Kondo effect in systems with dynamical symmetries

This paper is devoted to a systematic exposure of the Kondo physics in quantum dots for which the low energy spin excitations consist of a few different spin multiplets $|S_{i}M_{i}>$. Under certain conditions (to be explained below) some of the lowest energy levels $E_{S_{i}}$ are nearly degenerate. The dot in its ground state cannot then be regarded as a simple quantum top in the sense that beside its spin operator other dot (vector) operators ${\bf R}_{n}$ are needed (in order to fully determine its quantum states), which have non-zero matrix elements between states of different spin multiplets $\ne 0$. These "Runge-Lenz" operators do not appear in the isolated dot-Hamiltonian (so in some sense they are "hidden"). Yet, they are exposed when tunneling between dot and leads is switched on. The effective spin Hamiltonian which couples the metallic electron spin ${\bf s}$ with the operators of the dot then contains new exchange terms, $J_{n} {\bf s} \cdot {\bf R}_{n}$ beside the ubiquitous ones $J_{i} {\bf s}\cdot {\bf S}_{i}$. The operators ${\bf S}_{i}$ and ${\bf R}_{n}$ generate a dynamical group (usually SO(n)). Remarkably, the value of $n$ can be controlled by gate voltages, indicating that abstract concepts such as dynamical symmetry groups are experimentally realizable. Moreover, when an external magnetic field is applied then, under favorable circumstances, the exchange interaction involves solely the Runge-Lenz operators ${\bf R}_{n}$ and the corresponding dynamical symmetry group is SU(n). For example, the celebrated group SU(3) is realized in triple quantum dot with four electrons.Comment: 24 two-column page

Pricing dynamic solvency insurance and investment fund protection

Nella prima parte del nostro lavoro, il surplus di una compagnia d’assicurazione è modellizzato come un processo di Wiener. Consideriamo un contratto d’assicurazione dinamica di solvibilità. Secondo questo contratto, i pagamenti necessari sono effettutati istantaneamente, in modo che il surplus modificato non divenga mai negativo. Matematicamente, questo corrisponde ad introdurre una barriera riflettente in zero. Otteniamo così un’espressione esplicita per il premio netto di un tale contratto. In the first part of the paper the surplus of a company is modelled by a Wiener process. We consider a dynamic solvency insurance contract. Under such a contract the necessary payments are made instantaneously so that the modified surplus never falls below zero. This means mathematically that the modified surplus process is obtained from the original surplus process by introduction of a reflecting barrier at zero. Theorem 1 gives an explicit expression for the net single premium of such a contract. In the second part we consider an investment fund whose unit value is modelled by a geometric Brownian motion. Different forms of dynamic investment fund protection are examined. The basic form is a guarantee which provides instantaneously the necessary payments so that the upgraded fund unit value does not fall below a protected level. Theorem 2 gives an explicit expression for the price of such a guarantee. This result can also be applied to price a guarantee where the protected level is an exponential function of time. Moreover it is shown explicitly how the garantee can be generated by construction of the replicating portfolio. The dynamic investment fund garantee is compared to the corresponding put option and it is observed that for short time intervals the ratio of the prices is about 2. Finally the price of a more exotic protection is discussed, under which the guaranteed unit value at any time is a fixed fraction of the maximal upgraded unit value that has been observed until then. Several numerical and graphical illustrations show how the theoretical results can be implemented in practice

Water as the next commodity

Water is increasingly recognized as a commodity in its own right, as evidenced by the launching in 2006 of the first water index, World Water Index or WOWAX. For a number of factors ranging from the growth of the world population to global warming and potential extinction of lakes and seas, water is increasingly receiving the attention it deserves. Expensive techniques of adduction or desalination are put into place in those regions in the world where it is scarce and in great demand. Both in developed and developing countries, large capital investments are required to establish and maintain water systems that will altogether benefit populations and investors. The goal of this article is to illustrate the ways in which one can currently invest in water and to discuss the ineluctable emergence of water markets and water right

CONTINGENT CLAIMS VALUED AND HEDGED BY PRICING AND INVESTING IN A BASIS

Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black-Scholes geometric Brownian motion model. Data on S & P 500 options from the "Wall Street Journal" are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black-Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements. Copyright 1994 Blackwell Publishers.