Fully-dynamic risk-indifference pricing and no-good-deal bounds

Abstract

The seller's risk-indifference price evaluation is studied. We propose a dynamic risk-indifference pricing criteria derived from a fully-dynamic family of risk measures on the $L_p$-spaces for $p\in [1,\infty]$. The concept of fully-dynamic risk measures extends the one of dynamic risk measures by adding the actual possibility of changing the risk perspectives over time. The family is then characterised by a double time index. Our framework fits well the study of both short and long term investments. In this dynamic framework we analyse whether the risk-indifference pricing criterion actually provides a proper convex price system, for which time-consistency is guaranteed. It turns out that the analysis is quite delicate and necessitates an adequate setting. This entails the use of capacities and an extension of the whole price system to the Banach spaces derived by the capacity seminorms. Furthermore, we consider the relationship of the fully-dynamic risk-indifference price with no-good-deal bounds. Recall that no-good-deal pricing guarantees that not only arbitrage opportunities are excluded, but also all deals that are "too good to be true". We shall provide necessary and sufficient conditions on the fully-dynamic risk measures so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. The use of no-good-deal bounds also provides a method to select the risk measures and then construct a proper fully-dynamic risk-indifference price system in the $L_2$-spaces.