Using an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion, we obtain the first-and second-order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the firstand second-order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.

Abstract. This article investigates several properties related to densities of solutions (X t ) t∈[0,1] to differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/4. We first determine conditions for strict positivity of the density of X t . Then we obtain some exponential bounds for this density when the diffusion coefficient satisfies an elliptic type condition. Finally, still in the elliptic case, we derive some bounds on the hitting probabilities of sets by fractional differential systems in terms of Newtonian capacities.

The goal of this paper is to show that under some assumptions, for a d-dimensional fractional Brownian motion with Hurst parameter H > 1/2, the density of the solution of the stochastic differential equationUnder the framework of this present work, the Laplace method can be obtained in general hypoelliptic case and without imposing the structure equations on vector fields in Theorem 1.1. These two assumptions are imposed to obtain the correct Riemannian distance in the kernel expansion.Remark 1.3. When H > 1/2, to obtain a short-time asymptotic formula for the density of solution to equation (1.1) but with drift, one need to work on a version of Laplace method with fractional powers of ε, which will be very heavy and tedious in computation.Remark 1.4. When the present work was almost completed, we noticed that a proof for the Laplace method for stochastic differential equation driven by fractional Brownian motion with Hurst parameter 1/3 < H <

In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/3. We show that under some geometric conditions, in the regular case H > 1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H > 1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

In this work we study rough differential equations driven by a fractional Brownian motion with Hurst parameter H > 1 4 and establish Varadhan's small time estimates for the density of solutions of such equations under Hörmander's type conditions.