We discuss the class of “Quadratic Normal Volatility” (QNV) models, which have drawn much attention in the financial industry due to their analytic tractability and flexibility. We characterize these models as those that can be obtained from stopped Brownian motion by a simple transformation and a change of measure that depends only on the terminal value of the stopped Brownian motion. This exp...

An available method of modeling and predicting the economic time series is the use of stochastic differential equations, which are often determined as jump-diffusion stochastic differential equations in financial markets and underlier economic dynamics. Besides the diffusion term that is a geometric Brownian model with Wiener random process, these equations contain a jump term that follows Pois...

Effect of nonlinear thermal radiation on the unsteady magnetohydrodynamic slip flow of Casson fluid between parallel disks in the presence of thermophoresis and Brownian motion effects are investigated numerically. A similarity transformation is employed to reduce the governing partial differential equations into ordinary differential equations. Further, Runge-Kutta and Newton’s methods are ado...

This article introduces mixing theorems, which offer both a theoretical and computational approach to certain advanced option models. Before explaining them, we first review a little background about option pricing theory. The Black-Scholes-Merton family of models is a wellknown and sensible starting framework for understanding option prices. The framework relies on the assumption that the unde...

This paper deals with a new methodology to evaluate the real operating options embedded in a manufacturing system investment. In a single product framework, the demand is assumed as the main source of uncertainty, therefore as a stochastic variable following a Geometric Brownian Motion (GBM). Then, focusing on the real option to expand the capacity at a certain time in the future, we have devel...

This article deals with the study of the two-dimensional mixed convection magnetohydrodynamic (MHD) boundary layer of stagnation-point flow over a stretching vertical plate in porous medium filled with a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis in the presence of thermal radiation. The skin-friction coefficient, Nusselt number an...

A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condit...

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in ge...

The research considers the properties of a number of statistical measures of volatility, extending from the common standard deviation metric to less widely used range-based measures. Prior research in this field, which has typically featured the use of data series generated by Monte Carlo simulation within the theoretical framework of Geometric Brownian Motion, has tended towards the conclusion...

We prove the existence of equilibrium in a continuous-time nance model; our results include the case of dynamically incomplete markets as well as dynamically complete markets. In addition, we derive explicitly the stochastic process describing securities prices. The price process depends on the risk-aversion characteristics of the utility function, as well as on the presence of additional sourc...