This paper presents a discrete random-field model for forward prices driven by the multivariate normal inverse Gaussian distribution. The model captures the idiosyncratic risk and adequately addresses the heavy tails characterizing electricity forward prices. We fit the model to forward prices from the Nordic power exchange using a Markov chain Monte Carlo algorithm. This is then compared with Gaussian-based multifactor models in terms of goodness of fit to historical log returns. Our finding is that the proposed model offers a superior fit to the empirical distributions.