37,475 research outputs found
From Smile Asymptotics to Market Risk Measures
The left tail of the implied volatility skew, coming from quotes on
out-of-the-money put options, can be thought to reflect the market's assessment
of the risk of a huge drop in stock prices. We analyze how this market
information can be integrated into the theoretical framework of convex monetary
measures of risk. In particular, we make use of indifference pricing by dynamic
convex risk measures, which are given as solutions of backward stochastic
differential equations (BSDEs), to establish a link between these two
approaches to risk measurement. We derive a characterization of the implied
volatility in terms of the solution of a nonlinear PDE and provide a small
time-to-maturity expansion and numerical solutions. This procedure allows to
choose convex risk measures in a conveniently parametrized class, distorted
entropic dynamic risk measures, which we introduce here, such that the
asymptotic volatility skew under indifference pricing can be matched with the
market skew. We demonstrate this in a calibration exercise to market implied
volatility data.Comment: 24 pages, 4 figure
Neutrino Oscillations in the Early Universe with Nonequilibrium Neutrino Distributions
Around one second after the big bang, neutrino decoupling and -
annihilation distort the Fermi-Dirac spectrum of neutrino energies. Assuming
neutrinos have masses and can mix, we compute the distortions using
nonequilibrium thermodynamics and the Boltzmann equation. The flavor behavior
of neutrinos is studied during and following the generation of the distortion.Comment: accepted for publication in Physical Review
Mappings of least Dirichlet energy and their Hopf differentials
The paper is concerned with mappings between planar domains having least
Dirichlet energy. The existence and uniqueness (up to a conformal change of
variables in the domain) of the energy-minimal mappings is established within
the class of strong limits of homeomorphisms in the
Sobolev space , a result of considerable interest in the
mathematical models of Nonlinear Elasticity. The inner variation leads to the
Hopf differential and its trajectories.
For a pair of doubly connected domains, in which has finite conformal
modulus, we establish the following principle:
A mapping is energy-minimal if and only if
its Hopf-differential is analytic in and real along the boundary of .
In general, the energy-minimal mappings may not be injective, in which case
one observes the occurrence of cracks in . Nevertheless, cracks are
triggered only by the points in the boundary of where fails to be
convex. The general law of formation of cracks reads as follows:
Cracks propagate along vertical trajectories of the Hopf differential from
the boundary of toward the interior of where they eventually terminate
before making a crosscut.Comment: 51 pages, 4 figure
- …