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

2014 ◽  
Vol 33 ◽  
pp. 103-115 ◽  
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

AN APPROPRIATE APPROACH TO PRICING EUROPEAN-STYLE OPTIONS WITH THE ADOMIAN DECOMPOSITION METHOD

2018 ◽  
Vol 59 (3) ◽  
pp. 349-369
Author(s):  
ZIWIE KE ◽  
JOANNA GOARD ◽  
SONG-PING ZHU
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
European Options ◽  
Black Scholes Model ◽  
Adomian Decomposition ◽  
Black Scholes ◽  
Interest Rate Model ◽  
Vasicek Interest Rate Model ◽  
Digital Options ◽  
Continuous Estimation

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.

scholarly journals PREFERENCES, LÉVY JUMPS AND OPTION PRICING

2007 ◽  
Vol 03 (01) ◽  
pp. 0750001 ◽  
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.

scholarly journals Efficacy of Nifty Index Options through BSM Model

2019 ◽  
Vol 8 (3S2) ◽  
pp. 947-949
Keyword(s):  
Options Pricing ◽  
Price Determination ◽  
Strike Price ◽  
Index Options ◽  
Put Options ◽  
Black Scholes Model ◽  
Exercise Price ◽  
Black Scholes ◽  
Long Time ◽  
Optimum Price

Options are one of the products in financial derivatives, which gives the rights to buy and sell the product to an option holder in pre-fixed price which known as the strike price or exercise price at certain periods. Options contract was existed in various countries for long time, but it became very popular among the investors when the Fisher Black, Myron Scholes and Robert Merton were introduced the Black-Scholes Model in the year of 1973. This model was formerly developed by these three economists who were also receiving the Nobel prize for finding this innovative model. This model is mainly used to deal with the theoretical pricing challenge in options price determination. In India the trading in Index Options commenced on 4th June 2001 and Options on individual securities commenced on 2nd July 2001. There are many types in options contracts like stock options; Index options, weather options, real options and etc. This study has mainly been focusing on Nifty 50 index options which are effectively trade at NSE. This paper goes to describe about the importance of options pricing and how the BSM model has effectively used to find the optimum price of the theoretical value of call and put options.

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

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.

scholarly journals On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1563
Author(s):  
Jung-Kyung Lee
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
American Options ◽  
Closed Form Solution ◽  
Numerical Results ◽  
Put Options ◽  
Optimal Exercise Boundary ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Exercise Boundary

We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.

scholarly journals Accuracy Tests Of American Put Valuation Models For Pharmaceutical Equity Options

2011 ◽  
Vol 6 (2) ◽  
Author(s):  
Yong H. Kim ◽  
Sangwoo Heo ◽  
Peter Cashel-Cordo ◽  
Yong S. Jang
Keyword(s):  
Option Prices ◽  
Equity Options ◽  
Forecasting Accuracy ◽  
Put Options ◽  
True Value ◽  
Black Scholes Model ◽  
American Put Options ◽  
Black Scholes ◽  
Variance Estimate ◽  
American Put

This study compares the performance of the Macmillan (1986), Barone-Adesi and Whaley (1987) MBAW model, Ju and Zhong (1999) MQuad model, Black-Scholes model and Put-Call Parity in pricing American put options of pharmaceutical companies. These are evaluated using actual option prices for three companies over 2000 to 2005, as opposed to the previous use of generated binomial option pricing data. We compare the forecasting accuracy by maturity, moneyness, and variance estimate. Contrary to Ju and Zhong (1999), we find that the MBAW outperforms the other models for at-the-money, and out-of-the-money options. The MQuad model performs best for in-the-money options. However, in this case both the MBAW and MQuad models estimates are very similar. Our results are consistent irrespective of option maturities and volatility estimates. These findings raise questions regarding the practice of using actual prices as the true value, compared to the previous results that use simulated prices.

Exchange option valuation using Liu process

2021 ◽  
pp. 2150018
Author(s):  
Seema Uday Purohit ◽  
Prasad Narahar Lalit
Keyword(s):  
Financial Mathematics ◽  
Option Valuation ◽  
Normal Distributions ◽  
Black Scholes Model ◽  
Liu Process ◽  
Proposed Model ◽  
Black Scholes ◽  
The Difference ◽  
Log Normal ◽  
Exchange Option

Margrabe formula is an extension of the famous Black–Scholes model extended to two correlated stocks. In the stochastic financial mathematics approach, the difficulty of addressing this valuation lies in the fact that the difference between two log-normal distributions is not log-normal. We avoided this approach in this work and valued the European type exchange option using the Liu process, a Brownian motion’s fuzzy counterpart. The work compares the proposed model values with the simulated values obtained by the Margrabe formula.

scholarly journals ESTIMASI NILAI IMPLIED VOLATILITY MENGGUNAKAN SIMULASI MONTE CARLO

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.

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

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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