The asymptotic smile of a multiscaling stochastic volatility model

Abstract

Abstract. We consider a stochastic volatility model which captures relevant stylized facts of financial series, including the multi-scaling of moments. The volatility evolves according to a generalized Ornstein-Uhlenbeck processes with super-linear mean reversion.Using large deviations techniques, we determine the asymptotic shape of the implied volatility surface in any regime of small maturity t → 0 or extreme log-strike |κ| → ∞ (with bounded maturity). Even if the price has continuous paths, out-of-the-money implied volatility diverges for small maturity, producing a very pronounced smile.

General Smile Asymptotics with Bounded Maturity

“…Our results can also be applied to a stochastic volatility model, recently introduced in [ACDP12], which exhibits multiscaling of moments. Even though no closed expression is available for the moment generating function of the log-price, the tail probabilities can be estimated explicitly, as we show in a separate paper [CC15]. This leads to precise asymptotics for the implied volatility, thanks to Theorems 2.3, 2.4 and 2.7.…”

Asymptotic behaviour of randomised fractional volatility models

“…These outputs confirm that stochastic volatility models with a random starting point constitute a class of counterexamples to the long-standing belief, formulated by Gatheral [29,Chapter 5], that jumps in the stock price process are needed to produce steep short-dated implied volatility skews. Another example of broadly different design was provided by Caravenna and Corbetta [12]. In their 'multiscaling' model, the stock price process is continuous, while the volatility process has (carefully designed) jumps, and steepness of the smile is achieved with a heavy-tail distribution of the small-time distribution of the volatility.…”

The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

“…Large deviation techniques have been employed by a number of authors as well. Here, we mention Forde and Jacquier (), Forde, Jacquier, and Lee (), Caravenna and Corbetta (), Deuschel, Friz, Jacquier, and Violante (,b), and Friz, Gerhold, and Pinter () for the analysis of purely diffusive SV models, while Jacquier, Keller‐Ressel, and Mijatović () extend the large‐deviation approach to SV models allowing for jumps. Jump diffusions are also considered by Alòs, Léon, and Vives () and Benhamou, Gobet, and Miri () who employ Malliavin calculus, while Medvedev and Scaillet () provide short‐time implied volatility approximations by means of yet another method.…”