Asymptotic results for time-changed Lévy processes sampled at hitting times

Abstract

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

“…(4.47)), and t −1/Y X t D −→ Z 1 , where Z 1 is a strictly Y -stable random variable (cf. [36], Prop. 1).…”

Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility

“…in [5]. Concretely, using the density transformation P → P * and then change of variables [14]), which is the same as the distribution of Z 1 under P. For the first term in (A.6), we further decompose it as follows:…”

Asymptotically optimal discretization of hedging strategies with jumps

“…This implies that Y ε,1 and Y ε,2 also converge to Y * almost surely, where Y ε,1 t = Y ε t − tB 2 ε α−1 and Y ε,2 t = Y ε t + tB 2 ε α−1 . Now using that the application which to a function f in the Skorohod space associates its first hitting time of a constant barrier is continuous at almost all f which are sample paths of strictly stable processes (see Proposition VI.2.11 in [20] and its use in [22]), we obtain that σ ε i converges almost surely to σ * for i = 1, 2, 3, where σ ε i and σ * are defined through Y ε,1 , Y ε,2 , Y * in the same way as τ ε i and τ * through X ε,1 , X ε,2 , X * . Moreover, since σ ε 3 ≤ σ j,ε for all ε, we also have that σ ε 2 ∧ σ j,ε → σ * almost surely.…”

“…In the case where X is a Lévy process, the parameter α coincides with the Blumenthal-Getoor index of X; see [4]. − The assumption 1 < α < 2 implies that X has infinite variation and ensures that the local behavior of the process is determined by the jumps rather than by the drift part; see [22]. Note that in a recent statistical study on liquid assets [1], the jump activity index defined similarly to our parameter α was estimated between 1.4 and 1.7.…”