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...Lévy process. In this setup, particular emphasis has been put on the small-time asymptotic of C(t; τ ,K) for τ → 0 and K fixed at some level different from F (t), see for instance [70], [105], [46], =-=[48]-=- among many others. A typical result in this area of research is that (1.8) δC (t; τ ,K) ∼ τ , K 6= F (t), where ∼ indicates the leading order term as τ → 0. For this relation to be true, the correspo...

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...However, this approach is able to produce the correct expressions for the higher order coefficients a2(y), . . . only in the compound Poisson case (cf. [12]). The problem was subsequently resolved in =-=[13]-=- (see Section 6 therein as well as [12] for a preliminary related result), using a new approach, under the assumption that the Lévy density of the Lévy process {Zt}t≥0 is sufficiently smooth and bound...

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...ey options converges to the volatility of the diffusion component as the maturity tends to zero. Small-maturity asymptotics for models with jumps (including Lévy processes) have been investigated in =-=[2, 3, 48, 45, 44, 16]-=-. However, these asymptotics do not provide any information on the forward smile or forward-start-based payoffs, such as cliquets and forward-start options and the literature on this topic is sparse. ...

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...ation, qualitative model assessment, and parametrization design. The small-time behavior of the level of implied volatility in Lévy models (and generalizations) has been investigated in great detail =-=[5, 12, 13, 14, 24, 29]-=-. We, on the other hand, focus on the at-themoney slope of implied volatility, i.e., the strike derivative, and investigate its behavior as maturity becomes small. For diffusion models, there typicall...

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...e, with many applications. Several tools have been introduced, ranging from the theory of random dynamical systems [5, 6, 12–14], the study of moment equations [3, 15]. Unfortunately, no systematic method in the flavor of bifurcation theory for the analysis of the dynamics of nonlinear SDE drive Poisson noisy exists, and this is the central problem of the present paper. We focus here on the dynamics of stochastic Rayleighvan der Pol equations with jumps. This type of equations may appear, for example, in diffusion approximations of physical models [16] or the classical model for a stock price [5]. The question we address is how the interplay between the stochastic jump system and diffusion system affects the behavior of the system. An important contribution of RDS theory to the field of stochastic bifurcations is characterized 2 Abstract and Applied Analysis by a change in the shape of the invariant measure and stochastic Lyapunov exponent. We investigate the questions of stochastic stability and bifurcations, random attractors, and limits for stochastic Rayleigh-van der Pol equations with jumps, using combinations of appropriate tools such as Lyapunov direct method, Arnold asymptotic...

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.... 1 Introduction Stemming in part from its importance for model calibration and testing, small-time asymptotics of option prices have received a lot of attention in recent years (see, e.g., [3], [5], =-=[6]-=-, [10], [11], [12], [13], [15], and references therein). In the present paper, we study the small-time behavior for at-the-money (ATM) call (or equivalently, put) option prices [ Π(t) := E (St − S0) +...

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...ch has been applied during the last decade to obtain high-order asymptotic expansions for the transition distributions and densities of Lévy processes and other Markov processes with jumps (see [38], =-=[19]-=-, [20], and [21]). These small-time asymptotic results have found a wide scope of applications ranging from estimation methods based on high-frequency sampling observations of the process (see, e.g., ...

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...ch has been applied during the last decade to obtain high-order asymptotic expansions for the transition distributions and densities of Lévy processes and other Markov processes with jumps (see [45], =-=[24]-=-, [25], and [26]). These small-time asymptotic results have found a wide scope of applications ranging from estimation methods based on high-frequency sampling observations of the process (see, e.g., ...

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...troduction Stemming in part from its importance for model testing and calibration, small-time asymptotics of option prices have received a lot of attention in recent years (see, e.g., [3], [5], [10], =-=[11]-=-, [18], [19], [20], [22], [25], and references therein). The fact that option prices and implied volatilities exhibit sharply different behaviors under different model assumptions provides a natural t...