Geometric Brownian Motion Process Research Articles

Geometric stochastic ray propagation using the special Euclidean group.

This paper describes a stochastic model of ray trajectory propagation through a medium-such as the ocean-which has an uncertain sound speed profile. We frame ray propagation as a geometric fractal Brownian motion process on the special Euclidean group of dimension two, SE(2). The framing includes diffusion parameters to describe how the stochastic rays deviate from the expected rays, and these diffusion parameters are a function of the uncertainty in the sound speed profile. We demonstrate this framing for the classical Munk profile and a double-ducted profile in the Beaufort.

A fundamental approach to corporate bond options

It is well known that interest rates and credit spreads are negatively correlated. Any successful model for pricing callable corporate bonds has to take this correlation into account. A new approach for the analysis of corporate bond options is developed by using two correlated geometric Brownian motion (GBM) processes for the spread and interest rate components of corporate bond yields. Analysis of the results and comparison with market prices suggest that the traditional methods of calculating the option premium overestimate the value of the call option. Our analysis can also explain why risk neutral credit spreads are significantly wider than implied by default probability and in fact justify higher spreads.

Exact solutions for the probability density of various conditioned processes with an entrance boundary

The probability density is a fundamental quantity for characterizing diffusion processes. However, it is seldom known except in a few renowned cases, including Brownian motion and the Ornstein–Uhlenbeck process and their bridges, geometric Brownian motion, Brownian excursion, or Bessel processes. In this paper, we utilize Girsanov’s theorem, along with a variation of the method of images, to derive the exact expression of the probability density for diffusions that have one entrance boundary. Our analysis encompasses numerous families of conditioned diffusions, including the Taboo process and Brownian motion conditioned on its growth behavior, as well as the drifted Brownian meander and generalized Brownian excursion.

A sequential real options analysis for renewable power-to-hydrogen plants for Germany and California

The novelty of large-scale green hydrogen technology with its numerous applications, and the competition with well-entrenched conventional technologies, still makes it difficult to operate renewable power-to-hydrogen (P2H) plants profitably. In this study, sequential investments in the P2H plants are investigated using a Real Options approach. The analysis addresses the question of what the optimal timing is for sequential investments in P2H modules that enable hydrogen feed-in into the gas grid in Germany and California, both pioneers in the sustainable energy transition. Choosing this approach, the uncertain and largely irreversible character of the P2H investments is tackled. Since the focus is on substituting natural gas by adding green hydrogen to the gas grid, the underlying asset considered as stochastic is the price of natural gas. It is modeled by applying a geometric Brownian motion process. The analysis conducted reveals that for both California and Germany polymer electrolyte membrane (PEM) electrolyzer systems are uneconomical in the foreseeable future, at least based on the assumptions made in the case studies. The required natural gas price level that would enable economic viability of gas grid feed-in exceeds the level of today's natural gas prices many times over. Without policy support (e.g., subsidies or quotas for green hydrogen, sufficiently high carbon tax), or an increased willingness to pay for green fuel on the customers' side, it is unlikely that the required minimum hydrogen price level for profitable business will be reached any time soon.

Pricing of American Parisian option as executive option based on the least‐squares Monte Carlo approach

AbstractIn this study, we create a novel American double‐barrier Parisian call option contract that may be utilized as an executive option for listed companies to incentivize staff and replace the classic American option. We address the option pricing problem by developing state variables to identify the price state and using the least‐squares Monte Carlo approach. We present several Lévy processes to simulate the movement path of the underlying asset. We discover that geometric Brownian motion and normal inverse Gaussian (NIG) process have successful outcomes, and NIG process has greater calculation accuracy than variance gamma process. The barrier width and window length are positively connected with the price of an American Parisian option, whereas the strike price is negatively correlated with it. Increasing the number of discrete periods of the contract will enhance the pricing accuracy.

A Real Options Analysis model for generation expansion planning under uncertain demand

Generation expansion planning is finding an optimal solution for installing new generation units with technical and financial limits. This study proposes a Real Options Analysis (ROA) model for evaluating a generation system expansion plan where the electricity demand fluctuates with volatility. We construct a binomial lattice to map the demand following a geometric Brownian motion (GBM) process. We obtain the Locational Marginal Pricing (LMP) at buses representing communities from an Optimal Power Flow (OPF) problem following Kirchhoff’s laws. Subsequently, we re-solve the OPF problem with additional generation capacity and attain LMPs associated with the expanded electrical network. The difference between these two LMPs is the benefit provided by the generation expansion. Considering generation expansion as a real option, we construct the option value tree for the economic valuation and demonstrate how the value of this option can be obtained at the initial node. A large option value expresses a substantial need for added generation capacity. This framework can detect necessary expansions along with their optimal timing. This decision-making tool is based on LMP differences, so a valuable expansion option reduces system congestion. We illustrate the key features of this model via a numerical example and present managerial insights with economic implications.

Spectral Expansions for Credit Risk Modelling with Occupation Times

We study two credit risk models with occupation time and liquidation barriers: the structural model and the hybrid model with hazard rate. The defaults within the models are characterized in accordance with Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process) of the U.S. Bankruptcy Code. The models assume that credit events trigger as soon as the occupation time (the cumulative time the firm’s value process spends below some threshold level) exceeds the grace period (time allowance). The hazard rate model extends the structural occupation time models and presumes that other random factors may also lead to credit events. Both approaches allow the firm to fulfill its obligations during the grace period. We derive new closed-from pricing formulas for credit derivatives containing the (risk-neutral) probability of defaults and credit default swap (CDS) spreads as special cases, which are derived analytically via a spectral expansion methodology. Our method works for any solvable diffusion, such as the geometric Brownian motion (GBM) and several state-dependent volatility processes, including the constant elasticity of variance (CEV) model. It allows us to write the pricing formulas explicitly as infinite series that converges rapidly. We then calibrate our models (assuming that GBM governs the firm’s value) to market CDS spreads from the Total Energy company. Our calibration results show that the computations are fast, and the fit is near-perfect.

Valuation of guaranteed minimum maturity benefits under generalised regime-switching models using the Fourier Cosine method

This paper presents a flexible valuation approach for variable annuity (VA) contracts embedded with guaranteed minimum maturity benefit (GMMB) riders written on an underlying fund that evolves according to a general regime-switching framework. Unlike the classical regime-switching models which only allow model parameters to change upon regime switches, our framework allows, more importantly, model structures to vary. With mild assumptions on the characteristic function of the log-stock price, our model settings enable the study of fundamental features of the market dynamics, such as stochastic volatility and jumps, on the underlying fund value of GMMB in a unified framework. This novel idea is illustrated by a three-regime model whose environments can be characterised by either the geometric Brownian motion process, the Heston (1993) stochastic volatility process, or double exponential process. Three versions of the GMMB riders are considered; a fixed or roll-up guarantee, a geometric average guarantee and a ratchet guarantee. With the Fourier Cosine (COS) method which utilises characteristic functions, explicit valuation expressions for various contracts are derived, and numerical illustrations are performed to analyse the efficiency of the approach in terms of computational speed and accuracy. The paper makes a unique contribution by presenting regime-dependent bounds and an algorithm for determining the optimal grid points required for the COS method to achieve a specific level of accuracy. Numerical experiments for the valuation framework reveal that i) as the likelihood of regime shifts increases, the price difference of VA contracts with different initial regimes diminishes, which is consistent with financial intuition; ii) the generalised regime-switching models are more suitable for VA pricing when the time-to-maturity spans over several decades, during which structural changes, although infrequent, are inevitable.

Weighted average price management of sales under the given minimum volume of assets obligatory for realization

In this paper, we consider the construction of a sales management strategy for a highly liquid asset trading at the market. The proposed strategy goal is to maximize the weighted average price of sales. It is assumed that the market price follows a geometric Brownian motion process in which drift and volatility coefficients are random functions of time. The important feature of the given management is that the volumes of assets in the succession of buy and sell executed bargains are calculated exclusively on the basis of the observed market prices rather than on the model coefficient values. In contrast to the management proposed earlier in this study, obligatory realization of some minimum volume of assets on the given time period is demanded. Examples of real-world markets trade demonstrating the imposed constraints effect on the weighted average price values obtained within the constructed management are given.

Blockchain Traceability Valuation for Perishable Agricultural Products Under Demand Uncertainty

Various perishable agricultural products are recalled due to harmful health risks. Blockchain has been used to reduce the amount of such products wasted and disposed. Specifically, a supply chain with a wholesaler, a retailer, and customers is considered where the retailer decides when to switch from a conventional supply chain information management system (SCIMS) to a blockchain-based SCIMS. This article models the uncertain customers' demand as a geometric Brownian motion process and shows how to obtain the optimal demand threshold above which the switch occurs and the corresponding expected time. Next, the model is extended by incorporating two types of government subsidies (i.e., a fixed subsidy on the switching cost and a variable subsidy per unit demand). Through sensitivity analysis and numerical studies, the impacts of key parameters on the optimal demand threshold and expected time of switching are presented. Finally, managerial insights and policy implications are derived.

VALUATION OF EUROPEAN PUT OPTION BY USING THE QUADRATURE METHOD UNDER THE VARIANCE GAMMA PROCESS

Dynamic asset pricing model uses the Geometric Brownian Motion process. The Black-Scholes model known as standard model to price European option based on the assumption that underlying asset prices dynamic follows that log returns of asset is normally distributed. In this paper, we introduce a new stochastic process called levy process for pricing options. In this paper, we use the quadrature method to solve a numerical example for pricing options in the Indian context. The illustrations used in this paper for pricing the European style option. We also try to develop the pricing formula for European put option by using put-call parity and check its relevancy on actual market data and observe some underlying phenomenon.

Reliability assessment of scenarios generated for stock index returns incorporating momentum

AbstractStochastic programming models for portfolio optimization rely on scenario paths for returns derived from stochastic process models. This paper investigates a variant of the geometric Brownian motion process for stock index returns that incorporates index momentum. Based on this model, three different processes for generating scenarios on a rolling basis are devised, which differ according to how frequently the momentum parameter is updated and whether it is estimated according to a simple moving average or an exponentially weighted moving average of returns. The reliability of scenario sets generated for multiple instances is assessed by applying a recently developed statistical tool. Backtesting is conducted in case studies of the Standard & Poor's 500 Index and the Financial Times Stock Exchange 100 Index for two different historical periods. The numerical results show that the frequency with which the expected return is updated does not significantly influence the performance of the scenario generation procedure, whereas how the expected return is calculated affects the autocorrelation and dispersion of generated scenarios drastically. All scenario generation schemes are highly sensitive to the estimated volatility of the index returns. Among the three processes tested, the algorithms that incorporate momentum estimates according to an exponentially weighted moving average can generate reliable scenarios when the volatility estimation error is small.

A lattice method for jump-diffusion process applied to transmission expansion investments under demand and distributed generation uncertainties

Strategic decision making for rare events is considered to be an arduous course of actions as their impacts cannot accurately be modeled. Transmission of electrical power is a major challenge in this day and age due to the prevalent of rare events. Decision makers of generation units, distribution utilities, and transmission networks do not often cooperate, which creates severe uncertainties for all players. Growth of demand for electricity and installation or removal of distributed generation (DG), considered to be a rare event, are among uncertainties encountered by transmission network owners. Expansion decisions for transmission lines should be strategically executed because installations of DGs may create a stranded cost for transmission owners. In this study, we propose a real options framework that quantifies the values of transmission investments under demand and DG uncertainties to guide decision makers of transmission companies regarding how to adapt their expansions decisions. We model demand uncertainty as a geometric Brownian motion process and DG uncertainty as a Poisson arrival process. We devise a computationally-efficient lattice diagram to discretize both processes in a single grid, which can also be employed to model other types of rare events in various sectors. The proposed framework is demonstrated on a realistic transmission network. Numerical study shows that our proposed lattice model is computationally superior over existing diagrams. As for managerial insights, it shows that depending on the locations of DG installations, DG penetration may not reduce the value of a transmission network. If the center at which a DG is installed has a large-capacity local generator, the installation may not undervalue the transmission network.

Closed Form Optimal Exercise Boundary of the American Put Option

We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the American put option. The optimal exercise boundary satisfies nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary. All these results can be used as approximations to the standard model with constant parameters, i.e., geometric Brownian motion process.

Optimal Timing of Onshore Wind Repowering in Germany under Policy Regime Changes: A Real Options Analysis

In this paper, we investigate capacity expansion, the technological development of wind turbine generators, and the merits of onshore wind repowering in Germany. We analyze the regulatory framework that is presently in place and its impact on the profitability of the onshore wind parks commissioned under previous regulatory frameworks. The optimal timing of repowering under today’s market premium model is scrutinized. Repowering in a market without subsidies for onshore wind energy is also evaluated. In the two cases (regimes) investigated, the electricity price is modeled stochastically as a geometric Brownian motion process. More specifically, using a real options modeling framework for investments under a free market regime, we analyze how the regime change affects the optimal timing of repowering. Further, we check the results for their robustness via a sensitivity analysis for both analyzed regimes. We find that repowering well before the plant’s expected end of life (‘early repowering’) can be economically preferable under the German Renewable Energy Sources Act 2017 remuneration scheme and that, due to the electricity spot market price development, repowering under the market premium regime is more profitable than it is under the free market regime. The economic viability is strongly influenced by the duration of the initial tariff granted to the old installation, the expected digression factor of the future reference tariff, and the increase in electricity generation achieved through repowering.

A GIS-based green supply chain model for assessing the effects of carbon price uncertainty on plastic recycling

Recycling plastic can abate the environmental pollution as well as CO2 emissions by saving the carbon-intensive feedstock input. The uncertain carbon price places significant effects on the establishment and operation of the whole supply chain. This study develops a green supply chain model combined with geographic information system (GIS) to account for carbon price uncertainty and evaluate its effects on the closed-loop supply chain (CLSC) of plastic recycling. A two-stage stochastic programming model is constructed, in which the stochastic variable, CO2 price is modelled as a geometric Brownian motion process. Six scenarios are designed with respect to price expectation and volatility. A case study is performed with the GIS information of the plastic supply chain in Zhejiang province, China. The results illustrate that triggering the establishment of reverse logistics requires a carbon price threshold significantly beyond the current level. Lower price volatility would facilitate the decision-making of investment into the reverse logistics. Mechanisms to alleviate the market variation shall be introduced. A sound market condition is desired to obtain the optimal balance that encourages the CLSC without creating extra pressure on the firms. The proposed modelling framework can be easily applied to other sectors with similar characteristics.

European option pricing model with generalized Ornstein\u2013Uhlenbeck process under stochastic earning yield and stochastic dividend yield

This paper aims to examine and establish the models for European option pricing which include parameters of stochastic dividend yield and stochastic earning yield. We generalize the Ornstein–Uhlenbeck process and define it as generalized Ornstein–Uhlenbeck process. We have learned that the firm stocks, according to Black–Scholes–Merton structure, obey the geometric Brownian motion process. Under a stochastic earning yield, the dividend yield complies with the generalized Ornstein–Uhlenbeck process. The firm dividend randomly deviates from the earning yield flow because of the presence of stochastic components of dynamic Wiener process of generalized Ornstein–Uhlenbeck. In this study, we model the stock price with stochastic earning yield, and stochastic dividend yield to be taking account stochastic market price of risk parameter which is mean-reverting as well. We developed explicit formulae for European call option pricing calculations. From numerical simulation, we could evaluate the performance of our new model that could be compared with other notable option pricing models by using actual option price data. The outcomes prove that our new model performance is best when compared with others.

A Valuation Model for the Variable Rate Demand Obligation (VRDO)

In this paper, a valuation framework is developed for the variable rate demand obligation (VRDO). The VRDO is a class of floating rate note whose coupon rate changes on a regular basis and is 'puttable' by the bondholder, given notice to the issuer. We model the coupon rate as a geometric Brownian motion process and assume that the incidence of puts is Poisson distributed, across time. Put events are assumed to be brought about by factors such as a change in the liquidity and consumption preferences of investors or a change in a Municipal issuer's creditworthiness.

A lattice-based approach to option and bond valuation under mean-reverting regime-switching diffusion processes

Nowadays, the pricing of financial instruments under continuous-time Markov switching models have received a widespread attention from researchers and practitioners in the finance industry. Lattice-based approaches are amongst the most widely used approaches to solve the pricing problem in this context. Recently, Yuen and Yang (2010) have proposed a simple and fast recombining trinomial tree method to handle the case of regime-switching geometric Brownian motion processes. In this paper, we generalize their approach to the regime-switching (exponential) mean-reverting case with state-dependent switching rates and derive the necessary conditions for the positivity of conditional branching probabilities. We employ the Hull and White’s tree-building procedure to limit tree growth away from the long-run mean of the process. We use the proposed lattice framework to price contingent claims of European, American and barrier type, and demonstrate its applicability in pricing default-free zero-coupon bonds. In the proposed tree structure, the number of nodes is substantially decreased and the computational cost is effectively reduced compared to the usual approaches. Extensive numerical experiments illustrate the efficiency and flexibility of the proposed scheme.

A Valuation Model for Callable Eurobonds

This paper models the value of callable Eurobonds, using stochastic calculus, by assuming that the exchange rate follows a geometric Brownian motion process and the arrival time of an early redemption of the bond by the issuer conforms to a negative exponential distribution. The solution to the stochastic model shows that there is a relationship between the call premium and the expected time to the call which is consistent with traditional Black-Scholes pricing formulae. The magnitude of the call premium can be viewed as a signal to the market on a Government treasury’s or company’s expectations about the future level of interest rates and possible refinancing strategies. This paper is unique because as of July 2019 there exists no attempt at valuing Callable Eurobonds in the research literature.