264 research outputs found

### Martingale Option Pricing

We show that our generalization of the Black-Scholes partial differential
equation (pde) for nontrivial diffusion coefficients is equivalent to a
Martingale in the risk neutral discounted stock price. Previously, this was
proven for the case of the Gaussian logarithmic returns model by Harrison and
Kreps, but we prove it for much a much larger class of returns models where the
diffusion coefficient depends on both returns x and time t. That option prices
blow up if fat tails in logarithmic returns x are included in the market
dynamics is also explained

### Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets

Arguably the most important problem in quantitative finance is to understand
the nature of stochastic processes that underlie market dynamics. One aspect of
the solution to this problem involves determining characteristics of the
distribution of fluctuations in returns. Empirical studies conducted over the
last decade have reported that they arenon-Gaussian, scale in time, and have
power-law(or fat) tails. However, because they use sliding interval methods of
analysis, these studies implicitly assume that the underlying process has
stationary increments. We explicitly show that this assumption is not valid for
the Euro-Dollar exchange rate between 1999-2004. In addition, we find that
fluctuations in returns of the exchange rate are uncorrelated and scale as
power-laws for certain time intervals during each day. This behavior is
consistent with a diffusive process with a diffusion coefficient that depends
both on the time and the price change. Within scaling regions, we find that
sliding interval methods can generate fat-tailed distributions as an artifact,
and that the type of scaling reported in many previous studies does not exist.Comment: 12 pages, 4 figure

### Rhombic Patterns: Broken Hexagonal Symmetry

Landau-Ginzburg equations derived to conserve two-dimensional spatial symmetries lead to the prediction that rhombic arrays with characteristic angles slightly differ from 60 degrees should form in many systems. Beyond the bifurcation from the uniform state to patterns, rhombic patterns are linearly stable for a band of angles near the 60 degrees angle of regular hexagons. Experiments conducted on a reaction-diffusion system involving a chlorite-iodide-malonic acid reaction yield rhombic patterns in good accord with the theory.Energy Laboratory of the University of HoustonOffice of Naval ResearchU.S. Department of Energy Office of Basic Energy SciencesRobert A. Welch FoundationCenter for Nonlinear Dynamic

### Systematic derivation of a rotationally covariant extension of the 2-dimensional Newell-Whitehead-Segel equation

An extension of the Newell-Whitehead-Segel amplitude equation covariant under
abritrary rotations is derived systematically by the renormalization group
method.Comment: 8 pages, to appear in Phys. Rev. Letters, March 18, 199

### Emergence of Order in Textured Patterns

A characterization of textured patterns, referred to as the disorder function
\bar\delta(\beta), is used to study properties of patterns generated in the
Swift-Hohenberg equation (SHE). It is shown to be an intensive,
configuration-independent measure. The evolution of random initial states under
the SHE exhibits two stages of relaxation. The initial phase, where local
striped domains emerge from a noisy background, is quantified by a power law
decay \bar\delta(\beta) \sim t^{-{1/2} \beta}. Beyond a sharp transition a
slower power law decay of \bar\delta(\beta), which corresponds to the
coarsening of striped domains, is observed. The transition between the phases
advances as the system is driven further from the onset of patterns, and
suitable scaling of time and \bar\delta(\beta) leads to the collapse of
distinct curves. The decay of $\bar\delta(\beta)$ during the initial phase
remains unchanged when nonvariational terms are added to the underlying
equations, suggesting the possibility of observing it in experimental systems.
In contrast, the rate of relaxation during domain coarsening increases with the
coefficient of the nonvariational term.Comment: 9 Pages, 8 Postscript Figures, 3 gif Figure

### Universal bifurcation property of two- or higher-dimensional dissipative systems in parameter space: Why does 1D symbolic dynamics work so well?

The universal bifurcation property of the H\'enon map in parameter space is
studied with symbolic dynamics. The universal-$L$ region is defined to
characterize the bifurcation universality. It is found that the universal-$L$
region for relative small $L$ is not restricted to very small $b$ values. These
results show that it is also a universal phenomenon that universal sequences
with short period can be found in many nonlinear dissipative systems.Comment: 10 pages, figures can be obtained from the author, will appeared in
J. Phys.

- …