Abstract
We consider fundamental questions of arbitrage pricing arising when the uncertainty model incorporates volatility uncertainty. The resulting ambiguity motivates a new principle of preference-free valuation. By establishing a microeconomic foundation of sublinear price systems, the principle of ambiguity-neutral valuation imposes the novel concept of equivalent symmetric martingale measures. Such measures exist when the asset price with uncertain volatility is driven by Peng's G-Brownian motion.
Highlights
In this paper we study a fundamental assumption behind theoretical models in Finance, namely, the assumption of a known single probability measure
We introduce an uncertainty model described as a set of possibly mutually singular probability measures
Very recent developments in stochastic analysis have established a complete theory in this field, a major objective of which has been the sublinear expectation operator introduced by Peng (2007a)
Summary
In this paper we study a fundamental assumption behind theoretical models in Finance, namely, the assumption of a known single probability measure. In a general semimartingale framework, the notion of no free lunch with vanishing risk Delbaen and Schachermayer (1994) ensured the existence of an equivalent martingale measure in the given (continuous time) financial market All these considerations have in common that the uncertainty of the model is given by a single probability measure. Moving to models with multiple probability measures, pasting martingale measures introduces the intrinsic structure of dynamic convexity, see Riedel (2004) and Delbaen (2006) This type of time consistency is related to recursive equations, see Chen and Epstein (2002), which can result in nonlinear expectation and generates a rational updating principle. We discuss mathematical foundations such as the space of price systems and a collection of results of stochastic analysis and G-expectations
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