Geometric Brownian Motion Model Research Articles

“Quantum Equilibrium-Disequilibrium”: Asset price dynamics, symmetry breaking, and defaults as dissipative instantons

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the “Quantum Equilibrium-Disequilibrium” (QED) model.The QED model incorporates non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, “classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from the QED dynamics. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non-analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter.

Exact Solutions for a GBM‐Type Stochastic Volatility Model Having a Stationery Distribution

Alan Lewis discusses the background to the Extended Geometric Brownian Motion model introduced in this issue.

Contribution Indonesian Composite Index in PT Telekomunikasi Indonesia stock price model using 2-dimensional Geometric Brownian Motion

Theoretically, the movement of the Composite Stock Price Index (CSPI) is in line with the company’s stock price movements. Hence, it would be appropriate to measure the CSPI contribution to the company’s stock price regarding modeling the company’s stock price. 2-dimensional Geometric Brownian Motion is believed to be the most appropriate model in this case. Therefore, this paper aims to project the share price of PT Telekomunikasi Indonesia in 2018 by considering the CSPI movement. The resulted mean absolute percentage error (MAPE) calculations lead to a conclusion that Prediction with 2-dimensional Geometric Brownian Motion (GBM) is more accurate than the individual modeling of stock prices of PT Telekomunikasi Indonesia. PT Telekomunikasi Indonesia Tbk’s stock price modeling is more appropriately-used by taking heed on the movement of the Composite Stock Price Index. It is conclusive that the two dimensional geometric Brownian motion model provides an accurate prediction of PT Telekomunikasi Indonesia Tbk shares with MAPE is 1.980296%.

Evaluation of an iron ore price forecast using a geometric Brownian motion model

Abstract Mining projects are often budgeted in millions of dollars, making it of interest to the investor to measure a project’s uncertainties and risks, which include the changes in the commodity price. The simulation of asset prices is valid because it enables displaying, with a degree of certainty, the future behavior of a financial asset. One of the most commonly used mathematical forecast models is the Geometric Brownian Motion (GBM) model. This article aims at simulating the forecast price of an iron ore commodity with the GBM method based on the historical price series and comparing it to other forecast approaches used in the financial market. The results of this study indicate that the model has limitations, which should be considered. However, it is an overall good tool to support the economic evaluation of a project, if performed along with other approaches.

Testing of binary regime switching models using squeeze duration analysis

We have developed a statistical technique to test the model assumption of binary regime switching extension of the geometric Brownian motion (GBM) model by proposing a new discriminating statistics. Given a time series data, we have identified an admissible class of the regime switching candidate models for the statistical inference. By performing several systematic experiments, we have successfully shown that the sampling distribution of the test statistics differs drastically, if the model assumption changes from GBM to Markov modulated GBM, or to semi-Markov modulated GBM. Furthermore, we have implemented this statistics for testing the regime switching hypothesis with Indian sectoral indices.

APPROXIMATION METHODS FOR INHOMOGENEOUS GEOMETRIC BROWNIAN MOTION

We present an accurate and easy-to-compute approximation of the transition probabilities and the associated Arrow-Debreu (AD) prices for the inhomogeneous geometric Brownian motion (IGBM) model for interest rates, default intensities or volatilities. Through this procedure, dubbed exponent expansion, transition probabilities and AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated after the first few terms. For farther time horizons, the exponent expansion can be combined with a fast numerical convolution to obtain high-precision results.

Estimation of geometric Brownian motion model with a t-distribution–based particle filter

Orientation: Geometric Brownian motion (GBM) model basically suggests whether the distribution of asset returns is normal or lognormal. However, many empirical studies have revealed that return distributions are usually not normal. These studies, time and again, discover evidence of non-normality, such as heavy tails and excess kurtosis. Research purpose: This work was aimed at analysing the GBM with a sequential Monte Carlo (SMC) technique based on t -distribution and compares the distribution with normal distribution. Motivation for the study: The SMC or particle filter based on the t -distribution for the GBM model, which involves randomness, volatility and drift, can precisely capture the aforementioned statistical characteristics of return distributions and can predict the random changes or fluctuation in stock prices. Research approach/design and method: The particle filter based on the t -distribution is developed to estimate the random effects and parameters for the extended model; the mean absolute percentage error (MAPE) were calculated to compare distribution fit. Distribution performance was assessed through simulation study and real data. Main findings: Results show that the GBM model based on student’s t -distribution is empirically more successful than the normal distribution. Practical/managerial implications: The proposed model which is heavier tailed than the normal does not only provide an approximate solution to non-normal estimation problem. Contribution/value-add: The GBM model based on student’s t -distribution establishes an efficient structure for GBM and volatility modelling.

On the Effect of Inflation and Impact of Hedging on Pension Wealth Generation Strategies under the Geometric Brownian Motion Model

This work investigates the effect of Inflation and the impact of hedging on the optimal investment strategies for a prospective investor in a DC pension scheme, using inflation-indexed bond and inflation-linked stock. The model used here permits the plan member to make a defined contribution, as provided in the Nigerian Pension Reform Act of 2004. The pension plan member is allowed to invest in risk-free asset (cash), and two risky assets (i.e., the inflation-indexed bond and inflation-linked stock). A stochastic differential equation of the pension wealth that takes into account certain agreed proportions of the plan member’s salary, paid as contribution towards the pension fund, is constructed and presented. The Hamilton-Jacobi-Bellman (H-J-B) equation, Legendre transformation, and dual theory are used to obtain the explicit solution of the optimal investment strategies for CRRA utility function. Our investigation reveals that the inflation have significant negative effect on wealth investment strategies, particularly, the RRA(w) is not constant with the investment strategy, since the inflation parameters and coefficient of CRRA utility function have insignificant input on the investment strategies, and also the inflation-indexed bond and inflation-linked stock has a positive damping effect (hedging) on the severe effect of inflation.

Optimal Dynamic Futures Portfolio in a Regime-Switching Market Framework

We study the problem of dynamically trading futures in a regime-switching market. Modeling the underlying asset price as a Markov-modulated diffusion process, we present a utility maximization approach to determine the optimal futures trading strategy. This leads to the analysis of the associated system of Hamilton-Jacobi-Bellman (HJB) equations, which are reduced to a system of linear ODEs. We apply our stochastic framework to two models, namely, the Regime-Switching Geometric Brownian Motion (RS-GBM) model and Regime-Switching Exponential Ornstein-Uhlenbeck (RS-XOU) model. Numerical examples are provided to illustrate the investor's optimal futures positions and portfolio value across market regimes.

Joint dynamic modeling and option pricing in incomplete derivative-security market

In this study, we evaluate the option prices on two assets under stochastic interest rates when the stochastic process that underlying asset prices follow is depending on a correlated bivariate Markov-modulated geometric Brownian motion model with jump risks. More specifically, we conduct the joint dynamic modeling by identifying two independent compound Poisson processes with the log-normal jump sizes to describe both individual jumps and systematic cojumps. Facilitating the cojumping behavior this way with the time-inhomogeneity of the volatility, option pricing expressions are readily obtainable since the Gerber–Siu’s approach is employed to determine a pricing kernel. The empirical results and numerical illustrations are provided to show the impact of cojumps and stochastic volatilities on option prices.

Optimal risk-averse timing of an asset sale: trending versus mean-reverting price dynamics

This paper studies the optimal risk-averse timing to sell a risky asset. The investor’s risk preference is described by the exponential, power, or log utility. Two stochastic models are considered for the asset price – the geometric Brownian motion and exponential Ornstein-Uhlenbeck models – to account for, respectively, the trending and mean-reverting price dynamics. In all cases, we derive the optimal thresholds and certainty equivalents to sell the asset, and compare them across models and utilities, with emphasis on their dependence on asset price, risk aversion, and quantity. We find that the timing option may render the investor’s value function and certainty equivalent non-concave in price. Numerical results are provided to illustrate the investor’s strategies and the premium associated with optimally timing to sell.

Optimal exchange rates management using stochastic impulse control for geometric Lévy processes

In this paper, we consider the optimal impulse control problem of a currency with exchange rate dynamics whose state follows a geometric Levy process. The objective of the Central Bank is to keep the exchange rate as close as possible to a given target. Running costs associating with the difference between the actual exchange rate and the target are continuously incurred to the system. We suppose that, when the Central Bank intervenes in the system, it requires the fixed and proportional costs. Applying the theory of stochastic impulse controls, we find the optimal exchange rate at which interventions should be performed and the optimal sizes of the interventions, so as to minimize the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. Furthermore, numerical comparisons for the optimal intervention strategy between the geometric Levy process model and the geometric Brownian motion model are investigated in details.

GENERALIZED FRAMEWORK FOR APPLYING THE KELLY CRITERION TO STOCK MARKETS

We develop a general framework for applying the Kelly criterion to the stock market. By supplying an arbitrary probability distribution modeling the future price movement of a set of stocks, the Kelly fraction for investing each stock can be calculated by inverting a matrix involving only first and second moments. The framework works for one or a portfolio of stocks and the Kelly fractions can be efficiently calculated. For a simple model of geometric Brownian motion of a single stock we show that our calculated Kelly fraction agrees with existing results. We demonstrate that the Kelly fractions can be calculated easily for other types of probabilities such as the Gaussian distribution and correlated multivariate assets.

Stochastic wind speed modelling for estimation of expected wind power output

Increased wind energy penetration causes problems to the operation and system balancing of electric power systems. This in turn leads to the need for more detailed wind power modelling. The modelling and management of wind power involves two stages, neither of which is analytically tractable. In particular, the first stage involves stochastic variations in wind speed; wind speed typically presents noisy short-term variations, plus cyclicality over periods of 24 h and longer. The second stage refers to stochastic variations of the resulting wind power output, a non-linear function of wind speed. This paper proposes and tests an Ornstein-Uhlenbeck Geometric Brownian Motion model in continuous time to represent the wind speed, while including its longer-term daily cycle. It also illustrates a partial differential equation model of the wind speed and of the resulting wind power output, aiming at computing both their statistics. The proposed stochastic model has the potential to be used for various applications where wind speed or wind power are stochastic inputs, such as the optimal valuation of energy storage or system balancing. We verify by statistical tests that the results from the proposed model for the wind speed and the wind power match those from the empirical data of a wind farm located in Spain.

Optimal leverage ratio estimate of various models for leveraged ETFs to exceed a target: Probability estimates of large deviations

In this paper, we study the deviation probability estimate for a leveraged exchanged-traded fund (LETF). By large deviation principle, we derive explicitly the logarithmic limit of the tail probability when the price of a LETF exceeds a given reference asset, which allows us to compute the underlying leverage ratio. Then we apply our results to various existing models, including the geometric Brownian motion (GBM) model, generalized autoregressive conditional heteroskedasticity (GARCH) model, inverse GARCH model, extended Cox–Ingersoll–Ross (CIR) model, 3/2 model, as well as the Heston and 3/2 stochastic volatility models, and to present their corresponding optimal leverage ratios, respectively.

INDEX OPTIONS AND VOLATILITY DERIVATIVES IN A GAUSSIAN RANDOM FIELD RISK-NEUTRAL DENSITY MODEL

We propose a risk-neutral forward density model using Gaussian random fields to capture different aspects of market information from European options and volatility derivatives of a market index. The well-structured model is built in the framework of the Heath–Jarrow–Morton philosophy and the Musiela parametrization with a user-friendly arbitrage-free condition. It reduces to the popular geometric Brownian motion model for the spot price of the market index and can be intuitively visualized to have a better view of the market trend. In addition, we develop theorems to show how the model drives local volatility and variance swap rates. Hence, volatility futures and options can be priced taking the forward density implied by European options as the initialization input. The model can be accordingly calibrated to the market prices of these volatility derivatives. An efficient algorithm is developed for both simulating and pricing, and a numerical study is conducted using real market data.

Structural model for fluctuations in financial markets.

In this paper we provide a comprehensive analysis of a structural model for the dynamics of prices of assets traded in a market which takes the form of an interacting generalization of the geometric Brownian motion model. It is formally equivalent to a model describing the stochastic dynamics of a system of analog neurons, which is expected to exhibit glassy properties and thus many metastable states in a large portion of its parameter space. We perform a generating functional analysis, introducing a slow driving of the dynamics to mimic the effect of slowly varying macroeconomic conditions. Distributions of asset returns over various time separations are evaluated analytically and are found to be fat-tailed in a manner broadly in line with empirical observations. Our model also allows us to identify collective, interaction-mediated properties of pricing distributions and it predicts pricing distributions which are significantly broader than their noninteracting counterparts, if interactions between prices in the model contain a ferromagnetic bias. Using simulations, we are able to substantiate one of the main hypotheses underlying the original modeling, viz., that the phenomenon of volatility clustering can be rationalized in terms of an interplay between the dynamics within metastable states and the dynamics of occasional transitions between them.

Modeling stock prices in a portfolio using multidimensional geometric brownian motion

Modeling and forecasting stock prices of public corporates are important studies in financial analysis, due to their stock price characteristics. Stocks investments give a wide variety of risks. Taking a portfolio of several stocks is one way to minimize risk. Stochastic process of single stock price movements model can be formulated in Geometric Brownian Motion (GBM) model. But for a portfolio that consist more than one corporate stock, we need an expansion of GBM Model. In this paper, we use multidimensional Geometric Brownian Motion model. This paper aims to model and forecast two stock prices in a portfolio. These are PT. Matahari Department Store Tbk and PT. Telekomunikasi Indonesia Tbk on period January 4, 2016 until April 21, 2017. The goodness of stock price forecast value is based on Mean Absolute Percentage Error (MAPE). As the results, we conclude that forecast two stock prices in a portfolio using multidimensional GBM give less MAPE than using GBM for single stock price respectively. We conclude that multidimensional GBM is more appropriate for modeling stock prices, because the price of each stock affects each other.

Pricing derivatives in a regime switching market with time inhomogenous volatility

ABSTRACTThis paper studies pricing derivatives in a componentwise semi-Markov (CSM) modulated market. We consider a financial market where the asset price dynamics follows a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local Black–Scholes–Merton-type PDE. We establish existence and uniqueness of a classical solution to the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. An explicit expression of quadratic residual risk is also obtained.

Stock price prediction using geometric Brownian motion

Geometric Brownian motion is a mathematical model for predicting the future price of stock. The phase that done before stock price prediction is determine stock expected price formulation and determine the confidence level of 95%. On stock price prediction using geometric Brownian Motion model, the algorithm starts from calculating the value of return, followed by estimating value of volatility and drift, obtain the stock price forecast, calculating the forecast MAPE, calculating the stock expected price and calculating the confidence level of 95%. Based on the research, the output analysis shows that geometric Brownian motion model is the prediction technique with high rate of accuracy. It is proven with forecast MAPE value ≤ 20%.