Q-Fractional Brownian Motion in Infinite Dimensions with Application to Fractional Black–Scholes Market

2009 ◽  
Vol 27 (1) ◽  
pp. 149-175 ◽  
Author(s):  
W. Grecksch ◽  
C. Roth ◽  
V. V. Anh
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Infinite Dimensions ◽  
Black Scholes

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].

scholarly journals FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE

2003 ◽  
Vol 06 (01) ◽  
pp. 1-32 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Standard Brownian Motion ◽  
Stochastic Integration ◽  
European Option ◽  
No Arbitrage ◽  
Black Scholes ◽  
White Noise Calculus ◽  
Replicating Portfolio

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

Option pricing under multifractional Brownian motion in a risk neutral framework

2020 ◽  
Vol 38 (3) ◽  
Author(s):  
Fabrizio Di Sciorio
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Market Price ◽  
Option Price ◽  
Mathematical Framework ◽  
Call Option ◽  
Multifractional Brownian Motion ◽  
European Call Option ◽  
Black Scholes Model ◽  
Black Scholes

In this paper, we introduce a new method to compute the European Call Option price (ct) under multi-fractional Brownian motion (mBm) with deterministic Hurst function. We build a mathematical framework using a Lebovits et al. study to approximate mBm to fractional Brownian motion (fBm). As a result we obtain ct , through the simulation of the logarithmic price under mBm, using a Vasicek model for the discount factor. Finally, we compare the results with those computed with the Black Scholes model and Call market price (SPX).

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