In this thesis, we explore the links between the various volatility modelling concepts of stochastic, implied and local volatility that are found in mathematical finance. We follow two distinct routes to compute new terms for the representation of stochastic volatility in terms of an equivalent local volatility. In addition to this, we discuss a framework for pricing multi-asset options under stochastic volatility models, making use of the local volatility representations derived earlier in the thesis. Previous approaches utilised by the quantitative finance community to price multi-asset options have relied heavily on numerical methods, however we focus on obtaining a semi-analytical solution by making use of approximation techniques in our calculations, with the aim of reducing the time taken to price such financial instruments. We also discuss in some detail the effects that the correlation between assets has when pricing multi-asset options under stochastic volatility models, and show how affine methods may be used to simplify calculations in certain cases.