In this paper, we present the modeling benefits of using Levy processes and the fast Fourier transform (FFT) in the valuation of gas storage assets and, from a practitioner's perspective, in creating market-consistent valuations and hedging portfolios. This valuation methodology derives the storage asset value via stochastic backward dynamic programming, drawing on established FFT methods. We present a modification to this algorithm that removes the need for a dampening parameter and leads to an increase in valuation convergence. The use of the FFT algorithm allows us to employ a wide range of potential spot price models. We present the characteristic function of one such model: the mean-reverting variance-gamma (MRVG) process. We provide a rationale for using this model in fitting the implied volatility smile by comparing the process moments with the more common mean-reverting diffusion model. We next present the dynamics of the implied spot price under a general singlefactor Levy-driven forward-curve model; using these results, we go on to present the forward-curve-consistent conditional-characteristic function of the implied spot price model. We derive a transform-based swaption formula in order to calibrate our models to market-traded options, and we use these calibrated models to then value a stylized storage asset and calculate the hedging positions needed to monetize this value. We demonstrate how one can perform an informative scenario-based analysis on the relationship between the implied volatility surface and the asset value. Convergence results for the valuation algorithm are presented, along with a discussion on the potential for further increasing the computational efficiency of the algorithm. Finally, to provide increased confidence around the fit of the MRVG model proposed, we conduct a formal model-specification analysis of this model against a benchmark mean-reverting jump-diffusion model.