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Probability Theory
Published in Hugo D. Junghenn, Principles of Analysis, 2018
The phrase Brownian motion, in the classical sense, refers to a phenomenon discovered in 1827 by the Scottish botanist Robert Brown, who observed that microscopic particles suspended in a fluid (liquid or gas) exhibited highly irregular motion characterized by seemingly independent random movements. Later it was determined that this motion resulted from collisions of the particles with molecules in the ambient fluid. In 1905, Albert Einstein gave a physical interpretation of Brownian motion. A rigorous mathematical model of Brownian motion was developed in the 1920s by Norbert Wiener. The model, known as a Brownian motion process or a Wiener process, has come to play an indispensable role in many areas of pure and applied mathematics. For example, in pure mathematics the process has spawned the study of continuous time martingales and stochastic calculus. In applied mathematics the Wiener process is used as mathematical model of “white noise.” In mathematical finance, geometric Brownian motion is the fundamental component in the Black-Scholes model for option pricing (discussed in §18.9). In the current section we consider a one-dimensional version of (mathematical) Brownian motion, which may be viewed as a model for the motion of a Brownian particle projected onto a vertical axis.
Diffusion processes, stochastic differential equations and applications
Published in Henry C. Tuckwell, Elementary Applications of Probability Theory, 2018
As early as 1900, the French mathematician Bachelier proposed the modelling of stock market fluctuations using a Wiener process. It was subsequently recognized that because the prices of shares could not be negative, a modification to the simple Wiener process was necessary. One solution proposed was the adoption of so-called geometric Brownian motion or the geometric Wiener process to represent certain financial entities. Thus Y = ln(X) should be a Wiener process with drift - so that X = eY might represent a stock price. We then have Y∈(−∞, ∞) but X ∈(0, ∞).
Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Application: An important application of stochastic calculus is in quantitative finance, in which asset prices are often assumed to follow stochastic differential equations. In the Black-Scholes model, prices are assumed to follow geometric Brownian motion.
Experience with valuation methods for the creation of real options enabling diversity of nuclear fuel supply
Published in Journal of Nuclear Science and Technology, 2022
Marcus Seidl, Andreas Wensauer, Wolfgang Faber
In financial markets options are in widespread use to hedge price fluctuations of the underlying assets [6]. In the simplest cases the asset price movements are assumed to behave according to a stochastic process driven by geometric Brownian motion [7] and options are valued according to the Black-Scholes option formula [8]. An option may be out-of-the money at the time of underwriting – but due to the volatility of the price of the underlying and due to the finite time until expiry it still can have a positive inherent value. Because of the stochastic nature of the asset price movements and because of the non-linear relationship between option payoff and asset price, the expected payoff can be positive. This behavior is very similar for contracts which give an electric utility the right but not the obligation to order additional or alternative fuel assemblies at some point in the future.
Probabilistic life cycle cash flow forecasting with price uncertainty following a geometric Brownian motion
Published in Structure and Infrastructure Engineering, 2021
M. (Martine) van den Boomen, H. L. M (Hans) Bakker, D. F. J (Daan) Schraven, M. J. C. M (Marcel) Hertogh
The current research proposes to include market price escalation and its uncertainty in conventional practice for probabilistic life cycle cost forecasting of infrastructures with long service lives. The method of choice for uncertainty modelling of market prices is a Geometric Brownian Motion because it follows the general notion of compounding inflation or deflation for prices, additionally accounts for uncertainty and is easily applicable in practice. The uncertainty parameters representing the drift and volatility are obtained from registered historic price indices. This approach removes some subjectivity from the current practice for probabilistic cash flow forecasting in which uncertainty distributions are based on the assumption of a triangular distribution and its upper bound, mean and lower bound are estimated based on expert-judgement. The advantage of choosing a GBM for uncertainty modelling of market prices instead of a triangular distribution is that it builds on real data and accounts for time-variability. The cone of uncertainty widens in time. Moreover, multiple GBMs for prices are easily included in the current Monte Carlo Simulation which also combines the other uncertainties among which volume and timing.
Supply contracts for critical and strategic materials of high volatility and their ramifications for supply chains
Published in The Engineering Economist, 2020
K. Jo Min, Laura Lilienkamp, John Jackman, Chung-Hsiao Wang
A useful mathematical way of modeling price volatility is geometric Brownian motion (GBM). A seminal paper by Schwartz and Smith (2000) provides more background including a positive growth rate, and a measure of volatility, Black and Scholes (1973) used GBM to model underlying asset price for the Nobel prize-winning Black Scholes European option pricing formula. This mathematical model currently has numerous variations and modifications such as mean-reverting GBM and diffusion-jump processes (see e.g., Anderson, 2007) to more accurately reflect empirical observations.