This paper deals with the problem of pricing European currency options in the mixed fractional Brownian environment. Both the pricing formula and the mixed fractional partial differential equation for European call currency options are obtained. Some Greeks and the estimator of volatility are also provided. Empirical studies and simulation results confirm the theoretical findings and show that the mixed fractional Brownian pricing model is a reasonable one.

Pricing currency options in the mixed fractional Brownian motion

This paper deals with the problem of pricing equity warrants in a mixed fractional Brownian environment. Based on the quasi-conditional expectation and the Fourier transform, we present the pricing model for equity warrants. Moreover, a hybrid intelligent algorithm, which is based on the Genetic Algorithm, is employed to solve the nonlinear optimization problem. The performance of our model and the proposed algorithm have been illustrated with some numerical examples.

Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm

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.

http://www.prpi.upm.edu.my/IPR/Copyright/UPM6254_7981_copyright.pdf

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

In this paper, we discuss the valuation of equity warrants in the geometric fractional Brownian environment based on the equilibrium condition Using the conditional expectation we present a fractional pricing model for equity warrants and analyze the influence of the Hurst parameter Then we propose an optimization procedure to obtain the valuation of equity warrants Some numerical examples are given to demonstrate the pricing results by comparing different pricing models Furthermore, we provide an empirical study to show how to apply our model in realistic contexts, and these comparative results of different pricing models show that the pricing model proposed in this paper matches the actual price quite well

The valuation of equity warrants in a fractional Brownian environment

While the arbitrage opportunity in the Black–Scholes model driven by fractional Brownian motion has a long history, the arbitrage strategy in the Black–Scholes model driven by general fractional Gaussian processes is in its infancy. The development of stochastic calculus with respect to fractional Gaussian processes allowed us to study such models. In this paper, following the idea of Shiryaev (1998), an arbitrage strategy is constructed for the Black–Scholes model driven by fractional Gaussian processes, when the stochastic integral is interpreted in the Riemann–Stieltjes sense. Arbitrage opportunities in some fractional Gaussian processes, including fractional Brownian motion, sub-fractional Brownian motion, bi-fractional Brownian motion, weighted-fractional Brownian motion and tempered fractional Brownian motion, are also investigated.

Arbitrage with fractional Gaussian processes

This paper deals with the problem of discrete-time option pricing by the mixed Brownian–fractional Brownian model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. In particular, the minimal pricing c min ( t , s t ) of an option under transaction costs is obtained, which shows that timestep δ t and Hurst exponent H play an important role in option pricing with transaction costs. In addition, we also show that there exists fundamental difference between the continuous-time trade and discrete-time trade and that continuous-time trade assumption will result in underestimating the value of a European call option.

Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model

A model for option pricing of fractional version of the Merton model with ‘Hurst exponent’ H being in [ 1 / 2 , 1 ) is established with transaction costs. In particular, for H ∈ ( 1 / 2 , 1 ) the minimal price C min ( t , S t ) of an option under transaction costs is obtained, which displays that the timestep δ t and the ‘Hurst exponent’ H play an important role in option pricing with transaction costs.

Scaling and long-range dependence in option pricing III: A fractional version of the Merton model with transaction costs

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Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model

Scaling and long-range dependence in option pricing III: A fractional version of the Merton model with transaction costs