Pricing European options and currency options by time changed mixed fractional Brownian motion with transaction costs

2016 ◽  
Vol 03 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Foad Shokrollahi ◽  
Adem Kılıçman ◽  
Marcin Magdziarz
Keyword(s):  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Discrete Time ◽  
Empirical Studies ◽  
Time Step ◽  
Currency Option ◽  
Mixed Fractional Brownian Motion ◽  
Text Model ◽  
The Impact

This study investigates a new formula for option pricing with transaction costs in a discrete time setting. The value of the financial assets is based on time-changed mixed fractional Brownian motion [Formula: see text] model. The pricing method is obtained for European call option using the time-changed [Formula: see text] model in a discrete time setting. Particularly, the minimal value [Formula: see text] of an option respect to transaction costs is obtained. Furthermore, the new model for pricing currency option is presented by utilizing the time-changed [Formula: see text] model. In addition, the impact of time step [Formula: see text], Hurst parameter H and transaction costs [Formula: see text] are also investigated, which substantiate that these parameters play a significant role in our pricing formula. Finally, the empirical studies and the simulation findings corroborate the theoretical bases and indicate the time-changed [Formula: see text] is a satisfactory model.

scholarly journals Pricing of American Carbon Emission Derivatives and Numerical Method under the Mixed Fractional Brownian Motion

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yuling Wang ◽  
Jing Wang
Keyword(s):  
Brownian Motion ◽  
Numerical Method ◽  
Fractional Brownian Motion ◽  
Carbon Emission ◽  
Convergence Rates ◽  
Strongly Correlated ◽  
Mixed Fractional Brownian Motion ◽  
Financial Derivative ◽  
Volume Method ◽  
The Impact

This paper studies the pricing of American carbon emission derivatives and its numerical method under the mixed fractional Brownian motion. To capture the long memory properties such as self-similarity and long-range dependence in the price process, we proposed a model based on a fractional Black–Scholes, which is more in line with the actual characteristics of the option market. We have outlined a power penalty approach using parabolic variation inequality and linear complementarity (LCP) which arises from mixed fractional Brownian motion. In addition, we introduced a nonuniform grid-based modification of the fitted finite volume method for numerical solution. Numerically, we show the impact of Hurst exponent on the pricing and analyze the convergence rates of the proposed penalty method. In conclusion, since mfBm is a well-developed mathematical model of strongly correlated stochastic processes, this model will be an efficient model for pricing carbon financial derivative.

scholarly journals Lookback Option Pricing with Fixed Proportional Transaction Costs under Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jiao-Jiao Sun ◽  
Shengwu Zhou ◽  
Yan Zhang ◽  
Miao Han ◽  
Fei Wang
Keyword(s):  
Brownian Motion ◽  
Transaction Costs ◽  
Fractional Brownian Motion ◽  
Asset Price ◽  
Brownian Motion Process ◽  
Proportional Transaction Costs ◽  
Cost Rate ◽  
Lookback Option ◽  
Fixed Proportion ◽  
The Impact

The pricing problem of lookback option with a fixed proportion of transaction costs is investigated when the underlying asset price follows a fractional Brownian motion process. Firstly, using Leland’s hedging method a partial differential equation satisfied by the value of the lookback option is derived. Then we obtain its numerical solution by constructing a Crank-Nicolson format. Finally, the effectiveness of the proposed form is verified through a numerical example. Meanwhile, the impact of transaction cost rate and volatility on lookback option value is discussed.

scholarly journals Pricing Currency Option in a Mixed Fractional Brownian Motion with Jumps Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Foad Shokrollahi ◽  
Adem Kılıçman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Exchange Rate ◽  
Fractional Brownian Motion ◽  
Fractional Partial Differential Equation ◽  
Currency Option ◽  
Mixed Fractional Brownian Motion ◽  
Spot Exchange Rate ◽  
New Framework

A new framework for pricing the European currency option is developed in the case where the spot exchange rate fellows a mixed fractional Brownian motion with jumps. The jump mixed fractional partial differential equation is obtained. Some Greeks and properties volatility are discussed. Finally the numerical simulations illustrate that our model is flexible and easy to implement.

scholarly journals Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xia Zhou ◽  
Dongpeng Zhou ◽  
Shouming Zhong
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Fractional Brownian Motion ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Banach Fixed Point Theorem ◽  
Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

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