  We develop a comprehensive mathematical framework for polynomial
jump-diffusions in a semimartingale context, which nest affine jump-diffusions
and have broad applications in finance. We show that the polynomial property is
preserved under polynomial transformations and L\'evy time change. We present a
generic method for option pricing based on moment expansions. As an
application, we introduce a large class of novel financial asset pricing models
with excess log returns that are conditional L\'evy based on polynomial
jump-diffusions.
