Abstract
In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric G-Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.
Highlights
Though many choice situations show uncertainty, owing to the Ellsberg Parasox, the impacts of ambiguity aversion on economic decisions are established and Beissner [1] considered general equilibrium economies with a primitive uncertainty model that features ambiguity about continuoustime volatility
The decision theoretical setting of multiple priors was introduced by Gilboa and Schmeidler [2] and Artzner et al [3] adapted it to monetary risk measures
When these multiple priors are used in finance areas, they result in drift uncertainty for stock prices
Summary
Though many choice situations show uncertainty, owing to the Ellsberg Parasox, the impacts of ambiguity aversion on economic decisions are established and Beissner [1] considered general equilibrium economies with a primitive uncertainty model that features ambiguity about continuoustime volatility. Owing to the fact that the volatility uncertainty leads to additional source of risk, the classical definition of arbitrage will no longer be adequate For this reason, a new arbitrage definition is presented to adjust our multiple priors model with mutually singular priors which are shown in (3). A new arbitrage definition is presented to adjust our multiple priors model with mutually singular priors which are shown in (3) In this modified sense, we confirm that our volatility uncertain financial markets do not admit any arbitrage opportunity. We employ the G-framework including G-expectation, GBrownian motion, and the concept of arbitrage to study the financial markets with volatility uncertainty; we gain the interval of no arbitrage, which is different from that in Denis and Martini [12].
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