Abstract:
The main object of study in the paper is the distance from a point to a line in the Riemannian manifold associated with the Heston model. We reduce the problem of computing such a distance to certain minimization problems for functions of one variable over finite intervals. One of the main ideas in this paper is to use a new system of coordinates in the Heston manifold and the level sets associated with this system. In the case of a vertical line, the formulas for the distance to the line are rather simple. For slanted lines, the formulas are more complicated, and a more subtle analysis of the level sets intersecting the given line is needed. We also find simple formulas for the Heston distance from a point to a level set. As a natural application, we use the formulas obtained in the present paper to compute the small maturity limit of the implied volatility in the correlated Heston model.

Abstract:
We consider the asymptotic behavior of the implied volatility in stochastic asset price models with atoms. In such models, the asset price distribution has a singular component at zero. Examples of models with atoms include the constant elasticity of variance model, jump-to-default models, and stochastic models described by processes stopped at the first hitting time of zero. For models with atoms, the behavior of the implied volatility at large strikes is similar to that in models without atoms. On the other hand, the behavior of the implied volatility at small strikes is influenced significantly by the atom at zero. S. De Marco, C. Hillairet, and A. Jacquier found an asymptotic formula for the implied volatility at small strikes with two terms and also provided an incomplete description of the third term. In the present paper, we obtain a new asymptotic formula for the left wing of the implied volatility, which is qualitatively different from the De Marco-Hillairet-Jacquier formula. The new formula contains three explicit terms and an error estimate. We show how to derive the De Marco-Hillairet-Jacquier formula from our formula, and compare the performance of the two formulas in the case of the CEV model. The resulting graphs show that the new formula provides a notably better approximation to the smile in the CEV model than the De Marco-Hillairet-Jacquier formula.

Abstract:
The asymptotic behavior of the implied volatility associated with a general call pricing function has been extensively studied in the last decade. The main topics discussed in this paper are Lee's moment formulas for the implied volatility, and Piterbarg's conjecture, describing how the implied volatility behaves in the case where all the moments of the stock price are finite. We find various conditions guaranteeing the existence of the limit in Lee's moment formulas. We also prove a modified version of Piterbarg's conjecture and provide a non-restrictive sufficient condition for the validity of this conjecture in its original form. The asymptotic formulas obtained in the paper are applied to the implied volatility in the CEV model and in the Heston model perturbed by a compound Poisson process with double exponential law for jump sizes.

Abstract:
The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and the sub-Riemannian distance function in the Grushin plane. For the Grushin plane, we establish an explicit formula for the Legendre-Fenchel transform of the limiting cumulant generating function and prove a partial large deviation principle that is true only inside a special set.

Abstract:
In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions, and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps, and make a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the NIG law.

Abstract:
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 be determined by a subset of components of the Gaussian vector, and we identify the relevant components by relating the asymptotics to a tractable quadratic optimization problem. As a corollary, we characterize the limiting behavior of the conditional law of the Gaussian vector, given a linear combination of the exponentials of its components. Our results can be used either to estimate the probability of tail events directly, or to construct efficient variance reduction procedures for precise estimation of these probabilities by Monte Carlo methods. They lead to important insights concerning the behavior of individual stocks and portfolios during market downturns in the multidimensional Black-Scholes model.

Abstract:
We consider uncorrelated Stein-Stein, Heston, and Hull-White models and their perturbations by compound Poisson processes with jump amplitudes distributed according to a double exponential law. Similar perturbations of the Black-Scholes model were studied by S. Kou. For perturbed stochastic volatility models, we obtain two-sided estimates for the stock price distribution density and compare the tail behavior of this density before and after perturbation. It is shown that if the value of the parameter, characterizing the right tail of the double exponential law, is small, then the stock price density in the perturbed model decays slower than the density in the original model. On the other hand, if the value of this parameter is large, then there are no significant changes in the behavior of the stock price distribution density.

Abstract:
We obtain a first order extension of the large deviation estimates in the G\"{a}rtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special family of functions mentioned above is based on heat kernel expansions. Some of the ideas employed in the paper come from the theory of affine stochastic processes. For instance, we provide an explicit expansion with respect to the homogenization parameter of the rescaled cumulant generating function in the case of a generic continuous affine process. We also compute the coefficients in the homogenization expansion for the Heston model that is one of the most popular stock price models with stochastic volatility.

Abstract:
In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a model-free tail-wing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function.

Abstract:
We study the probability mass at the origin in the SABR stochastic volatility model, and derive several tractable expressions for it, in particular when time becomes small or large. In the uncorrelated case, tedious saddlepoint expansions allow for (semi) closed-form asymptotic formulae. As an application--the original motivation for this paper--we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage-free, allow us to quantify the impact of the mass at zero on currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous.