Are share prices in Brownian motion?

Are share prices in Brownian motion?

PurposeThe purpose of this paper is to determine if the US Treasury's at‐the‐market sales of 5.27 billion Citigroup shares in 2010 drove down the banks' share price. It attempts to use the evidence of Citigroup's stock returns to accept or reject competing hypotheses of larger stock sales.Design/methodology/approachThe paper uses a geometric Brownian motion model to test if there were abnormal returns at various points in the US Treasury's highly publicized stock sale that lasted from 26 April to 6 December 2010.FindingsThere was a weakly significant drop in the stock price at the announcement of the sale and a weakly significant rise in the stock price just after it ended. This is evidence that the demand curve for the stock had a negative slope.Practical implicationsThe evidence from this study will influence policy makers and investors in the upcoming privatizations of large bailed‐out firms such as American International Group, Ally Financial, Chrysler, and General Motors. The evidence indicates that slow at‐the‐market sales may temporarily depress stock prices more than quicker, underwritten secondary offerings. Patient investors may experience modest abnormal returns from providing liquidity to the US Treasury as it privatizes its holdings.Originality/valueThis is the only paper to study the stock price impacts of the US Treasury's liquidation of its 27 percent stake in Citigroup in 2010. Because the stock sales were delegated to a third party and highly publicized, unlike most other large stock sales, the Citigroup privatization is an unprecedented opportunity to test if the demand curve for common stocks is perfectly elastic.

Mine valuation under market and geological uncertainty is an active research area. Twenty years ago, a seminal paper by Brennan and Schwartz described the application of Real Option Theory to the valuation of mines where metal prices are volatile. The study focused mainly on the impact of metal price uncertainty on the value of a mine. Geological uncertainty was not considered. For a simple mine model, this paper describes the close analogy between the decision to process a mining block at a given date and the European call financial option. The value of the European call depends primarily on the share price model, the present share price, the price volatility and the time to expiry. A mining block is either processed when the metal price covers the processing costs or otherwise stockpiled as waste. Metal prices and technical variables like grades, recovery, and costs are uncertain. Using geostatistical simulations, the study shows that grade uncertainty may introduce asymmetries in the block value greater than metal price uncertainty. The asymmetries are more pronounced for blocks with larger uncertainty. Greater value is given presently to these blocks assuming the block grades are perfectly known at the time of mining. The extension of this concept from individual blocks to the mine scale is done by considering a mine panel as equivalent to a portfolio of European call options. Implications for strategic planning are illustrated with a gold mine panel-scheduling example. Gold price was modelled with a Geometric Brownian Motion process. The case study shows that the value of the panel and its development strategy depend on the level of geological uncertainty and price volatility. However, the example shows that the benefits of optimising the panel under geological uncertainty is an order of magnitude below the benefits of resolving the geological uncertainty.

The stock prices of publicly traded companies exhibit continuous and random fluctuations over time, necessitating the inclusion of a stochastic term in dynamic models to accurately capture this behavior. This study applies the geometric Brownian motion (GBM) model to analyse the stock prices of 20 randomly selected companies listed on the Dhaka Stock Exchange (DSE). The GBM model was resolved through Monte Carlo simulation to forecast stock prices over a trading horizon of approximately 30 to 35 days. Using historical data from the first four months of 2024, we predicted the share prices for the subsequent one-and-a-half months. The comparison between forecast and actual prices demonstrated a high level of concordance, with a mean absolute percentage error (MAPE) of less than 8%. These findings underscore the efficacy of the model in providing robust predictions of share prices for selected companies.

Asset allocation under stochastic interest rate with regime switching

The Survey and Review article in this issue is “The Narrow Escape Problem,” by D. Holcman and Z. Schuss. We are all familiar with the jagged plots formed by share prices and interest rates as they fluctuate over time. It is natural to ask questions like -when will my portfolio of Facebook shares reach $1 Million? -when will my bank's interest rate leave the 2% -- 3% range? These questions concern hitting times. Modeling the evolution as a stochastic process (a random variable that changes over time), we want to know the first occasion when the value reaches a specified level. More generally, for processes in a higher dimension, such as Robert Brown's pollen particles suspended in a fluid, we could ask when a tagged particle will first meet a specified region on the boundary of a given domain. The hitting time is itself a random variable, and we can reduce to a simpler deterministic quantity by asking for its expected value. It will then be no surprise to many readers that this deterministic value can be obtained by solving an appropriate partial differential equation. The topic of this article is a commonly arising special case which has wide practical significance and offers the added attraction of allowing mathematicians to introduce a small parameter. The narrow escape time (NET) relates to the first time that Brownian motion meets a small absorbing window on the boundary of a domain that is otherwise reflecting. Figure 1.3 illustrates the setting very clearly. A closely related problem is to determine when a particle traverses a narrow passage between almost disconnected regions, as illustrated in Figure 1.1. The article addresses the development of asymptotic methods that deal with various geometries, motivated by applications in cell biology. The underlying mathematical challenge can be viewed as solving a mixed Neumann--Dirichlet boundary value problem for the Poisson equation in a bounded domain, and the authors survey a range of developments in nonstandard asymptotics that have been brought to bear in recent years. This article will be a valuable resource for readers with interests at the interface between probability theory and partial differential equations, and those who study asymptotic analysis of singular perturbation problems and applications to physics, biochemistry, and neuroscience.

This paper presents both theoretical and empirical evidence about a probability distribution which describes the behavior of share price changes. Osborne's Brownian motion theory of share price changes is modified to account for the changing variance of the share market. This produces a scaled t-distribution which is an excellent fit to series of share price indices. This distribution is the only known simple distribution to fit changes in share prices. It provides a far better fit to the data than the stable Paretian, compound process, and normal distributions.

Investment is an activity of managing sources of funds with the goal of increasing profits within a certain period of time. The number of investors in the capital market, especially stock investments continue to increase. Stock movements result in returns that investors can obtain. Randomly fluctuating share prices make it difficult for investors to forecast share prices. This research helps investors in forecasting stock price movements based on PT. Gudang Garam Tbk. (GGRM) for the period 2022. This research aims to determine the level accuracy of the Geometric Brownian Motion (GBM) and Autoregressive Integrated Moving Average (ARIMA) methods in forecasting stock price movements. The accuracy level of the Mean Absolute Percentage Error (MAPE) for the GBM method is 1.68% and the ARIMA method forecasting results is 3.37%. The MAPE value of both methods is less than 10\%, so it can be said that both methods are best fitting and have a high level of accuracy in forecasting stock price movements. The GBM method is better at forecasting stock prices because it is more realistic for financial asset price models because it includes volatility in the model.

Stock portfolio investment gives investors the opportunity to engage in diversification among stocks which helps to achieve higher financial risk-adjusted return. A properly diversified stock portfolio should include stocks that have different economic variable such as interest rate, exchange rate and share price. It helps investors to achieve higher risk-adjusted return, nevertheless there are investors who love risk known as Risk Premium investors and Risk Averse investors who prefer lower risk. The risk value of stocks acts as an indicator in selecting stocks to be included in a portfolio. Risk value of stock portfolios is used in selecting stock portfolio that follows investors’ risk preference. This study forecast share price by using Geometric Brownian Motion and hence use the Variance Covariance to calculate Value at Risk of each stock and stock portfolio in future investment. The stock portfolios in this study are based on five sectors which are industrial-product, consumer, construction, technology, and trading and services. Thus, there are five stock portfolios that are produced for each type; Risk Averse and Risk Premium investors. Using the Value at risk of each stock portfolio, it is found that Stock portfolio from the Industrial Product Trading and Service sectors is the most suitable for Risk Averse and Risk Premium investors respectively. This study may serve as a guide for investors in creating and choosing stock portfolios for future investment based on their risk preference and also forecast its Value at Risk.

We propose a mathematical model of momentum risk-taking, which is essentially real-time risk management focused on short-term volatility. Its implementation, a fully automated momentum equity trading system, is systematically discussed in this paper. It proved to be successful in extensive historical and real-time experiments. Momentum risk-taking is one of the key components of general decision-making, a challenge for artificial intelligence and machine learning. We begin with a new mathematical approach to news impact on share prices, which models well their power-type growth, periodicity, and the market phenomena like price targets and profit-taking. This theory generally requires Bessel and hypergeometric functions. Its discretization results in some tables of bids, basically, expected returns for main investment horizons, the key in our trading system. A preimage of our approach is a new contract card game. There are relations to random processes and the fractional Brownian motion. The ODE we obtained, especially those of Bessel-type, appeared to give surprisingly accurate modeling of the spread of COVID-19.

The aim of this research is to compare the stock price model of PT. Ciputra Development Tbk. i.e. between the ABM (Arithmetic Brownian Motion) model and the GBM (Geometric Brownian Motion) model. The measures of the goodness of the model are used MAPE (Mean Absolute Percentage Error), RMSE (Root Mean Square Error) and MAE (Mean Absolute Error). The data used is stock prices from August 11, 2020 to March 26, 2021 and is used to predict stock prices for the next 59 days, namely from March 29, 2021 to June 28, 2021. Simulation studies with repetitions of B=1000 times are used to calculate MAPE, RMSE and MAE and used to test whether the results are significantly different or not (with a level of significance α = 5%). The results obtained if the ABM model is used are MAPE = 19.60 %, RMSE = 238.84 and MAE = 200.96, while when GBM is used, they are MAPE = 24.23 %, RMSE = 293.86 and MAE = 249.27. The significant difference test (t.test) of the two results showed that the two results were significantly different. As a result, in this case the ABM model is more suitable to be used for the stock price data. This research can be developed for comparison of other stock price models such as the GBM model with jump.

How fast do stock prices adjust to market efficiency? Evidence from a detrended fluctuation analysis

13 - Volatility forecasting in a tick data model

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let $$V(S_t, t)$$ be the price of an option at time t when the share price of the underlying stock is $$S_t$$ . See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond $$B_t$$ and a risky asset stock $$S_t$$ . They evolve as where r is the interest rate, $$\mu $$ is the drift, $$\sigma $$ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price $$B_t^{-1}V(S_t,t)$$ . By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure . Also, under the risk-neutral measure $$\mu =r$$ . We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.

An Artificial Neural Network (ANN) algorithms have been widely used in machine learning for pattern recognition, classifications and time series forecasting today; especially in financial applications with nonlinear and nonparametric modeling’s. The objective of this study is an attempt to develop a new hybrid forecasting approach based on back propagation neural network (BPN) and Geometric Brownian Motion (GBM) to handle random walk data patterns under the high volatility. The proposed methodology is successfully implemented in the Colombo Stock Exchange (CSE) Sri Lanka, the daily demands of the All Share Price Index (ASPI) price index from April 2009 to March 2017. The performances of the model are evaluated based on the best two forecast horizons of 75% and 85% training samples. According to the empirical results, 85% training samples have given highly accurate in their testing process. Furthermore, the results confirmed that the proposed hybrid methodology always gives the best performances under the high volatility forecasting compared to the separate traditional time series models.