Time-consistent and market-consistent evaluations

We consider evaluation methods for payoffs with an inherent financial risk as encountered for instance for portfolios held by pension funds and insurance companies. Pricing such payoffs in a way consistent to market prices typically involves combining actuarial techniques with methods from mathematical finance. We propose to extend standard actuarial principles by a new market-consistent evaluation procedure which we call "two-step market evaluation." This procedure preserves the structure of standard evaluation techniques and has many other appealing properties. We give a complete axiomatic characterization for two-step market evaluations. We show further that in a dynamic setting with continuous stock prices every evaluation which is time-consistent and market-consistent is a two-step market evaluation. We also give characterization results and examples in terms of g-expectations in a Brownian-Poisson setting.