A Lévy model combines a Brownian motion with drift and a purejump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of the process, is studied under a discrete-sampling scheme. In that case, the jumps are latent variables whose statistical properties can in principle be assessed when the frequency of observations increase to infinity. We propose nonparametric estimators for the Lévy density following Grenander's method of sieves. The associated problem of selecting a suitable approximating sieve is subsequently investigated using regular piece-wise polynomials as sieves and assuming standard smoothness conditions on the Lévy density. By sampling the process at a high enough frequency relative to the time horizon T , we show that it is feasible to choose the dimension of the sieve so that the rate of convergence of the risk of estimation off the origin is the best possible from a minimax point of view, and even if the estimation were based on the whole sample path of the process. The sampling frequency necessary to attain the optimal minimax rate is explicitly identified. The proposed method is illustrated by simulation experiments in the case of variance Gamma processes.

We derive a small-time expansion for out-of-the-money call options under an exponential Lévy model, using the small-time expansion for the distribution function given in , combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a non-zero volatility σ of the Gaussian component of the driving Lévy process is to increase the call price by (1)) as t → 0, where ν is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatilityσ 2 t (k) at log-moneyness k, which sharpens the first order estimateσ. Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed Lévy models. We also consider a small-time, small log-moneyness regime for the CGMY model, and apply this approach to the small-time pricing of at-the-money call options; we show that for Y ∈ (1, 2), limt→0 t −1/Y E(St − S0)+ = S0E * (Z+) and the corresponding at-the-money implied volatilityσt(0) satisfies limt→0σt (0), where Z is a symmetric Ystable random variable under P * and Y is the usual parameter for the CGMY model appearing in the Lévy density ν(x) = Cx −1−Y e −M x 1 {x>0} + C|x| −1−Y e −G|x| 1 {x<0} of the process.

Let X = (X t ) t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small-time expansions of arbitrary polynomial order in t are obtained for the tails P (X t ≥ y), y > 0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density p t of X t , an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order n in t are derived for the transition densities of the process. The conditions imposed on p t require that, away from the origin, its derivatives remain uniformly bounded as t → 0. Such conditions are then shown to be satisfied for symmetric stable Lévy processes as well as some tempered stable Lévy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.

The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel secondorder approximation for at-the-money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second-order term, in timet, is of the form d 2 t (3−Y)/2 , with d 2 only depending on Y, the degree of jump activity, on σ , the volatility of the continuous component, and on an additional parameter controlling the intensity of the "small" jumps (regardless of their signs). This extends the well-known result that the leading first-order term is σ t 1/2 / √ 2π. In contrast, under a pure-jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form d 1 t 1/Y . The second-order term is shown to be of the formd 2 t and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the logreturn process under the share measure and a suitable change of probability measure under which the pure-jump component of the log-return process becomes a Y-stable process. Our approach is sufficiently general to cover a wide class of Lévy processes, which satisfy the latter property and whose Lévy density can be closely approximated by a stable density near the origin. Our numerical results show that the first-order term typically exhibits rather poor performance and that the second-order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.KEY WORDS: exponential Lévy models, CGMY and tempered stable models, short-time asymptotics, at-the-money option pricing, implied volatility.We gratefully acknowledge the constructive and insightful comments provided by two anonymous referees, which significantly contributed to improve the quality of the manuscript. We also thank Victor Rivero for bringing our attention to the formula (3.9) as well as SveinnÓlafsson for pointing out some mistakes in a previous proof of a lemma and other helpful comments. José E.

No abstract