On Minimization of Shortfall Risk in Tradable and Non-Tradable Assets

This work studies the problem of shortfall risk minimization in incomplete financial markets where the payoff of a contingent claim depends on a non-tradable asset. The trading is allowed only in a correlated tradable asset. This setting naturally leads to market incompleteness and makes perfect hedging impossible.

We study the problem in a continuous-time Brownian framework under different informational structures. First, we consider the case where the non-tradable asset is fully observable. In this setting, the optimal trading strategy is derived using stochastic calculus techniques, including Itô’s formula and a change of probability measure. The resulting strategy incorporates a term depending on the market price of risk and a term reflecting the dependence of the payoff on the non-tradable asset.

Second, we study the case where the non-tradable asset is unobservable until the terminal time. Thus, the problem is reformulated by conditioning on the hidden factor. As a result, an optimal strategy is expressed in terms of conditional expectations with respect to this hidden factor. The conclusions demonstrate the impact of information availability on hedging performance.

The approach is further extended to a mixed model driven by both Brownian motion and fractional Brownian motion, with the Hurst parameter H>3/4, ensuring that the model remains within the semi-martingale framework.