Attainability of European Path-Independent Claims in Incomplete Markets

In this paper we consider the question which path-independent claims are attainable through self-financing trading strategies in an incomplete market. We show for continuous-time stochastic volatility models and for models exhibiting both stochastic volatility and jumps that from this special group of claims only affine linear payoffs can be replicated.We provide a simple proof for this proposition based on the requirement that the stock and the claim must be locally perfectly correlated and, in case of the stochastic volatility model, on the partial differential equation that any path-independent claim has to satisfy. An important application of our result is the quick derivation of bounds on European option prices which were previously deduced by other authors using very demanding techniques from probability theory. Furthermore, we show that there is no analogy for our result in models with discrete time and discrete state variables, i.e. in these models we can generate at least some non-linear path-independent claims by self-financing trading strategies