60F10 Large deviations
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- Affine processes (1)
- Generalized gradient structure (1)
- Implied volatility in the large maturity limit. (1)
- Large deviation principle (1)
- Stochastic volatility with jumps (1)
- energy-dissipation principle (1)
- evolutionary Γ-convergence (1)
- gradient system (1)
- large-deviation principle (1)
- relative entropy (1)
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.
Let $\sigma_t(x)$ denote the implied volatility at maturity t for a strike $K = S_0 e^{x t}$, where $x \in R$ and $S_0$ is the current value of the underlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity t tends to infinity, given by the formula \sigma_{\infty}(x) = \sqrt2 (h^*(x)^{1/2} + (h^*(x) − x)^{1/2}, for $x$ in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function $h^*$ is the convex dual of the limiting cumulant generating function $h$ of the scaled log-spot process. We express $h$ in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).