Abstract
We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain abstract (pointwise) fundamental theorem of asset pricing and pricing–hedging duality. Our results are general and, in particular, cover both the so-called model independent case as well as the classical probabilistic case of Dalang–Morton–Willinger. Our analysis is scenario-based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.
Highlights
The state preference model or asset pricing model underpins most mathematical descriptions of financial markets
We show that choosing a probability measure P on X is equivalent to fixing a suitable set of scenarios ΩP, and our results lead to probabilistic notions of arbitrage as well as the probabilistic version of the fundamental theorem of asset pricing
We work on a Polish space X and denote @X as its Borel sigma-algebra and 3 the set of all probability measures on (X, @X)
Summary
The state preference model or asset pricing model underpins most mathematical descriptions of financial markets. The idea of introducing a reference probability measure to select scenarios proved very fruitful in the case of a general X and was instrumental for the rapid growth of the modern financial industry It was pioneered by Samuelson [46] and Black and Scholes [7], who used it to formulate a continuous-time financial asset model with unique rational prices for all contingent claims. As special cases of our general FTAP, we recover results in Acciaio et al [1] and Burzoni et al [10] as well as the classical Dalang–Morton–Willinger theorem of Dalang et al [16] For the latter, we show that choosing a probability measure P on X is equivalent to fixing a suitable set of scenarios ΩP, and our results lead to probabilistic notions of arbitrage as well as the probabilistic version of the fundamental theorem of asset pricing.
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