  This paper introduces the class of volatility modulated L\'{e}vy-driven
Volterra (VMLV) processes and their important subclass of L\'{e}vy
semistationary (LSS) processes as a new framework for modelling energy spot
prices. The main modelling idea consists of four principles: First,
deseasonalised spot prices can be modelled directly in stationarity. Second,
stochastic volatility is regarded as a key factor for modelling energy spot
prices. Third, the model allows for the possibility of jumps and extreme spikes
and, lastly, it features great flexibility in terms of modelling the
autocorrelation structure and the Samuelson effect. We provide a detailed
analysis of the probabilistic properties of VMLV processes and show how they
can capture many stylised facts of energy markets. Further, we derive forward
prices based on our new spot price models and discuss option pricing. An
empirical example based on electricity spot prices from the European Energy
Exchange confirms the practical relevance of our new modelling framework.
