No-arbitrage, leverage and completeness in a fractional volatility model

• Mathematical consistency and no-arbitrage in a fractional volatility market model.
• Market not complete if the volatility process is independent from the price process.
• Market arbitrage free and complete when driven by a unique process.
• Arbitrage free complete market displays leverage properties as in empirical data.

When the volatility process is driven by fractional noise one obtains a model which is consistent with the empirical market data. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behaviors. Here, the no-arbitrage and completeness properties of the models are rigorously studied.