Financial markets with volatility uncertainty

We investigate financial markets under model risk caused by uncertain volatilities. To this end, we consider a financial market that features volatility uncertainty. We use the notion of G-expectation and its corresponding G-Brownian motion recently introduced by Peng (2007) to ensure a mathematically consistent framework. Our financial market consists of a riskless asset and a risky stock with price process modeled by geometric G-Brownian motion. We adapt the notion of arbitrage to this more complex situation, and consider stock price dynamics which exclude arbitrage opportunities. Volatility uncertainty results in an incomplete market. We establish the interval of no-arbitrage prices for general European contingent claims, and deduce explicit results in the Markovian case.