A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

PENGARUH PEMBAGIAN DIVIDEN MELALUI MODEL BLACK-SCHOLES

Stock trading has a risk that can be said to be quite large due to fluctuations in stock prices. In stock trading, one alternative to reduce the amount of risk is options. The focus of this research is on European options which are financial contracts by giving the holder the right, not the obligation, to sell or buy the principal asset from the writer when it expires at a predetermined price. The Black-Scholes model is an option pricing model commonly used in the financial sector. This study aims to determine the effect of dividend distribution through the Black-Scholes model on stock prices. The effect of dividend distribution through the Black-Scholes model on stock prices results in the stock price immediately after the dividend distribution being lower than the stock price shortly before the dividend distribution

On Comparative Analysis for the Black-Scholes Model in the Generalized Fractional Derivatives Sense via Jafari Transform

The Black-Scholes model is well known for determining the behavior of capital asset pricing models in the finance sector. The present article deals with the Black-Scholes model via the Caputo fractional derivative and Atangana-Baleanu fractional derivative operator in the Caputo sense, respectively. The Jafari transform is merged with the Adomian decomposition method and new iterative transform method. It is worth mentioning that the Jafari transform is the unification of several existing transforms. Besides that, the convergence and uniqueness results are carried out for the aforesaid model. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest its applicability for a variety of nonlinear evolutionary problems.

The Pricing and Hedging of an Attainable Claim in a Hybrid Black–Scholes Model under Regime Switching

This article formulates and dissects a Black–Scholes model with regime switching that can be used to describe the performance of a complete market. An explicit integrand formula ϕ t , ω is obtained when the T -claim F ω is given for an attainable claim in this complete market. In addition, some perfect results are presented on how to hedge an attainable claim for this Black–Scholes model, and the price p of the European call and the self-financing portfolio θ t = θ 0 t , θ 1 t are given explicitly. Finally, some concluding remarks are provided to illustrate the theoretical results.

Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

In this paper, we study the valuation of European vulnerable options where the underlying asset price and the firm value of the counterparty both follow the bifractional Brownian motion with jumps, respectively. We assume that default event occurs when the firm value of the counterparty is less than the default boundary. By using the actuarial approach, analytic formulae for pricing the European vulnerable options are derived. The proposed pricing model contains many existing models such as Black–Scholes model (1973), Merton jump-diffusion model (1976), Klein model (1996), and Tian et al. model (2014).

Pricing Exotic Options Using Some Lattice Procedures

In this work, we discuss some very simple and extremely efficient lattice models, namely, Binomial tree model (BTM) and Trinomial tree model (TTM) for valuing some types of exotic barrier options in details. For both these models, we consider the concept of random walks in the simulation of the path which is followed by the underlying stock price. Our main objective is to estimate the value of barrier options by using BTM and TTM for different time steps and compare these with the exact values obtained by the benchmark Black-Scholes model (BSM). Moreover, we analyze the convergence of these lattice models for these exotic options. All the results have been shown numerically as well as graphically. GANITJ. Bangladesh Math. Soc.41.1 (2021) 26-40

An analytical approximation formula for the pricing of credit default swaps with regime switching

We derive an analytical approximation for the price of a credit default swap (CDS) contract under a regime-switching Black–Scholes model. To achieve this, we first derive a general formula for the CDS price, and establish the relationship between the unknown no-default probability and the price of a down-and-out binary option written on the same reference asset. Then we present a two-step procedure: the first step assumes that all the future information of the Markov chain is known at the current time and presents an approximation for the conditional price under a time-dependent Black–Scholes model, based on which the second step derives the target option pricing formula written in a Fourier cosine series. The efficiency and accuracy of the newly derived formula are demonstrated through numerical experiments. doi:10.1017/S1446181121000274