حوالہ کا فقہی تصور اور اس کی جدید صورتیں

This article is divided in to two sections: the first one; is to study of Ḥawāla according to Fiqhi rules, i.e. Its definition, conditions, its qualities and its specifications. In the second part, Ḥawāla is discussed according to application of the Bank, and its practical forms in different institutions. Ḥawāla is a contract in which a party or a person transfers his debt from his risk to another party or another person. Ḥawāla is used for different purposes, for example, bill of exchange, Financial Papers and different Banking accounts. This paper also discusses the difference between Ḥawāla and wakālah, Ḥawāla and Kafāla, and also Ḥawāla and Debt. Therefore, analysis, about Ḥawāla is very essential and many studies have been done on it. Furthermore, it will also critically provide their textual evidence and rational arguments in order to reach a financial juristic judgment. Ḥawāla is used in credit card, discounting of bill of exchange, etc. At the end the paper discusses its conclusion and offers some requests and suggestions.

Numerical Techniques to Solve Black- Scholes Equation

Black-Scholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks, commodities and derivatives can be evaluated using this model. Option valuation, a part of derivative products, is extremely important to trade in the stocks. The numerical solutions of Black-Scholes equations are used to simulate these options and are addressed in this dissertation. In particular, the discontinuities in the domain are addressed. The discontinuities in options create oscillations near exercise price in the solution. A grid adaptive finite difference technique is developed to evaluate financial options using Black-Scholes equation. The grid is refined near the exercise price to resolve discontinuities in the option evaluation and a coarse grid is generated otherwise. To cope with these uneven space steps, an innovative numerical finite difference schemes are developed named as adaptive explicit, adaptive fully implicit and adaptive Crank Nicolson techniques. These techniques are used to cure oscillations produced by discontinuities in the digital and butterfly options. These techniques are also used to simulate multi-asset digital and butterfly options. The numerical experiments show that the adaptive finite difference method is much more efficient than the method with uniform spacing. The technique reduces the points drastically which in turn decreases the computational cost and makes the algorithms highly efficient