Trinomial Tree Research Articles

Numerical methods base on trinomial trees for option pricing

This paper investigates the approach to pricing European options, starting with one-step binomial tree pricing (a relatively simple way to calculate option value). In the next step, an additional possible rate of change of stock price is added to make the model more realistic, resulting in the one-step trinomial tree model. The model bounds the option price under the no-arbitrary principle. The paper then analyzes the circumstances under which options have a xed price by completing the market and giving the solution formula of option price through the model. Last, put-call parity is used to prove the rationality of one-step trinomial model so that the model eectively prevents the occurrence of risk-free arbitrage in the market. This helps traders to price options reasonably in the market and maintains the stability of the options market.

The Boyle–Romberg Trinomial Tree, a Highly Efficient Method for Double Barrier Option Pricing

Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option’s lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types.

Numerical Methods for Option Pricing

Nowadays, financial markets have become more complex and have given rise to more research opportunities, one of which focuses on research related to the pricing of various financial instruments used in options trading. In this essay, a gradual process of reflection is used to deepen the option pricing theory. Firstly, an effective mathematical method for pricing option contracts is using the binomial tree model, a relatively straightforward way of calculating the value of an option. However, it has only two upward and downward trends and has limitations that are the most different from the actual market conditions. That is why the study will have to continue in-depth to get as close as possible to the real situation in the market. The next step is to make the model more in line with market reality by adding a possible rate of change, resulting in the Trinomial Tree Model. The model incorporates the possibility of future price increases, decreases, or stabilization (the third change). The Trinomial Tree Model follows the no-arbitrage principle and removes the assumption of a risk-free market opportunity. This paper derives a method for constraining market prices under risk-neutral conditions. This is crucial for investors seeking profitable outcomes in options trading. The main objective of this research-based paper is to develop a complete theory of option pricing in a one-step trinomial tree model.

Business decision-making of power generators in competitive electricity market

This paper presents a theoretical framework for the business decision-making process of the power generators as price takers when considering the participation of energy storage. The framework assesses rational valuation, optimal sales strategies, and hedging options for power plants with and without a gross sales constraint. The valuation and optimal sales strategy problems are analyzed using a risk-neutral pricing approach, dynamic programming principles, and the trinomial tree model suitable for the regime switching model. The formulation of a price risk hedging scheme flexible and widely used over-the-counter electricity derivative, the electricity contract for difference, as a tool for hedging electricity spot price risk. The minimum variance hedge ratio and its corresponding hedging efficiency formula are derived. In the section of numerical simulations, we first use the EM algorithm to calibrate the electricity spot model based on electricity spot price data of Nord Pool. Numerical simulations are then conducted on the operational decision-making of power generators under three different forms of energy storage. The results of the simulations provide a basis for power generators to evaluate the real-time value of power plants, to select optimal real-time power sales, and to determine the optimal timing of power plant transfer and storage methods.

Investment Evaluation of CCUS Retrofitting for Coal-to-Liquid Industry in China

Coal-to-oil (CTL) combined with carbon capture, utilization and storage (CCUS) can significantly reduce the CO2 emissions generated in the production process to achieve clean coal utilization. Taking CTL enterprises as sources and deep saline aquifers and oil fields as sinks, this paper establishes a source–sink matching model, which is combined with a trinomial tree real-option model of carbon price fluctuation, and evaluates the investment decisions of CTL. The results show that 36 pipelines with an average transportation distance of 319.13 km and predominantly small diameters must be constructed for CO2 capture and storage combined with enhanced oil recovery (EOR). Under the current carbon price, 83.33% of enterprises can invest immediately when adopting EOR; when utilizing storage in a deep saline aquifer (DSF), even with a 50% subsidy and a decrease in costs due to the learning rate, enterprises still need to execute the deferred option investment. Government subsidies and technological advances can greatly increase the value of investment. The critical carbon price of CTL-CCUS projects is sensitive to government subsidies, technological advances, and CO2 transportation distances. Therefore, China should reasonably guide the development of the carbon market and give play to the role of the carbon market in emission reduction incentives. In addition, the Chinese government can provide direct financial support for the CTL-CCUS project to increase the enthusiasm of CTL enterprises for CCUS transformation and promote technological progress.

Application of UAV-Based Imaging and Deep Learning in Assessment of Rice Blast Resistance

Rice blast is regarded as one of the major diseases of rice. Screening rice genotypes with high resistance to rice blast is a key option for ensuring global food security. Unmanned aerial vehicles (UAV)-based imaging, coupled with deep learning, can high-throughput acquire imagery traits about rice blast infection. In this study, we developed a segmented detection model (RiceblastSegMask) for rice blast detection and resistance evaluation. The feasibility of different backbones and target detection models was further investigated. RiceblastSegMask was a two-stage instance segmentation model, which included an image denoising backbone network, feature pyramid and trinomial tree fine-grained feature extraction combination network, and image pixel codec module. The results showed that the model that combining the image-denoising and fine-grained feature extraction based on Swin Transformer and the feature pixels matching feature labels with the trinomial tree recursive algorithm presented the best performance. Overall accuracy for instance segmentation of RiceblastSegMask reached 97.56%. It achieved a satisfying result of 90.29% accuracy for grading unique resistance to rice blast. The results demonstrated that UAV low-altitude remote sensing, coupled with the proposed RiceblastSegMask model, can efficiently calculate the infected area of rice blast, providing a new phenotypic tool to evaluate the resistance to rice blast in rice at a field scale for the breeding program.

Investment feasibilities of CCUS technology retrofitting China's coal chemical enterprises with different CO2 geological sequestration and utilization approaches

CO2 capture, utilization, and storage (CCUS) technology provides an efficient alternative for decarbonizing the coal chemical industry under China's carbon neutrality goal. However, uncertainties such as geological conditions, technology level, carbon markets, subsidy policies, etc., will greatly affect investor confidence in the CCUS applicability in this industry. This study develops a trinomial tree real options based CCUS investment decision model to explore CCUS investment feasibilities for China's coal chemical enterprises with different CO2 geological sequestration and utilization approaches, i.e., CO2 enhanced oil recovery (CO2-EOR) projects, CO2 enhanced coalbed methane (CO2-ECBM) projects, and deep saline aquifer (DSA) projects. The findings show that 1) CO2-ECBM projects have a larger investment benefit compared to CO2-EOR and DSA projects, with a critical coalbed methane (CBM) price for the immediate investment of 1.3 CNY/m³ under the current technology and carbon price level. 2) CO2-EOR projects are immediately investable at a critical oil price of 200 CNY/BBL; DSA projects require the execution of a delayed option with a critical carbon price of 361.8 CNY/t. 3) Provincial investment benefits vary substantially depending on CO2 transport distance and product revenue, particularly for CO2-ECBM projects in Henan and Guizhou, which have significant investment benefits due to high local CBM prices and short CO2 transport distances.

A Fast VVC Intra Prediction Based on Gradient Analysis and Multi-Feature Fusion CNN

The Joint Video Exploration Team (JVET) has created the Versatile Video Coding Standard (VVC/H.266), the most up-to-date video coding standard, offering a broad selection of coding tools. The maturity of commercial VVC codecs can significantly reduce costs and improve coding efficiency. However, the latest video coding standards have introduced binomial and trinomial tree partitioning methods, which cause the coding units (CUs) to have various shapes, increasing the complexity of coding. This article proposes a technique to simplify VVC intra prediction through the use of gradient analysis and a multi-feature fusion CNN. The gradient of CUs is computed by employing the Sobel operator, the calculation results are used for predecision-making. Further decisions can be made by CNN for coding units that cannot be judged whether they should be segmented or not. We calculate the standard deviation (SD) and the initial depth as the input features of the CNN. To implement this method, the initial depth can be determined by constructing a segmented depth prediction dictionary. For the initial segmentation depth of the coding unit, regardless of its shape, it can also be determined by consulting the dictionary. The algorithm can determine whether to split CUs of varying sizes, decreasing the complexity of the CU division process and making VVC more practical. Experimental results demonstrate that the proposed algorithm can reduce encoding time by 36.56% with a minimal increase of 1.06% Bjøntegaard delta bit rate (BD-BR) compared to the original algorithm.

Real options analysis for urban flood mitigation under environmental change

As urban flooding becomes more frequent and severe due to climate change and accelerated urbanization, effective urban flood control measures are needed to mitigate urban flooding under environmental change. However, few studies consider both climate change and land use change and formulate urban flood control plans for the uncertainty of climate change. This study proposes a real options method based on trinomial tree model to design urban flood control measures under environmental changes. Future rainfall was predicted based on the SDSM model. Land use changes for 2035 and 2050 were predicted using the CA-Markov model. Considering the uncertainty of future climate change, future urban flood control measures based on the real options method were developed, and to compare and analyze them with fixed investment method. The research results show that rainfall and construction land in the study area will increase in the future. The benefit of real options retrofit scheme is higher than the fixed investment retrofit scheme and climate change uncertainty has a greater impact on flood control planning decisions. The research framework proposed in this paper provides an effective basis for the formulation of urban flood control strategies and the design of flood control measures under environmental change.

The Convergence Rate of Option Prices in Trinomial Trees

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1/n and 1/n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations.

Two Approaches for Option Pricing under Illiquidity

The paper focuses on option pricing under unusual behaviour of the market, when the price may not be changed for some time what is quite a common situation on the modern financial markets. There are some patterns that can cause permanent price gaps to form and lead to illiquidity. For example, global changes that have a negative impact on financial activity, or a small number of market participants, or the market is quite young and is just in the process of developing, etc.In the paper discrete and continuous time approaches for modelling market with illiquidity and evaluation option pricing were considered.Trinomial discrete time model improves upon the binomial model by allowing a stock price not only to move up, down but stay the same with certain probabilities, what is a desirable feature for the illiquid modelling. In the paper parameters for real financial data were identified and the backward induction algorithm for building call option price trinomial tree was applied.Subdiffusive continuous time model allows successfully apply the physical models for describing the trapping events to model financial data stagnation's periods. In this paper the Inverse Gaussian process IG was proposed as a subordinator for the subdiffusive modelling of illiquidity and option pricing. The simulation of the trajectories for subordinator, inverse subordinator and subdiffusive GBM were performed. The Monte Carlo method for option evaluation was applied.Our aim was not only to compare these two models each with other, but also to show that both models adequately describe the illiquid market and can be used for option pricing on this market. For this purpose absolute relative percentage (ARPE) and root mean squared error (RMSE) for both models were computed and analysed.Thanks to the proposed approaches, the investor gets a tools, which allows him to take into account the illiquidity.

Correction to: Trinomial tree based option pricing model in supply chain financing

Correction to: Trinomial tree based option pricing model in supply chain financing

Convergence Rate of the High-Order Finite Difference Method for Option Pricing in a Markov Regime-Switching Jump-Diffusion Model

The high-order finite difference method for option pricing is one of the most popular numerical algorithms. Therefore, it is of great significance to study its convergence rate. Based on the relationship between this algorithm and the trinomial tree method, as well as the definition of local remainder estimation, a strict mathematical proof is derived for the convergence rate of the high-order finite difference method for option pricing in a Markov regime-switching jump-diffusion model. The theoretical result shows that the convergence rate of this algorithm is O(Δτ) . Moreover, the results also hold in the case of Brownian motion and jump-diffusion models that are specialized forms of the given model.

Portfolio Real Option Based on Trigeminal Tree Model in Sustainable Utilization of Exploration Asset

Uncertainty of crude oil market causes the value of oil and gas exploration assets to fluctuate upward, remain unchanged, and fluctuate downward. So exploration project investment is usually chartered by multistages. This brings many challenges to investment decisions regarding the sustainable utilization of exploration assets. To avoid the investment risk of exploration projects, a portfolio option method based on the trigeminal tree model was proposed. By analysing the real options involved in oil and gas exploration assets, the investment process is divided into European options and American options. The pricing model of portfolio options is constructed by the trigeminal tree method, and sequential compound diagram and compound option diagram are built. Volatility is predicted based on the GARCH model, and then, the trinomial tree of the underlying asset value, evolution chart, sequential compound option chart, and portfolio option chart is established, respectively. Finally, the project option value is calculated. The application shows that the ROV of oil and gas exploration project of company A is 1.39 million US$ (USMM$), and the project value is 279.87 USMM$. Compared with the binary tree model, the model is superior to the traditional option pricing method in the application effect, which provides a scientific basis for investment decision-making.

The Convergence Rate of Option Prices in Trinomial Trees

The Convergence Rate of Option Prices in Trinomial Trees

Trinomial tree based option pricing model in supply chain financing

Trinomial tree based option pricing model in supply chain financing

Now or later? Optimal timing of mangrove rehabilitation under climate change uncertainty

Climate change poses a variety of threats to ecosystems, biodiversity, and human lives. Mangrove forest is one of the key nature-based solutions that address climate change and its impacts while providing socio-economic and ecological services. With the increasing frequency of extreme weather events, various national and local governments are considering mangrove rehabilitation as a natural defense to these threats. This study aims to assess the value of mangrove rehabilitation by focusing on the ecological goods and services it provides as well as the cost of the project at different implementation periods. We apply a real options-based trinomial tree model to evaluate the optimal timing of making an investment decision for the rehabilitation project under uncertainty in climate change damage cost. Validating the model using the Philippines as a case, the results found that the economic benefits from the mangrove forest, including the market value of forestry and fishery products, protection from climate-related damages, carbon sequestration, and ecotourism, exceeded the investment cost for the rehabilitation project. Considering the uncertainty in damage costs from natural disasters, the real option valuation gave a higher total investment value which described a more optimal decision to postpone the implementation of the mangrove rehabilitation. However, the results found conditions in the model to invest immediately when the frequency of natural disasters and their associated damage costs significantly increased. The implications of the results recommend government policies to enable the realization of mangrove rehabilitation as a nature-based solution addressing the Paris Agreement, Sustainable Development Goals, and the Disaster Risk Reduction global agendas of transition to low-carbon and climate-resilient economies.

Pricing Exotic Options Using Some Lattice Procedures

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

Optimal Selling Strategies under Regime-Switching Market Environment with Finite Expiry

This paper addresses an optimal stock liquidation problem over a finite-time horizon; to that end, we model it as an optimal stopping problem in a regime-switching market. The optimal stopping time is written as a solution to a system of Volterra type integral equations. Moreover, it reveals that when the risk-free interest rate is always lower than the return rate of the stock, it is never optimal to sell the stock early; otherwise, one should sell the stock in bear market if the stock price reaches a critical value and hold the stock in bull market until the maturity date. Finally, we present a trinomial tree method for numerical implementation. The numerical results are consistent with the theoretical findings.

Optimal Surrender Policy of Guaranteed Minimum Maturity Benefits in Variable Annuities with Regime-Switching Volatility

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.