Geometric Brownian Motion Process Research Articles

Improving Financial Performance with Hedging Via Forwards for Electric Power Generation Companies

In order to improve the financial performance of electric power generation companies, we show how additional on-peak forwards can be exploited to hedge against the risk exposure of nonpeak physical power, taking advantage of the competitive market outside the regulated service territory. Specifically, we model the evolution of the forward price as a geometric Brownian motion (GBM) process without drift and show how the optimal amount of additional on-peak forwards can be determined under the mean minus variance criterion via simulation. Furthermore, we show how the simulation approach and the analytical results may lead to a robust decision-making process.

Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus

We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion $B^H$ with Hurst parameter $H\in(\frac{1}{2},1)$ in the Wick--It\^{o} sense, including a geometric fractional Brownian motion. To this end, we apply a Donsker-type approximation of the fractional Brownian motion by disturbed binary random walks due to Sottinen. Moreover, we replace the rather complicated Wick products by their discrete counterpart, acting on the binary variables, in the corresponding systems of Wick difference equations. As the solutions of the SDEs admit series representations in terms of Wick powers, a key to the proof of our Euler scheme is an approximation of the Hermite recursion formula for the Wick powers of $B^H$.

The effect of mean reversion on entry and exit decisions under uncertainty

Many economic variables of interest exhibit a tendency to revert to predictable long-run levels. However, mean reverting processes are rarely used in investment models in the literature. In most models, geometric Brownian motion processes are used for tractability. In this paper, a firm's entry and exit decisions when the output equilibrium price follows an exogenous mean reverting process are examined, and then compared to the decisions of the firm under the usually employed assumption of lognormally distributed output price, presented in Dixit (1989a). By extending previous work by Sarkar (2003) to account for costly reversibility, we show that mean reversion has a significant effect, not only on firm-specific entry and exit decisions, but also on the balance of entering and exiting firms in an industry/market. Thus it would be erroneous to use the more tractable geometric Brownian motion process as an approximation for a mean-reverting process in models and investigations of aggregate industry investment.

Real R&D options and optimal activation of two-dimensional random controls

We value investments under uncertainty with embedded optional costly controls (impulse-type with uncertain outcome) that capture managerial intervention for value enhancement and/or information acquisition (exploration, R&D, advertising, marketing research, etc). Implementing real option models but neglecting such embedded managerial actions can severely underestimate investment opportunities and lead to erroneous investment decisions. Optimal decisions are solutions to a maximization problem where the trade-off between the control's cost and the value added by such actions is explicitly taken into consideration. In this paper, we generalize such a methodology from one dealing with the special case of actions affecting only one state-variable, to one with actions that affect several. Asset values follow geometric Brownian motion or jump-diffusion processes with multiple generating sources of jumps. The Markov-chain numerical methodology we provide can handle sequential controls. Although we report the results with open-loop policies, the approach can be readily extended to accommodate dependency among the controls.

Valuation of Discrete Dynamic Fund Protection Under Lévy Processes

This paper investigates the valuation of discrete dynamic fund protection (DFP) under Levy processes. Specifically, the analytical solution of discrete DFP under Lévy processes is obtained in terms of Fourier transforms. The derivation uses Spitzer’s formula and leads to a recursion on computing the characteristic function of the maximum protection-to-fund ratio using the Fourier inversion. DFP can then be valued efficiently and accurately via the fast Fourier transform. The pricing behavior of the discrete DFP is numerically examined using several Levy processes, such as geometric Brownian motion, jump-diffusion models, and variance gamma process. Numerical experiments confirm that the proposed approach produces highly accurate discrete DFP values within 1 second.

Modelling the number of customers as a birth and death process

Birth and death may be a better model than Brownian motion for many physical processes, which real options models will increasingly need to deal with. In this paper, we value a perpetual American call option, which gives the monopoly right to invest in a market in which the number of active customers (and hence the sales rate) follows a birth and death process. The problem contains a singular point, and we develop a mixed analytic/numeric method for handling this singular point, based on the method of Frobenius. The method may be useful for other cases of singular points. The birth and death model gives lower option values than the geometric Brownian motion model, except at very low volatilities, so that if a firm incorrectly assumes a geometric Brownian motion process in place of a birth and death process, it will invest too seldom and too late.

Valuing the switching flexibility of the ethanol-gas flex fuel car

Renewable energy sources are becoming more important as the world’s supply of fossil fuels decrease and also due to environmental concerns. Since 2003, when the ethanol-gasoline flex fuel car became commercially available in Brazil, the growth of this market has been significant, to the point where currently more than 50% of the fuel consumption of cars in Brazil is from renewable biofuels (ethanol). This has been made possible due to the success of the flex fuel car, which can run on ethanol, gasoline, or any mix of these in the same fuel tank, and which is sold at a premium over the non-flex models. Flex fuel cars, on the other hand, provide the owner with the flexibility to choose fuels at each refueling stop. Given the uncertainty on future prices of ethanol and gas, this option adds value to the owner since he can always opt for the cheaper fuel whenever he fills up his car. We use the Real Options method to analyze the value of the flex fuel option assuming both a Geometric Brownian Motion and Mean Reverting diffusion processes for the prices of gasoline and ethanol and compare the results arising from both methods. We conclude that the flex option value is significant using either method and twice as high as flex premium charged by the car manufacturers, which helps explain the success that this type of automobiles have gained in Brazil since 2003. Our results also indicate that consumers should be willing to purchase flex fuel cars even if manufacturers increase the flex premium.

Capacity expansion under a service-level constraint for uncertain demand with lead times

For a service provider facing stochastic demand growth, expansion lead times and economies of scale complicate the expansion timing and sizing decisions. We formulate a model to minimize the infinite horizon expected discounted expansion cost under a service-level constraint. The service level is defined as the proportion of demand over an expansion cycle that is satisfied by available capacity. For demand that follows a geometric Brownian motion process, we impose a stationary policy under which expansions are triggered by a fixed ratio of demand to the capacity position, i.e., the capacity that will be available when any current expansion project is completed, and each expansion increases capacity by the same proportion. The risk of capacity shortage during a cycle is estimated analytically using the value of an up-and-out partial barrier call option. A cutting plane procedure identifies the optimal values of the two expansion policy parameters simultaneously. Numerical instances illustrate that if demand grows slowly with low volatility and the expansion lead times are short, then it is optimal to delay the start of expansion beyond when demand exceeds the capacity position. Delays in initiating expansions are coupled with larger expansion sizes. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009

Market Risk VaR Historical Simulation Model with Autocorrelation Effect: A Note

The modern market risk model using Value at Risk (VaR) method in the banking area under the BASEL II Accord can take different forms of simulation. In this paper, historical simulation will be applied to the VaR model comparing the two different approaches of Geometric Brownian Motion (GBM) process and Bootstrapping methods. The analysis will use correlation plots and examine the effects of the autocorrelation function for stock returns.

On the Probabilities of Correlated Defaults: a First Passage Time Approach

This article investigates the joint probability of correlated defaults in the first passage time approach of credit risk subject to condition that the underlying firms’ assets values and the default boundaries follow geometric Brownian motion processes. The exact analytical expression of joint probability of two correlated defaults in the case of stochastic default boundaries is presented. Also, some properties of this solution are provided.

Investment in High-Tech Industries: An Example from the LCD Industry

This paper considers a representative firm taking investment decisions in a high-tech environment where different generations of products are invented over time. First, we develop a real options investment model in which, according to standard practice, the sales price and the unit production cost both satisfy a geometric Brownian motion (GBM) process. However, from real life data of the LCD industry it follows that output prices behave according to a crystal cycle that does not match a GBM. We proceed by conducting a thorough econometric analysis, leading to the conclusion that a vector autoregressive model (VAR) provides the best fit. Integrating this model with the real options machinery, we find that (i) at the moment of investment the increased production capacity goes along with increasing production cost and decreasing price, (ii) a management effect is present in the sense that a price drop is followed by a cost decrease due to management pushing harder on cost decreasing programs, and (iii) investing can be optimal while at the same time a GBM yields a negative net present value (NPV). We also find that investment decisions taken in practice are better supported by our V AR model than by the standard real options model based on GBM.

Real option valuation of free destination in long-term liquefied natural gas supplies

This paper presents a real option model for the valuation of destination flexibility in long-term LNG supplies. Stochastic price dynamics in the different markets is modelled through geometric Brownian motion processes. Mean reversion is considered as well as correlation between markets, but instead of the usual correlation in return shocks, a price convergence term is introduced representing the arbitrage streams between markets. Model parameters are estimated from market data on LNG prices by maximum log-likelihood. The goodness of the fit for the proposed model is tested as well as for two alternative models. Confidence intervals for the parameters are given. Results for the model are calculated by Monte Carlo simulation. Frequency distributions for the main results are plotted. The effect of the main parameters of the model is studied (i.e. price volatilities, price convergence, initial prices in the markets, mean reversion, extra transportation costs, number of alternative markets). The value of destination flexibility is found to be an important share of the value of LNG.

A stochastic process approach in setting the appropriate margin level for the TAIFEX stock index futures

PurposeThe main purpose of this paper is to compute the appropriate margin level for the stock index futures traded on the Taiwan Futures Exchange (TAIFEX) and, then, to examine the appropriateness of the real margin requirement set by the TAIFEX.Design/methodology/approachThis paper develops a new approach assuming the future's prices follow a geometric Brownian motion process. Compared with the extreme value theory that has been intensively used to determine the appropriate futures margin levels, one of the advantages of the present model is no need to specify the frequency at which extremes are taken.FindingsThe evidences indicate that the theoretical margins obtained by the proposed model can provide a more accurate and flexible margin level in accordance with the market volatility.Research limitations/implicationsThe main limitation of this approach is that the natural logarithm of the futures prices is assumed to follow a Brownian motion process. However, such an assumption might not be practical for financial returns.Practical implicationsThe research is helpful for the clearinghouse to set up its margins policy, especially under various conditions of volatility risks.Originality/valueThis paper proposes a theoretical procedure to set an appropriate futures margin for the TAIFEX. This paper also provides a better understanding of Taiwan's futures market that is newly launched and is useful for investors to hedge and speculate.

Evaluating alternative capacity strategies in semiconductor manufacturing under uncertain demand and price scenarios

Because product demand is very volatile and machine and process technologies are advanced at a rapid pace, capacity planning, and investment in the semiconductor industry is a challenging task. Many studies have addressed mid- and short-term machine portfolio planning tasks of capacity planning. However, capacity planning should be considered in a framework of strategy planning in order to address the whole problem. This paper describes a method of analysis for adapting capacity strategy to changing business environment. The paper has three major parts. In the first part, empirical data analysis of a case study company yields predictive formulas for production and capacity costs. The uncertainty of demand is also calibrated using the geometric Brownian motion process. In the second part, experiments are designed to simulate demand and profit scenarios for comparing two capacity strategies. In the third part, the resultant data of simulation are analysed to gain insights to the relative performance of the two strategies in various scenarios. The analysis method provides a framework for formulating capacity strategy and for integrating capacity planning with business planning.

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum-Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.

Multifractal Modeling of the Japanese Treasury Term Structure

This paper identifies Multifractal Models of Asset Returns (MMARs) for each of the eight nodal term structure series of monthly Japanese Treasury rates and, after proper synthesis, simulates those MMARs. All nodal rate series are found to be slightly anti-persistent, with the exception of the 20-year rate which is heavily anti-persistent and in the range of being chaotic. This contrasts with our earlier findings of the US term structure (Jamdee-Los, 2005), where the US interest rates are identified to be slightly persistent, although that model identification was based on daily data. The simulations of the identified Japanese interest rate MMARs are compared with the original empirical time series, and also with the simulated results from the corresponding Geometric Brownian Motion (GBM) and GARCH processes. All eight different maturity Japanese Treasury rates are found to be multifractal processes. Identified distributions of all simulated processes are compared with the empirical distributions in both snapshot and time-scale (-frequency) analyses. Using wavelet scalograms, we demonstrate that the MMAR does not clearly outperform either the GBM and GARCH(1, 1) in time-frequency comparisons. The simulated MMAR replicates all attributes of the short maturity empirical distributions, while the simulated GBM and GARCH(1, 1) processes perform better for the long maturity series. Nevertheless, these results are somewhat inconclusive since the MMAR experiences difficulty with the simulations of these anti-persistent,

On The Validity of The Geometric Brownian Motion Assumption

The geometric Brownian motion (GBM) process is frequently invoked as a model for such diverse quantities as stock prices, natural resource prices and the growth in demand for products or services. We discuss a process for checking whether a given time series follows the GBM process. Methods to remove seasonal variation from such a time series are also analyzed. Of four industries studied, the historical time series for usage of established services meet the criteria for a GBM; however, the data for growth of emergent services do not.

A SIMPLIFIED TREATMENT OF BROWNIAN MOTION AND STOCHASTIC DIFFERENTIAL EQUATIONS ARISING IN FINANCIAL MATHEMATICS

ABSTRACT Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.

The effect of mean reversion on investment under uncertainty

Mean-Reverting processes are appropriate for most “real option” investment models, yet Geometric Brownian Motion (GBM) processes are generally used for tractability. Hassett and Metcalf (J. Econom. Dyn. Control 19 (1995) 1471–1488) argue that using a GBM process is justified because mean-reversion has two opposing effects—it brings the investment trigger closer, and also reduces the conditional volatility (thereby lowering the likelihood of reaching the trigger)—and the overall effect on investment should be negligible for most reasonable parameter values. This paper extends the Hassett and Metcalf model by incorporating a third factor, the effect of mean-reversion on systematic risk. The main result is that mean reversion, in general, does have a significant impact on investment. Moreover, this effect could be either positive or negative, depending on various factors such as project duration, cost of investing, interest rate, etc. Thus it is generally inappropriate to use the GBM process to approximate a mean-reverting process.

Geometric Brownian motion as a model for river flows

AbstractLet X(t) be the flow of a certain river at time t. A geometric Brownian motion process is used as a model for X(t) and is found to give very good forecasts of future flows. The forecasted values generated by this one‐dimensional model are compared with those provided by a deterministic model that requires the evaluation of 18 entries. Based on two important criteria, the stochastic model is superior, on average, to the deterministic model for forecasts up to 4 days ahead. Copyright © 2002 John Wiley & Sons, Ltd.