Summary: We study the totally asymmetric exclusion process on a finite one-dimensional lattice with open boundaries, i.e. in contact with two reservoirs at different potentials. The total (time-integrated) current through the system is a random variable that scales linearly with time in the long-time limit. We conjecture a parametric representation for the generating function of the cumulants of the current, which is related to the large deviation function by Laplace transform. This formula is valid for all system sizes and for all values of the boundary coupling parameters.