  In this paper we derive a generic decomposition of the option pricing formula
for models with finite activity jumps in the underlying asset price process
(SVJ models). This is an extension of the well-known result by Alos (2012) for
Heston (1993) SV model. Moreover, explicit approximation formulas for option
prices are introduced for a popular class of SVJ models - models utilizing a
variance process postulated by Heston (1993). In particular, we inspect in
detail the approximation formula for the Bates (1996) model with log-normal
jump sizes and we provide a numerical comparison with the industry standard -
Fourier transform pricing methodology. For this model, we also reformulate the
approximation formula in terms of implied volatilities. The main advantages of
the introduced pricing approximations are twofold. Firstly, we are able to
significantly improve computation efficiency (while preserving reasonable
approximation errors) and secondly, the formula can provide an intuition on the
volatility smile behaviour under a specific SVJ model.
