Fractional Brownian Motion Model Research Articles

Valuation of Reset Option with Multiple Reset Features under Mixed Fractional Brownian Motion Model

Valuation of Reset Option with Multiple Reset Features under Mixed Fractional Brownian Motion Model

Research on effect of anode microstructures on mass transfer and electrochemical reaction in SOFCs based on a fractional Brownian motion model

Research on effect of anode microstructures on mass transfer and electrochemical reaction in SOFCs based on a fractional Brownian motion model

Geometric Asian power option pricing with transaction cost under the geometric fractional Brownian motion with [formula omitted] sources of risk in fuzzy environment

In this paper, we obtain an explicit formula to calculate the geometric Asian power option price with floating strike price and transaction cost under the fractional geometric Brownian motion model with w sources of risk and fuzzy parameters. First, by considering the Leland and Kabanov theorems, we derive a non-linear PDE with the transaction cost formula to obtain the option price. Then, using the Green function find a closed form solution for the PDE and achieve the price of the option under different amounts of the model and option parameters. Next, we consider the model’s parameters as fuzzy numbers and acquire a general formula to obtain intervals for the option price under different belief degrees and power option parameter amounts.

Option pricing in the illiquid markets under the mixed fractional Brownian motion model

This paper deals with the option pricing in the illiquid markets under the mixed fractional geometric Brownian motion model with jump process. We propose a general long memory financial model, where its featuring parameters are related to demand and supply by showing also the existence of some restrictions on them. Moreover, by using the delta Hedging strategy and replicating portfolio, we obtain an integro partial differential equation (PIDE) for the option price which is solved by the spectral numerical method with suitable diagonal functions and an infinite series. In particular, by using some operational matrices and Gauss–Hermite quadrature rule, we derive a linear system of algebraic equations solved by a standard collocation method. Moreover, we study the existence and uniqueness of the solution of PIDE and prove the convergence of the numerical scheme. The applicability and efficiency of the collocation method are shown on some nontrivial numerical examples.

A Noisy Fractional Brownian Motion Model for Multiscale Correlation Analysis of High-Frequency Prices

We analyze the multiscale behaviors of high-frequency intraday prices, with a focus on how asset prices are correlated over different timescales. The multiscale approach proposed in this paper is designed for the analysis of high-frequency intraday prices. It incorporates microstructure noise into the stochastic price process. We consider a noisy fractional Brownian motion model and illustrate its various statistical properties. This leads us to introduce new latent correlation and noise estimators. New numerical algorithms are developed for model estimation using empirical high-frequency data. For a collection of stocks and exchange-traded funds, examples are provided to illustrate the relationship between multiscale correlation and sampling frequency as well as the evolution of multiscale correlation over time.

Quantile Hedging in the complete financial market under the mixed fractional Brownian motion model and the liquidity constraint

This article proposes a pricing framework for European option that utilizes a Quantile hedging strategy in a complete financial market. The methodology involves applying the long memory Geometric Brownian motion model, utilizing the generalized mixed fractional Girsanov theorem, and incorporating relevant findings related to quasi-conditional expectation. The first step in this framework is to derive the option price formula using the equivalent martingale measure. This approach enables investors to use the entire option price without any restrictions to hedge their contingent claim. Then, we add the initial wealth condition to hedge and determine the option price with the Quantile hedging strategy. To do this, we find the maximum probability of success in complete risk hedging for different amounts of model parameters and then use the probability values to determine the option price. Moreover, we apply the minimum variance algorithms and the error correction method to find the hedge ratio. Finally, we illustrate the hedging strategy and provide numerical results that assist investors in determining the required capital for covering the option risk under market data. This calculation takes into account both the level of risk tolerance and the anticipated return on investment.

Pricing European Options under a Fuzzy Mixed Weighted Fractional Brownian Motion Model with Jumps

This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. By converting the partial differential equation (PDE) into a Cauchy problem, we derive explicit solutions for both European call options and European put options. The figures and tables demonstrating the effectiveness of the results highlight the suitability of the fuzzy mixed weighted fractional Brownian motion with jump model for option pricing.

Order estimation for a fractional Brownian motion model of glucose control

When a subject is at rest and meals have not been eaten for a relatively long time (e.g. during the night), presumably near-constant, zero-order glucose production occurs in the liver. Glucose elimination from the bloodstream may be proportional to glycemia, with an apparently first-order, linear elimination rate. Besides glycemia itself, unobserved factors (insulinemia, other hormones) may exert second and higher order effects. Random events (sleep pattern variations, hormonal cycles) may also affect glycemia. The time-course of transcutaneously, continuously measured glycemia (CGM) thus reflects the superposition of different orders of control, together with random system error. The problem may be formalized as a fractional random walk, or fractional Brownian motion. In the present work, the order of this fractional stochastic process is estimated on night-time CGM data from one subject.

An IMEX-based approach for the pricing of equity warrants under fractional Brownian motion models

In this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models. doi: 10.1017/S1446181123000159

Propagating Particle Tracking Uncertainty Defined by Fuzzy Numbers in Spatially Variable Velocity Fields

Accurate prediction of the trajectories of material drifting on the ocean surface is critical for risk assessment and responses to environmental emergencies. Prediction of these trajectories is subject to uncertainty arising from a number of sources, with a primary source being uncertainty in the modelled ocean surface currents and winds used as input to the trajectory model. This article presents a fuzzy number-based algorithm for propagating uncertainty through a particle tracking scheme in a time- and space-varying velocity field. The performance of the algorithm was tested by applying it to idealized, analytical velocity fields and scoring the results against the analytical solution. Both epistemic and aleatoric uncertainty were considered and combined using a fractional Brownian motion model for temporal autocorrelation of the uncertainty. In the evaluation of the algorithm, sensitivity was quantified with respect to parameters such as timestep size, resolution of the forcing velocity field, spatial and temporal gradients in the forcing, and resolution of the applied uncertainty. Parameter values optimizing uncertainty representation and computational cost were identified. The applied uncertainty was found to evolve in agreement with classical relative dispersion relationships.

Towards a Better Understanding of Fractional Brownian Motion and Its Application to Finance

The aim of this work is to first build the underlying theory behind fractional Brownian motion and applying fractional Brownian motion to financial market. By incorporating the Hurst parameter into geometric Brownian motion in order to characterize the long memory among disjoint increments, geometric fractional Brownian motion model is constructed to model S &P 500 stock price index. The empirical results show that the fitting effect of fractional Brownian motion model is better than ordinary Brownian motion.

Multiscale Volatility Analysis for Noisy High-Frequency Prices

We present a multiscale analysis of the volatility of intraday prices from high-frequency data. Our multiscale framework includes a fractional Brownian motion and microstructure noise as the building blocks. The proposed noisy fractional Brownian motion model is shown to possess a variety of volatility behaviors suitable for intraday price processes. Algorithms for numerical estimation from time series observations are then introduced, with a new Hurst exponent estimator proposed for the noisy fractional Brownian motion model. Using real-world high-frequency price data for a collection of U.S. stocks and ETFs, we estimate the parameters in the noisy fractional Brownian motion and illustrate how the volatility varies over different timescales. The Hurst exponent and noise level also exhibit an intraday pattern whereby the the noise ratio tends to be higher near market close.

Comparison of GBM, GFBM and MJD Models in Malaysian Rubber Prices Forecasting

This research studies three mathematical models, namely geometric Brownian motion (GBM), geometric fractional Brownian motion (GFBM) model which was developed by adding the Hurst parameter to GBM to characterize the long-memory phenomenon, and Merton jump-diffusion (MJD) model which captures shocks via GBM. This study sets out to forecast Malaysia rubber prices for the six months period beginning in January 2022 and ending in June 2022, which involves four main steps; calculating the logarithmic return of rubber prices; estimating the parameters for forecasting the rubber prices using the three models; simulating the rubber prices using the GBM, GFBM and MJD models via Monte Carlo simulation; and computing the mean absolute percentage errors (MAPE) and forecast accuracy. Simulation results show that the MJD model is the most accurate model in forecasting the rubber prices.

Variational inference of fractional Brownian motion with linear computational complexity.

We introduce a simulation-based, amortized Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. In the first step a graph neural network is trained on simulated data to learn optimized low-dimensional summary statistics of the random walk. In the second step an invertible neural network generates the posterior distribution of the parameters from the learned summary statistics using variational inference. We apply our method to infer the parameters of the fractional Brownian motion model from single trajectories. The computational complexity of the amortized inference procedure scales linearly with trajectory length, and its precision scales similarly to the Cramér-Rao bound over a wide range of lengths. The approach is robust to positional noise, and generalizes to trajectories longer than those seen during training. Finally, we adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.

AN IMEX-BASED APPROACH FOR THE PRICING OF EQUITY WARRANTS UNDER FRACTIONAL BROWNIAN MOTION MODELS

AbstractIn this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models.

A stochastic model to reproduce the star formation history of individual galaxies in hydrodynamic simulations

ABSTRACT The star formation history (SFH) of galaxies is critical for understanding galaxy evolution. Hydrodynamical simulations enable us to precisely reconstruct the SFH of galaxies and establish a link to the underlying physical processes. In this work, we present a model to describe individual galaxies’ SFHs from three simulations: TheThreeHundred, Illustris-1, and TNG100-1. This model divides the galaxy SFH into two distinct components: the ‘main sequence’ and the ‘variation’. The ‘main sequence’ part is generated by tracing the history of the SFR − M* main sequence of galaxies across time. The ‘variation’ part consists of the scatter around the main sequence, which is reproduced by fractional Brownian motions. We find that: (1) the evolution of the main sequence varies between simulations; (2) fractional Brownian motions can reproduce many features of SFHs; however, discrepancies still exist; and (3) the variations and mass-loss rate are crucial for reconstructing the SFHs of the simulations. This model provides a fair description of the SFHs in simulations. On the other hand, by correlating the fractional Brownian motion model to simulation data, we provide a ’standard’ against which to compare simulations.

Variable annuities valuation under a mixed fractional Brownian motion environment with jumps considering mortality risk

AbstractPricing a variable annuity (VA) is traditionally based on models with standard Brownian motion (BM). The assumption of independent increments in the BM is not always realistic as the price process of the risky assets exhibit a long‐range dependency. To model such assets, a fractional Brownian motion (FBM) with Hurst parameter is an appropriate model. However, due to arbitrage in an FBM model and jumps in stock returns, we consider in this work a mixed FBM (MFBM) model with jumps and evaluate variable annuities with different riders. We analyze the proposed model numerically to see the impact of mortality risk. We perform a comparison between eight stochastic models to obtain the best‐suited mortality model, for a dataset of the US male population. Finally, we obtain the price of VA guarantees using the forecasted values from the fitted mortality model.

Pricing Formula for European Option in Regime-Switching Mixed Fractional Brownian Motion Model with Jumps

Pricing Formula for European Option in Regime-Switching Mixed Fractional Brownian Motion Model with Jumps

Passive tracer advection in the equatorial Pacific region: statistics, correlations and a model of fractional Brownian motion

Abstract. Evaluating passive tracer advection is a common tool to study flow structures, particularly Lagrangian trajectories ranging from molecular scales up to the atmosphere and oceans. Here we report on numerical experiments in the region of the tropical Pacific (20∘ S–20∘ N), where 6600 tracer parcels are advected from a regular initial configuration (along a meridional line at 110∘ W between 15∘ S and 15∘ N) during periods of 1 year for 25 years altogether. We exploit AVISO surface flow fields and solve the kinematic equation for passive tracer movement in the 2D advection tests. We demonstrate that the strength of the advection defined by mean monthly westward displacements of the tracer clouds exhibit surprisingly large inter- and intra-annular variabilities. Furthermore, an analysis of cross-correlations between advection strength and the El-Niño and Southern Oscillation (SOI) indices reveal a significant anticorrelation between advection intensity and ONI (the Oceanic Niño Index) and a weaker positive correlation with SOI, both with a time lag of about 3 months (the two indices are strongly anticorrelated near real time). The statistical properties of advection (time-dependent mean squared displacement and first passage time distribution) suggest that the westward-moving tracers can be mapped into a simple 1D stochastic process, namely fractional Brownian motion. We fit the model parameters and show by numerical simulations of the fractional Brownian motion model that it is able to reproduce the observed statistical properties of the tracers' trajectories well. We argue that a traditional explanation based on the superposition of ballistic drift and a diffusion term yields different statistics and is incompatible with our observations.

Real-time welding condition monitoring by roughness information extracted from surface images

Monitoring the friction stir welding is an important part of intelligent systems. In this paper, we designed welding experiments based on the single factor variable method and studied the roughness of the weld surface in detail. By analysing the variation in the roughness, the theoretical formulas of roughness Ra and Rsm are obtained. The deviation between the measured value and the theoretical value of Ra can be used to evaluate the welding condition. An industrial camera was used to capture the texture images of the weld surface and process them to extract features. Based on the fractional Brownian motion model, the fractal dimension of statistical significance was obtained from the power spectrum of the gray distribution in texture images. The results show that the extracted fractal dimension can accurately reflect the changes in roughness Ra. This research is helpful to realize welding condition monitoring based on machine vision systems.