Basis risk arises when the writer of a contingent claim cannot trade in the underlying asset and must use a correlated proxy asset to hedge the contingent claim. Suppose the proxy asset is not perfectly correlated to the underlying. In that case, there is a risk that the hedge portfolio does not precisely track the contingent claim, which may lead to significant losses at maturity. There are several existing approaches to hedging and pricing of contingent claims in the presence of basis risk. The existing approaches considered in this dissertation are based on the quadratic and exponential utility functions. This dissertation compares these current approaches to a new policy that parameterises the hedge parameters as a recurrent neural network at each rebalancing date. This new approach is called Deep Hedging, and under this approach, the hedge parameters are determined in a model agnostic way. This is achieved using Long Short-Term Memory networks written in TensorFlow. This allows one to make the hedge parameters at each time point a function of current market data and previous hedging decisions. Deep Hedging is Greek-free and more easily allows for the incorporation of other market frictions, like transaction costs, compared to existing approaches. Lastly, we can find optimal hedging strategies under coherent risk metrics, like expected shortfall, using the Deep Hedging approach and given a price. By fixing the volatility and correlation parameters, Deep Hedging produces results that are comparable to the best existing strategies, in both complete and incomplete market settings, across a variety of moneyness levels.