Price as a choice under nonstochastic randomness in finance

Abstract

Arrow–Debreu state preference approach to derivatives pricing is embedded into a decision theoretical framework. Derivatives prices are considered as decision variables. Axiomatic decision theory, concerned with the attitude toward uncertainty and existence of closed in *-weak topology sets of finitely-additive probabilities is applied. A version of indifference pricing relation is obtained that extends classical relations for European contingent claims. The obtained structure happens to be a convenient way of addressing such traditional problems of mathematical finance as derivatives valuation in incomplete markets, portfolio choice and market microstructure modeling. An alternative interpretation of the closed sets of finitely-additive probabilities as statistical laws of statistically unstable (nonstochastic) random phenomena is discussed.