Riesenauswahl an Markenqualität. Folge Deiner Leidenschaft bei eBay! Über 80% neue Produkte zum Festpreis; Das ist das neue eBay. Finde Brownian Motion Heute bestellen, versandkostenfrei The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic processes. Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion. Brownian Motion in Finance F ive years before Einstein's miracle year paper, a young French mathematician named Louis Bachelier described a process very similar to that eventually described by Einstein, albeit in the context of asset prices in financial markets

Brownian Motion GmbH Bleichstrasse 55 DE-60313 Frankfurt am Main Phone: +49 (0)69 8700 50 940 Fax: +49 (0)69 8700 50 968 E-Mail: info @ brownianmotion. eu Repräsentanz Schweiz Tödistr. 60 CH-8002 Zürich Phone: +41 (0)44 283 610 * Brownian motion is a must-know concept*. They are heavily used in a number of fields such as in modeling stock markets, in physics, biology, chemistry, quantum computing to name a few. Additionally..

Fractional Brownian motion (fBm) was first introduced within a Hilbert space framework by Kolmogorov [1], and further studied and coined the name 'fractional Brownian motion' in the 1968 paper by Mandelbrot and Van Ness [2]. It has been widely used in various scientific fields, most notability in hydrology as first suggested in [3] Brownian motion is a simple continuous stochastic process that is widely used in physics and finance for modeling random behavior that evolves over time. Examples of such behavior are the random movements of a molecule of gas or fluctuations in an asset's price Brownian motion is furthermore Markovian and a martingale which represent key properties in finance. Brownian motion was first introduced by Bachelier in 1900. Samuelson then used the exponential of a Brownian motion (geometric Brownian motion) to avoid negativity for a stock price model Brownian motion or pedesis is the term used in physics to describe the random motion of particles (such as specs of dust) suspended in a fluid (such as a liquid or gas) resulting from collisions of such particles with the fast-moving molecules of the fluid Brownian motions have unbound variation. This means that if the sign of all negative gradients were switched to positive, then $B$ would hit infinity in an arbitrarily short time period. Brownian motions are continuous. Although Brownian motions are continuous everywhere, they are differentiable nowhere. Essentially this means that a Brownian motion has fractal geometry

Brownian motion, or pedesis (from Ancient Greek: πήδησις /pɛ̌ːdɛːsis/ leaping), is the random motion of particles suspended in a medium (a liquid or a gas). [2] This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain underlying Brownian motion and could drop in value causing you to lose money; there is risk involved here. 1.1 Lognormal distributions If Y ∼ N(µ,σ2), then X = eY is a non-negative r.v. having the lognormal distribution; called so because its natural logarithm Y = ln(X) yields a normal r.v. X has density f(x) = (1 xσ √ 2π e −(ln(x)−µ)

- Geometric Brownian motion (GBM) is a stochastic process. It is probably the most extensively used model in financial and econometric modelings. After a brief introduction, we will show how to apply GBM to price simulations. A few interesting special topics related to GBM will be discussed. Although a little math background is required, skipping the [
- Geometric Brownian motion is simply the exponential (this's the reason that we often say the stock prices grows or declines exponentially in the long term) of a Brownian motion with a constant drift. Therefore, you may simulate the price series starting with a drifted Brownian motion where the increment of the exponent term is a normal distribution
- Brownian Motion in Financial Practice: Normality and Path-Continuity of Returns. Path-continuity and normality of returns form the common core of the portfolio theory, option pricing theory, and fundamental asset pricing methods. Portfolio theory is considered a central factor in the making of finance into a scientific discipline (MacKenzie 2006)

- // Brownian Motion in Finance // Want more help from David Moadel? Contact me at davidmoadel @ gmail . comSubscribe t... About Press Copyright Contact us Creators Advertise Developers Terms.
- istration, Helleveien 30, N-5045, Bergen, Norway Abstrac
- Financial Brownian Motion March 27, 2018 • Physics 11, s36 Using data on the activity of individual financial traders, researchers have devised a microscopic financial model that can explain macroscopic market trends
- A stochastic process, S, is said to follow Geometric
**Brownian****Motion**(GBM) if it satisfies the stochastic differential equation where For an arbitrary starting value , the SDE has the analytical solutio - Brownian motion is used as building block in models for a number different applications e.g. financial markets, turbulence, seismology, fatigue, neuronal activity and hydrology. Usually these models are formulated as stochastic differential equations

- Für unseren Mandanten - einen Beratungsexperten im Bereich Finance und Insurance - suchen wir für den Standort München einen (Senior) Consultant/Business Analysten (m/w/x) Unser Kunde verbindet aktuelle Anforderungen und Neuerungen u.a. aus der Finanz- und Versicherungsbranche mit den modernsten IT-Technologien und verhilft seinen Kunden mit innovativen Lösungen zum Erfolg
- On the Generalized Brownian Motion and its Applications in Finance Finance Research Group. On the Generalized Brownian Motion and its Applications in Finance Esben P. H˝gy Aarhus School of Business Per Frederiksenz Nordea Markets Daniel Schiemertx Universit at Stuttgart November 200
- the geometric Brownian motion model. This model is one of the most mathematical models used in asset price modelling. According to the geometric Brownian motion model the future price of financial stocks has a lognormal probability distribution and their future value therefore can be estimated with a certain level of confidence
- Brownian Motion and Stochastic Calculus The modeling of random assets in nance is based on stochastic processes, which are families (X t) t2Iof random variables indexed by a time intervalI. In this chapter we present a description of Brownian motion and a construction of the associated It^o stochastic integral. 4.1 Brownian Motion
- But before going into Ito's calculus, let's talk about the property of Brownian motion a little bit because we have to get used to it. Suppose I'm using it as a model of a stock price. So I'm using--use Brownian motion as a model for stock price--say, daily stock price. The market opens at 9:30 AM. It closes at 4:00 PM. It starts at some price.

finance cran monte-carlo stock-market derivatives option option-pricing sde stochastic-differential-equations jump-diffusion stochastic-processes black-scholes computational-finance brownian-motion Updated Jun 7, 202 Brownian Motion and Einstein's explanation of it based on kinetic theory of gase • Brownian motion is nowhere diﬀerentiable despite the fact that it is continuous everywhere. • It is self-similar; i.e., any small piece of a Brownian motion tra-jectory, if expanded, looks like the whole trajectory, like fractals [5]. • Brownian motion will eventually hit any and every real value, no matter how large or how negative Setting Up Brownian Motion and Mean Reverting Time Series. I will use some time series equations to illustrate the various points relating to Brownian motion time series and mean reverting time series. Begin by understanding that the measured volatility is the standard deviation of the rate of return or the rate of change in price

Brownian motion is named after the Scottish Botanist Robert Brown, who first observed that pollen grains move in random directions when placed in water. An illustration describing the random movement of fluid particles (caused by the collisions between these particles) is provided below Some Properties of Brownian Motion (1) (Invariance under scaling.) If (W t) t≥0 is a standard Brownian motion, then so is (cW t/c2) t≥0 for any c 6= 0. This says that the process ( W t) t≥0 has the fractal-like property, that a typical path of the process will look similar if it is scaled up. For instance, the process (10W t/100) should look similar to the original process (W t) t≥ Brownian motion is used in finance to model short-term asset price fluctuation. Suppose the price (in dollars) of a barrel of crude oil varies according to a Brownian motion process; specifically, suppose the change in a barrel's price \(t\) days from now is modeled by Brownian motion \(B(t)\) with \(\alpha = .15\) So it's very interesting to see, that a concept as important as Brownian motion, which is used throughout the physical sciences and engineering was actually introduced by Bachelier in a financial context. Wiener, in the 1920s, was the first to show that it actually exists as a well defined mathematical entity The usual model for the time-evolution of an asset price S ( t) is given by the geometric Brownian motion, represented by the following stochastic differential equation: d S ( t) = μ S ( t) d t + σ S ( t) d B ( t) Note that the coefficients μ and σ, representing the drift and volatility of the asset, respectively, are both constant in this model

** Using geometric Brownian motion in tandem with your research, you can derive various sample paths each asset in your portfolio may follow**. This will give you an entire set of statistics associated with portfolio performance from maximum drawdown to expected return. There are uses for geometric

t,t ≥ 0) is a Brownian motion starting from 0 iﬀ (a) (B t) is a Gaussian process; (b) EB t = 0 and EB sB t = s∧t, for all s,t ≥ 0; (c) With probability one, t → B t is continuous. This deﬁnition is often useful in checking that a process is a Brownian motion, as in the transformations described by the following examples based on ( Brownian motion, binomial trees and Monte Carlo simulations. R Example 5.1 (Brownian motion): R commands to create and plot an approximate sample path of an arithmetic Brownian motion for given α and σ, over the time interval [0,T] and with n points

Once, during a job interview, I was asked to explain how to construct a Brownian motion. Let's make a quick recap of the theory behind it and then I will tell you how I missed the job, but also what I learned from it. Brownian Motion is the typical tool in finance to build stochastic diffusio be Brownian motion, that is, the increments must be normally distributed. This is analogous to the Poisson counting process which is the unique simple counting process that has both stationary and independent increments: the stationary and independent increments property forces the increments to be Poisson distributed

BROWNIAN MOTION 1. INTRODUCTION 1.1. Wiener Process: Deﬁnition. Deﬁnition 1. A standard (one-dimensional) Wiener process (also called Brownian motion) is a stochastic process fW tg t 0+ indexed by nonnegative real numbers twith the following properties: (1) W 0 = 0. (2)With probability 1, the function t!W tis continuous in t. (3)The process fW t The first application of Brownian motion in finance can be traced back to Louis Bachelier in 1900 in his doctoral thesis titled Theorie de la speculation. This chapter aims at providing the necessary background on Brownian motion to understand the Black‐Scholes‐Merton model and how to price and manage (hedge) options in that model

That is, fractional Brownian motion means that a security's price moves seemingly randomly, but with some external event sending it in one direction or the other. Farlex Financial Dictionary. © 2012 Farlex, Inc The Brownian motion of visible particles suspended in a fluid led to one of the first accurate determinations of the mass of invisible molecules. The name giver of Brownian motion, however, was completely unaware of molecules in their present meaning, namely compounds of atoms from the Periodic System. The Scottis In the second part of the past decade, the usage of fractional Brownian motion for financial models was stuck. The favorable time-series properties of fractional Brownian motion exhibiting long-range dependence came along with an apparently insuperable shortcoming: the existence of arbitrage ** Generate the Geometric Brownian Motion Simulation**. To create the different paths, we begin by utilizing the function np.random.standard_normal that draw $(M+1)\times I$ samples from a standard Normal distribution. To ensure that the mean is 0 and the standard deviation is 1 we adjust the generated values with a technique called moment matching Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. Geometric Brownian Motion 9:35. Taught By. Martin Haugh. Co-Director, Center for Financial Engineering. Garud Iyengar. Professor. Try the Course for Free

For standard Brownian motion, density function of X(t) is given by f. t (x) = 1 2ˇt. e. x. 2 =2t. 1.2 Hitting Time The rst time the Brownian motion hits a is called as hitting time. To show that PfT. a <1g= 1 and E(T. a) = 1for a6= 0 Consider, X(t) Normal(0;t) Let, T. a =First time the Brownian motion process hits a. When a>0, we will compute. Fractional Brownian motion in finance and queueing Tommi Sottinen Academic dissertation To be presented, with the premission of the Faculty of Science of the University of Helsinki, for public criticism in Auditorium XIV of the Main Building of the University, on March 29th, 2003, at 10 o'clock a.m. Department of Mathematics Faculty of Scienc Brownian motion process. The most important stochastic process is the Brownian motion or Wiener process.It was first discussed by Louis Bachelier (1900), who was interested in modeling fluctuations in prices in financial markets, and by Albert Einstein (1905), who gave a mathematical model for the irregular motion of colloidal particles first observed by the Scottish botanist Robert Brown in 1827 The BrownianMotion(X 0, Mu, Sigma) defines an n-dimensional Brownian motion with drift Mu and covariance Sigma. This process is defined by the SDE dX t = Μ dt + B dW ** This open access textbook is a precise and intuitive introduction to modern financial theory for Business and Economics Ph**.D. students. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but keeping the necessary mathematical formalism

Brownian motion, any of various physical phenomena in which some quantity is constantly undergoing small, random fluctuations. It was named for the Scottish botanist Robert Brown, the first to study such fluctuations (1827). If a number of particles subject to Brownian motion are present in a give Brownian motion is an important part of Stochastic Calculus. When you start developing quantitative trading strategies, pretty soon you will hit upon Brownian Motion. If you are interested in designing and developing algorithmic trading strategies than you should know stochastic calculus and Brownian motion. It will take some effort to learn stochastic calculus and Brownian [

Brownian motion is the random movement of particles in a fluid due to their collisions with other atoms or molecules. Brownian motion is also known as pedesis, which comes from the Greek word for leaping.Even though a particle may be large compared to the size of atoms and molecules in the surrounding medium, it can be moved by the impact with many tiny, fast-moving masses Abstract. In 1900, the mathematician Louis Bachelier proposed in his dissertation Théorie de la Spéculation to model the dynamics of stock prices as an arithmetic Brownian motion (the mathematical definition of Brownian motion had not yet been given by N. Wiener) and provided for the first time the exact definition of an option as a financial instrument fully described by its terminal. Simulating stock price dynamics using Geometric Brownian Motion. Thanks to the unpredictability of financial markets, simulating stock prices plays an important role in the valuation of many derivatives, such as options. Due to the aforementioned randomness in price movement, these simulations rely on stochastic differential equations (SDE) The Wiener process or Brownian motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. It is the most common stochastic building block for random walks in finance. Example. Pollen in water, smoke in a room, pollution in a river, are all examples of Brownian motion Brownian motion Finance Assignment & Project Help. Secret Techniques for Brownian Motion That Only a Few People Know About The Appeal of Brownian Motion. The algorithm has been successfully implemented and tested on a lot of examples involving different kinds of robots with diverse degrees of freedom

Spezifikationen: mu=drift factor [Annahme von Risikoneutralitaet] sigma: volatility in % T: time span dt: lenght of steps S0: Stock Price in t=0 W: Brownian Motion with Drift N[0,1] ''' T=1 mu=0.025 sigma=0.1 S0=20 dt=0.01 Steps=round(T/dt) t=(arange(0, Steps)) x=arange(0, Steps) W=(standard_normal(size=Steps)+mu*t)### standard brownian motion### X=(mu-0.5*sigma**2)*dt+(sigma*sqrt(dt)*W) ###geometric brownian motion#### y=S0*math.e**(X) plot(t,y) show( 2 Brownian Motion (with drift) Deﬂnition. A Brownian Motion (with drift) X(t) is the solution of an SDE with constant drift and diﬁusion coe-cients dX(t) = dt+¾dW(t); with initial value X(0) = x0. By direct integration X(t) = x0 +t+¾W(t) and hence X(t) is normally distributed, with mean x0 +t and variance ¾2t. Its density function i Brownian motion Finance Assignment & Project Help. Whatever They Told You About Brownian Motion Is Dead WrongAnd Here's Why What Brownian Motion Is - and What it Is Not . The motion is currently called Brownian movement. It is known as the Brownian motion, discovered by means of a scientist named Brown Quantitative Finance Interview Questions. Menu. Widgets. Search. Brownian Motion Stochastic Calculus: Brownian Motion. Round 1: Investment Bank Quantitative Research. Question 1: Name the three (3) properties of a standard Brownian Motion Brownian motion Brownian Motion is a continuous Stochastic process named in honor of Norbert Wiener. It is one of the best know Leavy Processes..

The Brownian Motion in Finance: An Epistemological Puzzle. Topoi, 2019. Christian Walter. Download PDF. Download Full PDF Package. This paper. A short summary of this paper. 37 Full PDFs related to this paper. READ PAPER. The Brownian Motion in Finance: An Epistemological Puzzle. Download BROWNIAN MOTION 1. INTRODUCTION 1.1. Wiener Process: Deﬁnition. Deﬁnition 1. A standard (one-dimensional) Wiener process (also called Brownian motion) is a stochastic process {Wt}t0+ indexed by nonnegative real numbers t with the following properties: (1) W0 =0. (2) The process {Wt}t0 has stationary, independent increments Part II consists of the articles themselves: [a] Sottinen, T. (2001) Fractional Brownian motion, random walks and binary market models. Finance Stoch. 5, no. 3, 343--355 Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references ** Fractional Brownian motion Fractional Brownian motion Z is a continu-ous and centered Gaussian process with sta-tionary increments and variance IEZ2 t = t 2H**. The parameter H allows us to model the statistical long-range dependence of the log-returns. In ﬁnancial modeling it is assumed that 1 2 < H < 1. Replace W with Z and consider the followin

Christian Bender, Lauri Viitasaari, Fractional Brownian Motion in Financial Modeling, Wiley StatsRef: Statistics Reference Online, 10.1002/9781118445112, (1-5), (2014). Wiley Online Library Zhidong Guo, Hongjun Yuan, Pricing European option under the time-changed mixed Brownian-fractional Brownian model, Physica A: Statistical Mechanics and its Applications, 10.1016/j.physa.2014.03.032, 406. Download PDF Abstract: A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs

Geometric Brownian motion (GBM) models allow you to simulate sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time GBM stochastic processes. Specifically, this model allows the simulation of vector-valued GBM processes of the for The Brownian motion is at the core of mathematical domains such as stochastic calculus and the theory of stochastic processes, but it is also central in applied fields such as quantitative finance, ecology, and neuroscience. In this recipe, we will show how to simulate and plot a Brownian motion in two dimensions

The textbook is excellent for economists and financial economists who want to understand a little deeper in the Brownian motion with this soft introduction. (Weiping Li, zbMATH 1426.91005, 2020) --This text refers to the paperback edition Fractional Brownian motion as a model in finance. 2001. Tommi Sottinen. Download PDF. Download Full PDF Package. This paper. A short summary of this paper. 37 Full PDFs related to this paper. READ PAPER. Fractional Brownian motion as a model in finance. Download Brownian motion is another widely-used random process. It has been used in engineering, finance, and physical sciences. It is a Gaussian random process and it has been used to model motion of particles suspended in a fluid, percentage changes in the stock prices, integrated white noise, etc. Figure 11.29 shows a sample path of Brownain motion 5.1 Expectation of a Geometric Brownian Motion In order to nd the expected asset price, a Geometric Brownian Motion has been used, which expresses the change in stock price using a constant drift and volatility ˙as a stochastic di erential equation (SDE) according to [5]: (dS(t) = S(t)dt+ ˙S(t)dW(t) S(0) = s (2

Brownian Motion has numerous applications like Physics, Engineerging, Finance, Economics, etc. Most notably, Albert Einstein used Brownian Motion in his work to prove the existence of atoms In regard to simulating stock prices, the most common model is geometric Brownian motion (GBM). GBM assumes that a constant drift is accompanied by random shocks Last Updated on Sun, 20 Dec 2020 | Brownian Motion Depository institutions are financial intermediaries that accept deposits. They include commercial banks (or simply banks), savings and loan associations (S&Ls), savings banks, and credit unions

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs Suppose R(t)=W(t) is a simple Brownian motion, what does $\int_0^t R(s)ds$ mean? Is it a Lebesgue integral? Or is it an Ito's integral? How to interpret it intuitively? This is from chapter 5 of Shreve's stochastic calculus for finance, equation 5.2.17 on page 215 Sure more or less. I don't think grains of pollen (video) are going to move around in a solid (at least not on the time scale of a PhD project), but some point. Geometric Brownian motion, and other stochastic processes constructed from it, are often used to model population growth, financial processes (such as the price of a stock over time), subject to random noise

Brownian Motion in the Stock Market 147 (NYSE) transaction for a given day. He is told that these data consti-tute a sample of approximately 1000 from some unknown population, together with some of their more important attributes or variables, eleven in all. The fact that these eleven were the most important, out of a muc Matlab → Simulation → Brownian Motion. The change in a variable following a Brownian motion during a small period of time is given by. where has a standardized normal distribution with mean 0 and variance 1. And, the change in the value of from time 0 to is the sum of the changes in in time intervals of length , where. That is Contact Department of Mathematics. David Rittenhouse Lab. 209 South 33rd Street Philadelphia, PA 19104-6395 Email: math@math.upenn.edu Phone: (215) 898-8178 & 898-8627 Fax: (215) 573-4063. Penn WebLogi ** Browse other questions tagged probability brownian-motion finance or ask your own question**. Featured on Meta Stack Overflow for Teams is now free for up to 50 users, forever. 27 votes · comment · stats Related. 12. Given particle undergoing Geometric Brownian Motion, want to. Creates and displays geometric Brownian motion (GBM) models, which derive from the cev (constant elasticity of variance) class. Geometric Brownian motion (GBM) models allow you to simulate sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time GBM stochastic processes

This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains **Brownian** **motion**, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field Brownian motion is the erratic, random movement of microscopic particles in a fluid, as a result of continuous bombardment from molecules of the surrounding medium. Whereas, diffusion is the movement of a substance from an area of high concentration to an area of low concentration Mathematically, the Brownian motion is a particular Markov continuous stochastic process. The Brownian motion is at the core of mathematical domains such as stochastic calculus and the theory of stochastic processes, but it is also central in applied fields such as quantitative finance, ecology, and neuroscience Now then, geometric Brownian motion is used in financial markets. It is a very similar model to the Brownian motion used in physics, hence the same name! In this case, the genral trend we use is the overall trend of data - e.g. a company's share value might be increasing (on average) by 5%/year to Brownian motion W: dSt = St(¹dt + ¾dWt);S0 > 0: The bond price is Bt = ert. Parameters ¹ 2 IR, r;¾ 2 IR + supposed to be known. Traditionally one assumes that there are no dividends, no transaction costs, same inter-est rate r for lending and saving on the bond and no limitations on short-selling of the stock.

It's easy to construct Brownian motion with drift and scaling from a standard Brownian motion, so we don't have to worry about the existence question. Relation to standard Brownian motion. Suppose that \(\bs{Z} = \{Z_t: t \in [0, \infty)\}\) is a standard Brownian motion, and that \(\mu \in \R\) and \(\sigma \in (0, \infty)\) A Geometric Brownian Motion simulator is one of the first tools you reach for when you start modeling stock prices. In particular, it's a useful tool for building intuition about concepts such as options pricing. Leveraging R's vectorisation tools, we can run tens of thousands of simulations in no time at all Finally, I actually get around to defining Brownian motion. Below, I state the result: Theorem (Dyson Brownian Motion): Let , and be the spectrum of eigen-values of the Hermitian matrix valued process . Then, we have: (8) for all , where and are independent Brownian motion processes Brownian motion in action. This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions As discussed by [2], a Geometric Brownian Motion (GBM) model is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion also known as Wiener process [10]. GBM is important in the modeling financial process mathematically