Hedging Conditional Value at Risk with Options

We present a method of hedging Conditional Value at Risk of a position in stock using put options. The result leads to a linear programming problem that can be solved to optimise risk hedging.Comment: 10 pages, 0 figure

Hyper-K{\"a}hler Hierarchies and their twistor theory

A twistor construction of the hierarchy associated with the hyper-K\"ahler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-K\"ahler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time ${\cal N}$ is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ${\cal N}$ is a moduli space of rational curves with normal bundle ${\cal O}(n)\oplus{\cal O}(n)$ in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ${\cal N}$ is shown to be foliated by four dimensional hyper-K{\"a}hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator.Comment: 23 pages, 1 figur

Geometric proof for normally hyperbolic invariant manifolds

We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a given neighbourhood. Our proof follows from a graph transform type method and is performed in the state space of the system. We do not require the map to be invertible. From our method follows also the smoothness of the established manifolds, which depends on the smoothness of the map, as well as rate conditions, which follow from bounds on the derivative of the map. Our method is tailor made for rigorous, interval arithmetic based, computer assisted validation of the needed assumptions.Comment: 64 pages, 4 figure

Cone Conditions and Covering Relations for Topologically Normally Hyperbolic Invariant Manifolds

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating H\'enon map

Transition Tori in the Planar Restricted Elliptic Three Body Problem

We consider the elliptic three body problem as a perturbation of the circular problem. We show that for sufficiently small eccentricities of the elliptic problem, and for energies sufficiently close to the energy of the libration point L2, a Cantor set of Lyapounov orbits survives the perturbation. The orbits are perturbed to quasi-periodic invariant tori. We show that for a certain family of masses of the primaries, for such tori we have transversal intersections of stable and unstable manifolds, which lead to chaotic dynamics involving diffusion over a short range of energy levels. Some parts of our argument are nonrigorous, but are strongly backed by numerical computations

Functional analysis of embolism induced by air injection in Acer rubrum and Salix nigra.

The goal of this study was to assess the effect of induced embolism with air injection treatments on the function of xylem in Acer rubrum L. and Salix nigra Marsh. Measurements made on mature trees of A. rubrum showed that pneumatic pressurization treatments that created a pressure gradient of 5.5 MPa across pit membranes (ΔP pit) had no effect on stomatal conductance or on branch-level sap flow. The same air injection treatments made on 3-year-old potted A. rubrum plants also had no effect on whole plant transpiration. A separate study made on mature A. rubrum trees showed that 3.0 and 5.5 MPa of ΔP pit values resulted in an immediate 100% loss in hydraulic conductance (PLC) in petioles. However, the observed change in PLC was short lived, and significant hydraulic recovery occurred within 5-10 min post air-pressurization treatments. Similar experiments conducted on S. nigra plants exposed to ΔP pit of 3 MPa resulted in a rapid decline in whole plant transpiration followed by leaf wilting and eventual plant death, showing that this species lacks the ability to recover from induced embolism. A survey that measured the effect of air-pressurization treatments on seven other species showed that some species are very sensitive to induction of embolism resulting in leaf wilting and branch death while others show minimal to no effect despite that in each case, the applied ΔP pit of 5.5 MPa significantly exceeded any native stress that these plants would experience naturally