  In this paper, we develop a 4/2 stochastic volatility plus jumps model,
namely, a new stochastic volatility model including the Heston model and 3/2
model as special cases. Our model is highly tractable by applying the Lie
symmetries theory for PDEs, which means that the pricing procedure can be
performed efficiently. In fact, we obtain a closed-form solution for the joint
Fourier-Laplace transform so that equity and realized-variance derivatives can
be priced. We also employ our model to consistently price equity and VIX
derivatives. In this process, the quasi-closed-form solutions for future and
option prices are derived. Furthermore, through adopting data on daily VIX
future and option prices, we investigate our model along with the Heston model
and 3/2 model and compare their different performance in practice. Our result
illustrates that the 4/2 model with an instantaneous volatility of the form
$(a\sqrt{V_t}+b/\sqrt{V_t})$ for some constants $a, b$ presents considerable
advantages in pricing VIX derivatives.
