scholarly journals Option pricing formulas under a change of numèraire

2020 ◽  
Vol 40 (4) ◽  
pp. 451-473
Author(s):  
Antonio Attalienti ◽  
Michele Bufalo
Keyword(s):  
Incomplete Markets ◽  
Forthcoming Paper ◽  
Call Option ◽  
European Options ◽  
Change Of Measure ◽  
Binomial Tree ◽  
Price Process ◽  
European Call Option ◽  
Black Scholes ◽  
Suitable Change

We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.

scholarly journals Analytical Solution of Black-Scholes Equation in Predicting Market Prices and Its Pricing Bias

2020 ◽  
pp. 17-23
Author(s):  
Azor, Promise Andaowei ◽  
Amadi, Innocent Uchenna
Keyword(s):  
Analytical Solution ◽  
Goodness Of Fit ◽  
Call Option ◽  
Option Prices ◽  
Goodness Of Fit Test ◽  
Market Prices ◽  
European Call Option ◽  
Black Scholes

This paper is geared towards implementation of Black-Scholes equation in valuation of European call option and predicting market prices for option traders. First, we explained how Black-Scholes equation can be used to estimate option prices and then we also estimated the BS pricing bias from where market prices were predicted. From the results, it was discovered that Black-Scholes values were relatively close to market prices but a little increase in strike prices (K) decreases the option prices. Furthermore, goodness of fit test was done using Kolmogorov –Sminorvov to study BSM and Market prices.

scholarly journals Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

2018 ◽  
Vol 10 (6) ◽  
pp. 108
Author(s):  
Yao Elikem Ayekple ◽  
Charles Kofi Tetteh ◽  
Prince Kwaku Fefemwole
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Call Option ◽  
Option Prices ◽  
Options Market ◽  
European Call Option ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Real Market ◽  
Independent Path

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

Pricing options on a mean-reverting asset by the analytical operator splitting method

2021 ◽  
pp. 2150002
Author(s):  
C. F. Lo ◽  
Y. W. He
Keyword(s):  
Operator Splitting ◽  
Splitting Method ◽  
Dynamical Symmetry ◽  
Call Option ◽  
Option Prices ◽  
Pricing Options ◽  
Operator Splitting Method ◽  
European Call Option ◽  
Price Formula ◽  
Black Scholes

In this paper, we propose an operator splitting method to valuate options on the inhomogeneous geometric Brownian motion. By exploiting the approximate dynamical symmetry of the pricing equation, we derive a simple closed-form approximate price formula for a European call option which resembles closely the Black–Scholes price formula for a European vanilla call option. Numerical tests show that the proposed method is able to provide very accurate estimates and tight bounds of the exact option prices. The method is very efficient and robust as well.

scholarly journals Estimasi Harga Multi-State European Call Option Menggunakan Model Binomial

CAUCHY ◽  
2011 ◽  
Vol 1 (4) ◽  
pp. 182
Author(s):  
Mila Kurniawaty, Endah Rokhmati ◽  
Endah Rokhmati
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Call Option ◽  
Expiration Date ◽  
Strike Price ◽  
Underlying Asset ◽  
Put Option ◽  
European Call Option ◽  
Black Scholes

Option merupakan kontrak yang memberikan hak kepada pemiliknya untuk membeli (call option) atau menjual (put option) sejumlah aset dasar tertentu (underlying asset) dengan harga tertentu (strike price) dalam jangka waktu tertentu (sebelum atau saat expiration date). Perkembangan option belakangan ini memunculkan banyak model pricing untuk mengestimasi harga option, salah satu model yang digunakan adalah formula Black-Scholes. Multi-state option merupakan sebuah option yang payoff-nya didasarkan pada dua atau lebih aset dasar. Ada beberapa metode yang dapat digunakan dalam mengestimasi harga call option, salah satunya masyarakat finance sering menggunakan model binomial untuk estimasi berbagai model option yang lebih luas seperti multi-state call option. Selanjutnya, dari hasil estimasi call option dengan model binomial didapatkan formula terbaik berdasarkan penghitungan eror dengan mean square error. Dari penghitungan eror didapatkan eror rata-rata dari masing-masing formula pada model binomial. Hasil eror rata-rata menunjukkan bahwa estimasi menggunakan formula 5 titik lebih baik dari pada estimasi menggunakan formula 4 titik.

scholarly journals A Finite Difference-Spectral Method for Solving the European Call Option Black–Scholes Equation

2021 ◽  
Vol 8 (2) ◽  
pp. 273-278
Author(s):  
Younes Talaei ◽  
Hasan Hosseinzadeh ◽  
Samad Noeiaghdam
Keyword(s):  
Finite Difference ◽  
Numerical Experiment ◽  
Spectral Method ◽  
Difference Method ◽  
Call Option ◽  
Algebraic Equations ◽  
Novel Technique ◽  
European Call Option ◽  
Black Scholes ◽  
Galerkin Spectral Method

In this paper, we present a novel technique based on backward-difference method and Galerkin spectral method for solving Black–Scholes equation. The main propose of this method is to reduce the solution of this problem to the solution of a system of algebraic equations. The convergence order of the proposed method is investigated. Also, we provide numerical experiment to show the validity of proposed method.

A MULTILEVEL APPROACH TO SOLVING THE BLACK–SCHOLES EQUATION

2010 ◽  
Vol 13 (03) ◽  
pp. 403-414 ◽  
Author(s):  
HEDLEY MORRIS ◽  
ALFONSO LIMON
Keyword(s):  
Multiresolution Analysis ◽  
Linear Equations ◽  
Coarse Grid ◽  
Call Option ◽  
Computational Resource ◽  
System Of Linear Equations ◽  
Multilevel Approach ◽  
European Call Option ◽  
Black Scholes ◽  
Derivative Information

In this manuscript, we develop a multilevel framework for the pricing of a European call option based on multiresolution techniques. In this approach, the Black–Scholes equation is transformed via finite differences into a system of linear equations, where the form of the implicit operator is used to construct coarse grid projectors. The reduction of the computational resource is achieved by truncating small wavelet coefficients. However, because traditional wavelets fail to prevent oscillations from developing in the Greeks, a multilevel approach is used to retain smoothness in Gamma by incorporating derivative information into the multiresolution analysis.

scholarly journals Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sangkwon Kim ◽  
Chaeyoung Lee ◽  
Wonjin Lee ◽  
Soobin Kwak ◽  
Darae Jeong ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Experiments ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Nonuniform Grid ◽  
Call Option ◽  
Splitting Scheme ◽  
European Call Option ◽  
Black Scholes

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-order α in the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

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