Asian Option Pricing with Monotonous Transaction Costs under Fractional Brownian Motion

Geometric-average Asian option pricing model with monotonous transaction cost rate under fractional Brownian motion was established. The method of partial differential equations was used to solve this model and the analytical expressions of the Asian option value were obtained. The numerical experiments show that Hurst exponent of the fractional Brownian motion and transaction cost rate have a significant impact on the option value.


Introduction
In 1989, Peters [1] firstly proposed that fractional Brownian motion could be used to describe the changes of asset prices.In 2000, the theory of stochastic integral about fractional Brownian motion was studied by Duncan et al. [2], and fractional Itô's formula and Girsanov theorem under the fractional Brownian motion were derived.The fractional Itô's integral had been further developed by Biagini et al. [3] for  ≥ 1/2.Equivalent definition of fractional Itô's integral was introduced by Alos et al. [4] and Bender [5].Necula [6] utilized the knowledge of fractal geometry and deduced the Black-Scholes option pricing formula under fractional Brownian motion, which was of great significance to the development of option pricing with fractional Brownian motion.The transaction cost is an important factor affecting the option pricing.Many scholars had studied the pricing problems of contingent claim with transaction costs.Leland [7] groundbreakingly proposed that a modified volatility should be applied to solving the problem of hedging error brought by transaction costs in Black-Scholes model.Barles and Soner [8] assumed that the investor's preference satisfied exponential utility function and provided a more complex model.The Black-Scholes option pricing model with transaction costs was given by Amster et al. [9].Liu and Chang [10] and Wang et al. [11] studied the European option pricing with transaction costs under the fractional Brownian motion.The pricing studies for the Asian option mostly are based on the standard Brownian motion, but the time-varying of Asian option pricing model under fractional Brownian motion had not been studied systematically.
Based on the previous references, the geometric Asian option pricing model with monotonous transaction rates under the fractional Brownian motion was presented, and the analytic expression of the Asian option value was derived.The influence of Hurst exponent and transaction cost on the Asian option value was discussed through the numerical calculations.This paper's outline is as follows.In Section 2, we studied the geometric-average Asian option pricing model under the fractional Brownian motion.The closed-form solution of the pricing model was presented in Section 3. In Section 4, the numerical examples were given.Section 5 serves as the conclusion of the whole paper.

Geometric-Average Asian Option Pricing
Model for a Fractional Brownian Motion under Monotonous Transaction Cost Rate If  = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion.
In this paper we will drive a geometric-average Asian option pricing model under the following assumptions.
(i) The price   of the underlying stock satisfies where   is the expected yields,  is the stock price volatility, and   () is the fractional Brownian motion with Hurst exponent.
(ii) The risk-free interest rate   is a deterministic function of time .
(iii) Suppose V  shares of the underlying stock are bought (V  > 0) or sold (V  < 0) at the price   ; then the transaction cost is given by ℎ(V  )|V  |  in either buying or selling; here ℎ(V  ) = −V  (,  > 0) is a monotone transaction cost rate.
(iv) The expected return rate of the portfolio equals the risk-free interest rate.
(v) The portfolio is revised every , where  is a finite and fixed small time step.
Let  = (,   ,   ) be the value of the geometric-average Asian call option on the time , where   =  (1/) ∫  0 ln    is the geometric average of the underlying asset price in the time period of [0, ].
Construct a portfolio Π  : long position of a Geometricaverage Asian call option and short position of Δ  shares of underlying asset.Then the value of the portfolio at current time  is In the period [, +], the change in the value of the portfolio is where Owing to we have hence we have By the assumption (iv) we have And then Owing to then Choosing Δ  = /  and substituting it into (10), the following formula can be obtained: Substituting these into (13), we get the following conclusions.

Option Pricing Formulas
Theorem 3. Supposing the underlying asset prices   satisfy (2), then at time  the value (,   ,   ) of Geometric-average Asian call option with transaction costs with expiration date  and exercise price  is as follows: where which satisfies the conditions () = () = () = 0; then we have Substituting ( 23) into (21), we have Let and combined with terminal conditions () = () = () = 0, we can get Thus (21) becomes According to the theory of classical heat conduction equation solution, we have Let By variable restored, we obtain that  where and the remaining symbols accord with Theorem 3.

Numerical Experiments
In this section, the influence of monotone transaction rate parameters and the Hurst exponent on Asian option value will be discussed through applying MATLAB software.The values of the parameters of geometric-average Asian options are assumed as follows: With the option pricing formulas (33) presented, the value of the option can be calculated.Figures 1 and 2 give the relationships between the price of the underlying assets and the value of Asian call option and put option with different Hurst exponent.From the figures, Hurst exponent is inversely proportional to the value of Asian option.Figures 3 and 4 demonstrate the changes of Asian call option with the stock price under the different parameter  and parameter .We can draw such a conclusion: the option value increases with the parameter  increasing and decreases with the parameter  increasing.This is mainly because transaction cost rate is a decreasing function of  and an increasing function of .

Discussions and Conclusions
In this paper, the problem of Asian option pricing with monotonous transaction cost rate under fractional Brownian motion was studied by using the portfolio technology and no arbitrage principle, and the pricing model was established.This model was solved by the method of partial differential equations, and the analytical expressions of the Asian option value were obtained.The numerical experiments showed that Hurst exponent of the fractional Brownian motion and transaction cost rates have a significant impact on the option value.

Figure 3 :
Figure 3: Asian put option pricing corresponding to the different .

Figure 4 :
Figure 4: Asian put option pricing corresponding to the different .