Fast and explicit European option pricing under tempered stable processes

We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential L\'evy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such as bilateral Gamma process and one-sided TS process. When compared to traditional Fourier pricing, our method has the advantage of being hyperparameter free. We also provide a detailed numerical analysis and show that our technique is competitive with state-of-the-art pricing methods.