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Given that the underlying assets in financial markets exhibit stylized facts such as leptokurtosis, asymmetry, clustering properties and heteroskedasticity effect, this paper presents a novel model for pricing American option under the assumptions that the stock prices processes are govern by time changed tempered stable Lévy process. As this model is constructed by introducing random time changes into tempered stable (TS) processes which specially refer to normal tempered stable (NTS) distribution as well as classical tempered stable (CTS) distribution, it permits infinite jumps as well as capturing random varying time in stochastic volatility, consequently taking into account the empirical facts such as leptokurtosis, skewness and volatility clustering behaviors. We employ the Fourier-cosine technique to calculate American option and propose the improved Particle Swarm optimization (IPSO) intelligent algorithm for model calibration. To demonstrate the advantage of the constructed model, we carry out empirical research on American index option in financial markets across wide ranges of models, with the time changing normal tempered stable distribution model yielding a superior performance than others.