On the impact of the number of vanishing moments on the dependence structures of compound Poisson motion and fractional Brownian motion in multifractal time

2010 ◽  
pp. 71-101 ◽  
Author(s):  
Béatrice Vedel ◽  
Herwig Wendt ◽  
Patrice Abry ◽  
Stéphane Jaffard
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Vanishing Moments ◽  
Compound Poisson ◽  
The Impact ◽  
Dependence Structures

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

OPTION PRICING UNDER THE FRACTIONAL STOCHASTIC VOLATILITY MODEL

2021 ◽  
pp. 1-20
Author(s):  
Y. HAN ◽  
Z. LI ◽  
C. LIU
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

Abstract We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented.

CONVERGENCE OF THE WEIERSTRASS-MANDELBROT PROCESS TO FRACTIONAL BROWNIAN MOTION

Fractals ◽  
2000 ◽  
Vol 08 (04) ◽  
pp. 369-384 ◽  
Author(s):  
VLADAS PIPIRAS ◽  
MURAD S. TAQQU
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Differentiable Function ◽  
General Type ◽  
Nowhere Differentiable Function ◽  
Dependence Structures

In this work, we develop Mandelbrot's idea that Weierstrass's nowhere differentiable function can be modified and randomized to approximate fractional Brownian motion (FBM). Our approach covers the convergence of processes of a more general type and allows us to consider different dependence structures in the above randomization.

scholarly journals Pricing of Equity Indexed Annuity under Fractional Brownian Motion Model

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Lin Xu ◽  
Guangjun Shen ◽  
Dingjun Yao
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Time Series ◽  
High Water ◽  
Self Similarity ◽  
Brownian Motion Model ◽  
Pricing Problems ◽  
Market Driven ◽  
Point To Point ◽  
The Impact

Fractional Brownian motion with Hurst exponentH∈(1/2,1)is a good candidate for modeling financial time series with long-range dependence and self-similarity. The main purpose of this paper is to address the valuation of equity indexed annuity (EIA) designs under the market driven by fractional Brownian motion. As a result, this paper presents an explicit pricing expression for point-to-point EIA design and bounds for the pricing of high-water-marked EIA design. Some numerical examples are given to illustrate the impact of the parameters involved in the pricing problems.

scholarly journals Pricing of American Carbon Emission Derivatives and Numerical Method under the Mixed Fractional Brownian Motion

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yuling Wang ◽  
Jing Wang
Keyword(s):  
Brownian Motion ◽  
Numerical Method ◽  
Fractional Brownian Motion ◽  
Carbon Emission ◽  
Convergence Rates ◽  
Strongly Correlated ◽  
Mixed Fractional Brownian Motion ◽  
Financial Derivative ◽  
Volume Method ◽  
The Impact

This paper studies the pricing of American carbon emission derivatives and its numerical method under the mixed fractional Brownian motion. To capture the long memory properties such as self-similarity and long-range dependence in the price process, we proposed a model based on a fractional Black–Scholes, which is more in line with the actual characteristics of the option market. We have outlined a power penalty approach using parabolic variation inequality and linear complementarity (LCP) which arises from mixed fractional Brownian motion. In addition, we introduced a nonuniform grid-based modification of the fitted finite volume method for numerical solution. Numerically, we show the impact of Hurst exponent on the pricing and analyze the convergence rates of the proposed penalty method. In conclusion, since mfBm is a well-developed mathematical model of strongly correlated stochastic processes, this model will be an efficient model for pricing carbon financial derivative.

Based on Fractional Brownian Motion and Fourier Descriptors of the Side Face Recognition Algorithm

2014 ◽  
Vol 596 ◽  
pp. 427-432
Author(s):  
Hong Wu ◽  
Gui Xin Tang ◽  
Tao Han ◽  
Bai Hao Jie
Keyword(s):  
Brownian Motion ◽  
Face Recognition ◽  
Fractional Brownian Motion ◽  
Recognition Algorithm ◽  
Fourier Descriptors ◽  
Feature Points ◽  
The Face ◽  
Side Face ◽  
Algorithm Base ◽  
The Impact

This paper proposes a kind of the side face recognition algorithm base on Fractional Brownian motion and Fourier descriptors. This method is mainly innovating for feature extraction and making up for a shortage of the side face recognition algorithm. Firstly, by Fractional Brownian motion of Hurst index will get silhouette extracted. Then we through Fourier descriptors to obtain the desired feature points. Further study of this algorithm can solve the side of the face recognition rotation, scaling, and translation transform the impact. Comparative characteristics will make distance classifier and BP neural network classifier. This article also processes the analysis of the algorithm and the future development.

scholarly journals Lookback Option Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jiao-Jiao Sun ◽  
Shengwu Zhou ◽  
Yan Zhang ◽  
Miao Han ◽  
Fei Wang
Keyword(s):  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Asset Price ◽  
Brownian Motion Process ◽  
Proportional Transaction Costs ◽  
Cost Rate ◽  
Lookback Option ◽  
Fixed Proportion ◽  
The Impact

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland’s hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, the effectiveness of the proposed form is verified through a numerical example. Meanwhile, the impact of transaction cost rate and volatility on lookback option value is discussed.

Pricing European options and currency options by time changed mixed fractional Brownian motion with transaction costs

2016 ◽  
Vol 03 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Foad Shokrollahi ◽  
Adem Kılıçman ◽  
Marcin Magdziarz
Keyword(s):  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Discrete Time ◽  
Empirical Studies ◽  
Time Step ◽  
Currency Option ◽  
Mixed Fractional Brownian Motion ◽  
Text Model ◽  
The Impact

This study investigates a new formula for option pricing with transaction costs in a discrete time setting. The value of the financial assets is based on time-changed mixed fractional Brownian motion [Formula: see text] model. The pricing method is obtained for European call option using the time-changed [Formula: see text] model in a discrete time setting. Particularly, the minimal value [Formula: see text] of an option respect to transaction costs is obtained. Furthermore, the new model for pricing currency option is presented by utilizing the time-changed [Formula: see text] model. In addition, the impact of time step [Formula: see text], Hurst parameter H and transaction costs [Formula: see text] are also investigated, which substantiate that these parameters play a significant role in our pricing formula. Finally, the empirical studies and the simulation findings corroborate the theoretical bases and indicate the time-changed [Formula: see text] is a satisfactory model.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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