133 research outputs found
Sieve-based confidence intervals and bands for L\'{e}vy densities
The estimation of the L\'{e}vy density, the infinite-dimensional parameter
controlling the jump dynamics of a L\'{e}vy process, is considered here under a
discrete-sampling scheme. In this setting, the jumps are latent variables, the
statistical properties of which can be assessed when the frequency and time
horizon of observations increase to infinity at suitable rates. Nonparametric
estimators for the L\'{e}vy density based on Grenander's method of sieves was
proposed in Figueroa-L\'{o}pez [IMS Lecture Notes 57 (2009) 117--146]. In this
paper, central limit theorems for these sieve estimators, both pointwise and
uniform on an interval away from the origin, are obtained, leading to pointwise
confidence intervals and bands for the L\'{e}vy density. In the pointwise case,
our estimators converge to the L\'{e}vy density at a rate that is arbitrarily
close to the rate of the minimax risk of estimation on smooth L\'{e}vy
densities. In the case of uniform bands and discrete regular sampling, our
results are consistent with the case of density estimation, achieving a rate of
order arbitrarily close to , where is the
number of observations. The convergence rates are valid, provided that is
smooth enough and that the time horizon and the dimension of the sieve
are appropriately chosen in terms of .Comment: Published in at http://dx.doi.org/10.3150/10-BEJ286 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias
We characterize the small-time asymptotic behavior of the exit probability of
a L\'evy process out of a two-sided interval and of the law of its overshoot,
conditionally on the terminal value of the process. The asymptotic expansions
are given in the form of a first-order term and a precise computable error
bound. As an important application of these formulas, we develop a novel
adaptive discretization scheme for the Monte Carlo computation of functionals
of killed L\'evy processes with controlled bias. The considered functionals
appear in several domains of mathematical finance (e.g., structural credit risk
models, pricing of barrier options, and contingent convertible bonds) as well
as in natural sciences. The proposed algorithm works by adding discretization
points sampled from the L\'evy bridge density to the skeleton of the process
until the overall error for a given trajectory becomes smaller than the maximum
tolerance given by the user.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ517 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
High-order short-time expansions for ATM option prices of exponential L\'evy models
In the present work, a novel second-order approximation for ATM option prices
is derived for a large class of exponential L\'{e}vy 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 time-, is of the form
, with only depending on , the degree of jump
activity, on , 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 . In contrast, under a
pure-jump model, the dependence on and on the separate intensities of
negative and positive small jumps are already reflected in the leading term,
which is of the form . The second-order term is shown to be of
the form and, therefore, its order of decay turns out to be
independent of . The asymptotic behavior of the corresponding Black-Scholes
implied volatilities is also addressed. Our approach is sufficiently general to
cover a wide class of L\'{e}vy processes which satisfy the latter property and
whose L\'{e}vy densitiy 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.Comment: 35 pages, 8 figures. This is an extension of our earlier submission
arXiv:1112.3111. To appear in Mathematical Financ
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