Random Binomial Tree Models and Options

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 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.

Under digital technology, the vigorous development of online education has also encountered challenges of different degrees, such as the high dropout rate of learners, the low completion rate of courses, and the loss of users. Learning engagement has not yet formed an effective assessment system. Based on an exploration of the core of learning activity engagement, this research evaluates the state of learning activity engagement utilizing learners’ adaptive adjustment processes of information exchange activities and a random trees model. A combined classifier is a random tree. Random trees are a combined classifier. Its main idea is to build multiple relatively independent decision tree classifiers based on two random processes, and then obtain the final prediction results by voting all decision trees. The traditional random trees model is improved by weighted calculation and aggregation calculation. After experimental analysis, it can be found that the highest can reach more than 80%, which proves that the improvement of the weighted value has a good reflection on the random trees model, and the accuracy rate is increased by 65.2% after the weighted improvement. Overall, the performance of the improved random trees model is improved by 67.3%.

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

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.

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

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.

Quantum walk option pricing model based on binary tree

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.

We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this model of random trees may be approximated geometrically by a Galton--Watson tree conditioned on having a large random size. This implies that a variety of results for the well-studied planted case also hold for unrooted trees, including Gromov--Hausdorff--Prokhorov scaling limits, tail-bounds for the diameter, distributional graph limits, and limits for the maximum degree. Our work complements results by Wang~(2016), who studied random unrooted plane trees whose diameter tends to infinity.

사회간접자본의 증가로 인해 신규건설투자 대비 유지관리 비용지출 비중이 확대되어가고 있어 유지관리 주체 입장에서 정확한 유지관리 비용 추정의 중요성이 강조되고 있다. 최근의 연구결과는 점진적이고 지속적인 기후변화에 의해 시설물에 축적되는 영향이 심각한 수준인 것으로 나타나고 있는데, 유지관리 비용추정에 있어 이를 고려한 연구가 미비한 실정이다. 본 연구에서는 중장기적 관점에서 연평균 기온변화의 도로시설 유지관리 비용 변동에의 영향을 추정하기 위해 이항분석모델을 활용한 비용변동 추정 체계를 제안하였다. 이를 위하여 IPCC (Intergovernmental Panel on Climate Change) 5차 보고서에서 도출된 기후변화 시나리오에 따른 연평균 기온변화를 도로시설 유지관리비용 변동에 적용하여, 기후변화의 영향이 고려된 유지관리 비용변동 추정을 위한 분석모델을 도출하였다. 이항모델 및 몬테칼로 시뮬레이션을 활용한 추정모델은 추후 유지관리 주체의 탄력적 의사결정에 다양하게 활용될 수 있을 것으로 판단된다. Due to the increasing trend of operation and maintenance cost (O&M cost) of infrastructure, the accurate estimation of O&M cost is crucial part to the government. Recent literatures pointed out that gradual climate changes such as average temperature changes, average precipitation changes, and etc. have significant impact on infrastructure O&M cost. This research is intended to develop a long-term O&M cost prediction model of road facilities by considering the impacts of average temperature changes. For this end, the climate change scenarios of Intergovernmental Panel on Climate Change (IPCC)'s <TEX>$5^{th}$</TEX> report are adopted to structure the impact of average temperature changes by using binomial lattice model. The proposed framework is expected to regional government in supporting decisions for road O&M cost.

In this expository paper, we illustrate the generality of the game-theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are naturally derived from game-theoretic probability protocols. Corresponding to any discrete distribution with finite support, we construct a finite-horizon discrete game, a replicating strategy of Skeptic, and a neutral forecasting strategy of Forecaster, such that the discrete distribution is derived from the game. Construction of a replicating strategy is the same as in the standard arbitrage arguments of pricing European options in binomial tree models. However the game-theoretic framework is advantageous because it eliminates the need for any a priori probabilistic assumption.

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.

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

BACKGROUND:The investment decision can be affected by changing levels of uncertainty and risk. The main objective of this research was to identify, characterize, and quantify the parameters which are essential in evaluation hospital construction projects and provide useful modeling techniques to give the best investment decisions for investors in Iran's health-care projects investment.MATERIALS AND METHODS:The methodology of this study was employing discounted cash flow (DCF) and real option valuation to investigate the feasibility investment in the public hospital construction project. The Islamshahr, Mashhad, and Fardis hospitals were included in the analysis. Economic indices of DCF methods were internal rate of return (IRR) and net present value.RESULTS:The economic evaluation of the Black–Scholes model was almost as same as the binomial tree model, but there was a significant difference between the real options model and traditional methods. According to the traditional methods, the profitability with IRR for Islamshahr, Mashhad, and Fardis hospital projects was 35%, 43%, and 26%, respectively. Black–Scholes model showed profitability only for Islamshahr and Mashhad hospitals, and there was no adequate profitability for investors of Fardis Hospital project during the study.CONCLUSIONS:The methods derived from the real options valuation could provide a more flexible and reliable indices for investors in dynamic and high revolution economic conditions. On the other hand, dynamic economic evaluation models can be applied to correctly evaluate the projects because of Iran's health revolution and its health plans.