257 research outputs found
Positive definite and related functions on hypergroups
In this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Lévy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups
Holomorphic transforms with application to affine processes
In a rather general setting of It\^o-L\'evy processes we study a class of
transforms (Fourier for example) of the state variable of a process which are
holomorphic in some disc around time zero in the complex plane. We show that
such transforms are related to a system of analytic vectors for the generator
of the process, and we state conditions which allow for holomorphic extension
of these transforms into a strip which contains the positive real axis. Based
on these extensions we develop a functional series expansion of these
transforms in terms of the constituents of the generator. As application, we
show that for multidimensional affine It\^o-L\'evy processes with state
dependent jump part the Fourier transform is holomorphic in a time strip under
some stationarity conditions, and give log-affine series representations for
the transform.Comment: 30 page
Continuous Equilibrium in Affine and Information-Based Capital Asset Pricing Models
We consider a class of generalized capital asset pricing models in continuous
time with a finite number of agents and tradable securities. The securities may
not be sufficient to span all sources of uncertainty. If the agents have
exponential utility functions and the individual endowments are spanned by the
securities, an equilibrium exists and the agents' optimal trading strategies
are constant. Affine processes, and the theory of information-based asset
pricing are used to model the endogenous asset price dynamics and the terminal
payoff. The derived semi-explicit pricing formulae are applied to numerically
analyze the impact of the agents' risk aversion on the implied volatility of
simultaneously-traded European-style options.Comment: 24 pages, 4 figure
Yield Curve Shapes and the Asymptotic Short Rate Distribution in Affine One-Factor Models
We consider a model for interest rates, where the short rate is given by a
time-homogenous, one-dimensional affine process in the sense of Duffie,
Filipovic and Schachermayer. We show that in such a model yield curves can only
be normal, inverse or humped (i.e. endowed with a single local maximum). Each
case can be characterized by simple conditions on the present short rate. We
give conditions under which the short rate process will converge to a limit
distribution and describe the limit distribution in terms of its cumulant
generating function. We apply our results to the Vasicek model, the CIR model,
a CIR model with added jumps and a model of Ornstein-Uhlenbeck type
On small-noise equations with degenerate limiting system arising from volatility models
The one-dimensional SDE with non Lipschitz diffusion coefficient is widely
studied in mathematical finance. Several works have proposed asymptotic
analysis of densities and implied volatilities in models involving instances of
this equation, based on a careful implementation of saddle-point methods and
(essentially) the explicit knowledge of Fourier transforms. Recent research on
tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and
S.~Violante. Marginal density expansions for diffusions and stochastic
volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with
the rescaled variable : while
allowing to turn a space asymptotic problem into a small- problem
with fixed terminal point, the process satisfies a SDE in
Wentzell--Freidlin form (i.e. with driving noise ). We prove a
pathwise large deviation principle for the process as
. As it will become clear, the limiting ODE governing the
large deviations admits infinitely many solutions, a non-standard situation in
the Wentzell--Freidlin theory. As for applications, the -scaling
allows to derive exact log-asymptotics for path functionals of the process:
while on the one hand the resulting formulae are confirmed by the CIR-CEV
benchmarks, on the other hand the large deviation approach (i) applies to
equations with a more general drift term and (ii) potentially opens the way to
heat kernel analysis for higher-dimensional diffusions involving such an SDE as
a component.Comment: 21 pages, 1 figur
Analysis of Fourier transform valuation formulas and applications
The aim of this article is to provide a systematic analysis of the conditions
such that Fourier transform valuation formulas are valid in a general
framework; i.e. when the option has an arbitrary payoff function and depends on
the path of the asset price process. An interplay between the conditions on the
payoff function and the process arises naturally. We also extend these results
to the multi-dimensional case, and discuss the calculation of Greeks by Fourier
transform methods. As an application, we price options on the minimum of two
assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ
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