Geometric Brownian Motion Process Research Articles

Flexible Capacity Investments and Product Mix: Optimal Decisions and Value of Postponement Options

In this article, we study a firm's interdependent decisions in investing in flexible capacity, capacity allocation to individual products, and eventual production quantities and pricing in meeting uncertain demand. We propose a three‐stage sequential decision model to analyze the firm's decisions, with the firm being a value maximizer owned by risk‐averse investors. At the beginning of the time horizon, the firm sets the flexible capacity level using an aggregate demand forecast on the envelope of products its flexible resources can accommodate. The aggregate demand forecast evolves as a Geometric Brownian Motion process. The potential market share of each product is determined by the Multinomial Logit model. At a later time and before the end of the time horizon, the firm makes a capacity commitment decision on the allocation of the flexible capacity to each product. Finally, at the end of the time horizon, the firm observes the demand and makes the production quantity and pricing decisions for end products. We obtain the optimal solutions at each decision stage and investigate their optimal properties. Our numerical study investigates the value of the postponed capacity commitment option in supplying uncertain operation environments.

Product Remanufacturing: A Real Options Approach

In this paper, we consider a firm that leases a single product to a service provider under the assumption that this firm can decide when to replace the leased product and remanufacture it. The cost of remanufacturing is assumed to follow a geometric Brownian motion process, which is characterized by its drift and volatility. Under these assumptions, we construct and analyze a basic real options model for product remanufacturing. Specifically, we derive the remanufacturing cost threshold at which the remanufacturing option is exercised. We also perform extensive sensitivity analysis to examine how the remanufacturing cost uncertainty and other factors influence the remanufacturing decision. Furthermore, we present an extension where a remanufacturing subsidy and a disposal fee by the government are considered. The impacts of these economic instruments on the remanufacturing decision as well as on the environment are investigated.

What do market-calibrated stochastic processes indicate about the long-term price of crude oil?

Stochastic process models of commodity prices are important inputs in energy investment evaluation and planning problems. In this paper, we focus on modeling and forecasting the long-term price level, since it is the dominant factor in many such applications. To provide a foundation for our modeling approach we first evaluate the empirical characteristics of crude oil price data from 1990 to 2013 using unit root and variance ratio tests. Statistical evidence from these tests shows only weak support for the applicability of stationary mean-reverting type processes up through 2004, with non-stationary Brownian motion type processes becoming more plausible when the data from 2005 to 2013 is added. We then apply a Kalman filtering method with maximum likelihood approach to estimate the model parameters for both a single-factor Geometric Brownian motion (GBM) process as well as the two-factor Schwartz and Smith (2000) process. The latter process decomposes the spot price into unobservable factors for the long-term equilibrium level and short-term deviation, and it accommodates aspects of both a GBM process and a mean-reverting process. Both empirical and simulated data are analyzed with these models, and we quantify the increases in both the drift rate and volatility of these processes that result from developments in the crude oil markets since the middle of the last decade. We conclude by comparing and contrasting both historical accuracy and forecasts from the parameterized models, and show that when the emphasis is on the long-term expectations, a single factor GBM forecast may be sufficient.

Product Remanufacturing and Replacement Decisions Under Operations and Maintenance Cost Uncertainties

In recent years, there has been substantial interest in remanufacturing because it conserves new natural resources and reduces wastes. In this article, we aim to contribute to a deeper understanding of this remanufacturing in the context of maintenance and replacement policies. Specifically, we assume that a product consists of durable and nondurable parts with uncertain, yet different, operations and maintenance (O&M) costs, and we aim to determine the optimal remanufacturing and replacement policies for the nondurable parts and the product. The evolution of the O&M costs is assumed to follow a geometric Brownian motion process, and the corresponding modeling and analysis use a real options approach. Managerial insights and economic implications are obtained via mathematical and numerical analyses.

Probabilistic characterisation of metal-loss corrosion growth on underground pipelines based on geometric Brownian motion process

This article describes a geometric Brownian motion process-based model to characterise the growth rate of the depth of corrosion defects on underground steel pipelines based on inspection data subjected to measurement uncertainties. To account for the uncertainties from different sources, the hierarchical Bayesian method is used to formulate the growth model, and the Markov Chain Monte Carlo simulation techniques are used to numerically evaluate the probabilistic characteristics of the model parameters. The growth model considers the bias and random scattering error associated with the in-line inspection (ILI) tool as well as the correlations between the random scattering errors associated with different ILI tools. The application of the growth model is illustrated through an example involving real ILI data collected from an in-service pipeline in Canada. The results indicate that the model in general can predict the growth of corrosion defects reasonably well. The proposed model can be used to facilitate the development and application of reliability-based pipeline corrosion management.

Web software reliability modeling with random impulsive shocks

As the web-server based business is rapidly developed and popularized, how to evaluate and improve the reliability of web-servers has been extremely important. Although a large number of software reliability growth models (SRGMs), including those combined with multiple change-points (CPs), have been available, these conventional SRGMs cannot be directly applied to web software reliability analysis because of the complex web operational profile. To characterize the web operational profile precisely, it should be realized that the workload of a web server is normally non-homogeneous and often observed with the pattern of random impulsive shocks. A web software reliability model with random impulsive shocks and its statistical analysis method are developed. In the proposed model, the web server workload is characterized by a geometric Brownian motion process. Based on a real data set from IIS server logs of ICRMS website (www.icrms.cn), the proposed model is demonstrated to be powerful for estimating impulsive shocks and web software reliability.

Implications of a lowered damage trajectory for mitigation in a continuous-time stochastic model

We provide counterexamples to the idea that mitigation of greenhouse gas emissions, and adaptation to climate change, are always substitutes. We consider optimal mitigation policy when climate damages follow a geometric Brownian motion process with positive drift and mitigation is lumpy. Climate damages can be affected by adaptation in two main ways: 1) reduced proportionately for given climate impact; or 2) their growth path down-shifted. In either case expectation and variance of the climate damage are both reduced by adaptation. In case 1, the variance effect (which leads to more rapid mitigation as the option value of waiting is reduced) may dominate over the expectation effect (which reduces mitigation), thus on balance increasing mitigation when damages are reduced. Mitigation and adaptation are then complements. A family of functions relating climate damage to adaptation cost in this way includes the Cobb–Douglas specification. In case 2, mitigation and adaptation are always substitutes.

Challenges in the Evaluation of Ultra-Long-Lived Projects: Risk Premia for Projects with Eternal Returns or Costs

The economic evaluation of ultra-long-lived investment projects is not only challenging due to the choice of the planning horizon but also due to the discounting of future uncertain cash flows. Thus, for real world investment decisions a better understanding of the project’s risks and their effect on the project’s value is crucial. If long-term investments are modeled, stochastic processes may be used to reflect the uncertain development of future prices and cash flows. The choice of the stochastic process is consequently an essential assumption in the modeling process. This paper critically discusses the risk of ultra-long-lived investment projects implied if future pay-off’s are assumed to follow geometric Brownian motion processes. In our analysis, we distinguish between projects driven by costs and such driven by revenues. For both kind of projects we compare the value at risk with the returns of a risk-free asset. Therein, the value at risk describes the threshold value of the confidence levels of the uncertain cash flow’s probability density function. The comparison for long time horizons shows that the lower confidence interval exceeds the returns of a risk-free asset used as a benchmark for any choice of the confidence level, which implies that the returns of a “worst-case” scenario (within the assumed confidence interval) will still exceed the returns of a risk-free asset in the long-term perspective. For the case of uncertain future cost, the risk measure is defined as the difference between the expected value and the boundary of the confidence interval. This value is also found to become negative in the long-term perspective.

Real options approach to renewable energy investments in Mongolia

Developing nations are seeking alternative energy for electricity, and one attractive alternative is renewable energy. This research analyzes changing investment environment for renewable energy with real options approach, and explores its potential in developing economies through studying the case of Mongolia under coal price uncertainty. To evaluate comparative attractiveness of either continuing to use coal-based infrastructures or switching to renewable energy, we formulate social revenue functions for the two environments, assuming that renewable energy has lower external costs, and coal prices follow geometric Brownian motion (GBM) or geometric mean-reverting (GMR) processes. We find the optimal trigger coal prices for switching technologies with some scenarios in electricity price and externality; characterize when renewable energy investments become attractive. In contrast to conventional wisdom, we identify some situations where the value of having more decision opportunities does not exceed that of a now-or-never decision for switching technologies, and welfare losses are incurred. The optimal trigger prices are higher in GBM than in GMR, and our result raises the possible risks for waiting to switch energy. To avoid welfare losses in Mongolia, the government should increase electricity prices or switch to renewable energy earlier, especially when people pay more for the removal of externalities.

Sequential real rainbow options

We develop two models to value European sequential rainbow options. The first model is a sequential option on the better of two stochastic assets, where these assets follow correlated geometric Brownian motion processes. The second model is a sequential option on the mean-reverting spread between two assets, which is applicable if the assets are co-integrated. We provide numerical solutions in the form of finite difference frameworks and compare these with Monte Carlo simulations. For the sequential option on a mean-reverting spread, we also provide a closed-form solution. Sensitivity analysis provides the interesting results that in particular circumstances, the sequential rainbow option value is negatively correlated with the volatility of one of the two assets, and that the sequential option on the spread does not necessarily increase in value with a longer time to maturity. With given maturity dates, it is preferable to have less time until expiry of the sequential option if the current spread level is way above the long-run mean.

Montecarlo Simulation of Long‐Term Dependent Processes: A Primer

As a natural extension to Leon and Vivas (2010) and Leon and Reveiz (2010) this paper briefly describes the Cholesky method for simulating Geometric Brownian Motion processes with long term dependence, also referred as Fractional Geometric Brownian Motion (FBM). Results show that this method generates random numbers capable of replicating independent, persistent or antipersistent time‐series depending on the value of the chosen Hurst exponent. Simulating FBM via the Cholesky method is (i) convenient since it grants the ability to replicate intense and enduring returns, which allows for reproducing well‐documented financial returns’ slow convergence in distribution to a Gaussian law, and (ii) straightforward since it takes advantage of the Gaussian distribution ability to express a broad type of stochastic processes by changing how volatility behaves with respect to the time horizon. However, Cholesky method is computationally demanding, which may be its main drawback. Potential applications of FBM simulation include market, credit and liquidity risk models, option valuation techniques, portfolio optimization models and payments systems dynamics. All can benefit from the availability of a stochastic process that provides the ability to explicitly model how volatility behaves with respect to the time horizon in order to simulate severe and sustained price and quantity changes. These applications are more pertinent than ever because of the consensus regarding the limitations of customary models for valuation, risk and asset allocation after the most recent episode of global financial crisis.

Monte Carlo Simulation of Long-Term Dependent Processes: A Primer

WilmottVolume 2012, Issue 60 p. 48-57 Technical Paper Monte Carlo Simulation of Long-Term Dependent Processes: A Primer Carlos Léon, Carlos LéonSearch for more papers by this authorAlejandro Reveiz, Alejandro ReveizSearch for more papers by this author Carlos Léon, Carlos LéonSearch for more papers by this authorAlejandro Reveiz, Alejandro ReveizSearch for more papers by this author First published: 27 September 2012 https://doi.org/10.1002/wilm.10131AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume2012, Issue60July 2012Pages 48-57 RelatedInformation

Three dimensional distribution of Brownian motion extrema

This paper describes the joint distributions of minima, maxima and endpoint values for a three dimensional (3D) Wiener process. In particular, the results provide the joint cumulative distributions for the maxima and/or minima of the components of the process. The method of images is used to derive explicit expressions for the densities; the analysis can only be carried out for special correlation structures and requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used in several applied fields such as financial mathematics to obtain analytical expressions for prices of options for the 3D geometric Brownian motion process.

Demand modeling of stochastic product diffusion over the life cycle

Planning during the product life cycle (PLC) poses a number of challenges for managers due to the pace of change and uncertainties in the marketplace. The ability to better understand, predict, and make decisions based on manifestations of demand forms a set of important operational problems that ultimately affect the profitability of enterprises. This paper models the stochastic diffusion of a product in the market as a geometric Brownian motion (GBM) process that has a time-varying drift rate. The model is calibrated such that model parameters are able to feature different product types and diffusion conditions. Imperfect information on the expected peak demand is treated as model uncertainty, and a Bayesian approach is employed to update knowledge on it. The demand model demonstrates robust performance over a wide range of conditions despite model uncertainty. It provides both qualitative and quantitative information for manufacturers and service providers to design strategies for stochastic PLC conditions as well as dynamic production planning.

On valuing and hedging European options when volatility is estimated directly

We quantify the effects on contingent claim valuation of using an estimator for the unknown volatility σ of a geometric Brownian motion (GBM) process. The theme of the paper is to show what difficulties can arise when failing to account for estimation risk. Our narrative uses a direct estimator of volatility based on the sample standard deviation of increments of the underlying Brownian motion. After replacing the direct estimator into the GBM, we derive the resulting distribution function of the approximated GBM for any time point. This allows us to present post-estimation distributions and valuation formulae for an assortment of European contingent claims that are in accord with many of the basic properties of the underlying risk-neutral process, and yet better reflect the additional uncertainties and risks that exist in the Black–Scholes–Merton paradigm.

Modeling subject‐specific phase‐dependent effects and variations in longitudinal responses via a geometric Brownian motion process

We address statistical issues regarding modeling a collection of longitudinal response trajectories characterized by the presence of subject-specific phase-dependent effects and variation. To accommodate these two time-varying individual characteristics, we employ a geometric stochastic differential equation for modeling based on a Brownian motion process and develop a two-step paradigm for statistical analysis. This paradigm reverses the order of statistical inference in random effects model. We first extract individual information about phase-dependent treatment effects and volatility parameters for all subjects. Then, we derive the association relationship between the parameters characterizing the individual longitudinal trajectories and the corresponding covariates by means of multiple regression analysis. The stochastic differential equation model and the two-step paradigm together provide significant advantages both in modeling flexibility and in computational efficiency. The modeling flexibility is due to the easy adaptation of temporal change points for subject-specific phase transition in treatment effects, whereas the computational efficiency benefits in part from the independent increment property of Brownian motion that avoids high-dimensional integration. We demonstrate our modeling approach and statistical analysis on a real data set of longitudinal measurements of disease activity scores from a rheumatoid arthritis study.

Earthquake insurance portfolio analysis of wood-frame houses in south-western British Columbia, Canada

Earthquake disasters affect many structures and infrastructure simultaneously and collectively, and cause tremendous tangible and intangible loss. In particular, catastrophic earthquakes impose tremendous financial stress on insurers who underwrite earthquake insurance policies in a seismic region, resulting in possible insolvency. This study develops a stochastic net worth model of an insurer undertaking both ordinary risk and catastrophic earthquake risk, and evaluates its solvency and operability under catastrophic seismic risk. The ordinary risk is represented by a geometric Brownian motion process, whereas the catastrophic earthquake risk is characterized by an earthquake-engineering-based seismic loss model. The developed model is applied to hypothetical 4000 wood-frame houses in south-western British Columbia, Canada, to investigate the impact of key insurance portfolio parameters to insurer’s ruin probability and business operability. The analysis results indicate: (i) the physical effects of spatially correlated ground motions and local soil conditions at insured properties are significant; (ii) the insurer’s earthquake risk exposure depends greatly on insurance arrangement (e.g. deductible and cap); and (iii) the maintenance of sufficient initial surplus is critical in keeping insurer’s insolvency potential reasonably low, while volatility of non-catastrophic risk is the key for insurer’s business stability. The results highlight the importance of adequate balance between business stability under normal conditions and solvency under extreme conditions for efficient earthquake risk management. Flexibility for determining an insurance arrangement would be beneficial for insurers to enhance their portfolio performance and to offer more affordable coverage to their clients.

The game analyses to price the target enterprise of merger and acquisition based on the perspective of real options under stochastic surroundings

Although several approaches have been developed to evaluate the target enterprise of merger and acquisition (M&A) such as classical net present value (NPV) evaluation model and real options techniques, the logic of pricing processes in all these approaches is still faulty. The classical approach of NPV perspective to price the target enterprise of M&A has been replaced by integration of contemporary perspectives such as NPV perspective, real options perspective, game perspective, and so on. In this paper, the dynamic analyses model to price the target enterprise of M&A is developed from the perspective of real options integrated with game theory under stochastic surroundings. The synergy multiplication coefficient is introduced into the evaluation model to reflect the synergy management process of M&A, and the equilibrium price formula is proposed by applying the famous offer–counteroffer Rubinstein theorem, which is improved to fit for the stochastic surroundings. The price formula is further discussed on condition that the operating cost is more than or less than profit flow, which is assumed to follow geometric Brownian motion process. The numeric simulations show that the proposed formulas in this paper can perfectly well reflect the realistic practice of M&A.

A Generalization Of Lattice Specifications For Currency Options

<p class="MsoNormal" style="text-align: justify; margin: 0in 0.5in 0pt; mso-pagination: none;"><span style="font-family: Times New Roman; font-size: x-small;">This article revisits the topic of two-state pricing of currency options.<span style="mso-spacerun: yes;">  </span>It examines the models developed by Cox, Ross, and Rubinstein, Rendleman and Bartter, and Trigeorgis, and presents two alternative binomial models based on the continuous and discrete time Geometric Brownian Motion processes respectively.<span style="mso-spacerun: yes;">  </span>This work generalizes the standard binomial approach incorporating the main existing models as particular cases.<span style="mso-spacerun: yes;">  </span>The proposed models are straightforward, flexible, accommodate any drift condition and afford additional insights into binomial trees and lattice models in general.<span style="mso-spacerun: yes;">  </span>Further, the alternative parameterizations are free of the negative aspects associated with the Cox, Ross, and Rubinstein model.</span></p>

Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift — Their characteristics and applications

In Łochowski (2008) [9] we defined truncated variation of Brownian motion with drift, W t = B t + μ t , t ≥ 0 , where ( B t ) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c > 0 . We prove that truncated variation is a random variable with finite moment-generating function for any complex argument. We also define two closely related quantities — upward truncated variation and downward truncated variation. The defined quantities may have interpretations in financial mathematics. The exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process. We calculate the Laplace transform with respect to the time parameter of the moment-generating functions of the upward and downward truncated variations. As an application of the formula obtained we give an exact formula for the expected values of upward and downward truncated variations. We also give exact (up to universal constants) estimates of the expected values of the quantities mentioned.