Path dependent volatility

We propose a general class of non-constant volatility models with dependence on the past. The framework includes path-dependent volatility models such as that by Hobson and Rogers and also path dependent contracts such as options of Asian style. A key feature of the model is that market completeness is preserved. Some empirical analysis, based on the comparison with standard local volatility and Heston models, shows the effectiveness of the path dependent volatility. In particular, it turns out that, when large market movements occur, the tracking errors of Heston minimum-variance hedging are up to twice the hedging errors of a path dependent volatility model.