In this study, the decomposition method for solving the linear heat equation and nonlinear Burgers equation is implemented with appropriate initial conditions. The application of the method demonstrated that the partial solution in the x-direction requires more computational work when compared...

In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This...

In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a...

This article is the second one in a series on the use of scaling invariance in finance. In the first paper, we introduced a new formalism for the pricing of derivative securities, which focusses on tradable objects only, and which completely avoids the use of martingale techniques. In this...

Bick (1987,1990) and He and Leland (1993) demonstrated that not every arbitrage-free Markovian diffusion price process is consistent with an equilibrium approach. We propose a unified framework for these results and we derive a new martingale characterization of equilibrium.

Prices of tradables can only be expressed relative to each other at any instant of time. This fundamental fact should therefore also hold for contingent claims, i.e. tradable instruments, whose prices depend on the prices of other tradables. We show that this property induces a local scaling...

The method of calculating numerically the Lyapunov exponent of partial differential equations is stated. The mean Lyapunov exponent and the local Lyapunov exponent are introduced in order to characterize the systems described by the partial differential equations. It is shown that the mean...

The characteristics of a system described by Kuramoto–Sivashinsky equation are obtained through the statistics of a mean Lyapunov exponent. This mean Lyapunov exponent takes large values and fluctuates large when the system is disordered temporally and spatially. This behavior of the spatially...