Trinomial Tree Research Articles

Evaluating the effect of a subsidy policy on carbon capture and storage (CCS) investment decision-making in China — A perspective based on the 45Q tax credit

A trinomial tree model based on the delay real option is developed to evaluate the carbon capture and storage (CCS) retrofitting investment for existing coal-fired power plants in the context of the 45Q tax credit. The uncertainties regarding the carbon price, the CCS retrofitting investment cost, the operation and maintenance (O&M) cost, the CCS investment subsidy scenarios, and the allocation ratio of the carbon dioxide (CO2) storage subsidy between the coal-fired power plants and CO2 storage enterprises are taken into consideration. The results show that if the allocation ratio of the CO2 storage subsidy for coal-fired power plants is zero, the full government subsidy for the initial CCS investment cost and clean electricity tariff (0.015 CNY/kWh) are not sufficiently attractive for the coal-fired power plants to invest in CCS and the critical allocation ratio is 17.8% in this case. The critical allocation ratio increases to 26.4% if the government subsidy for the initial CCS investment cost is zero. Moreover, if the government subsidy for the initial CCS investment cost and clean electricity tariff are both canceled, the coal-fired power plants need to receive at least 33.3% of the CO2 storage subsidy to invest in CCS. The results provide theoretical support for the decision-making regarding CCS retrofitting investment and CCS subsidy policy-making.

Non-recombining trinomial tree pricing model and calibration for the volatility smile

Abstract In this paper, we consider the non-recombining trinomial tree pricing model under the volatility, which is a function of time, establish the option pricing model and give the convergence rates of the non-recombining trinomial tree method. In addition, we research the calibration problem of volatility and adopt an exterior penalty method to transform this problem into a nonlinear unconstrained optimization problem. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S&P 500 index. The results show the effectiveness of the non-recombining trinomial tree pricing model.

Optimal design of carbon tax to stimulate CCS investment in China's coal‐fired power plants: A real options analysis

AbstractCarbon tax is an important policy instrument to control greenhouse gases. How to design a suitable carbon tax rate is of significance for policy makers. This study aims to address this issue from a microeconomic perspective by exploring the decision behavior of a potential investor. A trinomial tree model based on real options theory is presented to evaluate carbon capture and storage (CCS) investment in coal‐fired power plants considering uncertain factors. The model is then applied to a case study and the main findings are as follows: (1) carbon tax policy will have a positive effect on the CCS investment. If the carbon price is not high enough to trigger CCS investment, a hybrid mechanism integrating carbon trading and carbon tax to stimulate CCS investment is required; (2) in applying the trinomial tree real options model, the optimal carbon tax rate should be 145.3 RMB/ton for supercritical pulverized coal (SCPC) (or 79.54 RMB/ton for integrated gasification combined cycle (IGCC)) plants under current market condition and at current technological levels; (3) the optimal carbon tax rate is sensitive to uncertainties. Specifically, the optimal tax rate depends largely on not only the cost of the CCS investment but also the portfolio of carbon price volatility and the initial carbon price level. These conclusions provide the theoretical foundations for decision making regarding CCS investment and related policy making. © 2018 Society of Chemical Industry and John Wiley & Sons, Ltd.

Abandonment Decision-Making of Overseas Oilfield Project Coping with Low Oil Price

The abandonment option of an operating oil project refers to the right to shut down or transfer the project. As a kind of American real option, it minimizes the impact of bad operating conditions, thus increases the initial project value. Meanwhile, as a put option, it maximizes the management flexibility in unfavorable environment, especially in the current low oil prices. This article uses the trinomial tree, rather than the binomial tree widely practiced in finance, to value the option. Its lattice structure shows flexibility and intelligibility, and improves computational efficiency and accuracy. In this article, the abandonment option value incorporates uncertainties of oil price, exchange rate, political environment and taxation policy. The risk-neutral based decisions are relatively objective for oil companies. The case study indicates that the relative relationship between the abandonment option value and the project scrap value or selling price is the key to the decision-making results. A novel conclusion from the risk-neutral prospective is that, the project is more likely to be sold at higher risk scenario or with higher profit requirement. Moreover, export duty and mineral extraction tax have a greater impact on the abandonment timing than corporate income tax. This decision-making model can be introduced with modifications to other investments with increasing risk of falling asset price.

Assessing real options in urban surface water flood risk management under climate change

Developing an adaptation option is challenging for long-term engineering decisions due to uncertain future climatic conditions; this is especially true for urban flood risk management. This study develops a real options approach to assess adaptation options in urban surface water flood risk management under climate change. This approach is demonstrated using a case study of Waterloo in London, UK, in which three Sustainable Drainage System (SuDS) measures for surface water flood management, i.e., green roof, bio-retention and permeable pavement, are assessed. A trinomial tree model is used to represent the change in rainfall intensity over future horizons (2050 s and 2080 s) with the climate change data from UK Climate Projections 2009. A two-dimensional Cellular Automata-based model CADDIES is used to simulate surface water flooding. The results from the case study indicate that the real options approach is more cost-effective than the fixed adaptation approach. The benefit of real options adaptations is found to be higher with an increasing cost of SuDS measures compared to fixed adaptation. This study provides new evidence on the benefits of real options analysis in urban surface water flood risk management given the uncertainty associated with climate change.

Enhancing binomial and trinomial equity option pricing models

We extend the classical Cox–Ross–Rubinstein binomial model in two ways. We first develop a binomial model with time-dependent parameters that equate all moments of the pricing tree increments with the corresponding moments of the increments of the limiting Itô price process. Second, we introduce a new trinomial model in the natural (historical) world, again fitting all moments of the pricing tree increments to the corresponding geometric Brownian motion. We introduce the risk-neutral trinomial tree and derive a hedging strategy based on an additional perpetual derivative used as a second asset for hedging at any node of the trinomial pricing tree.

Analysis of a multiple year gas sales agreement with make-up, carry-forward and indexation

A typical gas sales agreement, also called a gas swing contract, is an agreement between a supplier and a purchaser for the delivery of variable daily quantities of gas, between specified minimum and maximum daily limits, over a certain number of years at a strike price. The main constraint of such an agreement that makes them difficult to value is that there is a minimum volume of gas (termed the take-or-pay or minimum bill) for which the buyer will be charged at the end of the year (or penalty date), regardless of the actual quantity of gas taken. We propose a framework for pricing such swing contracts where both the gas price and strike price (an index) are stochastic processes. With the help of a two-dimensional trinomial tree, we are able to price such swing contracts with both so-called make-up and carry-forward provisions; find optimal daily decisions and optimal yearly usage of both the make-up bank and the carry-forward bank. With the help of a number of numerical examples, we also provide a detailed analysis, not only of the different features these contracts have, but also how different model parameters will affect both the optimal value and the optimal decisions.

An Adjusted Trinomial Lattice for Pricing Arithmetic Average Based Asian Option

An adjusted trinomial model for pricing both European and American arithmetic average-based Asian options is proposed. The Kamrad and Ritchken trinomial tree governs the underlying asset evolution. The algorithm selects a subset of the true averages realized at each node to serve as the representative averages. The option prices are then computed via backward induction and interpolation. The results show that the trinomial method produces more accurate prices especially in the case of European style Asian options.

Quasi-Monte Carlo in Tree Pricing Models

Quasi-Monte Carlo sampling methods for the pricing of path dependent derivatives have been shown to outperform random sampling when used in a variety of different interest rate models. In certain applications it is desirable to perform Monte Carlo simulations on a tree based interest rate model, however there is a lack of literature describing how to perform QMC on a tree. This paper describes a path compensation algorithm that allows QMC sampling based on the Sobol sequence in tree pricing applications. Solutions are provided for single factor Hull-White trinomial trees as well as the Black- Karasinski model. This algorithm can be adapted to binomial tree models. Path construction by the Brownian bridge is compared to a fast discrete cosine transform (DCT) approximation of principal component analysis (PCA). Both QMC methods are shown to outperform MC.

FFT network for interest rate derivatives with Lévy processes

This paper extends the fast Fourier transform (FFT) network to interest derivative valuation under the Hull–White model driven by a Levy process. The classical trinomial tree for the Hull–White model is a widely adopted approach in practice, but fails to accommodate the change in the driving stochastic process. Recent finance research supports the use of a Levy process to replace Brownian motion in stochastic modeling. The FFT network overcomes the drawback of the trinomial approach but maintains its advantages in super-calibration to the term structure of interest rate and efficient computation to various kinds of interest rate derivatives under Levy processes. The FFT network only requires knowledge of the characteristic function of the Levy process driving the interest rate process, but not of the interest rate process itself. The numerical comparison between the closed-form solutions of interest rate caps and swaptions and those from FFT network confirms that the proposed network is accurate and efficient. We also demonstrate its use in pricing Bermudan swaptions and other American-style options. Finally, the FFT network is expanded to accommodate path-dependent variables, and is applied to interest rate target redemption notes and a range of accrual notes.

A simple trinomial lattice approach for the skew-extended CIR models

In this paper, we introduce a simple and efficient trinomial lattice tree approach for the skew Cox-Ingersoll-Ross (CIR) model and the doubly skewed CIR model. Suffering from the terms of local times and non-constant volatility, we apply two transforms to the skew-extended CIR processes. Then we construct a modified trinomial tree for the transformed processes which are piecewise tractable diffusions with constant volatility. As a result, the tree for the original skew-extended CIR processes can be easily obtained by using the inverse transform. Results of applications to zero-coupon bonds, European and American options demonstrate that our simple tree approach is efficient and satisfactory.

EFFICIENT PIECEWISE TREES FOR THE GENERALIZED SKEW VASICEK MODEL WITH DISCONTINUOUS DRIFT

In this paper, we explore two new tree lattice methods, the piecewise binomial tree and the piecewise trinomial tree for both the bond prices and European/American bond option prices assuming that the short rate is given by a generalized skew Vasicek model with discontinuous drift coefficient. These methods build nonuniform jump size piecewise binomial/trinomial tree based on a tractable piecewise process, which is derived from the original process according to a transform. Numerical experiments of bonds and European/American bond options show that our approaches are efficient as well as reveal several price features of our model.

The Informational Content of Infinite Maturity Bonds

This paper investigates how the market implicitly prices very long-dated cash-flows. We do this empirically, using daily market data on the 3½% War Loan, a UK infinite maturity coupon bond. We price the War Loan as an American option with 100 years of maturity. To this end, we perform daily calibrations to observed swaption prices using the Hull-White (1990) one-factor term structure model. The mean-reversion and volatility parameters resulting from the calibrations are then used in a finite difference method, which is equivalent to the Hull-White (1994 and 1996) trinomial tree procedure. We calculate the constant long-term yield beyond the last observed yield, which equates the results of the finite difference procedure with the observed prices of the War Loan. We find that prior to the financial crisis the constant long-term yield was situated above the last observed yield, whereas after the crisis it was situated below the last observed yield.

Implementation of Short-Rate Models - A Case Study of the Black-Derman-Toy Model of Interest Rate

In order to value a bond with one or more embedded options, one must consider the volatility of interest rates, as changes in volatility will affect the possibility of the option being exercised. The valuation of bonds with embedded options can be achieved by constructing a binomial or trinomial interest rate tree that models the random evolution of future interest rates and then by using backward-induction to generate the value of the bond at each node. No-arbitrage models such as the Black-Derman-Toy (BDT) model are more commonly used in practice to price financial instruments with embedded options or contingent claims. In this paper therefore, we explore how one of the no-arbitrage short rate models, the BDT model, can be implemented in a practical and reproducible manner. This is the first in our series of discussions on short-rate models. In future essays, we will discuss the implementation of other short-rate models namely the Hull-White, Ho-Lee, Kalotay-Fabozzi-Williams and the Black-Karansinski models of interest rates.

Investment opportunity in China's overseas oil project: An empirical analysis based on real option approach

China's overseas oil investments uphold national energy security. Located in the complex international economic and political environment, the benefits of overseas oil projects are affected by various uncertainties. Oil companies call for a set of evaluation method dealing with these uncertainties, especially when encountering low-oil-price conditions. It is much more crucial to answer when to optimally invest rather than whether to invest. This paper analyzes the investment opportunity of an oil project in the development and production phase considering uncertainty, irreversibility and management flexibility. The decision-making process combines Discounted Cash Flow (DCF) method and the trinomial tree model of Real Option Approach (ROA). For practical purposes, tools of hurdle price, cumulative probability and decision tree are adopted. In addition, results of the case project show the economic feasibility at present and optimal start-up timing at the end of 2016. Besides, the lower cost parameters generally make it easier to meet the opportunity. The scenario analysis suggests the higher risk contributes to an earlier start-up. Furthermore, the issues of method applicability, investment signals and decision-lag effects are discussed.

Connection between trinomial trees and finite difference methods for option pricing with state-dependent switching rates

ABSTRACTTree approaches (binomial or trinomial trees) are very popularly used in finance industry to price financial derivatives. Such popularity stems from their simplicity and clear financial interpretation of the methodology. On the other hand, PDE (partial differential equation) approaches, with which standard numerical procedures such as the finite difference method (FDM), are characterized with the wealth of existing theory, algorithms and numerical software that can be applied to solve the problem. For a simple geometric Brownian motion model, the connection between these two approaches is studied, but it is lower-order equivalence. Moreover such a connection for a regime-switching model is not so clear at all. This paper presents the high-order equivalence between the two for regime-switching models. Moreover the convergence rates of trinomial trees for pricing options with state-dependent switching rates are first proved using the theory of the FDMs.

The Equation of Real Option Value under Trinomial Tree Model

Trinomial tree model is excelled than binomial tree model in precision and calculation from an example. Triple Tree pricing error is much smaller, the result is simpler. By careful analysis, we conclude the real option value under trinomial tree model.

The Effects of Negative Nominal Rates on the Pricing of American Calls: Some Theoretical and Numerical Insights

The article investigates the effects played on options pricing by negative risk-free rates when the underlying is an equity with null dividends. In such anomalous conditions, in fact, the fair value at early exercise of the American Call would not match the value of the European Call with the same financial features. We originally motivate this assumption with theoretical arguments. We then move to an empirical investigation where we put at work some quasi-closed formulas for pricing an American option and the stochastic trinomial trees algorithm. We then draw the conclusion that from a numerical viewpoint, the bias between the fair value of the American Call and the value of the corresponding. European Call is mainly due to approximation errors, which can be mitigated when Trinomial Stochastic Trees are used.

Wykorzystanie drzew trójmianowych i teorii cyklu życia produktu do wyceny piłkarzy

Nowadays, football is not just only an amateur sport, but also a big and a very interesting branch of industry for investors and gamblers. Football clubs start to act like an ordinary company on the market. One of the most important assets in a club’s balance are footballers’ performance rights. Every year thousands of players are the subject of football club transfers (in 2013 there were over 12.000 of such transfers with a total value of over USD 37 billion). It means that the monitoring of a footballer’s value could protect a club before the risk of a sudden disadvantageous change. This article indicates the possibilities of using quantitative methods in players’ performance rights value monitoring. Authors have raised the hypothesis that joins methods of the trinomial tree and the product life cycle that could give the managers of clubs more powerful tools in managing a football club company. The trinomial tree is usually used to value the option price giving three paths of price movement. In this approach, the method was used to determine the future hypothetical value of footballers’ performance rights. Using the product life cycle we could observe and analyse the moment in the life cycle of a football player. In this part of the research, non-linear models were used to approximate the best-fitting function of the market value of a player. Such information indicates the direction of movement on the trinomial tree. In this article, we use the historical data of the market values of chosen football players from the web site http://transfermarkt.de. To make a proper analysis, we chose three kinds of players: in the phase of growth, in the phase of stabilization and the phase of dropping down in value. Such an assumption causes the necessity of choosing players with adequate features, for example: age, skills and the team played for. Therefore, the length of the time series depends on the years and the performances of the footballer. The opportunities and threats of such an attempt are shown in this article.

Study on the investment value and investment opportunity of renewable energies under the carbon trading system

ABSTRACTChina has promised to start the national carbon trading system in 2017. In the carbon trading system, the renewable energy projects may obtain additional benefits through the Certified Carbon Emission Reduction (CCER) trade. As the carbon price fluctuates along with the market conditions, such fluctuation enables the renewable power projects to acquire the rights of an option, i.e. it may contain an even higher value due to the uncertainties in the future. While making an investment decision, the renewable power companies may choose to make the investment immediately, or postpone the investment and accumulate more information to increase the return of investment; and for immediate investments, the return must be sufficient to exceed the potential value of a waiting option. To study the investment in renewable power projects subject to the fluctuation of carbon price, this paper adopts the trinomial tree model of real options to estimate the net present value (NPV) and real option value (ROV) of th...