Geometric Brownian Motion Model Research Articles

Variance gamma for stock model performance with excess kurtosis

The Geometric Brownian Motion (GBM) model is used widely for model the dynamic of asset price movement. One of the company’s assets is a stock. The distribution of stock data that is normally distributed can be modeled using the Geometric Brownian Motion model. However, the distribution of stock data showed excess kurtosis and tail when using the Brown Geometry Motion model was less precise. One model for data showing excess kurtosis and tail was Variance Gamma (VG). In this research, the sample used was the stock data of PT Bank Danamon Indonesia Tbk for the period April 25 th , 2018 to April 24 th , 2020. The data sample was divided into two parts, namely training data and testing data. Based on the result of the stock description statistics, the value of skewness = -2.105417 and kurtosis = 22.16438 was obtained, while hypothesis testing concluded that the stock distribution did not spread normally. The resulting parameters for the VG model were σ = 0.08071, v = 8.00500 and θ = 0.01976. Based on the results of testing on the last 38 observations, the MAPE value was = 6.97560%. These results gave the conclusion that the VG model provided excellent forecasting results.

European option under a skew version of the GBM model with transaction costs by an RBF method

The principal aim of this paper is to calculate the price of the European option under a skew version of the geometric Brownian motion model. To do this, we use the Leland and Kabanov strategy to remove arbitrage opportunities by the delta hedging strategy and adding transaction costs to a portfolio that contains an option and a share of the related stock. This idea generates a partial differential equation (PDE) problem to calculate the option price. To solve the obtained PDE, we use a radial basis functions (RBFs) method and apply the operational matrix of derivative for multiquadric-RBF to reduce the problem to a set of algebraic equations. Finally, we present some numerical results to show the efficiency of the model and the method by considering the Newton–Raphson method and the Bayes information criterion.

A parameter based approach to single factor stochastic process selection for real options applications

The single factor stochastic diffusion processes most commonly used for Real Options Valuation are the Geometric Brownian Motion and Mean Reversion Models. Nonetheless, the choice of process to model asset price dynamics is still one of the main challenges for researchers and practitioners in the field. Particularly, in investment projects where there is significant managerial flexibility, the project value and the investment rule may depend in large part on the process used to model the underlying uncertainties. In this article, we develop an approach based on the parameter values of the model, which has some advantages over the methods currently used for stochastic process selection. We use the half-life and normalized variance of the time series to be modeled to determine the optimal choice and discuss related theoretical as well as practical issues concerning the application of this approach to real options valuation. Numerical examples are used to illustrate the method, and a guideline for implementation of this approach is provided.

An empirical assessment of symmetric and asymmetric jump-diffusion models for the Nigerian stock market indices

We examine empirically, the suitability of three stock price models viz: geometric Brownian motion, symmetric and asymmetric jump-diffusion models, on the empirical log-returns of the Nigerian All-Share Index. 5334 daily observed data from January 2, 1998, to February 21, 2020, were utilized. Using a non-parametric jump-test method, our results show that jumps are present in the empirical log-returns of the stock market price. The results obtained for the optimal parameters in the models indicate high jump intensity, more upward jumps, and a positively skewed jump process. However, the parameters in the asymmetric jump-diffusion model were found to be more sensitive to the varied threshold of jumps in the log-returns than the symmetric jump-diffusion model. The suitability analysis results show that the symmetric jump-diffusion model fits the market indices better. Therefore, it can be used for future predictions of the market price.

Cat bond valuation using Monte Carlo and quasi Monte Carlo method

The frequency of natural disasters is very small, but the loss of damage and the risks that must be borne is of enormous value. Recently, Catastrophe bond (CAT bond) has grown rapidly in financial markets to increase the coverage of environmental disasters and the resulting economic losses, insurance companies and reinsurers are barely covered. Catastrophe bond (CAT Bond) is one of the insurance-linked financial securities instruments that aim to transfer risk by insuring natural disaster events to the capital market. The model of natural disaster loss without a jump-diffusion process followed a model of geometric Brownian motion. We estimate and match each parameter in each model by using grid search and then we use the method of Monte Carlo and quasi-Monte Carlo to obtain numerical results for the CAT bonds pricing formulas. From this simulation, the Monte Carlo method has good enough accuracy and efficiency to valuates the CAT bond. Less then 500 iterations Monte Carlo have reached convergence in 1280.49 seconds. And quasi-Monte Carlo more efficient with less than 400 iterations have reached convergence in 30.89 seconds. Both methods have reached small enough MAPE. Based on the simulations carried out through this research, it was showed that the quasi-Monte Carlo method is better than the Monte Carlo method based on the value of Mean Absolute Percentage Error (MAPE) and running time.

The voice of Physics in finance: A glance on the theoretical application of heat equation to stock price diffusions

Stock price volatility is considered the main matter of concern within the investment grounds. However, the diffusivity of these prices should as well be considered. As such, proper modelling should be done for investors to stay healthy-informed. This paper suggest to model stock price diffusions using the heat equation from physics. We hypothetically state that, our model captures and model the diffusion bubbles of stock prices with a better precision of reality. We compared our model with the standard geometric Brownian motion model which is the wide commonly used stochastic differential equation in asset valuation. Interestingly, the models proved to agree as evidenced by a bijective relation between the volatility coefficients of the Brownian motion model and the diffusion coefficients of our heat diffusion model as well as the corresponding drift components. Consequently, a short proof for the martingale of our model is done which happen to hold.

Adaptive Sampling using a Geometric Brownian Motion Model to Predict MD Trajectory Mobility on a Free Energy Surface

We present an enhanced adaptive sampling method for swarms of Molecular Dynamics (MD) trajectories based on a mathematical model of Geometric Brownian Motion. Adaptive sampling protocols often determine the direction of trajectories in MD swarm simulation from the time evolution of collective variables (CVs) of interest extracted from multiple MD trajectories. Using dimensionality reduction projections such onto spaces such as defined by time-independent component analysis (tICA), the data are then projected onto a 2D space composed of two vectors with the slowest motion along the CVs.

The Optimal Limit Prices of Limit Orders under an Extended Geometric Brownian Motion with Bankruptcy Risk

In the Black and Scholes system, the underlying asset price model follows geometric Brownian motion (GBM) with no bankruptcy risk. While GBM is a commonly used model in financial markets, bankruptcy risk should be considered in the case of a severe economic crisis, such as that caused by the COVID-19 pandemic. The omission of bankruptcy risk could considerably influence the setting of a trading strategy. In this article, we adopt an extended GBM model that considers the bankruptcy risk and study its optimal limit price problem. A limit order is a classical trading strategy for investing in stocks. First, we construct the explicit expressions of the expected discounted profit functions for sell and buy limit orders and then derive their optimal limit prices. Furthermore, via sensitivity analysis, we discuss the influence of the omission of bankruptcy risk in executing limit orders.

Geometric fractional Brownian motion model for commodity market simulation

The geometric Brownian motion (GBM) model is a mathematical model that has been used to model asset price paths. By incorporating Hurst parameter to GBM to characterize long-memory phenomenon, the geometric fractional Brownian motion (GFBM) model was introduced, which allows its disjoint increments to be correlated. This paper investigates the accuracy of GBM and GFBM in modelling Malaysia’s crude palm oil price simulation, and to see display of persistent or anti-persistent behaviour across different periods. Results show that the GFBM model is more accurate than the GBM model in simulating future price path for the given data set.

Estimation of a term structure model of carbon prices through state space methods: The European Union emissions trading scheme

AbstractThis study models the term structure of the European Union Emissions Trading Scheme. The one‐factor geometric Brownian motion model of Abadie and Chamorro is replicated using the data now available and then compared with a two‐factor short‐term/long‐term (STLT) stochastic model. The STLT model has the better statistical fit to the term structure of European Union Allowances (EUAs). A real options analysis of the value of the option to retrofit carbon capture and storage shows that forecasting phase four EUAs with the STLT model almost triples the estimated project net present value and lowers investment trigger prices by approximately 24 percent.

EFFICIENT ESTIMATORS FOR GEOMETRIC FRACTIONAL BROWNIAN MOTION PERTURBED BY FRACTIONAL ORNSTEIN-UHLENBECK PROCESS

This paper discusses an enhanced model of geometric fractional Brownian motion where its volatility is assumed to be stochastic volatility model obey fractional Ornstein-Uhlenbeck process. The method of estimation for all parameters in this model are derived. After, simulation experiments are conducted to examine the performance of the proposed estimators. The result shows that the proposed method provides efficient estimates for the parameters. Thus, the proposed model is promising and can apply in real financial environments.

A Time Series Framework for Pricing Guaranteed Lifelong Withdrawal Benefit

In this work, the pricing problem of a variable annuity (VA) contract embedded with a guaranteed lifelong withdrawal benefit (GLWB) rider has been considered. VAs are annuities whose value is linked with a sub-account fund consisting of bonds and equities. The GLWB rider provides a series of regular payments to the policyholder during the policy duration when he is alive irrespective of the portfolio performance. Also, the remaining fund value is given to his nominee, at the time of death of the policyholder. The appropriate modelling of fund plays a crucial role in the pricing of VA products. In the literature, several authors model the fund value in a VA contract using a geometric Brownian motion (GBM) model with a constant variance. However, in real life, the financial assets returns are not Normal distributed. The returns have non-zero skewness, high kurtosis, and leverage effect. This paper proposes a discrete-time model for annuity pricing using generalized autoregressive conditional heteroscedastic (GARCH) models, which overcome the limitations of the GBM model. The proposed model is analyzed with numerical illustration along with sensitivity analysis.

Optimal Distributional Trading Gain: State Price Density Equilibrium and Bayesian Statistics

This paper considers multiple market agents who have distinct distributional opinions about the state price density. We first determine the optimal trading positions of a utility maximizing market taker who trades Arrow-Debreu securities for prices set by the market maker. We use calculus of variations to determine the solution of this problem for a general utility function. The choice of the logarithmic utility function leads to a solution in terms of a likelihood ratio of the densities corresponding to the market taker and the market maker and the resulting optimal utility is the Kullback-Leibler divergence. In particular, we obtain a trivial solution for Merton's portfolio problem in the traditional geometric Brownian motion model and and we show its immediate extension to the multivariate case. A further extension gives a solution for the market driven by a geometric Poisson process. In a market without the market maker, the distributional opinions of market takers reach an equilibrium in the form of the linear mixture of the distributions. We show that when the the result of the outcome is observed, the profit and loss from trading updates agents' bankrolls in a Bayesian fashion, which provides one to one correspondence for the logarithmic utility maximazers' profits and Bayesian statistics. We extend these results to the continuous time setup and show that the bankrolls of agents following Merton optimal portfolio strategy evolve as a posterior distribution.

Conditional Davis pricing

We study the set of Davis (marginal utility-based) prices of a financial derivative in the case where the investor has a non-replicable random endowment. We give a new characterisation of the set of all such prices, and provide an example showing that even in the simplest of settings – such as Samuelson’s geometric Brownian motion model –, the interval of Davis prices is often a non-degenerate subinterval of the set of all no-arbitrage prices. This is in stark contrast to the case with a constant or replicable endowment where non-uniqueness of Davis prices is exceptional. We provide formulas for the endpoints of these intervals and illustrate the theory with several examples.

Hurst Exponent Analysis on the Ghana Stock Exchange

This paper talks about the application of Hurst Index on the Ghana Stock Exchange (GSE). The aim of the paper was to find out, whether GSE daily returns have some autocorrelation (long-term dependency) and multifractality using the Hurst Index analysis. Hurst Index of daily returns of some selected stocks in the period of January 2018 to December 2018 constituting 247 trading days were computed using Rescale Range Method and the Periodogram Method. The findings show that, 91.7% of the stocks considered possess long-term dependency and only 8.3% shows multifractality. Besides, the average percentage error of the geometric fractional Brownian motion (GFBM) model was 16.68% with an efficiency accuracy of 83.32% whilst that of the geometric Brownian motion (GBM) model percentage error is 20.90% with an accuracy of 79.10%. This indicates that, the GFBM model yielded better predicting accuracy than GBM in the long-run and par predicting accuracy in the short-run.

Data-driven prediction for volatile processes based on real option theories

This paper presents a new prediction model for time series data by integrating a time-varying Geometric Brownian Motion model with a pricing mechanism used in financial engineering. Typical time series models such as Auto-Regressive Integrated Moving Average assumes a linear correlation structure in time series data. When a stochastic process is highly volatile, such an assumption can be easily violated, leading to inaccurate predictions. We develop a new prediction model that can flexibly characterize a time-varying volatile process without assuming linearity. We formulate the prediction problem as an optimization problem with unequal overestimation and underestimation costs. Based on real option theories developed in finance, we solve the optimization problem and obtain a predicted value, which can minimize the expected prediction cost. We evaluate the proposed approach using multiple datasets obtained from real-life applications including manufacturing, and finance. The numerical results demonstrate that the proposed model shows competitive prediction capability, compared with alternative approaches.

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.

“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%.