Price Systems for Random Amounts

Internal consistency—or “coherence”—of a price system is the basis of several key concepts in many fields, such as subjective probability (in Probability Theory), no-arbitrage pricing, and risk measures (in Mathematical Finance). Furthermore, Actuarial Mathematics uses coherence to describe the analyti- cal form of risk premia, and an analogous approach has recently been proposed for firms’ valuation. Technically, it amounts to a characterisation of functionals with particular properties (a typical goal in Functional Analysis), which translates into a numerical representation of preferences along the tradi- tional guidelines of Decision Theory, whose analogies with Mathematical Finance are numerous and really impressive. This is explored in this chapter.