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

New set of measures to analyze non-equilibrium structures

We introduce a set of statistical measures that can be used to quantify non-equilibrium surface growth. They are used to deduce new information about spatiotemporal dynamics of model systems for spinodal decomposition and surface deposition. Patterns growth in the Cahn-Hilliard Equation (used to model spinodal decomposition) are shown to exhibit three distinct stages. Two models of surface growth, namely the continuous Kardar-Parisi-Zhang (KPZ) model and the discrete Restricted-Solid-On-Solid (RSOS) model are shown to have different saturation exponents

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