Optimal Management of a Variable Annuity Invested in a Black–Scholes Market Driven by a Multidimensional Fractional Brownian Motion

2010 ◽  
Vol 29 (1) ◽  
pp. 61-77
Author(s):  
Alexandros A. Zimbidis
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Optimal Management ◽  
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 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.

ARBITRAGE IN FRACTAL MODULATED BLACK–SCHOLES MODELS WHEN THE VOLATILITY IS STOCHASTIC

2005 ◽  
Vol 08 (03) ◽  
pp. 283-300 ◽  
Author(s):  
ERHAN BAYRAKTAR ◽  
H. VINCENT POOR
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Time Change ◽  
Quadratic Variation ◽  
Model Driven ◽  
Black Scholes ◽  
Arbitrage Strategy ◽  
Adapted Process

In this paper an arbitrage strategy is constructed for the modified Black–Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant "volatility" and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black–Scholes models does not carry the same meaning as in the classic Black–Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black–Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.

scholarly journals Itô's formula with respect to fractional Brownian motion and its application

1996 ◽  
Vol 9 (4) ◽  
pp. 439-448 ◽  
Author(s):  
W. Dai ◽  
C. C. Heyde
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Itô’S Formula ◽  
Black Scholes Model ◽  
Itô's Formula ◽  
Ito’S Formula ◽  
Itô Calculus ◽  
Black Scholes ◽  
Special Case ◽  
Ito's Formula

Fractional Brownian motion (FBM) with Hurst index 1/2<H<1 is not a semimartingale. Consequently, the standard Itô calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2<H<1. In this paper we derive a version of Itô's formula for fractional Brownian motion. Then, as an application, we propose and study a fractional Brownian Scholes stochastic model which includes the standard Black-Scholes model as a special case and is able to account for long range dependence in modeling the price of a risky asset. This article is dedicated to the memory of Roland L. Dobrushin.

scholarly journals THE VALUATION OF EUROPEAN OPTION UNDER SUBDIFFUSIVE FRACTIONAL BROWNIAN MOTION OF THE SHORT RATE

2020 ◽  
Vol 23 (04) ◽  
pp. 2050022
Author(s):  
FOAD SHOKROLLAHI
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Pricing Model ◽  
European Option ◽  
Put Options ◽  
Merton Model ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Pricing Formula

In this paper, we propose an extension of the Merton model. We apply the subdiffusive mechanism to analyze European option in a fractional Black–Scholes environment, when the short rate follows the subdiffusive fractional Black–Scholes model. We derive a pricing formula for call and put options and discuss the corresponding fractional Black–Scholes equation. We present some features of our model pricing model for the cases of [Formula: see text] and [Formula: see text].

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