A risky asset model with strong dependence through fractal activity time

1999 ◽  
Vol 36 (04) ◽  
pp. 1234-1239 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Brownian Motion ◽  
Strong Dependence ◽  
Geometric Brownian Motion ◽  
Risky Asset ◽  
Data Sets ◽  
Activity Time ◽  
Scaling Properties ◽  
Model Based ◽  
Black Scholes Model ◽  
Black Scholes

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

A risky asset model with strong dependence through fractal activity time

1999 ◽  
Vol 36 (4) ◽  
pp. 1234-1239 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Brownian Motion ◽  
Strong Dependence ◽  
Geometric Brownian Motion ◽  
Risky Asset ◽  
Data Sets ◽  
Activity Time ◽  
Scaling Properties ◽  
Model Based ◽  
Black Scholes Model ◽  
Black Scholes

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

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.

scholarly journals Alternative methods to derive the Black-Scholes-Merton equation

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Geometric Brownian Motion ◽  
Alternative Methods ◽  
Fixed Income ◽  
Trivial Case ◽  
Wiener Processes ◽  
Black Scholes ◽  
Basic Ideas

We investigate the derivation of option pricing involving several assets following the Geometric Brownian Motion (GBM). First, we propose some derivations based on the basic ideas of the assets. Next, we consider the trivial case where we have n assets. Finally, we consider different drifts, volatilities and Wiener processes but now from n stochastic assets taking into account a fixed-income.

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.

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