Simplest differential equation of stock price, its solution and relation to assumption of black-scholes model

2003 ◽  
Vol 24 (6) ◽  
pp. 654-658 ◽  
Author(s):  
Yun Tian-quan ◽  
Lei Guang-long
Keyword(s):  
Differential Equation ◽  
Stock Price ◽  
Black Scholes Model ◽  
Black Scholes

The two best ways to derive the Black–Scholes PDE

2019 ◽  
Vol 10 (2) ◽  
pp. 168-174
Author(s):  
Paul Wilmott
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Design Methodology ◽  
Content Type ◽  
Technical Requirements ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Business Perspective ◽  
Big Picture ◽  
Complex Methods

Purpose The purpose of this paper is to explain the Black–Scholes model with minimal technical requirements and to illustrate its impact from a business perspective. Design/methodology/approach The paper employs simple accounting concepts and an argument part based on business need. Findings The Black–Scholes partial differential equation can be derived in many ways, some easy to understand, some hard, some useful and others not. The two methods in this paper are extremely insightful. Originality/value The paper takes a big-picture view of derivatives valuation. As such, it is a simple accompaniment to more complex methods and aims to keep modelling grounded in reality.

Valuation Method of Equity Incentives of Listed Companies Based on the Black-Scholes Model

2020 ◽  
Vol 12 (2) ◽  
pp. 36-49
Author(s):  
Zhongwen Liu ◽  
Yifei Chen
Keyword(s):  
Stock Price ◽  
Turnover Rate ◽  
Garch Model ◽  
Listed Companies ◽  
Valuation Method ◽  
Black Scholes Model ◽  
Black Scholes ◽  
S Model ◽  
The Garch Model ◽  
Equity Incentive

This article applies the classic Black-Scholes model (i.e. B-S model) and turnover rate adapted B-S model (revised B-S model) to equity incentive valuation of listed companies. Unlike other studies on equity incentive valuation which generally adopt historical volatility, this article applies the GARCH model to equity incentive valuation. The volatility of stock price is estimated by the GARCH model to improve the accuracy of equity incentive valuation. The turnover rate has an important impact on the equity incentive valuation of listed companies. Considering the turnover rate can improve the accuracy of the equity incentive valuation and reduce the error of equity incentive valuation. Through the case study of the equity incentive valuation of Infinova, the practicality of the equity incentive valuation method is further verified.

scholarly journals The Extended Black-Scholes Model with-LAGS-and “Hedging Errors”

2003 ◽  
Author(s):  
Mondher Bellalah
Keyword(s):  
Stock Price ◽  
Standard Form ◽  
Small Time ◽  
Market Model ◽  
Short Term ◽  
Hedging Strategy ◽  
Underlying Asset ◽  
Elapsed Time ◽  
Black Scholes Model ◽  
Black Scholes

The Black-Scholes model is derived under the assumption that heding is done instantaneously. In practice, there is a “small” time that elapses between buying or selling the option and hedging using the underlying asset. Under the following assumptions used in the standard Black-Scholes analysis, the value of the option will depend only on the price of the underlying asset S, time t and on other Variables assumed constants. These assumptions or “ideal conditions” as expressed by Black-Scholes are the following. The option us European, The short term interest rate is known, The underlying asset follows a random walk with a variance rate proportional to the stock price. It pays no dividends or other distributions. There is no transaction costs and short selling is allowed, i.e. an investment can sell a security that he does not own. Trading takes place continuously and the standard form of the capital market model holds at each instant. The last assumption can be modified because in practice, trading does not take place in-stantaneouly and simultaneously in the option and the underlying asset when implementing the hedging strategy. We will modify this assumption to account for the “lag”. The lag corresponds to the elapsed time between buying or selling the option and buying or selling - delta units of the underlying assets. The main attractions of the Black-Scholes model are that their formula is a function of “observable” variables and that the model can be extended to the pricing of any type of option. All the assumptions are conserved except the last one.

scholarly journals Pricing Exotic Options Using Some Lattice Procedures

2021 ◽  
Vol 41 (1) ◽  
pp. 26-40
Author(s):  
Sadia Anjum Jumana ◽  
ABM Shahadat Hossain
Keyword(s):  
Stock Price ◽  
Lattice Models ◽  
Exotic Options ◽  
Barrier Options ◽  
Tree Model ◽  
Binomial Tree ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Binomial Tree Model ◽  
Trinomial Tree Model

In this work, we discuss some very simple and extremely efficient lattice models, namely, Binomial tree model (BTM) and Trinomial tree model (TTM) for valuing some types of exotic barrier options in details. For both these models, we consider the concept of random walks in the simulation of the path which is followed by the underlying stock price. Our main objective is to estimate the value of barrier options by using BTM and TTM for different time steps and compare these with the exact values obtained by the benchmark Black-Scholes model (BSM). Moreover, we analyze the convergence of these lattice models for these exotic options. All the results have been shown numerically as well as graphically. GANITJ. Bangladesh Math. Soc.41.1 (2021) 26-40

scholarly journals Options valuation analysis of the shares of Ecopetrol and Pacific Exploration between June 2013 and July 2016

2017 ◽  
Vol 15 (40) ◽  
pp. 41-53
Author(s):  
Álvaro Javier Cangrejo ◽  
Christian Camilo Cortés
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Lower Risk ◽  
Fixed Time ◽  
Oil Companies ◽  
Black Scholes Model ◽  
Black Scholes ◽  
The Future ◽  
Options Valuation ◽  
Box Cox Transformation

In this paper, we analyze the environment and the dynamics of the Black-Scholes model starting from a stochastic differential equation that explains the evolution of the future prices of an asset. With these defined guidelines, the data obtained by the daily closing prices between June 2013 and June 2016 of the shares of Ecopetrol and Pacific Exploration are normalized, by means of a Box-Cox transformation, to determine the volatility of each of them and apply this model to calculate the value of the asset with fixed time, and thus determine which of the two oil companies have a lower risk at the time of investing.

Option Pricing using Black Scholes Model (Case-Tata Motors and Reliance)

2015 ◽  
Vol 9 (1and2) ◽  
Author(s):  
Ms. Mamta Shah
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Safety Net ◽  
Volatility Smile ◽  
Model Case ◽  
Underlying Asset ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Strategies ◽  
Additional Income

The power of options lies in their versatility. It enables the investors to adjust position according to any situation that arises. Options can be speculative or conservative. This means investor can do everything from protecting a position from a decline to outright betting on the movement of a market or index. Options can enable the investor to buy a stock at a lower price, sell a stock at a higher price, or create additional income against a long or short stock position. One can also uses option strategies to profit from a movement in the price of the underlying asset regardless of market direction. the responsible act and safe thing to do. Options provide the same kind of safety net for trades and investments already committed, which is known as hedging. The research paper is based on Black Scholes Model. The study includes the Implied Volatility Test and Volatility Smile Test. This study also includes the solver available in MS Excel. This study is based on stock price of Reliance and Tata Motors.

scholarly journals Fractional Black-Scholes Model and Technical Analysis of Stock Price

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Song Xu ◽  
Yujiao Yang
Keyword(s):  
Stock Returns ◽  
Stock Prices ◽  
Stock Price ◽  
Technical Analysis ◽  
Stochastic Volatility Model ◽  
Black Scholes Model ◽  
Dependent Process ◽  
Large Numbers ◽  
Volatility Model ◽  
Black Scholes

In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameterH>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

The Black-Scholes Model and Stock Price Forecasting

1980 ◽  
Vol 15 (2) ◽  
pp. 13-22
Author(s):  
Alan Gleit ◽  
Ben Branch
Keyword(s):  
Stock Price ◽  
Price Forecasting ◽  
Stock Price Forecasting ◽  
Black Scholes Model ◽  
Black Scholes
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