From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery's work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note: text overlap with arXiv:0909.491

Parametric arbitrage-free models for implied smile dynamics

Based on the theory of Tangent Levy model [1] developed by R. Carmona and S. Nadtochiy, this thesis gives a paramatrized realization of dynamic implied smile.\ud \ud After specifying a Dirac style Levy measure, we give argument about the consistency issue of our model with the Tangent Levy Model. A corresponding no arbitrage drift condition is derived for the parameters. Numerical setup under our model for option pricing and parameter estimation for calibration is given. Implementation results are illustrated in detail and in the end we provide with simulation results of one day ahead implied smile

Applications of Least Mean Square (LMS) Algorithm Regression in Time-Series Analysis

In this paper we present a very brief description of least mean square algorithm with applications in time-series analysis of economic and financial time series. We present some numerical applications; forecasts for the Gross Domestic Product growth rate of UK and Italy, forecasts for S&P 500 stock index returns and finally we examine the day of the week effect of FTSE 100 for a short period. A full programming routine written in MATLAB software environment is provided for replications and further research applications.LMS; Least Mean Square Algorithm; MATLAB; time-series; stock returns; gross domestic product; forecast