Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model

We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility. We study the resulting European call price and its implied volatility for a range of parameters consistent with daily Dow Jones Index data.Comment: 26 pages, 6 colored figure

SPI observations of positron annihilation radiation from the 4th galactic quadrant: sky distribution

During its first year in orbit the INTEGRAL observatory performed deep exposures of the Galactic Center region and scanning observations of the Galactic plane. We report on the status of our analysis of the positron annihilation radiation from the 4th Galactic quadrant with the spectrometer SPI, focusing on the sky distribution of the 511 keV line emission. The analysis methods are described; current constraints and limits on the Galactic bulge emission and the bulge-to-disk ratio are presented.Comment: 4 pages, 2 figures, accepted for publication in the proceedings of the 5th INTEGRAL worksho

Detection of gamma-ray lines from interstellar 60Fe by the high resolution spectrometer SPI

It is believed that core-collapse supernovae (CCSN), occurring at a rate about once per century, have seeded the interstellar medium with long-lived radioisotopes such as 60Fe (half-life 1.5 Myr), which can be detected by the gamma rays emitted when they beta-decay. Here we report the detection of the 60Fe decay lines at 1173 keV and 1333 keV with fluxes 3.7 +/- 1.1 x 10(-5) ph cm(-2) s(-1) per line, in spectra taken by the SPI spectrometer on board INTEGRAL during its first year. The same analysis applied to the 1809 keV line of 26Al yielded a line flux ratio 60Fe/26Al = 0.11 +/- 0.03. This supports the hypothesis that there is an extra source of 26Al in addition to CCSN.Comment: 4pp., 5 Figs., accepted by Astronomy & Astrophysics (letter), ref.'s comments include

Probability distribution of returns in the exponential Ornstein-Uhlenbeck model

We analyze the problem of the analytical characterization of the probability distribution of financial returns in the exponential Ornstein-Uhlenbeck model with stochastic volatility. In this model the prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of an hidden variable Y governed by a mean-reverting process. We derive closed-form expressions for the probability distribution and its characteristic function in two limit cases. In the first one the fluctuations of Y are larger than the volatility normal level, while the second one corresponds to the assumption of a small stationary value for the variance of Y. Theoretical results are tested numerically by intensive use of Monte Carlo simulations. The effectiveness of the analytical predictions is checked via a careful analysis of the parameters involved in the numerical implementation of the Euler-Maruyama scheme and is tested on a data set of financial indexes. In particular, we discuss results for the German DAX30 and Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data and the theoretical description.Comment: 26 pages, 9 figures and 3 tables. New section with real data analysis and related references added, some minor typos corrected. Accepted for publication on JSTA

Risk measures with non-Gaussian fluctuations

Reliable calculations of financial risk require that the fat-tailed nature of prices changes is included in risk measures. To this end, a non-Gaussian approach to financial risk management is presented, modeling the power-law tails of the returns distribution in terms of a Student-$t$ (or Tsallis) distribution. Non-Gaussian closed-form solutions for Value-at-Risk and Expected Shortfall are obtained and standard formulae known in the literature under the normality assumption are recovered as a special case. The implications of the approach for risk management are demonstrated through an empirical analysis of financial time series from the Italian stock market. Detailed comparison with the results of the widely used procedures of quantitative finance, such as parametric normal approach, RiskMetrics methodology and historical simulation, as well as with previous findings in the literature, are shown and commented. Particular attention is paid to quantify the size of the errors affecting the risk measures obtained according to different methodologies, by employing a bootstrap technique.