style Asian option under jump-diffusion process - Laham, Mohamed Faris (2024)
This thesis addresses evaluation of American-style Asian options within a jumpdiffusion framework, an extension of the traditional Black-Scholes model to capture real-world financial market behaviors better. The primary objective of this research is to develop an evaluation framework for pricing American-style Asian options, where the strike price is dependent on the average path of the underlying asset prices. To address these challenges, this research develops a comprehensive pricing model, comparing the well-established Black-Scholes model with the Merton jump-diffusion model. Through this comparison, it is demonstrated that the Merton model offers a more accurate representation of market behaviors such as price jumps and volatility clustering. A decision rule for initial parameter estimation using maximum likelihood estimation (MLE) is proposed, affirming the Merton model’s suitability for real-world stock price behavior. Utilizing theories of conditioned expectations and martingales, the research addresses the free boundary problem associated with optimal early exercise. The Monte Carlo simulation method is adapted to accommodate the complexity of the early exercise boundary in this study. Further, the study develops numerical methods for solving the nonlinear partial differential equations (PDEs) and variational inequalities that arise in the valuation process. A penalty method is employed to approximate the nonlinear complementarity problem (NCP) resulting from discretizing the free boundary problem. Overall, significant results highlight the differences in option valuation with and without the incorporation of jumps. In scenarios where jumps are present, the model reflects larger and more sudden price changes, leading to significantly different option prices compared to models that assume smooth, continuous price movements. Without jumps, the valuation follows more predictable patterns, but it fails to capture extreme market behaviors, which can lead to pricing inaccuracies, especially in volatile markets. Numerical experiments conducted via the modified Monte Carlo simulation and the penalty method underscore each method’s strengths and limitations, showcasing their potential applications in practical scenarios in option pricing.