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

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.

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Asian Option Pricing with Transaction Costs and Dividends under the Fractional Brownian Motion Model

Journal of Applied Mathematics ◽

10.1155/2014/652954 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Yan Zhang ◽

Di Pan ◽

Sheng-Wu Zhou ◽

Miao Han

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Brownian Motion ◽

Transaction Costs ◽

Fractional Brownian Motion ◽

Asian Option ◽

Partial Differential ◽

No Arbitrage ◽

Geometric Average ◽

Brownian Motion Model

The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper. The partial differential equation satisfied by the option’s value is presented on the basis of no-arbitrage principle and fractional formula. Then by solving the partial differential equation, the pricing formula and call-put parity of the geometric average Asian option with dividend payment and transaction costs are obtained. At last, the influences of Hurst index and maturity on option value are discussed by numerical examples.

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

International Journal of Financial Engineering ◽

10.1142/s2424786316500031 ◽

2016 ◽

Vol 03 (01) ◽

pp. 1650003 ◽

Cited By ~ 9

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.

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