Numerical solution of the time fractional Black–Scholes model governing European options

We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results.

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PREFERENCES, LÉVY JUMPS AND OPTION PRICING

Annals of Financial Economics ◽

10.1142/s2010495207500017 ◽

2007 ◽

Vol 03 (01) ◽

pp. 0750001 ◽

Cited By ~ 1

Author(s):

CHENGHU MA

Keyword(s):

Option Pricing ◽

Contingent Claims ◽

Economic Variable ◽

European Options ◽

Jump Risk ◽

Representative Agent ◽

Potential Explanation ◽

Black Scholes Model ◽

Black Scholes ◽

Lévy Jumps

This paper derives an equilibrium formula for pricing European options and other contingent claims which allows incorporating impacts of several important economic variable on security prices including, among others, representative agent preferences, future volatility and rare jump events. The derived formulae is general and flexible enough to include some important option pricing formulae in the literature, such as Black–Scholes, Naik–Lee, Cox–Ross and Merton option pricing formulae. The existence of jump risk as a potential explanation of the moneyness biases associated with the Black–Scholes model is explored.

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A Note on Numerical Solution of a Linear Black-Scholes Model

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v33i0.17664 ◽

2014 ◽

Vol 33 ◽

pp. 103-115 ◽

Cited By ~ 3

Author(s):

Md. Kazi Salah Uddin ◽

Mostak Ahmed ◽

Samir Kumar Bhowmilk

Keyword(s):

Time Integration ◽

Weighted Average ◽

Financial Mathematics ◽

European Options ◽

Average Scheme ◽

Put Options ◽

Weighted Average Method ◽

Black Scholes Model ◽

Black Scholes ◽

Solution Methods

Black-Scholes equation is a well known partial differential equation in financial mathematics. In this article we discuss about some solution methods for the Black Scholes model with the European options (Call and Put) analytically as well as numerically. We study a weighted average method using different weights for numerical approximations. In fact, we approximate the model using a finite difference scheme in space first followed by a weighted average scheme for the time integration. Then we present the numerical results for the European Call and Put options. Finally, we investigate some linear algebra solvers to compare the superiority of the solvers. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 103-115 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17664

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Bifractional Black-Scholes Model for Pricing European Options and Compound Options

Journal of Systems Science and Information ◽

10.21078/jssi-2020-346-10 ◽

2020 ◽

Vol 8 (4) ◽

pp. 346-355

Author(s):

Feng Xu

Keyword(s):

Empirical Studies ◽

Asset Price ◽

European Options ◽

Hedging Strategy ◽

Underlying Asset ◽

Bifractional Brownian Motion ◽

Compound Options ◽

Black Scholes Model ◽

Black Scholes ◽

Delta Hedging

AbstractRecent empirical studies show that an underlying asset price process may have the property of long memory. In this paper, it is introduced the bifractional Brownian motion to capture the underlying asset of European options. Moreover, a bifractional Black-Scholes partial differential equation formulation for valuing European options based on Delta hedging strategy is proposed. Using the final condition and the method of variable substitution, the pricing formulas for the European options are derived. Furthermore, applying to risk-neutral principle, we obtain the pricing formulas for the compound options. Finally, the numerical experiments show that the parameter HK has a significant impact on the option value.

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Numerical solution of generalized Black–Scholes model

Applied Mathematics and Computation ◽

10.1016/j.amc.2017.10.004 ◽

2018 ◽

Vol 321 ◽

pp. 401-421

Author(s):

S. Chandra Sekhara Rao ◽

Manisha

Keyword(s):

Numerical Solution ◽

Black Scholes Model ◽

Black Scholes

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ESTIMASI NILAI IMPLIED VOLATILITY MENGGUNAKAN SIMULASI MONTE CARLO

E-Jurnal Matematika ◽

10.24843/mtk.2018.v07.i03.p209 ◽

2018 ◽

Vol 7 (3) ◽

pp. 239

Author(s):

MAKBUL MUFLIHUNALLAH ◽

KOMANG DHARMAWAN ◽

NI MADE ASIH

Keyword(s):

Monte Carlo ◽

Capital Market ◽

Implied Volatility ◽

Option Price ◽

European Options ◽

Price Variation ◽

Black Scholes Model ◽

Black Scholes ◽

Alternative Investment ◽

The Capital Market

Investing among investors is an exciting activity to gain profit in the financial world. The development of investment in the financial world affects the number of alternative investment instruments that can be offered to investors in the capital market. The management of instruments in finance depends on the accuracy of forecasting of variables for example volatility. Volatility is a statistic of the degree of price variation in one period to the next which is expressed by ?. Volatility values can be estimated using Implied Volatility. Implied Volatility is the volatility used in determining the price of European options obtained by equalizing the price of the theoretical options, the price obtained from the Black-Scholes model, with the option price in the market. In this research will discuss how to estimate Implied Volatility value using the option obtained from simulation with Monte Carlo.

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The Performance of Skewness and Kurtosis Adjusted Option Pricing Model in Emerging Markets

International Journal of Finance & Banking Studies (2147-4486) ◽

10.20525/ijfbs.v5i3.285 ◽

2016 ◽

Vol 5 (3) ◽

pp. 70-84

Author(s):

Özge Sezgin Alp

Keyword(s):

Emerging Markets ◽

Option Pricing ◽

Option Prices ◽

Pricing Model ◽

European Options ◽

Black Scholes Model ◽

Black Scholes ◽

Option Pricing Model ◽

Skewness And Kurtosis ◽

Pricing Formula

In this study, the option pricing performance of the adjusted Black-Scholes model proposed by Corrado and Su (1996) and corrected by Brown and Robinson (2002), is investigated and compared with original Black Scholes pricing model for the Turkish derivatives market. The data consist of the European options written on BIST 30 index extends from January 02, 2015 to April 24, 2015 for given exercise prices with maturity April 30, 2015. In this period, the strike prices are ranging from 86 to 124. To compare the models, the implied parameters are derived by minimizing the sum of squared deviations between the observed and theoretical option prices. The implied distribution of BIST 30 index does not significantly deviate from normal distribution. In addition, pricing performance of Black Scholes model performs better in most of the time. Black Scholes pricing Formula, Carrado-Su pricing Formula, Implied Parameters

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