This thesis introduces a general framework for model-free discretisation-invariant swaps. In the first main chapter a novel design for swap contracts is developed where the realised leg is modified such that the fair value is independent of the monitoring partition. An exact swap rate can then be derived from the price aportfolio of vanilla out-of-the-money options without any discrete-monitoring or jump errors. In the second main chapter the P&Ls on discretisation-invariant swaps associated with the variance, skewness and kurtosis of the log return distribution are used as estimators for the corresponding higher-moment risk premia. An empirical study on the S&P 500 investigates the factors determining these risk premia for different sampling frequencies and contract maturities. In the third main chapter the dynamics of conventional and discretisation-invariant variance swaps and variance risk premia are compared in an affine jump-diffusion setting. The ideas presented in this thesis set the ground for many interesting and practically relevant applications.