One of the ways of managing the risk that can arise from the changes in the relationship between assets and liabilities is by asset-liability management. Recently, Value-at-risk (VaR) and tail conditional expectation (TCE) have also emerged as standard tools for measuring and controlling the risk of trading portfolios. The limits of TCE can be transformed into the limits of VaR and conversely in some dynamical setting, TCE is preferable to VaR for being coherent. In this paper we obtain a portfolio selection model for an institution's assets-liabilities under the TCE with consumption cost and transaction cost. A set of partial differential equations are derived and closed form solution proffered, when there is no transaction cost. Résumé. Jusqu'à récemment, les mesures de risque dénommées Valeur Au Risque (VaR) et l'Espérance Conditionnelle de Queue (ECQ),étaient utilisées pourévaluer et controler les risques en gestion de portefeuille. Ces deux mesures sontéquivalentes dans les cas limites. De plus, dans une approche dynamique, l'ECQ est préférableà la VaR en raison notamment de sa cohérence. Une autre approche de la gestion des portefeuille concerne l'évaluation du risque potentiellement dû aux changements qui peuvent intervenir dans la relation entre les actifs et les passifs, communément dénommée gestion de l'actif-passif. Dans ce papier, nous proposons un modèle de sélection de portefeuille d'actifs-passifs d'une institution sous contrôle du TCE en présence des coûts de consommation et de transaction. Une famille d'équations différentielles partielles est proposée et une solution fermée est trouvée, lorsqu'il n'y a pas de coût de transaction.

In this paper, we considered the different strategies that generate the optimal wealth on investment. The strategy examine depends on the utility function an investor is willing to adopt, say H * at time N in every 2n possible states; in an N period setting. Negative exponential, logarithm, square root and power utility functions were established, as the market structures changed according to a Markov chain through a martingale approach. The problem of maximization is solved via Lagrange method. The performance of the investment from day-today is driven by the ratio of the risk neutral probability and the probability of rising to falling.

Highlights• Construction of a hybrid inertial algorithm.• Theoretical proof of the constructed algorithm.• Numerical illustration of the applicability of the algorithm.

We introduce a new iterative scheme and prove a strong convergence theorem for it. This iterative scheme finds a common point in the set of fixed points of a finite family of Bregman quasi-nonexpansive mappings and the common solution set of the variational inequality problem for continuous monotone mappings.

Abstract. In this paper, we propose a single economic investor whose asset follows a geometric Brownian motion process. Our objective therefore is to obtain the fair price and the present market value of the asset with an infinitely horizon expected discounted investment output. We apply dynamic programming principle to derive the Hamilton Jacobi Bellman (HJB)-equation associated with the problem which is found to be equivalent to the famous Black-Scholes Model under no risk neutrality. In addition, for a complete market under equilibrium, we obtained the value of the present asset with risk neutrality and its fair price. IntroductionThe study that Asset Prices follow a simple diffusion process was first proposed in the historical work of Bachelier in 1900. His work is rather remarkable in that it addressed the problem of option pricing and his main aim was to actually provide a fair price for the European call option [3]. Bachelier assumed stock price dynamics with a Brownian motion without drift (resulting in a normal distribution for the stock prices), and no time-value of money. The formula provided may be used to valuate a European style call option [10]. Later on, [4] obtained the same results as Bachelier. As pointed out by [5] and [9], this approach allows negative realizations for both stock and option prices. Moreover, the option price may exceed the price of its underlying asset.According to [10], Black -Scholes Model is often regarded as either the end or the beginning of the option valuation history. Using two different approaches for the valuation of European style options, they present a general equilibrium solution that is a function of "observable" variables only, making therefore the model subject to direct empirical tests. The stock price dynamics is described by a geometric Brownian motion with drift as a more refined market model for which prices cannot be negative, the volatility can be viewed as the diffusion coefficient of this random walk [7]. The manifest characteristic of the final valuation formula is the parameters it does not depend on. The option price does not depend on the expected return rate of the stock or the risk preferences of the investors. It is not assumed that the investors agree on the expected return rate of the stock. It is expected that investors may have quite different estimates for current and future returns. However, the option price depends on the risk-free interest rate and on the variance of the return rate of the stock.To make simplier normally distributed financial asset returns, on assumption that its distribution is difficult, [11] concludes that for a distribution model to reproduce the properties of empirical evidence it must possess certain parameters as: a location parameter; a scale parameter; an asymmetry parameter and a parameter describing the decay of the distribution since financial returns distribution is difficult to determine since normally distributed financial asset returns is not supported by empirical evidence.We suppose the movement of a...