We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < 1/2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t 1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

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We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.Keywords Stein-Stein model · Heston model · Mixing distribution density · Stock price · Bessel processes · Ornstein-Uhlenbeck processes · CIR processes · Asymptotic formulas · Implied volatility where W t and Z t are standard Brownian motions. We will assume that µ ∈ R, a ≥ 0, b < 0, and c > 0. The initial conditions for X t and Y t are denoted by x 0 and y 0 , respectively. The volatility equation in (2) can be 0 A. Gulisashvili rewritten in the mean-reverting form. This giveswhere r = −b and m = − a b . The volatility equation in (2) is uniquely solvable in the strong sense, and the solution Y t is a positive stochastic process. This process is called a Cox-Ingersoll-Ross process (a CIRprocess). This process was studied in [6]. Interesting results concerning the Heston model were obtained in [7].It will be assumed throughout the present paper that the models described by (1) and (2) are uncorrelated. This means that the Brownian motions W t and Z t driving the stock price equation and the volatility equation in (1) and (2) are independent. In the analysis of the probability distribution of the stock price X t , the mean-square averages of the volatility process over finite time intervals play an important role. For the Stein-Stein model, we set α t = 1 t t 0

We study fractional stochastic volatility models in which the volatility process is a positive continuous function σ of a continuous Gaussian process B. Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function σ is globally Hölder-continuous and the process B is fractional Brownian motion. In the present paper, we prove a similar small-noise large deviation principle under weaker restrictions on σ and B. We assume that σ satisfies a mild local regularity condition, while the process B is a Volterra type Gaussian process. Under an additional assumption of the self-similarity of the process B, we derive a large deviation principle in the small-time regime. As an application, we obtain asymptotic formulas for binary options, call and put pricing functions, and the implied volatility in certain mixed regimes.AMS 2010 Classification: 60F10, 60G15, 60G18, 60G22, 41A60, 91G20.

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment s+ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: σBS(k, T ) 2 T ∼ Ψ(s+ − 1) × k (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drȃgulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443-453], and then show the validity of a refined expansion of the type σBS(k, T ) 2 T = (β1k 1/2 + β2 + . . . ) 2 , where all constants are explicitly known as functions of s+, the Heston model parameters, spot vol and maturity T . In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287-315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of log ST (equivalently: Mellin transform of ST ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter ("critical slope"), defined in a model free manner, which drives the second and higher order terms in tail-and implied volatility expansions.

Abstract. In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails of stock price distributions in uncorrelated Hull-White, Stein-Stein, and Heston models and find asymptotic formulas with error estimates for call pricing functions in these models.

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.

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