Cheng Ouyang

Asymptotics of Implied Volatility in Local Volatility Models

Using an expansion of the transition density function of a 1-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 first order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusions using probabilistic methods, i.e. the so called principle of not feeling the boundary.

Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

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.

On probability laws of solutions to differential systems driven by a fractional Brownian motion

This article investigates several properties related to densities of solutions (Xt)t∈[0,1](Xt)t∈[0,1] to differential equations driven by a fractional Brownian motion with Hurst parameter H>1/4H>1/4. We first determine conditions for strict positivity of the density of XtXt. 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.

On Small Time Asymptotics for Rough Differential Equations Driven by Fractional Brownian Motions

We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.

Gradient Bounds for Solutions of Stochastic Differential Equations Driven by Fractional Brownian Motions

We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts formulas on the path space of a fractional Brownian motion.