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 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

scholarly journals VOLATILITY SMILE AT THE RUSSIAN OPTION MARKET

2006 ◽  
Vol 7 (1) ◽  
pp. 9-15
Author(s):  
D. Golembiovsky ◽  
I. Baryshnikov
Keyword(s):  
Normal Distribution ◽  
Option Price ◽  
Underlying Asset ◽  
Basic Model ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Basic Purpose ◽  
Log Normal ◽  
In The Beginning ◽  
Log Normal Distribution

The main derivative exchange in Russia is FORTS (Futures and Options in RTS) which is a division of Russian Trade System (RTS). The underlying assets of option contracts are futures on Russian companies’ shares: OJSC “EES"1, OJPC “Lukoil"2 and OJSC “Gazprom"3. A basic model for estimation of fair option price is Black‐Scholes model, developed in the beginning of 70‐s’ years of the last century. This model defines the option premium as a cost of its hedging by underlying asset. It uses a number of assumptions: prices of underlying assets follow log‐normal distribution; hedging is accomplished continuously; an underlying asset is infinitely divisible; a volatility is constant on all period of option life. However, according to practice, prices of shares and futures do not follow normal or log‐normal distribution, a volatility can change during a life of option, and hedging is a discrete process. Thus, Black‐Scholes model can yield inexact results in real markets, especially it concerns deeply “in the money” or deeply “out of the money” options. The basic purpose of the paper is to investigate opportunities to apply Black‐Scholes model for an estimation of option premiums in the Russian market.

scholarly journals Log-normal Distribution Type Symmetry Model for Square Contingency Tables with Ordered Categories

2018 ◽  
Vol 47 (3) ◽  
pp. 39-48
Author(s):  
Kiyotaka Iki
Keyword(s):  
Normal Distribution ◽  
Contingency Tables ◽  
Ordered Categories ◽  
Symmetry Model ◽  
Proposed Model ◽  
Type Symmetry ◽  
The Difference ◽  
Log Normal ◽  
Distribution Type ◽  
Log Normal Distribution

For the analysis of square contingency tables with the same row and column ordinal classications, this article proposes a new model which indicates that the log-ratios of symmetric cell probabilities are proportional to the difference between log-row category and log-column category. The proposed model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate log-normal distribution. Also, this article gives the decomposition of the symmetry model using the proposed model with the orthogonality of test statistics. Examples are given. The simulation studies based on bivariate log-normal distribution are given.

scholarly journals An Analytic Solution For Hedge Ratios On Stocks With Dividends And The Accuracy Of Black-Scholes Hedge Ratios

2011 ◽  
Vol 1 (7) ◽  
Author(s):  
Kanwal Sachdeva ◽  
William Sterk
Keyword(s):  
Analytic Solution ◽  
Stock Volatility ◽  
Market Prices ◽  
Hedge Ratio ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Hedge Ratios ◽  
Option Pricing Model ◽  
Complicated Model ◽  
The Difference

The Black-Scholes model continues to be the standard option pricing model discussed in virtually all corporate finance and investments texts and continues to be widely used in practice. The models associated hedge ratio has also been widely used for hedging purposes. The associated hedge ratio (or delta) is determined as part of calculating the Black-Scholes option value. However, the original model assumes no dividends on the underlying stock. The model has been modified to allow for dividends, but the modification does not lead to values as precise as other models, such as the Roll-Geske-Whaley model that specifically account for dividends. Empirical research has shown that the RGW model values are closer to actual market prices than the modified Black-Scholes values. This paper is primarily concerned with the hedge ratio. We derive an analytic solution for a more accurate hedge ratio based on the RGW model. The paper is then concerned with how large the errors are associated with using the BS approximation rather than the more complicated model that specifically accounts for dividends. We find that although there are times when the BS approximation can be accurate, at other times the differences can be significant. These differences are related to the size of the dividend, the difference between the time to expiration and the time to ex-dividend, the rate of interest, the stock volatility, and the degree to which the option is in-the-money.

scholarly journals Bayesian Confidence Intervals for Coefficients of Variation of PM10 Dispersion

2021 ◽  
Vol 5 (2) ◽  
pp. 139-154
Author(s):  
Warisa Thangjai ◽  
Sa-Aat Niwitpong ◽  
Suparat Niwitpong
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Confidence Intervals ◽  
Bayesian Approach ◽  
Normal Distributions ◽  
Coefficients Of Variation ◽  
The Difference ◽  
Log Normal ◽  
Log Normal Distribution ◽  
The Bayesian Approach

Herein, we propose the Bayesian approach for constructing the confidence intervals for both the coefficient of variation of a log-normal distribution and the difference between the coefficients of variation of two log-normal distributions. For the first case, the Bayesian approach was compared with large-sample, Chi-squared, and approximate fiducial approaches via Monte Carlo simulation. For the second case, the Bayesian approach was compared with the method of variance estimates recovery (MOVER), modified MOVER, and approximate fiducial approaches using Monte Carlo simulation. The results show that the Bayesian approach provided the best approach for constructing the confidence intervals for both the coefficient of variation of a log-normal distribution and the difference between the coefficients of variation of two log-normal distributions. To illustrate the performances of the confidence limit construction approaches with real data, they were applied to analyze real PM10 datasets from the Nan and Chiang Mai provinces in Thailand, the results of which are in agreement with the simulation results. Doi: 10.28991/esj-2021-01264 Full Text: PDF

scholarly journals Confidence Intervals for the Signal-to-Noise Ratio and Difference of Signal-to-Noise Ratios of Log-Normal Distributions

Stats ◽  
2019 ◽  
Vol 2 (1) ◽  
pp. 164-173 ◽  
Author(s):  
Warisa Thangjai ◽  
Sa-Aat Niwitpong
Keyword(s):  
Confidence Interval ◽  
Confidence Intervals ◽  
Coverage Probability ◽  
Average Length ◽  
Signal To Noise Ratio ◽  
Signal To Noise ◽  
Normal Distributions ◽  
Noise Ratio ◽  
The Difference ◽  
Log Normal

In this article, we propose approaches for constructing confidence intervals for the single signal-to-noise ratio (SNR) of a log-normal distribution and the difference in the SNRs of two log-normal distributions. The performances of all of the approaches were compared, in terms of the coverage probability and average length, using Monte Carlo simulations for varying values of the SNRs and sample sizes. The simulation studies demonstrate that the generalized confidence interval (GCI) approach performed well, in terms of coverage probability and average length. As a result, the GCI approach is recommended for the confidence interval estimation for the SNR and the difference in SNRs of two log-normal distributions.

scholarly journals The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 129 ◽  
Author(s):  
Panumart Sawangtong ◽  
Kamonchat Trachoo ◽  
Wannika Sawangtong ◽  
Benchawan Wiwattanapataphee
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Financial Market ◽  
Caputo Fractional Derivative ◽  
Homotopy Perturbation Method ◽  
Homotopy Perturbation ◽  
Black Scholes Model ◽  
Proposed Model ◽  
Black Scholes ◽  
The Laplace Transform

It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method.

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