Price as a Choice Under Nonstochastic Randomness in Finance

Abstract

Closed sets of finitely-additive probabilities are statistical laws of statistically unstable random phenomena. Decision theory, adapted to such random phenomena, is applied to the problem of valuation of European options. Embedding of the Arrow-Debreu state preference approach to options pricing into decision theoretical framework is achieved by means of considering option prices as decision variables. A version of indifference pricing relation is proposed that extends classical relations for European contingent claims to statistically unstable random behavior of the underlying. A static hedge is proposed that can be called either the model specification hedge or the uncertainty hedge or the generalized Black-Scholes delta. The obtained structure happens to be a convenient way to address such traditional problems of mathematical finance as derivatives valuation in incomplete markets, portfolio choice and market microstructure modeling; and, beyond that, as shown in the appendix, an opportunity to introduce in mathematical finance an alternative interpretation of closed families of finitely-additive probabilities as statistical laws of statistically unstable random phenomena.

Keywords: Statistical instability, Randomness, Finitely-additive measures, Decision theory, Uncertainty profiling, Derivatives Valuation, Portfolio choice, Bid-Ask Spread