Direct and indirect excitons in $\mathrm{Ga}\mathrm{As}\text{\ensuremath{-}}{\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ coupled double quantum wells, under growth-direction applied electric and magnetic fields, have been theoretically investigated within a variational procedure in the effective-mass and parabolic-band approximations. The exciton hydrogenic $1s$-like envelope wave function is obtained through a variational procedure and an appropriate expansion in trigonometric functions of the electron and hole wave functions. The applied electric field produces a polarization of the exciton by pushing the electron and hole away from each other, whereas the magnetic field contracts the exciton by pushing the electron and hole closer to each other. Intersubband mixing produced by the Coulomb interaction of electron-hole pairs is taken into account and a detailed analysis of the properties of direct- and indirect-exciton states in $\mathrm{Ga}\mathrm{As}\text{\ensuremath{-}}{\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ coupled double quantum wells is presented, with theoretical results in good agreement with available experimental measurements.