Financial markets with volatility uncertainty

Abstract

We investigate financial markets under model risk caused by uncertain volatilities. For this purpose we consider a financial market that features volatility uncertainty. To have a mathematical consistent framework we use the notion of G-expectation and its corresponding G-Brownian motion recently introduced by Peng (2007). Our financial market consists of a riskless asset and a risky stock with price process modeled by a geometric G-Brownian motion. We adapt the notion of arbitrage to this more complex situation and consider stock price dynamics which exclude arbitrage opportunities. Due to volatility uncertainty the market is not complete any more. We establish the interval of no-arbitrage prices for general European contingent claims and deduce explicit results in a Markovian setting.

References in corpus (4)

• Martingale Representation Theorem for the G-expectation
• Some properties on $G$-evaluation and its applications to $G$-martingale decomposition
• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
• Properties of hitting times for $G$-martingale

Cited by in corpus (5)

• Coherent Price Systems and Uncertainty-Neutral Valuation
• Reflected Solutions of BSDEs Driven by G-Brownian Motion
• A note on pricing of contingent claims under G-expectation
• Finance Without Probabilistic Prior Assumptions
• A Stochastic Maximum Principle for Processes Driven by G-Brownian Motion and Applications to Finance