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

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

Real options - realistic valuation

In valuation of real options, a widely accepted assumption is that the underlying real asset is perfectly correlated with a financial one. As a result, valuation techniques from the financial world can be used. Since this assumption is in general unrealistic and may lead to substantial mispricing, even if the correlation is very high but not perfect, we argue that a different approach is more adequate. It is based on a simple principle of invariance of the market price of risk computed for certain portfolios involving the underlying asset and the options. This is illustrated on a simple model where we can clearly see the relations between the prices obtained by various methods

Predicting exchange rates via a futures market

Predicting future spot exchange rates has always been useful for companies trading internationally. Now finding future exchange rates is essential for countries that are to join common currency zones (Eurosystem) and need to set reference rates for the ERM II. This paper presents a model that attempts to determine exchange rates and, unlike others, is based on an analysis of the futures market. The model is based on the assumption that the futures market is dominated by two categories of traders: arbitrageurs and fundamental traders. The divergence of the futures rate from its theoretical value is gauged and then considered to be an indication of the direction and strength of the two forces in the market. The arbitrageurs' influence is filtered out and thus the model outputs the rate based on the fundamental traders' expectations

Implications of contrarian and one-sided strategies for the fair-coin game

We derive some results on contrarian and one-sided strategies by Skeptic for the fair-coin game in the framework of the game-theoretic probability of Shafer and Vovk \cite{sv}. In particular, concerning the rate of convergence of the strong law of large numbers (SLLN), we prove that Skeptic can force that the convergence has to be slower than or equal to $O(n^{-1/2})$. This is achieved by a very simple contrarian strategy of Skeptic. This type of result, bounding the rate of convergence from below, contrasts with more standard results of bounding the rate of SLLN from above by using momentum strategies. We also derive a corresponding one-sided result

Company valuation : value, structure, risk

This book, for a few different reasons, was focused on the DCF method, the main of the reasons being that the DCF method captures best the value of prof- itable, economically sound companies – it works for all firms which have real expertise. The main purpose of this book was to explain the inner workings of the DCF method, especially the variant in which capital structure constantly af- fects cost of equity, as it does in reality. The focus then was on the valuation model which integrates the three com- ponents that elsewhere are often treated separately: cash flows, the cost of capi- tal and the discounting process itself. The book revolved around these three is- sues. For example, it is commonly known that if the value of equity changes, the capital structure changes too. At the same time the change may affect the cost of capital, which in turn will influence the value of equity itself. The re- cursive approach to company valuation that was presented in the book relied on solving such logical loops that appeared both at each time period and along time periods. Performing the company valuation is such a way is much more complicated than assuming no links between for example the value of equity and the cost of capital, but leads to a much more reliable and methodically f law- less valuation

Geometric Proof of Strong Stable/Unstable Manifolds, with Application to the Restricted Three Body Problem

We present a method for establishing invariant manifolds for saddle--center fixed points. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs, and does not require rigorous integration of the vector field in order to prove the existence of the invariant manifolds. We apply our method to the restricted three body problem and show that for a given choice of the mass parameter, there exists a homoclinic orbit to one of the libration points.Comment: 34 pages, 6 figure

A Stochastic Process Approach of the Drake Equation Parameters

The number N of detectable (i.e. communicating) extraterrestrial civilizations in the Milky Way galaxy is usually done by using the Drake equation. This equation was established in 1961 by Frank Drake and was the first step to quantifying the SETI field. Practically, this equation is rather a simple algebraic expression and its simplistic nature leaves it open to frequent re-expression An additional problem of the Drake equation is the time-independence of its terms, which for example excludes the effects of the physico-chemical history of the galaxy. Recently, it has been demonstrated that the main shortcoming of the Drake equation is its lack of temporal structure, i.e., it fails to take into account various evolutionary processes. In particular, the Drake equation doesn't provides any error estimation about the measured quantity. Here, we propose a first treatment of these evolutionary aspects by constructing a simple stochastic process which will be able to provide both a temporal structure to the Drake equation (i.e. introduce time in the Drake formula in order to obtain something like N(t)) and a first standard error measure.Comment: 22 pages, 0 figures, 1 table, accepted for publication in the International Journal of Astrobiolog

Characterising blenders via covering relations and cone conditions

A blender is an invariant hyperbolic set of a diffeomorphism with the property that its stable or unstable manifold has a dimension larger than expected from the underlying hyperbolic splitting. We present a characterisation of a blender based on the correct topological alignment of sets in combination with the propagation of cones. It is applicable to multidimensional blenders in ambient phase spaces of any dimension. The required conditions can be verified by checking properties of a single iterate of the diffeomorphism, which is achieved by positioning the required sets in such a way that they form a suitable sequence of coverings. This setup is flexible and allows for a rigorous, interval arithmetic based, computer assisted validation. As a demonstration, we apply our approach to obtain a computer-assisted proof of the existence of a blender in a three-dimensional H{\'e}non-like family of diffeomorphisms over a considerable range of the relevant parameter.Comment: 29 pages, 15 figure