arXiv research

Reflected geometric Brownian motion models are not arbitrage-free.

Study bounds for Brownian motion on manifolds with sticky boundary conditions.

Estimates spectral gap for Brownian motion on sticky-reflecting domains.

Upper bounds on constants for Brownian motion with sticky boundary.

We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by R. Pinsky, who treated the case in which the ambient space is flat, our result r…

Researchers calculate the Laplace transform of a geometric Brownian motion integral.

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …

Innovative extensions to option pricing models using asymmetric Brownian motion and random walk approaches.

Solves optimal liquidation problem for stock price following geometric Brownian motion.

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…

This paper develops the first method for the exact simulation of reflected Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The running time of generating exact samples of non-stationary RBM at any time $t$ is uniformly bounded by $\mathcal{O}(1/\barγ^2)$ where $\barγ$ is the average drift of…

Paper compares stock price prediction models using Heston and Geometric Brownian Motion.

We consider a limit order book, where buyers and sellers register to trade a security at specific prices. The largest price buyers on the book are willing to offer is called the market bid price, and the smallest price sellers on the book are willing to accept is called the market ask price. Market ask price is always …

The evolution of prices on ideal market is given by geometrical Brownian motion, where Gaussian white noise describes fluctuations. We study the effect of correlations introduced by a color noise.

Unified framework for Brownian motion distances on specific geometric manifolds.

New method calculates geometric Brownian motion with affine drift and its integral.

An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, sub…

Unified geometric framework for Brownian motion on various manifolds.

While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessaril…

The paper extends Merton's problem by adding benchmark tracking, finding optimal strategies.

We introduce a novel description of the dynamics of the order book of financial markets as that of an effective colloidal Brownian particle embedded in fluid particles. The analysis of a comprehensive market data enables us to identify all motions of the fluid particles. Correlations between the motions of the Brownian…

The paper proposes estimators for bid-ask spreads with and without serial dependence.

Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.

Paper offers a fast method to assess DeFi liquidation risk.

Modeling price dynamics in AMMs with fees using geometric Brownian motion.

On a multi-assets Black-Scholes economy, we introduce a class of barrier options. In this model we apply a generalized reflection principle in a context of the finite reflection group acting on a Euclidean space to give a valuation formula and the semi-static hedge.

In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. W…

Study optimal consumption with relaxed benchmarks and drawdown constraints.

We construct a binomial tree model fitting all moments to the approximated geometric Brownian motion. Our construction generalizes the classical Cox-Ross-Rubinstein, the Jarrow-Rudd, and the Tian binomial tree models. The new binomial model is used to resolve a discontinuity problem in option pricing.

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps.…

Asymptotic Laplace transform for geometric Brownian motion applied to bond pricing.

To convert standard Brownian motion $Z$ into a positive process, Geometric Brownian motion (GBM) $e^{βZ_t}, β>0$ is widely used. We generalize this positive process by introducing an asymmetry parameter $α\geq 0$ which describes the instantaneous volatility whenever the process reaches a new low. For our new process, …

Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian m…

GMMNs model cross-sectional dependence for better option pricing and simulation.

The paper prices long-term options with a reflecting barrier model.

Entropy corrections improve GBM's predictive accuracy for non-log-normal distributions.

We study how resetting affects geometric Brownian motion, showing it becomes stationary but remains non-ergodic.

Researchers develop a generalised geometric Brownian motion for better asset pricing.

Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.

The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stoc…

The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent $H>0.5$, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with $10^4$ geometric fractional Brownian motion realisations is performed as extensions of historical data. The accuracy of statistical inferen…

Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A com…

Estimates returns for dollar cost averaging using geometric Brownian motion.

Geometric Brownian motion simulates stock prices for Brazilian small caps index.

Study investigates ruin probability with random premiums and risky investments.

Develops non-Markovian couplings for sub-Riemannian Brownian motions.

Continuous time models in the theory of real options give explicit formulas for optimal exercise strategies when options are simple and the price of an underlying asset follows a geometric Brownian motion. This paper suggests a general, computationally simple approach to real options in discrete time. Explicit formulas…

We extend martingale transport results to weak martingale transport.