Multi-purpose binomial model: Fitting all moments to the underlying geometric Brownian motion

This chapter presents the binomial tree approach to the option pricing problem. We first illustrate the basic ideas of option pricing by considering the one-period binomial tree model and then extend to a multi-period binomial tree model. We then show that, by taking limits in an appropriate way, the binomial expression for the option price converges to the Black–Scholes option price and pricing equation. Alternatively, the continuous time model can be discretised in a way that yields the same expressions as obtained by the binomial tree approach.

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.

In this paper we introduce and study the simplest random binomial tree model. Usual binomial tree model is prescribed by pair of numbers (u, d), where u denotes the increase rate of the stock over the fixed period of time and d denotes the decrease rate, with 0< d < 1< u. We call the pair (u, d) an environment of the binomial tree model. A pair (Un, Dn), where {Un} and {Dn} are the sequences of independent, identically distributed random variables with 0< Dn < 1< Un for all n, is called a random environment and binomial tree model with random environment is called random binomial tree model. In this paper we define and study European put option for such models.

Quantum walk option pricing model based on binary tree

This paper presents a comprehensive comparative analysis of Monte Carlo simulation and binomial tree models for pricing geometric Asian options. In the process of evaluations, the Tesla (TSLA) stock is the underlying asset for the options, and the modified Black-Scholes model serves as the benchmark for comparisons of pricing accuracy, Greeks calculation. Besides, this research also analyzes the difference between the two methods in terms of computational efficiency. According to the simulations, the binomial tree model shows a higher accuracy compared to the Monte Carlo simulation. For Greeks calculations, both methods illustrate consistency in Delta and Gamma values, and the binomial tree model shows almost perfect Vega precision. At the same time, the Monte Carlo simulation offers computational advantages with near-linear complexity, making it ideal for real-time applications. The binomial tree model, despite exponential complexity growth, remains ideal for high-precision requirements. Therefore, these results offer a guideline for the application of these two methods, in which one can determine the usage based on the tradeoff between accuracy and computational speed.

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.

This paper discusses a comprehensive analysis of the pricing of two typed of options and early exercise premiums under the binomial tree model. We find the differences between European and American options, with particular emphasis on the conditions governing optimal early exercise strategies.We verify that early exercise is never the best option for American call options on stocks that pay no dividends, making their pricing the same as their European counterparts.Moreover, in the single-period model, the exercise of American put options depends on K = uq r+q S0, while in contrast, the relative positioning of the strike price and the ultimate stock value influences the exercise decisions in the two-period model.Our analysis indicates that the early exercise premium primarily manifests in multiperiodwhere holders can determine their own exercise timingcreating additional value relative to European put options.

European Options under Proportional Transaction Costs: An Algorithmic Approach to Pricing and Hedging

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

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.

A fuzzy analogue of the binomial option pricing tree model introduced by Cox et.al [1] was proposed by Yoshida [2] to study fuzzy American put option model in an uncertain environment using discrete time fuzzy stochastic process. Later, Muzzioli [3] used non overlapping triangular and trapezoidal fuzzy numbers to model the jump factors in American put option pricing model admitting impreciseness only in volatility while Xcaojian Yu [4] persumed impreciseness in both risk-free interest rate and volatility of the underlying stock involving non-overlapping trapezoidal fuzzy numbers. K. Meenakshi et al. [5] defined new fuzzy risk-neutral probability measures using general trapezoidal fuzzy numbers to study "Problem of Pricing American Fuzzy Put Option Buyer’s Model". The two up and down jump factors and the risk-free interest rate fluctuates oftenly and hence are uncertain in nature. When financial investors come across a high volatile or low volatile (up and down jump factors) fuzzy stocks, non-overlapping type of fuzzy numbers will not be sufficient to predict the underlying fuzzy stock prices as the fuzzy stock prices would go up only in the up state and would go down only in the down state of the fuzzy binomial tree. The uncertainty associated in the above stated fuzzy option pricing parameters could not be captured completely here. Such a scenario needs attention. To handle such a situation, we need to consider fuzzy numbers which are not only non - overlapping but partially or fully overlapping and/ or contained in too. Also not much consideration had been given to American fuzzy put option model involving fuzzy martingales. This paper will consider the study of the fuzzy analogue of martingale pricig theory in the context of fuzzy option pricing theory. In this study, we discuss American Fuzzy Put Option Seller’s Model (AFPOSM) based on fuzzy future contract involving general linear octagonal fuzzy numbers (GLOFN); general in the sense that they could be either overlapping or non-overlapping partially or completely and/or contained in using the fuzzy risk-neutral probability measures introduced by us. We record a computational procedure to obtain the fuzzy profit and loss (PL) values of sellers using two-period fuzzy binomial tree model wherein the fuzzy stock price and the fuzzy future price following discrete time fuzzy stochastic process. The same is performed by deploying the real stock market data obtained from the website [6] that includes American style Microsoft Corporation (MSFT) shares of the year 2018 under the bullish (the constant initial fuzzy stock price goes above the constant fuzzy strike price) scenario. Utilizing the PL values, we estimate an optimal exercise time and an optimal exercise price of sellers. Also the discounted fuzzy intrinsic values of AFPOSM on fuzzy future contract are validated. The computations carried out in AFPOSM are performed using MATLAB 2016a software. &nbsp; &nbsp;

This paper presents a risk neutral binomial process as an alternative approach for the derivation of analytic pricing equation called “Black - Scholes Partial differential Equation” in the theory of option pricing. Binomial option pricing is a powerful technique that can be used to solve many complex option pricing problems. In contra st to the Black - Scholes model and other option pricing models that require solutions to stochastic differential equations, the binomial model is mathematically simple. Binomial model is based on the assumption of no arbitrage. The assumption of no arbitra ge implies that all risk - free investments earn the risk - free rate of return and no investment opportunity exists that requires zero amounts of investment but yield positive returns.

In this paper a new method is designed to account price of military software, which is based on option pricing models. Firstly, it is introduced about the basic knowledge of option pricing model. And then the military software pricing model is provided based on Black-Scholes option model and binomial tree model. Several variables in reference to military software price are redefined in the related models. Then, in order to calculate the software price, we build up system of equations, of which one is Black-Scholes equation and the other one is binomial tree equation. Lastly, a case study is given to demonstrate the feasibility of the model. The study results and sensitivity analysis show the factors that influence the real price of software and the consistency of two classic option pricing models in practical application. In addition, one of the factors—volatility plays an important role in military software pricing based on Black-Scholes model, and there is a further research to make it clear and objective.

This paper considers the introduction of stock options in an (dynamically) incomplete securities market made up of a riskless bond and the stock. The stock price follows a geometric Brownian motion with constant drift. However, there is incomplete information about the unknown stochastic volatility. The option price is determined by a uniform-price auction. Thus an option pricing formula results from the interaction of market participants relying on private information on the unknown stochastic volatility under an explicit market structure. This paper incorporates market microstructure considerations into an extended Black-Scholes model with incomplete information on the underlying volatility. It relies on the growing importance of auctionlike trading rules in financial markets.

This paper is mainly about the research object of early exercise of American options by two-period binomial models. In this work, we use the two-period binomial model to analyze the price of American options and European options, decide whether to exercise early or not, and compare the differences between American options and European options. What is more, we use Python to help us with our calculations. Options are a kind of financial derivatives that help one to arbitrage, speculate, and hedge. To make more profits, people can use the binomial model to calculate the price of options. Through this analysis, we got to know each price of options in different situations, and we know that the price of American options is always equal to or larger than the price of European options. This work can help us to calculate the price of the option and decide if it is exercised early, and make sure to get more profits and apply it to daily life.