<p>This study introduces a novel methodology for pricing options with early exercise features, specifically American and Bermudan options, using the discrete cosine transform (DCT) within the binomial tree model framework. The research begins by addressing the limitations of traditional binomial tree methods when applied to complex models, such as Lévy processes, which have been perceived as inefficient for option pricing. We outline a systematic approach that incorporates the DCT to enhance computational efficiency and accuracy in option pricing. The procedure involves first establishing the binomial tree model, followed by the integration of the DCT to estimate option prices rapidly. We demonstrate the effectiveness of this method by applying it to various models, including the classic Black-Scholes model, as well as jump-diffusion and exponential Lévy process models, such as the exponential CGMY and exponential normal inverse Gaussian models. The key highlights of our research include the significant improvement in pricing speed and precision, as well as the versatility of the DCT in adapting to both standard and complex financial models. This study not only expands the applicability of the binomial tree model but also paves the way for future research in option pricing methodologies.</p>