A diffusive model with a price dependent diffusion coefficient was recently proposed to explain the occurrence of non-Gaussian price return distributions observed empirically in real markets
[J.L. McCauley and G.H. Gunaratne, Physica A 329, 178 (2003)].
Depending on the functional form of the diffusion coefficient, the exactly solved continuum limit of the model can produce either an exponential distribution, or a "fat-tailed" power-law distribution of returns. Real markets, however, are discrete, and, in this paper, the effects of discreteness on the model are explored. Discrete distributions from simulations and from numerically exact calculations are presented and compared to the corresponding distributions of the continuum model. A type of phase transition is discovered in discrete models that lead to fat-tailed distributions in the continuum limit, sheading light on the nature of such distributions. The transition is to a phase in which infinite price changes can occur in finite time.
A new theory for pricing of options is presented. It is based on the assumption that successive movements depend on the value of the return. The solution to the Fokker-Planck equation is shown to be an asymmetric exponential distribution, similar to those observed in intra-day currency markets. The "volatility smile", used by traders to correct the Black-Scholes pricing is shown to be a heuristic mechanism to implement options pricing formulae derived from our theory.
A superconducting quantum interference grating (SQUIG), consisting of many Josephson junctions in parallel, exhibits quantum interference analogous to the optical interference pattern produced by a diffraction grating, just as the dc SQUID is the magnetic analog of the double slit in optics. The field- dependent critical current of a SQUIG with sufficiently low inductance exhibits a series of sharp peaks, similar to the position-dependent intensity of a multiple-slit interference pattern. Such a device has the potential for improved sensitivity as a magnetic sensor because of its enhanced flux-to-voltage transfer coefficient. The behavior is quite different, however, when the inductance is large. In this limit, the field-dependent critical current becomes extremely hysteretic and exhibits self- organized phase coherence, in which a peak in the critical current (indicating coherence of the junction phases) is observed after each reversal of the field sweep. We have observed this phenomenon in large inductance arrays with up to ten YBa2Cu3O7 (YBCO) bi-crystal grain boundary junctions in parallel. We have modeled the behavior using a Frenkel-Kontorova model generalized to include global interactions due to mutual inductance. According to our model calculations, self-organized phase coherence occurs when the system relaxes from its critical state, causing the phases to simultaneously relax to minima in the periodic potential representing the Josephson coupling energy, and thus to become coherent modulo 2(pi) .
Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks.
You are receiving this notice because your organization may not have SPIE eBooks access.*
*Shibboleth/Open Athens users─please
sign in
to access your institution's subscriptions.
To obtain this item, you may purchase the complete book in print or electronic format on
SPIE.org.
INSTITUTIONAL Select your institution to access the SPIE Digital Library.
PERSONAL Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.