Large Deviations for the Extended Heston Model: The Large-Time Case

Abstract

Abstract. We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of [13]) and deduce a closed-form formula for the large-maturity implied volatility smile.We concentrate on (rescaled) strikes around the money, which are the most common in practice, and extend the results in [4] and [8].

Large-time option pricing using the Donsker–Varadhan LDP—correlated stochastic volatility with stochastic interest rates and jumps

“…leading order) implied volatility smile for the well known Heston model in the so-called large-time, large log-moneyness regime, under a mild restriction on the model parameters, and the rate function is computed numerically as a Fenchel-Legendre transform which is just a one-dimensional root-finding exercise. [GJ11] show that the asymptotic smile can actually be computed in closed-form via the SVI parameterization and [FJM10] compute the correction term to this smile using saddlepoint methods; [Forde14] derives a similar result for the Stein-Stein model and [JM12] derive a similar result for a displaced Heston model (and relax the aforementioned condition on the parameters). [JKRM12] extended the results in [FJ11] to a general class of affine stochastic volatility models (with jumps), which includes the Heston, Bates and the Barndorff-Nielsen-Shephard model, and under mild assumptions, they show that the limiting smile necessarily corresponds to the smile generated by an exponential Lévy model.…”

Asymptotics of Forward Implied Volatility

“…Extreme strike asymptotics arose with the seminal paper by Roger Lee [43] and have been further extended by Benaim and Friz [4,5] and in [30,31,23,15]. Comparatively, large-maturity asymptotics have only been studied in [55,19,36,35,21] using large deviations and saddlepoint methods. Fouque et al [22] have also successfully introduced perturbation techniques in order to study slow and fast mean-reverting stochastic volatility models.…”

Large-Maturity Regimes of the Heston Forward Smile

“…Roger Lee [55] was the first to study extreme strike asymptotics, and further works on this have been carried out by Benaim and Friz [6,7] and in [39,40,41,31,23,19]. Large-maturity asymptotics have only been studied in [67,27,46,45,29] using large deviations and saddlepoint methods. Fouque et al [30] have also successfully introduced perturbation techniques in order to study slow and fast mean-reverting stochastic volatility models.…”

“…It is clear (see also [27] and [45]) that the function V is infinitely differentiable, strictly convex and essentially smooth on the open interval (u − , u + ) and that V (0) = 0. Furthermore V (1) = 0 if and only if ρ ≤ κ/ξ.…”