Extreme volatility, price spikes, and complex nonlinear dynamics are characteristics often shown in the electricity market, particularly the Australian national electricity market (NEM). These attributes are usually driven by supply-demand imbalances, renewable energy integration, and regulatory interventions, which make traditional pricing models, such as the Black-Scholes framework, inadequate for capturing key market behaviors like long-memory effects, heavy tails, and volatility clustering. This study developed and applied a fractional Black-Scholes model that integrated space-fractional derivatives to account for memory effects, anomalous diffusion, and extreme price movements. The proposed model was calibrated using historical NEM data, demonstrating significant improvements in the pricing of electricity options. The results suggested that it effectively captured key market behaviors. Overall, the findings offered valuable insights and provided a more accurate representation of the Australian electricity market's distinct dynamics.