We derived a model-free analytical approximation of the price of a multi-asset option defined over an arbitrary multivariate process, applying a semi-parametric expansion of the unknown risk-neutral density with the moments. The analytical expansion termed as the Multivariate Generalised Edgeworth Expansion (MGEE) is an infinite series over the derivatives of the known continuous time density. The expected value of the density expansion is calculated to approximate the option price. The expansion could be used to enhance a Monte Carlo pricing methodology incorporating the information about moments of the risk-neutral distribution. The numerical efficiency of the approximation is tested over a jump-diffusion density. For the known density, we tested the multivariate lognormal (MVLN), even though arbitrary densities could be used, and we provided its derivatives until the fourth-order. The MGEE relates two densities and isolates the effects of multivariate moments over the option prices. Results show that a calibrated approximation provides a good fit when the difference between the moments of the risk-neutral density and the auxiliary density are small relative to the density function of the former, and the uncalibrated approximation has immediate implications over risk management and hedging theory. The possibility to select the auxiliary density provides an advantage over classical Gram–Charlier A, B and C series approximations. The density approximation and the methodology can be applied to other fields of finance like asset pricing, econometrics, and areas of statistical nature.