Pricing the implicit contracts in the Paris Club debt buybacks

In 2005, more than 20 billion dollars were bought back by Paris Club debtors: Russia USD 15 billion Poland USD 5.4 billion and Peru USD 1.5 billion. During the first half of 2006, more than USD 30 billion in buybacks was announced: Russia USD 22 billion, Algeria USD 8 billion dollars, Brazil USD 1.5 billion. The buybacks consisted of the prepayment of debts at par with no penalties. These transactions were carried out at a discount of more than 20% compared to their net present value. The total loss incurred by creditors in the three buybacks is estimated at more than USD 10 billion. This raises the question as to why the Paris Club creditors agreed to the buybacks voluntarily. It appears that these buybacks are the result of the exercise of specific contracts previously agreed with the debtors in the 1990s, without receiving any compensation for this and without assessing the consequences. These implicit contracts make it possible to formalise the respective interests for creditors and debtors. Their pricing requires the use of financial mathematics tools (derivatives) and stochastic models for interest rates (Vasicek), but applied in the Paris Club framework.buyback; Paris Club; par value; Vasicek model; creditor cartel

Sub-Poissonian laser emission from a single-electron permanently interacting with a single-mode cavity

Quiet (or sub-Poissonian) oscillators generate a number of dissipation events whose variance is less than the mean. It was shown in 1984 by Golubev and Sokolov that lasers driven by regular pumps are quiet in that sense. The purpose of this paper is to show that, as long as the laser-detector system is strictly stationary, quantization of the optical field is not required to explain such phenomena. The theory presented here is semi-classical, yet exact. Previous theories considering excited-state atoms regularly-injected in resonators, on the other hand, do require in principle light quantization. Specifically, we consider a laser involving a single electron permanently interacting with the field and driven by a constant-potential battery, and point out a similarity with reflex klystrons. The detected noise is found to be only 7/8 of the shot-noise level. It is therefore sub-Poissonian. Our calculations are related to resonance-fluorescence treatments but with different physical interpretations.Comment: 7 pages, submitted to Phys Rev

Quiet Lasers

We call "quiet laser" a stationary laser that generates in detectors regular photo-electrons (sub-Poisson statistics). It follows from the law of conservation of energy that this is so when the laser power supply does not fluctuate. Various configurations are analyzed on the basis of the Planck (1907) semi-classical concept: "I am not seeking the meaning of light quanta in the vacuum but rather in places where emission and absorption occur, and I assume that what happens in the vacuum is rigorously described by Maxwell's equations". Exact agreement with Quantum Optics results is noted. Comments welcome!Comment: 186 page

Semi-classical theory of quiet lasers. I: Principles

When light originating from a laser diode driven by non-fluctuating electrical currents is incident on a photo-detector, the photo-current does not fluctuate much. Precisely, this means that the variance of the number of photo-electrons counted over a large time interval is much smaller that the average number of photo-electrons. At non-zero Fourier frequency $\Omega$ the photo-current power spectrum is of the form $\Omega^2/(1+\Omega^2)$ and thus vanishes as $\Omega\to 0$, a conclusion equivalent to the one given above. The purpose of this paper is to show that results such as the one just cited may be derived from a (semi-classical) theory in which neither the optical field nor the electron wave-function are quantized. We first observe that almost any medium may be described by a circuit and distinguish (possibly non-linear) conservative elements such as pure capacitances, and conductances that represent the atom-field coupling. The theory rests on the non-relativistic approximation. Nyquist noise sources (in which the Planck term $\hbar\omega/2$ is being restored) are associated with positive or negative conductances, and the law of average-energy conservation is enforced. We consider mainly second-order correlations in stationary linearized regimes.Comment: 116 pages Second draft of a book project. To be completed by a part II incuding extended details on application of the theor

Statistics of non-interacting bosons and fermions in micro-canonical, canonical and grand-canonical ensembles: A survey

The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, particularly by Khinchin in 1950. Others were obtained only recently, often motivated by experimental results relating to atomic confinement. A number of formulas are reported for the first time in the present paper. Formulas applicable to the case where the system may exchange energy but not particles with a reservoir (canonical ensemble) are derived from the micro-canonical ensemble expressions. The differences between the three ensembles tend to vanish in the so-called Thermodynamics limit, that is, when the number of particles and the volume go to infinity while the particle number density remains constant. But we are mostly interested in systems of moderate size, often referred to as being mesoscopic, where the grand-canonical formalism is not applicable. The mathematical results rest primarily on the enumeration of partitions of numbers.Comment: 18 pages, submitted to J. Phys.

Modified logarithmic Sobolev inequalities and transportation inequalities

We present a class of modified logarithmic Sobolev inequality, interpolating between Poincar\'e and logarithmic Sobolev inequalities, suitable for measures of the type \exp(-|x|^\al) or more complex \exp(-|x|^\al\log^\beta(2+|x|)) (\al\in]1,2[ and \be\in\dR) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincar\'e inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities

A simple quantum heat engine

Quantum heat engines employ as working agents multi-level systems instead of gas-filled cylinders. We consider particularly two-level agents such as electrons immersed in a magnetic field. Work is produced in that case when the electrons are being carried from a high-magnetic-field region into a low-magnetic-field region. In watermills, work is produced instead when some amount of fluid drops from a high-altitude reservoir to a low-altitude reservoir. We show that this purely mechanical engine may in fact be considered as a two-level quantum heat engine, provided the fluid is viewed as consisting of n molecules of weight one and N-n molecules of weight zero. Weight-one molecules are analogous to electrons in their higher energy state, while weight-zero molecules are analogous to electrons in their lower energy state. More generally, fluids consist of non-interacting molecules of various weights. It is shown that, not only the average value of the work produced per cycle, but also its fluctuations, are the same for mechanical engines and quantum (Otto) heat engines. The reversible Carnot cycles are approached through the consideration of multiple sub-reservoirs.Comment: RevTeX 9 pages, 4 figures, paper shortened, improved presentatio

A simple model for Carnot heat engines

We present a (random) mechanical model consisting of two lottery-like reservoirs at altitude $E_h$ and $E_l<E_h$, respectively, in the earth's gravitational field. Both reservoirs consist of $N$ possible ball locations. The upper reservoir contains initially $n_h\le N$ weight-1 balls and the lower reservoir contains initially $n_l\le N$ weight-1 balls. Empty locations are treated as weight-0 balls. These reservoirs are being shaken up so that all possible ball configurations are equally likely to occur. A cycle consists of exchanging a ball randomly picked from the higher reservoir and a ball randomly picked from the lower reservoir. It is straightforward to show that the efficiency, defined as the ratio of the average work produced to the average energy lost by the higher reservoir is $\eta=1-E_l/E_h$. We then relate this system to a heat engine. This thermal interpretation is applicable only when the number of balls is large. We define the entropy as the logarithm of the number of ball configurations in a reservoir, namely $S(n)=\ln[N!/n!(N-n)!]$, with subscripts $h,l$ appended to $S$ and to $n$. When $n_l$ does not differ much from $n_h$, the system efficiency quoted above is found to coincide with the maximum efficiency $\eta=1-T_l/T_h$, where the $T$ are absolute temperatures defined from the above expression of $S$. Fluctuations are evaluated in Appendix A, and the history of the Carnot discovery (1824) is recalled in Appendix B. Only elementary physical and mathematical concepts are employed.Comment: To appear in American Journal of Physic