Path dependent volatility

Abstract

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&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 the performance of standard local volatility and Heston models, shows the effectiveness of the path dependent volatility.

Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options

“…However, there are significant classes of American options, commonly traded in financial markets, whose corresponding diffusion process X is associated with Kolmogorov type operators which are not uniformly elliptic-parabolic, i.e., in particular m < N . Two such examples are provided by American Asian style options, see [1], and by American options priced in the stochastic volatility suggested in [13], see also [7] and [11]. Furthermore, as noted in [9] a general (mathematical) theory for American options in these settings is not yet available and the bulk of the literature focus mainly on numerical issues.…”

Free boundary and optimal stopping problems for American Asian options

“…We consider an extension of the local volatility model in which the volatility is defined as a function of the whole trajectory of the underlying asset, not only in terms of the spot price. Path dependent volatility was first introduced by Hobson & Rogers [15] and then generalized by Foschi and one of the authors [11]: the main feature is that it generally leads to a complete market model. We refer to [11] for an empirical analysis which shows the effectiveness of the model and compares the hedging performance with respect to standard stochastic volatility models.…”

“…Path dependent volatility was first introduced by Hobson & Rogers [15] and then generalized by Foschi and one of the authors [11]: the main feature is that it generally leads to a complete market model. We refer to [11] for an empirical analysis which shows the effectiveness of the model and compares the hedging performance with respect to standard stochastic volatility models. Let ψ be an average weight that is a non-negative, piecewise continuous and integrable function on ] − ∞, T ].…”

Intrinsic Taylor formula for Kolmogorov-type homogeneous groups

“…In mathematical finance, Kolmogorov equations arise in models incorporating some sort of dependence on the past: typical examples are Asian options (see, for instance, Ingersoll (1987), Barucci et al (2001), Pascucci (2008), Frentz et al (2010)) and some volatility models (see, for instance, Hobson and Rogers (1998) and Foschi and Pascucci (2008)). …”