Trinomial Lattice Research Articles

Solar Energy Investment Valuation With Intuitionistic Fuzzy Trinomial Lattice Real Option Model

Under the uncertain and the flexibility required conditions, the conventional capital budgeting techniques are insufficient to decide the most feasible investment due to their static nature. Classical real option valuation techniques have been used for various investment options such as deferment and abandonment. Although these techniques help to evaluate the projects in uncertain environments, they still follow the static, certain Boolean logic assumptions. Therefore, further improvement in the real option valuation has been introduced with the usage of fuzzy logic. Fuzzy logic is applied by binomial and Black–Scholes real option valuation methods up to date, but not with the trinomial lattice. Moreover, Types-1 and 2 fuzzy numbers are used; however, the intuitionistic fuzzy numbers are not analyzed in detail in the real option valuation. In this article, we propose an intuitionistic fuzzy trinomial lattice real option valuation model for a solar energy investment project. While valuing the deferment option, expert opinions, two different volatility methods, and a tailor-made electricity price-forecasting model are used. According to our results, the deferment option is valuable and delaying the start of the project provides a much higher expanded net present value than an immediate start. The contribution of this article is that the uncertainty in the real option valuation is modeled from a broader perspective with intuitionistic fuzzy numbers by taking the hesitancy of the decision-maker and the market into account. Consequently, the proposed method would increase the accuracy of the real option valuation in uncertain environments such as renewable energy sector.

Efficient and Robust Combinatorial Option Pricing Algorithms on the Trinomial Lattice for Polynomial and Barrier Options

Options can be priced by the lattice model, the results of which converge to the theoretical option value as the lattice’s number of time steps n approaches infinity. The time complexity of a common dynamic programming pricing approach on the lattice is slow (at least O n 2 ), and a large n is required to obtain accurate option values. Although O n -time combinatorial pricing algorithms have been developed for the classical binomial lattice, significantly oscillating convergence behavior makes them impractical. The flexibility of trinomial lattices can be leveraged to reduce the oscillation, but there are as yet no linear-time algorithms on trinomial lattices. We develop O n -time combinatorial pricing algorithms for polynomial options that cannot be analytically priced. The commonly traded plain vanilla and power options are degenerated cases of polynomial options. Barrier options that cannot be stably priced by the binomial lattice can be stably priced by our O n -time algorithm on a trinomial lattice. Numerical experiments demonstrate the efficiency and accuracy of our O n -time trinomial lattice algorithms.

Real options analysis of revenue risk sharing in post-disaster housing reconstruction

PurposePost-disaster housing reconstruction (PDHR) demands a considerable percentage of global property investment, yet the post-disaster environment presents intricate challenges to reconstruction financing for governments and at the same time, revenue uncertainty for private investors. The purpose of this study is to develop a methodology for tackling land shortage and the financial challenges of PDHR in the aftermath of a disaster.Design/methodology/approachThis study developed a methodology based on a combined minimum revenue guarantee and maximum revenue cap model using a well-established real options analysis (ROA) for revenue risk sharing in PDHR projects and land readjustment (LR) for finance. The applicability of the purported model is demonstrated through an illustrative example.FindingsThe results show that flexibility in the options could increase the PDHR contractor’s risk profile by increasing the expected value of the contractor investment and reducing the probability of investment loss. On the other side, a cap on the contractor revenue stream would allow the government to benefit from any excess in revenue and would counterbalance the value of the option.Practical implicationsThe framework proposed in this study could serve as a practical risk-revenue sharing in PDHR projects. Governments and policymakers could use the findings to enable the successful delivery of PDHR projects and consequently bring the quality of life of affected people to pre-disaster conditions.Originality/valueThis study can be considered as a first attempt toward the use of the Australian barrier style options structure, and the trinomial lattice valuation model in PDHR projects, which incorporates LR, public-private partnerships, governmental guarantees and PDHR concepts in one ROA-based framework.

Funciones isoelásticas de utilidad, rejillas trinomiales, volatilidad y aversión al riesgo cambiante: Un modelo de valuación con opciones reales de empresas de base tecnológica

La valoración de inversiones en I&D, intangibles y empresas de base tecnológica (ebt) requiere de modelos que valoren: flexibilidad estratégica, volatilidad cambiante según ciclo de vida y riesgos sin activos financieros gemelos. En mercados incompletos e imperfectos, los riesgos deben valorarse con funciones de utilidad del inversor. Por lo expuesto, se propone un modelo de valuación de opciones reales con rejillas trinomiales, volatilidad cambiante, funciones isoelásticas de utilidad y aversión al riesgo variable. Mediante un análisis de caso y estudio de sensibilidad sobre valuación de ebt, es ilustrado su funcionamiento. Los resultados obtenidos demuestran su versatilidad. El modelo captura riesgos “privados” y flexibilidad estratégica, como herramienta de valuación para este tipo de inversiones dentro de mercados ambiguos, imperfectos e incompletos.

Real Option Valuation of the R&D Investment in Renewable Energy Considering the Effects of the Carbon Emission Trading Market: A Korean Case

A carbon market was introduced for the first time in January 2005, when the EU assigned carbon emission allowances to approximately 15,000 enterprises in 25 countries and established a market for emissions trading. In Korea, the carbon emission trading system started from January 2015 with three phases running up to 2025. As many countries have introduced carbon markets, new evaluation models that consider not only fossil energy prices but also carbon emission costs are necessary because additional costs of using fossil energy might have been incurred due to carbon emissions. The purpose of this paper is to develop a real option model that considers not only the uncertainty of existing fossil energy prices, but also the uncertainty of carbon emission rights prices, in evaluating the economic value of renewable energy R&D. Using the real option model, we attempted to assess the effects of the uncertainty of newly appearing carbon markets on the economic value of renewable energy R&D in Korea empirically. Furthermore, we derived an optimal decision path according to the uncertain future situations of fossil energy and carbon markets by analyzing a trinomial lattice model in which the optimal timings of R&D and deployment are identified.

Pricing Options Using Trinomial Lattice Method

How much to spend on an option contract is the main problem at the task of pricing options. This become more complex when it comes to projecting the future possible price of the option. This is attainable if one knows the probabilities of prices either increasing, decreasing or remaining the same. Every investor wishes to make profit on whatever amount they put in the stock exchange and thus the need for a good formula that give a very good approximations to the market prices. This paper aims at introducing the concept of pricing options by using numerical methods. In particular, we focus on the pricing of a European put option which lead us to having American put option curve using Trinomial lattice model. In Trinomial method, the concept of a random walk is used in the simulation of the path followed by the underlying stock price. The explicit price of the European put option is known. Therefore at the end of the paper, the numerical prices obtained by the Black Scholes equation will be compared to the numerical prices obtained using Trinomial and Binomial methods.

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.

Firm-specific credit risk estimation in the presence of regimes and noisy prices

Security prices are important inputs for estimating credit risk. Yet, to obtain an accurate firm-specific credit risk assessment, one needs a reliable model and a methodology that filters the elements unrelated to the firm’s fundamentals from market prices.In this article, we introduce a hybrid credit risk model defined in a Markov-switching environment. It captures firm-specific changes in the leverage uncertainty during crises. We also propose a new efficient method to estimate the model, and a numerical scheme based on trinomial lattices to price credit derivatives. The estimation is finally performed on more than 200 firms using maximum likelihood estimation.

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.

A lattice-based model to evaluate variable annuities with guaranteed minimum withdrawal benefits under a regime-switching model

We consider the problem of evaluating variable annuities with a guaranteed minimum withdrawal benefit under a regime-switching model. We propose a trinomial lattice model to approximate the evolution of the investment fund value and the policy value at inception is computed through a backward induction scheme. Finally, the insurance fee is computed as the solution of the equation that makes the contract actuarially fair. Numerical results are reported to illustrate the consistency of the proposed model.

A copula-based approach for generating lattices

Discrete approximations such as binomial and trinomial lattices have been developed to model the intertemporal dynamics of variables in a way that also allows contingent decisions to be included at the appropriate increments in time. In this paper we present an approach for developing these types of models based on copulas. In addition to ease of implementation, a primary benefit of this approach is its generality, and we show that various binomial and trinomial approximation methods for valuing contingent claim securities in the literature are special cases of this approach, each based on a choice of a particular set of probability and/or branching parameters. Because this approach encompasses these and other cases as feasible solutions, we also show how it can be used to optimize the construction of lattices so that discretization error is minimized, and we demonstrate its application for an option pricing example.

Valuation of Wind Energy Projects: A Real Options Approach

We address the valuation of an operating wind farm and the finite-lived option to invest in it under different reward/support schemes: a constant feed-in tariff, a premium on top of the electricity market price (either a fixed premium or a variable subsidy such as a renewable obligation certificate or ROC), and a transitory subsidy, among others. Futures contracts on electricity with ever longer maturities enable market-based valuations to be undertaken. The model considers up to three sources of uncertainty: the electricity price, the level of wind generation, and the certificate (ROC) price where appropriate. When analytical solutions are lacking, we resort to a trinomial lattice combined with Monte Carlo simulation; we also use a two-dimensional binomial lattice when uncertainty in the ROC price is considered. Our data set refers to the UK. The numerical results show the impact of several factors involved in the decision to invest: the subsidy per MWh generated, the initial lump-sum subsidy, the maturity of the investment option, and electricity price volatility. Different combinations of variables can help bring forward investments in wind generation. One-off policies, e.g., a transitory initial subsidy, seem to have a stronger effect than a fixed premium per MWh produced.

Modifying the Black-Scholes-Merton model to calculate the cost of employee stock options

Purpose – More than 80 percent of S&P 500 firms that issue ESOs use the Black-Scholes-Merton (BSM) model and substitute the estimated average term for the contractual expiration to calculate ESO expense. This simplification systematically overprices ESOs, which worsens as the stock's volatility increases. The purpose of this paper is to present a modification of the BSM model to explicitly incorporate the rates of forfeiture pre- and post-vesting and the rate of early exercise. Design/methodology/approach – The paper demonstrates the model's usefulness by employing historical exercise and forfeiture data for 127 separate ESO grants and 1.31 billion ESOs to calculate the exercise and forfeiture parameters and value ESOs for nine firms. Findings – The modified BSM model is just as accurate but easier to use than the more computationally intensive utility maximization and trinomial lattice models, and it avoids the ASC 718 BSM model's overpricing bias. Originality/value – If firms prefer the BSM model over more mathematically elegant alternatives, they should at least use a BSM model that is free of overpricing bias.

Inflation Effect on Value Calculation of Asian Real Option

This paper establishes a degree of influence between an inflation rate and a risk-free investment rate on the precision of estimated value of an Asian real option. It has been shown with an example that in the event of advancing by the inflation rate beyond profitability of risk-free investments, which is typical for Russia, an option valuation in a trinomial lattice will be lower than that in a binomial lattice. The result serves a useful purpose for analysts considering the fact that a trinomial model is a more accurate discrete model than a binomial model.

On the Minimal Entropy Martingale Measure and Multinomial Lattices with Cumulants

In this article, we describe with relevant examples based on empirical data how to use the minimal entropy martingale measure (MEMM) to price European and American Options in multinomial lattices which take into account cumulants information. For trinomial lattices, we show that minimal entropy prices are close to results obtained using the Black and Scholes option pricing formula. For pentanomial lattices, minimal entropy prices are close to results obtained under the mean-correcting martingale measure using the discrete Fourier transform. The MEMM is very easy to compute and is therefore a good candidate for option pricing in multinomial lattices.

Real option valuation of public sector R&D investments with a down-and-out barrier option

This paper presents a real options approach for valuing public-sector research and development projects, using a down-and-out barrier option. Specifically, it considers the potential savings to the tax payer for investing in technology to be purchased by a national government. The valuation is performed with stretched trinomial lattices. Government-driven demand for this technology is equated with the underlying asset, and valuation measured in terms of potential government savings. Two variables, volatility of demand for the technology and unit cost, are treated as uncertain. A Monte Carlo simulation is performed to understand the effects of these variables on the valuation. Other variables are estimated, and a parametric analysis is performed to understand the effects of these variables. To illustrate how this approach could be used, the development of a new sensor, to be used in large networks that track greenhouse gas fluxes, is considered as an example.

Efficient and high accuracy pricing of barrier options under the CEV diffusion

Binomial and trinomial lattices are popular techniques for pricing financial options. These methods work well for European and American options, but for barrier options, the need to place a tree node very close to a barrier brings difficulties in their implementation and a large number of time steps are usually required when the barrier is close to the current asset price. A finite difference implementation is simpler and we propose a fourth-order numerical scheme for continuously and discretely monitored barriers. We demonstrate the superior performance of our technique over existing procedures for the Black–Scholes model and we then price barriers under constant elasticity of variance (CEV) diffusion. Continuously monitored barriers under CEV admit an analytical solution but evaluation via this formula is not straightforward. Furthermore, discretely monitored barriers have to be priced numerically. Our main contribution is therefore a highly accurate and efficient numerical scheme for barrier options under CEV and we provide several numerical examples to illustrate the merit of the new technique.

Generalized Multinomial Lattice Method for One-Factor Models

In this paper, we present recombining multinomial lattice method that is a generalization of widely used trinomial lattice. We develop a practical method of calculating jump probabilities of K-nomial lattice, K ≥ 3, when the local means are state dependent. With the jump probabilities, we match K moments of the probability distribution of demeaned process at each node of the lattice. Applying the method, we find the range of the standardized local means that a K-nomial lattice supports and develop a streaming lattice scheme to extend the supported range to handle convection dominant processes without reducing the time intervals. We devise a scheme to paste smoothly the jump probabilities of the branches belonging to the streaming lattices with overlapping range, and wind up with a lattice with potential 2K − 1 branches, of which only up to K+1 contiguous branches have positive jump probabilities. Finally, we compare the numerical results from multinomial lattice implementation against the closed form solutions. We find that the finer granularity of spatial grids of multinomial lattice improve numerical properties.

Effects of Temperature Uncertainty on the Valuation of Geothermal Projects: A Real Options Approach

In this paper we investigate the economic viability of geothermal district heating networks using both net present value (NPV) and real options analysis (ROA). We give an introduction to geothermal energy technologies and review the relevant ROA literature focusing on applications in similar fields, such as oil projects. The similarities and differences of oil and geothermal projects are discussed and summarized. The investment structure of geothermal projects is analyzed concerning costs and uncertainties in the different investment stages. We develop a method to assess the impact of temperature uncertainty with ROA. Investments in geothermal projects are evaluated applying a binomial and a trinomial lattice approach. A novel real options model for the evaluation of normally distributed uncertainty is developed for binomial lattices. Using data obtained from a Dutch project in The Hague, we find positive option values for both lattice approaches, compared to negative values for the common NPV calculation. Drilling and production costs are found to have a significant impact on the option value.

Recombining Trinomial Tree for Real Option Valuation with Changing Volatility

This paper presents a recombining trinomial tree for valuing real options with changing volatility. The trinomial tree presented in this paper is constructed by simultaneously choosing such a parameterization that sets a judicious state space while having sensible transition probabilities between the nodes. The volatility changes are modeled with the changing transition probabilities while the state space of the trinomial tree is regular and has a fixed number of time and underlying asset price levels. The presented trinomial lattice can be extended to follow a displaced diffusion process with changing volatility, allowing also taking into account the level of the underlying asset price. The lattice can also be easily parameterized based on a cash flow simulation, using ordinary least squares regression method for volatility estimation. Therefore, the presented recombining trinomial tree with changing volatility is more flexible and robust for practice use than common lattice models while maintaining their intuitive appeal.