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

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

‘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

Individual and collective stock dynamics: intra-day seasonalities

We establish several new stylised facts concerning the intra-day seasonalities of stock dynamics. Beyond the well known U-shaped pattern of the volatility, we find that the average correlation between stocks increases throughout the day, leading to a smaller relative dispersion between stocks. Somewhat paradoxically, the kurtosis (a measure of volatility surprises) reaches a minimum at the open of the market, when the volatility is at its peak. We confirm that the dispersion kurtosis is a markedly decreasing function of the index return. This means that during large market swings, the idiosyncratic component of the stock dynamics becomes sub-dominant. In a nutshell, early hours of trading are dominated by idiosyncratic or sector specific effects with little surprises, whereas the influence of the market factor increases throughout the day, and surprises become more frequent.Comment: 9 pages, 7 figure

Volatility Effects on the Escape Time in Financial Market Models

We shortly review the statistical properties of the escape times, or hitting times, for stock price returns by using different models which describe the stock market evolution. We compare the probability function (PF) of these escape times with that obtained from real market data. Afterwards we analyze in detail the effect both of noise and different initial conditions on the escape time in a market model with stochastic volatility and a cubic nonlinearity. For this model we compare the PF of the stock price returns, the PF of the volatility and the return correlation with the same statistical characteristics obtained from real market data.Comment: 12 pages, 9 figures, to appear in Int. J. of Bifurcation and Chaos, 200

Some Open Points in Nonextensive Statistical Mechanics

We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.Comment: 30 pages including 6 figures. Invited paper to appear in the International Journal of Bifurcation and Chao

Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution

The nonlinear diffusion equation $\frac{\partial \rho}{\partial t}=D \tilde{\Delta} \rho^\nu$ is analyzed here, where $\tilde{\Delta}\equiv \frac{1}{r^{d-1}}\frac{\partial}{\partial r} r^{d-1-\theta} \frac{\partial}{\partial r}$, and $d$, $\theta$ and $\nu$ are real parameters. This equation unifies the anomalous diffusion equation on fractals ($\nu =1$) and the spherical anomalous diffusion for porous media ($\theta=0$). Exact point-source solution is obtained, enabling us to describe a large class of subdiffusion ($\theta > (1-\nu)d$), normal diffusion ($\theta= (1-\nu)d$) and superdiffusion ($\theta < (1-\nu)d$). Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.Comment: 3 pages, 2 eps figure

Goodness-of-Fit tests with Dependent Observations

We revisit the Kolmogorov-Smirnov and Cram\'er-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not distribution-free but rather depend on all the lagged bivariate copulas. These objects, that we call "self-copulas", encode all the non-linear temporal dependences. We introduce a specific, log-normal model for these self-copulas, for which a number of analytical results are derived. An application to financial time series is provided. As is well known, the dependence is to be long-ranged in this case, a finding that we confirm using self-copulas. As a consequence, the acceptance rates for GoF tests are substantially higher than if the returns were iid random variables.Comment: 26 page

Liquidity and the multiscaling properties of the volume traded on the stock market

We investigate the correlation properties of transaction data from the New York Stock Exchange. The trading activity f(t) of each stock displays a crossover from weaker to stronger correlations at time scales 60-390 minutes. In both regimes, the Hurst exponent H depends logarithmically on the liquidity of the stock, measured by the mean traded value per minute. All multiscaling exponents tau(q) display a similar liquidity dependence, which clearly indicates the lack of a universal form assumed by other studies. The origin of this behavior is both the long memory in the frequency and the size of consecutive transactions.Comment: 7 pages, 3 figures, submitted to Europhysics Letter

Pricing Exotic Options in a Path Integral Approach

In the framework of Black-Scholes-Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price European path dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the cases of Asian, barrier knock out, reverse cliquet and basket call options, evaluating prices and Greeks. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at-the-money and out-of-the-money options, the path integral approach exhibits competitive performances.Comment: 21 pages, LaTeX, 3 figures, 6 table