A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

Abstract

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes.

Nonlinearities and Chaos: A New Analysis of CEE Stock Markets

“…Nevertheless, noteworthy studies have shown that stock returns do not follow random walks, being characterized by nonlinearities and chaos (for a review of the literature see [2]). Consequently, recent developments in investigating the nonlinear dependence and deterministic chaos of financial variables altered the traditional view of their erratic behavior [3,4]. If the stock market returns are characterized by nonlinearities and chaos, then the market is inefficient.…”

Tangent nonlinear equation in context of fractal fractional operators with nonsingular kernel

“…Many new fractional operators [ 6 ] and Mittag–Leffler [ 7 , 8 ] kernel as Atangana–Baleanu (AB) derivative in Caputo sense [ 9 ] appear to enrich the existing fractional derivatives (FVs) as the Caputo derivative (CV) [ 10 , 11 ] and the Rieman–Liouville derivative [ 10 , 11 ] and their existing generalizations [ 12 , 13 ]. Many applications related to the FVs operators (FDOs) have already been addressed in the literature, in physics [ 14 , 15 ], in mechanical fluids [ 16 ], in finance models[ 17 – 19 ], in diffusion equations[ 20 ] and others. It is important to mention that there exist many FVs, which are not cited above, derived from the CV and the Riemann–Liouville (RL) derivatives.…”

Finite Difference Method for the Multi-Asset Black–Scholes Equations

“…The HHW model applies to pricing options arising from the probabilistic nature of asset prices, volatility, and risk-free interest rates. In [9], Ankudinova and Ehrhardt presented numerical methods for the nonlinear BS equation [20].…”