Bilinear Integrable soliton solutions and carbon emission rights pricing

Pricing carbon emission rights and other financial assets using the soliton theory is a pioneering attempt. In this study, we investigated the pricing of carbon emission rights according to the basic attributes of solitons, whose amplitude and velocity remain unchanged after a collision. First, we showed that the price fluctuation in the sequence of carbon emission rights possesses the characteristics of a soliton, such as non-dispersion while spreading and being stable after a collision. With a variation in the time scale, the waveform and velocity of the carbon price movement did not change with its translation in the same direction. Second, we demonstrated that the carbon soliton equation passes the $Painlev\acute{e}$ test for integrability. Moreover, at the resonance point, there exists an arbitrary function ${u}_j(t)$ of $t$ in which the compatibility condition always holds. This indicates the existence of soliton solutions to the carbon soliton equation. Third, the exact solutions of single-soliton, two-soliton and three-soliton equations were obtained by using a nonlinear evolution equation constructed with a bilinear method. In the three soliton solutions, only the single-soliton solution is the central value of the carbon emission rights and its theoretical value is 13 Euro/tCO2e.