Comparison: Binomial model and Black Scholes model

The establishment of option prices is one of the crucial aspects in derivative trade. Black-Scholes model (BSM) is one of the most popular models of option price establishment. The option exchanges such as CBOE, HKEX, and NSE use this model to determine the price options. Black-Scholes Model is modelled with stock price movement as a stochastic process. Another popular model is binomial model (BSM), originated from stock exchange movement model which divides interval time [0, T] into n equal length step. It holds several models to determine the value of up-move, down-move, and probability. Binomial model is categorised as Cox-Ross-Rubinstein, Jarrow-Rudd, and Leisen-Reimer. There are numerous literatures which discuss the relation between BM and BSM, including the convergence of binomial model and BSM. The former’s model which is often put side-by-side with BSM is Cox-Ross-Rubinstein. Even though this model is quite simple, it requires a lot, even thousands of steps to render Cox-Ross-Rubinstein to converge with BSM. It certainly takes a lot of time to calculate. Therefore, in this study, with limited steps, Jarrow-Rudd and Leisen-Reimer models are compared to BSM with the Cox-Ross-Rubinstein model. It aims to check on which binomial model is more convergent to BSM with limited steps in the same period. The data collected were secondary data from finance.yahoo.com. Judgemental sampling was used for technique sampling and several shares, with large market capitalization in Hongkong, India, and Indonesia are chosen. By calculating the MAFE error value from option price of BSM and BM, it is discovered that Leisen-Reimer with 101 steps is more convergent to BSM.

This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solution, while it can only be used for European options with stock prices regarded as a continuous process. For the CRR binomial model, although it can only give numerical solutions and requires lots of calculation because of too many branches, but it can solve the American option prices with early exercises and the European option prices with stock prices regarded as a discrete process.

In recent years, quantitative researchers used a wide range of models to price options, from the Black-Scholes model to more complex models such as the Heston model. This paper aims to analyze the effectiveness of the Black-Scholes model and the Binomial Tree model by using them to price Berkshire Hathaway’s equity options and European-style S&P 100 index options. The method used in this paper is gathering the market data of the options first. Second, using the data gathered to price the options by applying the Black-Scholes and Binomial Tree models. Third, comparing the derived theoretical price with the market price by getting the Sum of Square Errors. Lastly, determining the best model for each type of option. Through this research, the author found that comparing the two models, the Binomial Tree model derives a smaller Sum of Square Errors when pricing European-style index options, and the Black-Scholes model derives a smaller Sum of Square Errors when pricing American equity options. Thus, the Binomial Tree model is a better model to price European-style index options, and the Black-Scholes model is more effective when pricing American equity options.

This paper reviews the binomial and trinomial option pricing models and their convergence to the Black-Scholes model result. These models are generalized for the European and American options. The trinomial models are said to be more accurate than the binomial when fewer steps are modelled. These models are widely used for the usual vanilla option types, European or American options, that respectively can be exercised only at the expiration date and at any time before the expiration date. The results are supportive of the conventional wisdom that trinomial option pricing models such as the Kamrad-Ritchken model and the Boyle model are converging faster than the binomial models. When binomial models are compared in terms of convergence, the most efficient model is the Jarrow-Rudd model. This paper concludes that improved binomial models such as the Haahtela model are converging faster to the BS model result. After some trials, binomial distribution follows log-normal distribution assumed by the Black-Scholes model.

Employee stock options are widely used. SFAS 123(R), Share-Based Payment, issued in December 2004 requires that all entities use a fair-value-based option pricing model, adjusted for the unique characteristics of the options, to recognize expenses for employee stock options at the grant date. The Cox-Ross-Rubinstein (CRR) model – a binomial lattice model that converges to the Black-Scholes (BS) model when the number of periods to expiration approaches infinity - meets the requirements of SFAS 123(R). The CRR model factors in the often complex characteristics of employee stock options and always gives a lower option value than the BS model. This paper demonstrates the implementation of the CRR model using Microsoft Excel and shows that constant dividend yield, early exercise, and cliff vesting can be easily factored into the CRR model. Keywords SFAS 123(R), Employee Stock Options, Fair-Value-Based Option Pricing Model, CRR Model

In this paper, artificial neural nets are applied to pricing the call warrants in the Taiwan stock market. Warrants were initialized in Taiwan in 1997 and hence a still very new product. It, therefore, may provide us a chance to test whether artificial neural nets, as a data-driven tool, can be more effective than the model-driven tools in dealing with this emerging derivative market. The data employed in this paper are the two earliest listed stock call warrants, namely, Yageo's and Pacific Electric Wire and Cable's warrants, ranging from September 4, 1997 to September 2, 1998. 24 neural nets, covering different inputs, numbers of hidden nodes and transfer functions, were attempted. Each neural net was trained for 20 independent runs. Based on the average of the in-sample performance, the best neural net was selected to compete with the Black-Scholes model and binomial model in the post-sample data. The post-sample performance of each model was evaluated by statistics. We found that the neural net model outperformed both the Black-Scholes model and the binomial model in almost all criteria.

Options are crucial to global financial markets. However, there is rarely existing literature combining a specific option and a certain company, with simultaneous analysis and conclusions of the option. Based on the ordinary least square (OLS) regression and Black-Scholes (BS) model, we design two simulated options for Morgan Stanley, which are European options and power options. Then the option's value is identified in this article. Besides, through the sensitivity analysis of the two options and their related parameters, we get some results and conclusions about the characteristics of the two options. In general, the value of options is affected by independent variables and parameters. Compared with a European option, a power option has a higher risk and higher future expected return. This paper provides better advice for investors, on the premise of fully considering various possible situations, compare the advantages and disadvantages of the two options, and make their own investment choices.

Comparison: Binomial model and Black Scholes model

The common practice of using different volatilities for options of different strikes in the Black-Scholes (1973) model imposes inconsistent assumptions on underlying securities. The phenomenon is referred to as the volatility smile. This paper addresses this problem by replacing the Brownian motion or, alternatively, the Geometric Brownian motion in the Black-Scholes model with a two-piece quadratic or linear function of the Brownian motion. By selecting appropriate parameters of this function we obtain a wide range of shapes of implied volatility curves with respect to option strikes. The model has closed-form solutions for European options, which enables fast calibration of the model to market option prices. The model can also be efficiently implemented in discrete time for pricing complex options. G1

Some Thoughts About Data Type, Distribution, and Statistical Significance

This paper examines the reliability of option fair value estimates in the presence of transaction costs. The Black Scholes Merton (BSM) framework assumes zero transaction costs and thus might not provide a reasonable approximation in this context. We investigate the model adjustments companies make to their BSM models to deal with these transaction costs. We specifically examine Employee Stock Option (ESO) plans listed on the French stock exchange, as detailed disclosure on modeling is available for these ESOs. Our analysis questions the reliability of these model adjustments, especially their bias and the extent to which they provide a faithful representation of option fair values. Holding parameter values constant, we find that the model adjustments lead to a median understatement of 52% compared to the BSM model price, higher than the discount we observe for the opportunistic determination of model parameters (below 20%). The paper contributes to the fair value literature by highlighting model risk in the fair valuation of options. This model risk stems from assumptions made about the size of transaction costs and complements the notion of parameter risk analyzed in previous literature. As a result, the model itself might be a possible channel for fair value management.

Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite process. We set forth an approximation of the Hermite Black–Scholes model by random walks in the Skorokhod topology, and apply the Donsker type approximation to the Hermite process as the Hurst index is greater than 12. We find that the binary model approximation of the Black–Scholes model driven by Hermite processes also admits arbitrage opportunities. Several numerical examples of the Hermite binomial model are presented as demonstration. Moreover, we provide an option pricing model when the geometric Hermite processes drives the price fluctuations of the underlying asset.

Option pricing revisited: The role of price volatility and dynamics

Option pricing is essential for market participants as it facilitates risk management and price discovery. This study provides a comprehensive elucidation of diverse option pricing models. The author delineates the development of option pricing models from the fundamental Black-Scholes-Merton model (BSM) to more intricate methodologies. The importance of comprehending option pricing for hedging strategies and risk management is crucial in financial mathematics. The study analyses several models, including the BS model, Jump Diffusion model (JDM), Binomial Tree model (BTM), Monte Carlo simulation (MCS), and stochastic volatility models (SVM), while addressing their applications, limitations, advantages, and disadvantages. This work delineates the history and distinctions between other models and the Black-Scholes-Merton model. The Black-Scholes-Merton model exerts the most substantial influence on financial markets. The alternative models exhibit greater flexibility and enhanced accuracy when confronting complicated and uncertain market trends. The trend of future development is bound to be the continuous updating of models to adapt to more complex situations.

The share repurchase decision is strategic for large corporations, yet comparative studies integrating mathematical modeling, financial options, and portfolio optimization remain limited. This study addresses this gap by proposing a quantitative analysis of share repurchase decisions using multiple approaches. The research compares the profitability of a Brazilian bank’s share repurchase program with the allocation of the same capital into an efficient portfolio, constructed based on DEA BCC models, Markowitz, Monte Carlo simulation, and financial option theory. Binomial modeling and the Black-Scholes model are applied to value repurchase as a financial option, incorporating risk, flexibility, and volatility. Data were processed in Python using historical time series from B3. The results show that, in the analyzed scenario, the optimized portfolio financially outperforms repurchase, providing a more efficient alternative. The study offers practical and technical evidence to support complex corporate decisions in highly uncertain environments.