BFS 2002

Contributed Talk

Properties of options on several underlying assets

Johan Tysk, Svante Janson

It is well-known that prices, expressed in units of constant value, of European options on one underlying asset decay with time and are convex in the underlying asset if the contract function is convex. In this article, options on several underlying assets are studied and we prove that if the volatility matrix is independent of time, then the options prices decay with time if the contract function is convex. However, the option prices are no longer necessarily convex in the underlying assets. If a time-dependent volatility is allowed we note that the option prices do not necessarily decay with time. We then formulate conditions on the volatility matrix for convexity to be preserved. These conditions show that even if the price processes are independent, convexity is for instance not preserved if the volatilities are of the form used in the constant elasticity of variance model.