  In this paper we show how to relate European call and put options on multiple
assets to certain convex bodies called lift zonoids. Based on this, geometric
properties can be translated into economic statements and vice versa. For
instance, the European call-put parity corresponds to the central symmetry
property, while the concept of dual markets can be explained by reflection with
respect to a plane. It is known that the classical univariate log-normal model
belongs to a large class of distributions with an extra property, analytically
known as put-call symmetry. The geometric interpretation of this symmetry
property motivates a natural multivariate extension. The financial meaning of
this extension is explained, the asset price distributions that have this
property are characterised and their further properties explored. It is also
shown how to relate some multivariate asymmetric distributions to symmetric
ones by a power transformation that is useful to adjust for carrying costs. A
particular attention is devoted to the case of asset prices driven by L\'evy
processes. Based on this, semi-static hedging techniques for multiasset barrier
options are suggested.
