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Price dynamics in stock market is modelled by a statistical physics systems: Ising model. A comparative analysis of the historical dynamics of stock returns between the US, UK, and French markets is given. Since the Ising model requires binary inputs, the effect of binarization is studied. Then, using the TAP approximation method, external fields and coupling strengths are calculated. The fluctuation cycles of coupling strengths have a remarkable corresponding relationship with the important period of the financial market. The highlight of this paper is to verify the phase transition can also occur in the stock market and it reveals the transformation of the market state. The numerical solution in this paper is consistent with the exact solution obtained by Lars Onsager. Our findings can help to discover the economic cycles and provide more possibilities for studying financial markets using physical models. Graphic abstract

In this study, we create a novel American double‐barrier Parisian call option contract that may be utilized as an executive option for listed companies to incentivize staff and replace the classic American option. We address the option pricing problem by developing state variables to identify the price state and using the least‐squares Monte Carlo approach. We present several Lévy processes to simulate the movement path of the underlying asset. We discover that geometric Brownian motion and normal inverse Gaussian (NIG) process have successful outcomes, and NIG process has greater calculation accuracy than variance gamma process. The barrier width and window length are positively connected with the price of an American Parisian option, whereas the strike price is negatively correlated with it. Increasing the number of discrete periods of the contract will enhance the pricing accuracy.