Before revision:  By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action  whose potential term is an exponential function , i. e. the Liouville Hamiltonian, which can be explicitly solved .  After working out some auxiliary integrations involving Bessel and Whittaker functions, we arrive at the spectral expansion expression of the value of an Asian option .
After revision:  By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action whose potential term is an exponential function , i. e. the Liouville Hamiltonian, which can be explicitly solved . After working out some auxiliary integrations involving Bessel and Whittaker functions, we arrive at the spectral expansion expression of the value of Asian options .