We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.
IntroductionLarge deviations are used extensively in Physics (thermodynamics, statistical mechanics) as well as in Mathematics (information theory, stochastic analysis, mathematical finance) to estimate the exponential decay of probability measures of rare events. Varadhan [61], Schilder [58], Freidlin and Wentzell [32] proved, in different degrees of generality, large deviations principles (in R n and on path space) for solutions of stochastic differential equations with small noise, and the monographs by Dembo and Zeitouni [22] and Deuschel-Stroock [25] provide a precise account of those advances (at least up to the mid-1990s). In the past decade, this set of techniques and results has been adopted by the mathematical finance community: finite-dimensional large deviations (in the sense of Gärtner-Ellis) have been used to prove small-and large-time asymptotics of implied volatility in affine models [28,41], sample-path LDP (à la Freidlin-Wentzell [32]) have proved efficient to determine importance sampling changes of probability [38,39,57], and heat kernel expansions (following Ben Arous [10] and Bismut [13]), have led to a general understanding of small-time and tail behaviour of multi-dimensional diffusions [6,7,23,24]. These asymptotics have overall provided a deeper understanding of the behaviour of models, and, ultimately, allow for better calibration of real data; a general overview can be found in [55].Motivated by financial applications, we derive here asymptotic small-time and tail behaviours of the solution to a generalised version of the Stein-Stein stochastic volatility model, originally proposed in [59,60]. We in particular consider two important (in light of the recent trends in the literature proposed models) extensions:(i) the SDE driving the instantaneous volatility process is started from a random distribution; this so-called 'randomised' type of models was recently proposed in [46,47,51], in particular to understand the behaviour of the so-called 'forward volatility'; (ii) the volatility process is driven by a fractional Brownian motion. Fractional stochastic volatility models, originally proposed by Comte and Renault [18,19] with H ∈ (1/2, 1), have recently been extended to the case H ∈ (0, 1/2), and a recent flourishing activity in this area [2,8,33,37,42] has established these models as the go-to standards for estimation and calibration.
