1,501 research outputs found

### A Non-Gaussian Option Pricing Model with Skew

Closed form option pricing formulae explaining skew and smile are obtained
within a parsimonious non-Gaussian framework. We extend the non-Gaussian option
pricing model of L. Borland (Quantitative Finance, {\bf 2}, 415-431, 2002) to
include volatility-stock correlations consistent with the leverage effect. A
generalized Black-Scholes partial differential equation for this model is
obtained, together with closed-form approximate solutions for the fair price of
a European call option. In certain limits, the standard Black-Scholes model is
recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and
Ross. Alternative methods of solution to that model are thereby also discussed.
The model parameters are partially fit from empirical observations of the
distribution of the underlying. The option pricing model then predicts European
call prices which fit well to empirical market data over several maturities.Comment: 37 pages, 11 ps figures, minor changes, typos corrected, to appear in
Quantitative Financ

### Microscopic dynamics underlying the anomalous diffusion

The time dependent Tsallis statistical distribution describing anomalous
diffusion is usually obtained in the literature as the solution of a non-linear
Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347
(1995)]. The scope of the present paper is twofold. Firstly we show that this
distribution can be obtained also as solution of the non-linear porous media
equation. Secondly we prove that the time dependent Tsallis distribution can be
obtained also as solution of a linear FP equation [G. Kaniadakis and P.
Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the
velocity, that describes a generalized Brownian motion. This linear FP equation
is shown to arise from a microscopic dynamics governed by a standard Langevin
equation in presence of multiplicative noise.Comment: 4 pag. - no figures. To appear on Phys. Rev. E 62, September 200

### Delta Hedged Option Valuation with Underlying Non-Gaussian Returns

The standard Black-Scholes theory of option pricing is extended to cope with
underlying return fluctuations described by general probability distributions.
A Langevin process and its related Fokker-Planck equation are devised to model
the market stochastic dynamics, allowing us to write and formally solve the
generalized Black-Scholes equation implied by dynamical hedging. A systematic
expansion around a non-perturbative starting point is then implemented,
recovering the Matacz's conjectured option pricing expression. We perform an
application of our formalism to the real stock market and find clear evidence
that while past financial time series can be used to evaluate option prices
before the expiry date with reasonable accuracy, the stochastic character of
volatility is an essential ingredient that should necessarily be taken into
account in analytical option price modeling.Comment: Four pages; two eps figure

### Ferromagnetic material in the eastern red-spotted newt notophthalmus viridescens

Behavioral results obtained from the eastern red-spotted newt (Notophthalmus viridescens) led to the suggestion of a hybrid homing system involving inputs from both a light-dependent and a non-light-dependent mechanism. To evaluate the possible role of a receptor based on biogenic magnetite in this animal, we performed magnetometry experiments on a set of newts previously used in behavioral assays. The natural remanent magnetization (NRM) carried by these newts was strong enough to be measured easily using a direct-current-biased superconducting quantum interference device functioning as a moment magnetometer. Isothermal remanent magnetizations were two orders of magnitude higher than the NRM, suggesting that ferromagnetic material consistent with magnetite is present in the body of the newt. The NRM has no preferential orientation among the animals when analyzed relative to their body axis, and the demagnetization data show that, overall, the magnetic material grains are not aligned parallel to each other within each newt. Although the precise localization of the particles was not possible, the data indicate that magnetite is not clustered in a limited area. A quantity of single-domain magnetic material is present which would be adequate for use in either a magnetic intensity or direction receptor. Our data, when combined with the functional properties of homing, suggest a link between this behavioral response and the presence of ferromagnetic material, raising the possibility that magnetite is involved at least in the map component of homing of the eastern red-spotted newt

### Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance

We show by explicit closed form calculations that a Hurst exponent H that is
not 1/2 does not necessarily imply long time correlations like those found in
fractional Brownian motion. We construct a large set of scaling solutions of
Fokker-Planck partial differential equations where H is not 1/2. Thus Markov
processes, which by construction have no long time correlations, can have H not
equal to 1/2. If a Markov process scales with Hurst exponent H then it simply
means that the process has nonstationary increments. For the scaling solutions,
we show how to reduce the calculation of the probability density to a single
integration once the diffusion coefficient D(x,t) is specified. As an example,
we generate a class of student-t-like densities from the class of quadratic
diffusion coefficients. Notably, the Tsallis density is one member of that
large class. The Tsallis density is usually thought to result from a nonlinear
diffusion equation, but instead we explicitly show that it follows from a
Markov process generated by a linear Fokker-Planck equation, and therefore from
a corresponding Langevin equation. Having a Tsallis density with H not equal to
1/2 therefore does not imply dynamics with correlated signals, e.g., like those
of fractional Brownian motion. A short review of the requirements for
fractional Brownian motion is given for clarity, and we explain why the usual
simple argument that H unequal to 1/2 implies correlations fails for Markov
processes with scaling solutions. Finally, we discuss the question of scaling
of the full Green function g(x,t;x',t') of the Fokker-Planck pde.Comment: to appear in Physica

### ‘Fixed-axis’ magnetic orientation by an amphibian: non-shoreward-directed compass orientation, misdirected homing or positioning a magnetite-based map detector in a consistent alignment relative to the magnetic field?

Experiments were carried out to investigate the earlier prediction that prolonged exposure to long-wavelength (>500 nm) light would eliminate homing orientation by male Eastern red-spotted newts Notophthalmus viridescens. As in previous experiments, controls held in outdoor tanks under natural lighting conditions and tested in a visually uniform indoor arena under full-spectrum light were homeward oriented. As predicted, however, newts held under long-wavelength light and tested under either full-spectrum or long-wavelength light (>500 nm) failed to show consistent homeward orientation. The newts also did not orient with respect to the shore directions in the outdoor tanks in which they were held prior to testing. Unexpectedly, however, the newts exhibited bimodal orientation along a more-or-less `fixed' north-northeast—south-southwest magnetic axis. The orientation exhibited by newts tested under full-spectrum light was indistinguishable from that of newts tested under long-wavelength light, although these two wavelength conditions have previously been shown to differentially affect both shoreward compass orientation and homing orientation. To investigate the possibility that the `fixed-axis' response of the newts was mediated by a magnetoreception mechanism involving single-domain particles of magnetite, natural remanent magnetism (NRM) was measured from a subset of the newts. The distribution of NRM alignments with respect to the head—body axis of the newts was indistinguishable from random. Furthermore, there was no consistent relationship between the NRM of individual newts and their directional response in the overall sample. However, under full-spectrum, but not long-wavelength, light, the alignment of the NRM when the newts reached the 20 cm radius criterion circle in the indoor testing arena (estimated by adding the NRM alignment measured from each newt to its magnetic bearing) was non-randomly distributed. These findings are consistent with the earlier suggestion that homing newts use the light-dependent magnetic compass to align a magnetite-based `map detector' when obtaining the precise measurements necessary to derive map information from the magnetic field. However, aligning the putative map detector does not explain the fixed-axis response of newts tested under long-wavelength light. Preliminary evidence suggests that, in the absence of reliable directional information from the magnetic compass (caused by the 90° rotation of the response of the magnetic compass under long-wavelength light), newts may resort to a systematic sampling strategy to identify alignment(s) of the map detector that yields reliable magnetic field measurements

### Markov vs. nonMarkovian processes A comment on the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims
made in the paper Stochastic feedback, nonlinear families of Markov processes,
and nonlinear Fokker-Planck equations by T. D. Frank. Our comment centers on
the claims of a nonlinear Markov process and a nonlinear Fokker-Planck
equation. First, memory in transition densities is misidentified as a Markov
process. Second, Frank assumes that one can derive a Fokker-Planck equation
from a Chapman-Kolmogorov equation, but no proof was given that a
Chapman-Kolmogorov equation exists for memory-dependent processes. A nonlinear
Markov process is claimed on the basis of a nonlinear diffusion pde for a
1-point probability density. We show that, regardless of which initial value
problem one may solve for the 1-point density, the resulting stochastic
process, defined necessarily by the transition probabilities, is either an
ordinary linearly generated Markovian one, or else is a linearly generated
nonMarkovian process with memory. We provide explicit examples of diffusion
coefficients that reflect both the Markovian and the memory-dependent cases. So
there is neither a nonlinear Markov process nor nonlinear Fokker-Planck
equation for a transition density. The confusion rampant in the literature
arises in part from labeling a nonlinear diffusion equation for a 1-point
probability density as nonlinear Fokker-Planck, whereas neither a 1-point
density nor an equation of motion for a 1-point density defines a stochastic
process, and Borland misidentified a translation invariant 1-point density
derived from a nonlinear diffusion equation as a conditional probability
density. In the Appendix we derive Fokker-Planck pdes and Chapman-Kolmogorov
eqns. for stochastic processes with finite memory

### Option Pricing from Wavelet-Filtered Financial Series

We perform wavelet decomposition of high frequency financial time series into
large and small time scale components. Taking the FTSE100 index as a case
study, and working with the Haar basis, it turns out that the small scale
component defined by most ($\simeq$ 99.6%) of the wavelet coefficients can be
neglected for the purpose of option premium evaluation. The relevance of the
hugely compressed information provided by low-pass wavelet-filtering is related
to the fact that the non-gaussian statistical structure of the original
financial time series is essentially preserved for expiration times which are
larger than just one trading day.Comment: 4 pages, 1 figur

### Geometry of escort distributions

Given an original distribution, its statistical and probabilistic attributs
may be scanned by the associated escort distribution introduced by Beck and
Schlogl and employed in the formulation of nonextensive statistical mechanics.
Here, the geometric structure of the one-parameter family of the escort
distributions is studied based on the Kullback-Leibler divergence and the
relevant Fisher metric. It is shown that the Fisher metric is given in terms of
the generalized bit-variance, which measures fluctuations of the crowding index
of a multifractal. The Cramer-Rao inequality leads to the fundamental limit for
precision of statistical estimate of the order of the escort distribution. It
is also quantitatively discussed how inappropriate it is to use the original
distribution instead of the escort distribution for calculating the expectation
values of physical quantities in nonextensive statistical mechanics.Comment: 12 pages, no figure

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