8,101 research outputs found
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model
We provide a general method to compute a Taylor expansion in time of implied
volatility for stochastic volatility models, using a heat kernel expansion.
Beyond the order 0 implied volatility which is already known, we compute the
first order correction exactly at all strikes from the scalar coefficient of
the heat kernel expansion. Furthermore, the first correction in the heat kernel
expansion gives the second order correction for implied volatility, which we
also give exactly at all strikes. As an application, we compute this asymptotic
expansion at order 2 for the SABR model.Comment: 27 pages; v2: typos fixed and a few notation changes; v3: published
version, typos fixed and comments added. in Large Deviations and Asymptotic
Methods in Finance, Springer (2015) 37-6
Principal Boundary on Riemannian Manifolds
We consider the classification problem and focus on nonlinear methods for
classification on manifolds. For multivariate datasets lying on an embedded
nonlinear Riemannian manifold within the higher-dimensional ambient space, we
aim to acquire a classification boundary for the classes with labels, using the
intrinsic metric on the manifolds. Motivated by finding an optimal boundary
between the two classes, we invent a novel approach -- the principal boundary.
From the perspective of classification, the principal boundary is defined as an
optimal curve that moves in between the principal flows traced out from two
classes of data, and at any point on the boundary, it maximizes the margin
between the two classes. We estimate the boundary in quality with its
direction, supervised by the two principal flows. We show that the principal
boundary yields the usual decision boundary found by the support vector machine
in the sense that locally, the two boundaries coincide. Some optimality and
convergence properties of the random principal boundary and its population
counterpart are also shown. We illustrate how to find, use and interpret the
principal boundary with an application in real data.Comment: 31 pages,10 figure
The Schrodinger Wave Functional and Vacuum State in Curved Spacetime II. Boundaries and Foliations
In a recent paper, general solutions for the vacuum wave functionals in the
Schrodinger picture were given for a variety of classes of curved spacetimes.
Here, we describe a number of simple examples which illustrate how the presence
of spacetime boundaries influences the vacuum wave functional and how physical
quantities are independent of the choice of spacetime foliation used in the
Schrodinger approach despite the foliation dependence of the wave functionals
themselves.Comment: 26 pages, 4 figures, LATE
Ultraviolet Limit of Open String Theory
We confirm the intuition that a string theory which is perturbatively
infrared finite is automatically perturbatively ultraviolet finite. Our
derivation based on the asymptotics of the Selberg trace formula for the Greens
function on a Riemann surface holds for both open and closed string amplitudes
and is independent of modular invariance and supersymmetry. The mass scale for
the open strings stretched between Dbranes suggests a natural world-sheet
ultraviolet regulator in the string path integral, preserving both T-duality
and open-closed string world-sheet duality. Note added (Jan 2005): Comments and
related references added.Comment: 22 pages, LaTeX. Note added (Jan 2005): comments and related ref
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