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We present an approach for pricing American put options with a regime-switching volatility. Our method reveals that the option price can be expressed as the sum of two components: the price of a European put option and the premium associated with the early exercise privilege. Our analysis demonstrates that, under these conditions, the perpetual put option consistently commands a higher price during periods of high volatility compared to those of low volatility. Moreover, we establish that the optimal exercise boundary is lower in high-volatility regimes than in low-volatility regimes. Additionally, we develop an analytical framework to describe American puts with an Erlang-distributed random-time horizon, which allows us to propose a numerical technique for approximating the value of American puts with finite expiry. We also show that a combined approach involving randomization and Richardson extrapolation can be a robust numerical algorithm for estimating American put prices with finite expiry.

This research studies the development of the evolving dynamic system model and explores the important elements or factors and what detailed attributes are the main influences model in achieving the success of a business, industry and management. It also identifies the real and major differences between static and dynamic business management models and the detailed factors that influence them. Later, this research investigates the benefits/advantages and limitations/disadvantages of some research studies. The studies conducted in this research put more emphasis on the capabilities of system dynamics (SD) in modeling and the ability to measure, analyse and capture problems in business, industry, manufacturing etc.

The research presented in this work is a qualitative research based on a literature review. Publicly available research publications and reports have been used to create a research foundation, identify the research gaps and develop new analyses from the comparative studies. As the literature review progressed, the scope of the literature search was further narrowed down to the development of SD models. Often, references to certain selected literature have been examined to find other relevant literature. To do so, a supporting tool (that connects related articles) provided by Google Scholar, Scopus, and particular journals has been used.

The dynamic business and management model is very different from the static business model in complexity, formality, flexibility, capturing, relationships, advantages, innovation model, new goals, updated information, perspective and problem-solving abilities. The initial approach of a static system was applied in the canvas business model, but further developments can be continued with a dynamic system approach.

Based on this study, which shows that businesses are developing more towards digitalisation, wanting the ability to keep up with the era that is moving so fast and the desire to increase profits, an instrument is needed that can help describe the difficulties of the needs and developments of the future world. This instrument, or tool of SD, is also expected to assist in drawing future models and in building a business with complex variables that can be predicted from the beginning.

This study will contribute to the SD study for many business incubator research studies. Many practical in business incubator management to have a benefit how to achieve the business performance management (BPM) in SD review.

The significant differences between static and dynamics to be used for business research and strategic performance management. This comparative study analyses some SD models from many authors worldwide. Their goals behind their strategic business models and encounter for their respective progress.