We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics underlying this class of models as well as identification constraints, and compute standard and extended transforms relevant to asset pricing. We also show that the LQJD class can be embedded into the affine class through use of an augmented state vector. We further establish that an equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model significantly reduces pricing errors, and further addition of a jump component in the stock price largely improves goodness-of-fit for in-the-money calls but less for out-of-the-money ones.