Basis risk arises whenever one hedges a derivative using an instrument different from the underlying asset. Recent literature has shown that this risk can significantly impair hedging effectiveness. In this work, we derive new semi-explicit expressions for discrete-time local and global quadratic hedging strategies under basis risk. The resulting solutions cover a wide range of derivatives and asset dynamics and are based on the inverse Laplace transform representation of the derivative. Moreover, we investigate whether quadratic hedging performs better at mitigating basis risk when compared to naive delta hedging strategies that are often used in practice. Finally, a sensitivity analysis is performed to evaluate the impact of the correlation coefficient, the option maturity and of the rebalancing frequency on the hedging performance.