Continuous time Black–Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime

2012 ◽  
Vol 391 (3) ◽  
pp. 750-759 ◽  
Author(s):  
Jun Wang ◽  
Jin-Rong Liang ◽  
Long-Jin Lv ◽  
Wei-Yuan Qiu ◽  
Fu-Yao Ren
Keyword(s):  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Continuous Time ◽  
Black Scholes

scholarly journals Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

2017 ◽  
Vol 22 (1) ◽  
pp. 161-180 ◽  
Author(s):  
Christoph Czichowsky ◽  
Rémi Peyre ◽  
Walter Schachermayer ◽  
Junjian Yang
Keyword(s):  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Portfolio Optimisation ◽  
Shadow Prices

A note on super-replicating strategies

1994 ◽  
Vol 347 (1684) ◽  
pp. 485-494 ◽  
Keyword(s):  
Option Pricing ◽  
Transaction Costs ◽  
Continuous Time ◽  
Black Scholes ◽  
Alternative Approaches

The standard Black-Scholes option pricing methodology fails in the presence of transaction costs because portfolios that exactly replicate the option pay-off no longer exist. Several alternative approaches have been proposed; our purpose is to examine one of them which is based on the idea of ‘super-replicating’ portfolios. It is argued that this approach does not lead to a viable theory of option pricing in continuous time.

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.

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