We propose a highly efficient and accurate valuation method for exotic-style options based on the novel Shannon wavelet inverse Fourier technique (SWIFT). Specifically, we derive an efficient pricing method for power options under a more realistic double exponential jump model with stochastic volatility and jump intensity. The inclusion of such innovations may accommodate for the various stylised facts observed in the prices of financial assets, and admits a more realistic pricing framework as a result. Following the derivation of our SWIFT pricing method for power options, we perform extensive numerical experiments to analyse both the method’s accuracy and efficiency. In addition, we investigate the sensitivities in the resulting prices, as well as the inherent errors, to changes in the underlying market conditions. Our numerical results demonstrate that the SWIFT method is not only more efficient when benchmarked to its closest competitors, such as the Fourier-cosine (COS) and the widely-acclaimed fast-Fourier transform (FFT) methods, but it is also robust across a range of different market conditions exhibiting exponential error convergence.