This paper tests the pricing accuracy and the hedging performance of the stochastic volatility with random jumps model in markets extended to contain swap contracts whose payoffs depend on the realized higher moments of the state variable. Using a two-step iterative approach, latent model variables are first filtered and then used to estimate the model parameters. The tests on European options and variance swaps written on the S&P 500 index show superior pricing accuracies in-sample and out-of-sample and jump risk is priced.