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

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 760 ◽  
Author(s):  
Seda Gulen ◽  
Catalin Popescu ◽  
Murat Sari
Keyword(s):  
Current Method ◽  
Strong Stability ◽  
Computational Effort ◽  
Academic Community ◽  
Third Order ◽  
New Approach ◽  
European Option Pricing ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Linear And Nonlinear

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.

scholarly journals Pricing Basket Options by Polynomial Approximations

2021 ◽  
Author(s):  
Pablo Olivares ◽  
Alexander Alvarez
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Closed Form ◽  
Computational Effort ◽  
Basket Options ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Power Moments ◽  
Exponential Power ◽  
Closed Form Approximation

We propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponential-power moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort

scholarly journals Pricing Basket Options by Polynomial Approximations

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Pablo Olivares ◽  
Alexander Alvarez
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Closed Form ◽  
Computational Effort ◽  
Basket Options ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Power Moments ◽  
Exponential Power ◽  
Closed Form Approximation

We propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponential-power moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort.

scholarly journals European Option Pricing Formula in Risk-Aversive Markets

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Shujin Wu ◽  
Shiyu Wang
Keyword(s):  
Risk Measure ◽  
European Option ◽  
Underlying Asset ◽  
No Arbitrage ◽  
European Option Pricing ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Risk Neutral ◽  
Arbitrage Opportunity ◽  
Weaker Conditions

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

scholarly journals Application of the Trefftz method for option pricing

2020 ◽  
Vol 2020 (146) ◽  
pp. 37-49
Author(s):  
Katarzyna BRZOZOWSKA-RUP ◽  
◽  
Sylwia HOŻEJOWSKA ◽  
Leszek HOŻEJOWSKI ◽  
◽  
...  
Keyword(s):  
Option Pricing ◽  
Design Methodology ◽  
Number Of Solutions ◽  
Trefftz Method ◽  
New Approach ◽  
Pricing Options ◽  
Black Scholes Model ◽  
Starting Point ◽  
Black Scholes ◽  
Computational Simplicity

Purpose: Option pricing is hardly a new topic, however, in many cases they lack an analytical 11 solution. The article proposes a new approach to option pricing based on the semi-analytical 12 Trefftz method. 13 Design/methodology/approach: An appropriate transformation makes it possible to reduce the 14 Black-Scholes equation to the heat equation. This admits the Trefftz method (which has shown 15 its effectiveness in heat conduction problems) to be employed. The advantage of such 16 an approach lies in its computational simplicity and in the fact that it delivers a solution 17 satisfying the governing equation. 18 Findings: The theoretical option pricing problem being considered in the paper has been solved 19 by means of the Trefftz method, and the results achieved appear to comply with those taken 20 from the Black-Scholes formula. Numerical simulations have been carried out and compared, 21 which has confirmed the accuracy of the proposed approach. 22 Originality/value: Although a number of solutions to the Black-Scholes model have appeared, 23 the paper presents a thoroughly novel idea of implementation of the Trefftz method for solving 24 this model. So far, the method has been applied to problems having nothing in common with 25 finance. Therefore the present approach might be a starting point for software development, 26 competitive to the existing methods of pricing options.

Comparison of European Option Pricing Models at Multiple Periods

2021 ◽  
pp. 149-166
Author(s):  
Amir Ahmad Dar ◽  
N. Anuradha ◽  
Ziadi Nihel
Keyword(s):  
Option Pricing ◽  
Graphical Representation ◽  
European Option ◽  
Pricing Models ◽  
European Option Pricing ◽  
Black Scholes Model ◽  
Time Period ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Anderson Darling

The point of this chapter is to think about the correlation of two well-known European option pricing models – Black Scholes Model and Binomial Option Pricing Model. The above two models not statistically significant at one period. In this examination, it is shown how the above two European models are statistically significant when the time period increases. The independent paired t-test is utilized with the end goal to demonstrate that they are statistically significant to vary from one another at higher time period and the Anderson Darling test being used for the normality test. The Minitab and Excel programming has been utilized for graphical representation and the hypothesis testing.

scholarly journals Pricing Basket Options by Polynomial Approximations

2021 ◽  
Author(s):  
Pablo Olivares ◽  
Alexander Alvarez
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Closed Form ◽  
Computational Effort ◽  
Basket Options ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Power Moments ◽  
Exponential Power ◽  
Closed Form Approximation

We propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponential-power moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort

Block-pulse operational matrix method for solving fractional Black-Scholes equation

2017 ◽  
Vol 44 (3) ◽  
pp. 489-502 ◽  
Author(s):  
Farshid Mehrdoust ◽  
Amir Hosein Refahi Sheikhani ◽  
Mohammad Mashoof ◽  
Sabahat Hasanzadeh
Keyword(s):  
Design Methodology ◽  
Operational Matrix ◽  
Initial Condition ◽  
European Option ◽  
Content Type ◽  
Operational Matrix Method ◽  
European Option Pricing ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Alternative Algorithm

Purpose The purpose of this paper is to evaluate a European option using the fractional version of the Black-Scholes model. Design/methodology/approach In this paper, the authors employ the block-pulse operational matrix algorithm to approximate the solution of the fractional Black-Scholes equation with the initial condition for a European option pricing problem. Findings The fractional derivative will be described in the Caputo sense in this paper. The authors show the accuracy and computational efficiency of the proposed algorithm through some numerical examples. Originality/value This is the first paper that considers an alternative algorithm for pricing a European option using the fractional Black-Scholes model.

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