Merton's model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence

2005 ◽  
Vol 37 (3) ◽  
pp. 585-598 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Short Range ◽  
Optimal Portfolio ◽  
Black Scholes ◽  
Market Driven

OPTIMAL CONSUMPTION AND PORTFOLIO IN A BLACK–SCHOLES MARKET DRIVEN BY FRACTIONAL BROWNIAN MOTION

2003 ◽  
Vol 06 (04) ◽  
pp. 519-536 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL ◽  
AGNÈS SULEM
Keyword(s):  
Mathematical Model ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Consumption Rate ◽  
Hurst Parameter ◽  
Optimal Consumption ◽  
Standard Brownian Motion ◽  
Power Type ◽  
Black Scholes ◽  
Market Driven

We present a mathematical model for a Black–Scholes market driven by fractional Brownian motion BH(t) with Hurst parameter [Formula: see text]. The interpretation of the integrals with respect to BH(t) is in the sense of Itô (Skorohod–Wick), not pathwise (which is known to lead to arbitrage). We find explicitly the optimal consumption rate and the optimal portfolio in such a market for an agent with utility functions of power type. When H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

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 On the Solution of Fractional Option Pricing Model by Convolution Theorem

2019 ◽  
pp. 143-157
Author(s):  
A. I. Chukwunezu ◽  
B. O. Osu ◽  
C. Olunkwa ◽  
C. N. Obi
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Long Memory ◽  
Hurst Exponent ◽  
Convolution Theorem ◽  
Pricing Model ◽  
One Dimensional ◽  
Black Scholes ◽  
Option Pricing Model

The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.

scholarly journals Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

2019 ◽  
Vol 15 (2) ◽  
pp. 81 ◽  
Author(s):  
Herry Pribawanto Suryawan
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Short Range ◽  
Probability Space ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Self Similarity ◽  
Sample Paths

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

scholarly journals Solving Black-Schole Equation Using Standard Fractional Brownian Motion

2019 ◽  
Vol 11 (2) ◽  
pp. 142
Author(s):  
Didier Alain Njamen Njomen ◽  
Eric Djeutcha
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Implied Volatility ◽  
Classical Model ◽  
Hurst Index ◽  
European Option ◽  
Black Scholes Model ◽  
Black Scholes ◽  
The Cost

In this paper, we emphasize the Black-Scholes equation using standard fractional Brownian motion BHwith the hurst index H ∈ [0,1]. N. Ciprian (Necula, C. (2002)) and Bright and Angela (Bright, O., Angela, I., & Chukwunezu (2014)) get the same formula for the evaluation of a Call and Put of a fractional European with the different approaches. We propose a formula by adapting the non-fractional Black-Scholes model using a λHfactor to evaluate the european option. The price of the option at time t ∈]0,T[ depends on λH(T − t), and the cost of the action St, but not only from t − T as in the classical model. At the end, we propose the formula giving the implied volatility of sensitivities of the option and indicators of the financial market.

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