Asian Option Pricing with Monotonous Transaction Costs under Fractional Brownian Motion

Abstract

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.

1. 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 𝐻≥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.

2. Geometric-Average Asian Option Pricing Model for a Fractional Brownian Motion under Monotonous Transaction Cost Rate

Let (Ω,𝐹,𝑃) be a complete probability space carrying a fractional Brownian motion {𝐵_𝐻(𝑡),𝑡≥0} with Hurst exponent 𝐻(0<𝐻<1), that is, a continuous, centered Gaussian process with covariance function [12] 𝐶ov𝐵_𝐻(𝑡),𝐵_𝐻(𝑠)=12|𝑡|^2𝐻+|𝑠|^2𝐻−|𝑡−𝑠|^2𝐻,𝑠,𝑡∈𝑅. (1)

If 𝐻=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 𝑆_𝑡 of the underlying stock satisfies 𝑑𝑆_𝑡=𝑟_𝑡𝑆_𝑡𝑑𝑡+𝜎𝑆_𝑡𝑑𝐵_𝐻(𝑡), (2) where 𝑟_𝑡 is the expected yields, 𝜎 is the stock price volatility, and 𝐵_𝐻(𝑡) is the fractional Brownian motion with Hurst exponent.(ii)The risk-free interest rate 𝑟_𝑡 is a deterministic function of time 𝑡.(iii)Suppose 𝑣_𝑡 shares of the underlying stock are bought (𝑣_𝑡>0) or sold (𝑣_𝑡<0) at the price 𝑆_𝑡; then the transaction cost is given by ℎ(𝑣_𝑡)|𝑣_𝑡|𝑆_𝑡 in either buying or selling; here ℎ(𝑣_𝑡)=𝑎−𝑏𝑣_𝑡(𝑎,𝑏>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 𝛿𝑡, where 𝛿𝑡 is a finite and fixed small time step.

Let 𝑉=𝑉(𝑡,𝐽_𝑡,𝑆_𝑡) be the value of the geometric-average Asian call option on the time 𝑡, where 𝐽_𝑡=𝑒^{(1/𝑡)∫𝑡0ln𝑆𝜏𝑑𝜏} is the geometric average of the underlying asset price in the time period of [0,𝑡].

Construct a portfolio Π_𝑡: long position of a Geometric-average Asian call option and short position of Δ_𝑡 shares of underlying asset. Then the value of the portfolio at current time 𝑡 is Π_𝑡=𝑉_𝑡−Δ_𝑡𝑆_𝑡. (3)

In the period [𝑡,𝑡+𝛿𝑡], the change in the value of the portfolio is 𝛿Π_𝑡=𝛿𝑉_𝑡−Δ_𝑡𝛿𝑆_𝑡−𝑎−𝑏𝜈_𝑡𝜈_𝑡𝑆_𝑡=𝜕𝑉𝜕𝑡+𝐻𝜎²𝑆²_𝑡𝑡^2𝐻−1𝜕²𝑉𝜕𝑆²_𝑡𝛿𝑡+𝜕𝑉𝜕𝑆_𝑡−Δ_𝑡𝛿𝑆_𝑡+𝜕𝑉𝜕𝐽_𝑡𝛿𝐽_𝑡−𝑎−𝑏𝜈_𝑡𝜈_𝑡𝑆_𝑡, (4) where 𝑣_𝑡=𝜕𝑉𝜕𝑆_{𝑡+𝛿𝑡}−𝜕𝑉𝜕𝑆_𝑡=𝜕²𝑉𝜕𝑆²_𝑡𝛿𝑆_𝑡+𝜕²𝑉𝜕𝑆_𝑡𝜕𝐽_𝑡𝛿𝐽_𝑡()=𝜕+𝑂𝛿𝑡²𝑉𝜕𝑆²_𝑡𝜎𝑆_𝑡𝛿𝐵_𝐻(𝑡)+𝑂(𝛿𝑡). (5) Owing to 𝐸𝑎𝜈_𝑡𝑆_𝑡=2𝜋𝑎𝜎𝑆²_𝑡(𝛿𝑡)^𝐻𝜕²𝑉𝜕𝑆²_𝑡+𝑂(𝛿𝑡), (6) we have 𝐸𝑏𝜈_𝑡²𝑆_𝑡=𝑏𝜎²𝑆³_𝑡𝜕²𝑉𝜕𝑆²_𝑡²Var𝛿𝐵_𝐻(𝑡)−𝐸𝛿𝐵_𝐻(𝑡)²+𝑂𝛿𝑡²=𝑏𝜎²𝑆³_𝑡𝜕²𝑉𝜕𝑆²_𝑡²(𝛿𝑡)^2𝐻+𝑂𝛿𝑡²; (7) hence we have 𝐸𝑎−𝑏𝜈_𝑡𝜈_𝑡𝑆_𝑡=𝐸𝑎𝜈_𝑡𝑆_𝑡−𝐸𝑏𝜈_𝑡²𝑆_𝑡=2𝜋𝑎𝜎𝑆²_𝑡(𝛿𝑡)^𝐻𝜕²𝑉𝜕𝑆²_𝑡−𝑏𝜎²𝑆³_𝑡𝜕²𝑉𝜕𝑆²_𝑡²(𝛿𝑡)^2𝐻. (8) By the assumption (iv) we have 𝐸𝑑Π_𝑡=𝑟Π_𝑡𝑑𝑡. (9) And then 𝑟𝑉−Δ_𝑡𝑆_𝑡=𝛿𝑡𝜕𝑉𝜕𝑡+𝐻𝜎²𝑆²_𝑡𝑡^2𝐻−1𝜕²𝑉𝜕𝑆²_𝑡𝛿𝑡+𝜕𝑉𝜕𝐽_𝑡𝛿𝐽_𝑡−2𝜋𝑎𝜎𝑆²_𝑡(𝛿𝑡)^𝐻𝜕²𝑉𝜕𝑆²_𝑡+𝑏𝜎²𝑆³_𝑡𝜕²𝑉𝜕𝑆²_𝑡²(𝛿𝑡)^2𝐻. (10) Owing to 𝐽_𝑡=𝑒^{(1/𝑡)∫𝑡0ln𝑆𝜏𝑑𝜏}, (11) then 𝛿𝐽_𝑡=𝐽𝛿𝑡^𝑡ln𝑆_𝑡/𝐽_𝑡𝑡. (12) Choosing Δ_𝑡=𝜕𝑉/𝜕𝑆_𝑡 and substituting it into (10), the following formula can be obtained: 𝜕𝑉𝜕𝑡+𝐻𝜎²𝑆²_𝑡𝑡^2𝐻−1𝜕²𝑉𝜕𝑆²_𝑡+𝑟𝑆_𝑡𝜕𝑉𝜕𝑆_𝑡−2𝜋𝑎𝜎𝑆²_𝑡(𝛿𝑡)^𝐻−1𝜕²𝑉𝜕𝑆²_𝑡+𝑏𝜎²𝑆³_𝑡𝜕²𝑉𝜕𝑆²_𝑡²(𝛿𝑡)^2𝐻−1+𝜕𝑉𝜕𝐽_𝑡𝐽_𝑡ln𝑆_𝑡/𝐽_𝑡𝑡−𝑟𝑉=0. (13) Let Le(𝐻)=2𝜋𝑎𝜎(𝛿𝑡)^𝐻−1,𝜎²=2𝜎²𝐻𝑡^2𝐻−1−Le(𝐻)sign𝑉_𝑆𝑆+𝑏𝑆_𝑡𝑉_𝑆𝑆(𝛿𝑡)^2𝐻−1. (14) Substituting these into (13), we get the following conclusions.

Theorem 1. If the underlying asset price 𝑆_𝑡 satisfies fractional Brownian motion (2), then the value of Geometric-average Asian call option with monotonous transaction cost rate at the time 𝑡(0≤𝑡≤𝑇) satisfies the following mathematical model: 𝜕𝑉+1𝜕𝑡2𝜎²𝑆²_𝑡𝜕²𝑉𝜕𝑆²_𝑡+𝑟𝑆_𝑡𝜕𝑉𝜕𝑆_𝑡+𝜕𝑉𝜕𝐽_𝑡𝐽_𝑡ln𝑆_𝑡/𝐽_𝑡𝑡𝑉−𝑟𝑉=0,𝑇,𝐽_𝑇,𝑆_𝑇=𝐽_𝑇−𝐾⁺. (15)

Remark 2. For a single European option with long position, the yield at expiration date is (𝐽_𝑇−𝐾)⁺ or (𝐾−𝐽_𝑇)⁺. Because they are convex functions, 𝑉_𝐽𝐽>0. Due to 𝐽_𝑡=𝑒^{(1/𝑡)∫𝑡0ln𝑆𝜏𝑑𝜏}, thus 𝑉_𝑆𝑆>0. Then (14) can be represented as 𝜎²=2𝜎²𝐻𝑡^2𝐻−1−2𝜋𝑎𝜎(𝛿𝑡)^𝐻−1+𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1. (16)

3. Option Pricing Formulas

Theorem 3. Supposing the underlying asset prices 𝑆_𝑡 satisfy (2), then at time 𝑡 the value 𝑉(𝑡,𝐽_𝑡,𝑆_𝑡) of Geometric-average Asian call option with transaction costs with expiration date 𝑇 and exercise price 𝐾 is as follows: 𝑉𝑡,𝐽_𝑡,𝑆_𝑡=𝐽^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡(1/𝑇)}𝑒^{𝑟∗(𝑇−𝑡)−∫𝑇𝑡𝑟𝜃𝑑𝜃+(𝜎∗2/2)(𝑇2𝐻−𝑡2𝐻)}×𝑁𝑑₁−𝐾𝑒^{−∫𝑇𝑡𝑟𝜃𝑑𝜃}𝑁𝑑₂, (17) where 𝑑₁=ln𝐽^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}/𝐾+𝑟^∗(𝑇−𝑡)+𝜎^∗2𝑇^2𝐻−𝑡^2𝐻𝜎^∗𝑇^2𝐻−𝑡^2𝐻,𝑑₂=𝑑₁−𝜎^∗𝑇^2𝐻−𝑡^2𝐻,𝑟^∗=∫^𝑇_𝑡𝑟_𝜃((𝑇−𝜃)/𝑇)𝑑𝜃−𝜎𝑇−𝑡²𝑇^2𝐻−𝑡^2𝐻2()+𝑇−𝑡𝐻𝜎²𝑇^2𝐻+1−𝑡^2𝐻+1𝑇()+12𝐻+1)(𝑇−𝑡2𝐿𝑒(𝐻)𝜎²𝑇−𝑡𝑇−12𝜎²𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1𝑇−𝑡𝑇,𝜎^∗𝑇=𝜎1−4𝐻^2𝐻+1−𝑡^2𝐻+1𝑇(2𝐻+1)𝑇^2𝐻−𝑡^2𝐻+𝐻𝑇^2𝐻+2−𝑡^2𝐻+2𝑇²(𝐻+1)𝑇^2𝐻−𝑡^2𝐻(𝐻)×()−2𝐿𝑒𝑇−𝑡³3𝑇²𝑇^2𝐻−𝑡^2𝐻+2𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1(𝑇−𝑡)³3𝑇²𝑇^2𝐻−𝑡^2𝐻1/2𝐿𝑒(𝐻)=2𝜋𝑎𝜎(𝛿𝑡)^𝐻−1,𝑁(𝑥)=^𝑥_−∞1√𝑒2𝜋^−(𝑡2/2)𝑑𝑡. (18)

Proof. Let 𝜉_𝑡=1𝑇𝑡ln𝐽_𝑡+(𝑇−𝑡)ln𝑆_𝑡,𝑉𝑡,𝐽_𝑡,𝑆_𝑡=𝑈𝑡,𝜉_𝑡; (19) then 𝜕𝑉=𝐽𝜕𝑡ln_𝑡/𝑆_𝑡𝑇𝜕𝑈𝜕𝜉_𝑡+𝜕𝑈,𝜕𝑡𝜕𝑉𝜕𝑆_𝑡=𝑇−𝑡𝑇𝑆_𝑡𝜕𝑈𝜕𝜉_𝑡,𝜕²𝑉𝜕𝑆²_𝑡=𝑇−𝑡𝑇𝑆_𝑡²𝜕²𝑈𝜕𝜉²_𝑡−𝑇−𝑡𝑇𝑆²_𝑡𝜕𝑈𝜕𝜉_𝑡,𝜕𝑉𝜕𝐽_𝑡=𝑡𝑇𝐽_𝑡𝜕𝑈𝜕𝜉_𝑡. (20) Hence model (15) can be transformed into 𝜕𝑈+𝑟𝜕𝑡^𝑡−𝜎²2𝑇−𝑡𝑇𝜕𝑈𝜕𝜉_𝑡+𝜎²2𝑇−𝑡𝑇²𝜕²𝑈𝜕𝜉²_𝑡−𝑟_𝑡𝑈𝜉𝑈=0,_𝑇,𝑇=𝑒^𝜉𝑇−𝐾⁺. (21) Let 𝜏=𝛾(𝑡),𝜂_𝜏=𝜉_𝑡(𝑡),𝑊+𝛼𝜏,𝜂_𝜏=𝑈𝑡,𝜉_𝑡𝑒^𝛽(𝑡), (22) which satisfies the conditions 𝛼(𝑇)=𝛽(𝑇)=𝛾(𝑇)=0; then we have 𝜕𝑈𝜕𝑡=𝑒^{−𝛽(𝑡)}𝜕𝑊𝛾𝜕𝜏^′(𝑡)−𝛽^′(𝑡)𝑊+𝜕𝑊𝜕𝜂_𝜏𝛼^′(𝑡),𝜕𝑈𝜕𝜉_𝑡=𝑒^{−𝛽(𝑡)}𝜕𝑊𝜕𝜂_𝜏,𝜕²𝑈𝜕𝜉²_𝑡=𝑒^{−𝛽(𝑡)}𝜕²𝑊𝜕𝜂²_𝜏. (23)
Substituting (23) into (21), we have 𝛾^′(𝑡)𝜕𝑊+𝜎𝜕𝜏2²𝑇−𝑡𝑇²𝜕²𝑊𝜕𝜂²_𝜏+𝑟_𝑡−𝜎²2𝑇−𝑡𝑇+𝛼^′(𝑡×)𝜕𝑊𝜕𝜂_𝜏−𝑟_𝑡+𝛽^′(𝑡)𝑊=0. (24) Let 𝑟_𝑡−𝜎²2𝑇−𝑡𝑇+𝛼^′(𝑡)=0,𝑟_𝑡+𝛽^′(𝑡)𝛾=0,_′(𝑡)𝜎=−²2𝑇−𝑡𝑇², (25) and combined with terminal conditions 𝛼(𝑇)=𝛽(𝑇)=𝛾(𝑇)=0, we can get 𝛼(𝑡)=^𝑇_𝑡𝑟_𝜃𝑇−𝜃𝑇𝑑𝜃−^𝑇_𝑡12𝜎²𝑇−𝜃𝑇=𝑑𝜃_𝑇𝑡𝑟_𝜃𝑇−𝜃𝑇𝑑𝜃+𝐻𝜎²𝑇^2𝐻+1−𝑡^2𝐻+1𝑇()−𝜎2𝐻+1₂𝑇^2𝐻−𝑡^2𝐻2+12Le(𝐻)𝜎²(𝑇−𝑡)²𝑇−12𝜎²𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1(𝑇−𝑡)²𝑇𝛾(𝑡)=^𝑇_𝑡12𝜎²𝑇−𝜃𝑇²=𝐻𝑑𝜃𝑇²𝜎²𝑇^2𝐻−𝑡^2𝐻𝑇2𝐻²𝑇−2𝑇^2𝐻+1−𝑡^2𝐻+1+𝑇2𝐻+1_2𝐻+2−𝑡^2𝐻+2(𝐻)𝜎2𝐻+2−Le²(𝑇−𝑡)³3𝑇²+𝑏𝑆_𝑡𝜎²(𝛿𝑡)^2𝐻−1(𝑇−𝑡)³3𝑇²𝛽(𝑡)=^𝑇_𝑡𝑟_𝜃𝑑𝜃. (26) Thus (21) becomes 𝜕𝑊=𝜕𝜕𝜏²𝑊𝜕𝜂²_𝜏,𝑊𝜂₀,0=𝑒^𝜂0−𝐾⁺. (27) According to the theory of classical heat conduction equation solution, we have 𝑊𝜂_𝜏=1,𝜏2√𝜋𝜏^+∞_ln𝐾𝑒^𝑦−𝐾𝑒^{−((𝑦−𝜂𝜏)2/4𝜏)}𝑑𝑦=𝑒^{𝜂𝜏+𝜏}𝑁2𝜏+𝜂_𝜏−ln𝐾√𝜂2𝜏−𝐾𝑁^𝜏−ln𝐾√.2𝜏 (28) Let 𝑟^∗=∫^𝑇_𝑡𝑟_𝜃((𝑇−𝜃)/𝑇)𝑑𝜃−𝜎𝑇−𝑡²𝑇^2𝐻−𝑡^2𝐻2()+𝑇−𝑡𝐻𝜎²𝑇^2𝐻+1−𝑡^2𝐻+1𝑇()+12𝐻+1)(𝑇−𝑡2Le(𝐻)𝜎²𝑇−𝑡𝑇−12𝜎²𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1𝑇−𝑡𝑇,𝜎^∗𝑇=𝜎1−4𝐻^2𝐻+1−𝑡^2𝐻+1𝑇(2𝐻+1)𝑇^2𝐻−𝑡^2𝐻+𝐻𝑇^2𝐻+2−𝑡^2𝐻+2𝑇²(𝐻+1)𝑇^2𝐻−𝑡^2𝐻(𝐻)()−2Le𝑇−𝑡³3𝑇²𝑇^2𝐻−𝑡^2𝐻+2𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1(𝑇−𝑡)³3𝑇²𝑇^2𝐻−𝑡^2𝐻1/2; (29) then 2𝜏+𝜂_𝜏−ln𝐾√=𝐽2𝜏ln^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}/𝐾+𝑟^∗(𝑇−𝑡)+𝜎^∗2𝑇^2𝐻−𝑡^2𝐻𝜎^∗𝑇^2𝐻−𝑡^2𝐻≜𝑑₁,𝜂_𝜏−ln𝐾√=𝐽2𝜏ln^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}/𝐾+𝑟^∗(𝑇−𝑡)𝜎^∗𝑇^2𝐻−𝑡^2𝐻=𝑑₁−𝜎^∗𝑇^2𝐻−𝑡^2𝐻≜𝑑₂. (30) By variable restored, we obtain that 𝑊𝜂_𝜏=𝐽,𝜏_𝑡𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}𝑒^{𝑟∗(𝑇−𝑡)+(𝜎∗2/2)(𝑇2𝐻−𝑡2𝐻)}𝑁𝑑₁−𝐾𝑁𝑑₂. (31) So the value of geometric-average Asian call option on the time 𝑡 is 𝑉𝑡,𝐽_𝑡,𝑆_𝑡=𝑈𝜉_𝑡,𝑡=𝑊𝑒^{−𝛽(𝑡)}=𝐽^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}𝑒^{𝑟∗(𝑇−𝑡)−∫𝑇𝑡𝑟𝜃𝑑𝜃+(𝜎∗2/2)(𝑇2𝐻−𝑡2𝐻)}𝑁𝑑₁−𝐾𝑒^{−∫𝑇𝑡𝑟𝜃𝑑𝜃}𝑁𝑑₂. (32)

Corollary 4. If the riskless interest 𝑟 and the volatility 𝜎 are constant, then the values 𝑉_𝐶 and 𝑉_𝑝 of Geometric-average Asian call option and put option respectively with monotonous transaction cost at time 𝑡 with expiration date 𝑇 and exercise price 𝐾 are 𝑉_𝐶𝑡,𝐽_𝑡,𝑆_𝑡=𝐽^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}𝑒^{(𝑟∗−𝑟)(𝑇−𝑡)+(𝜎∗2/2)(𝑇2𝐻−𝑡2𝐻)}𝑁𝑑₁−𝐾𝑒^{−𝑟(𝑇−𝑡)}𝑁𝑑₂,𝑉_𝑃𝑡,𝐽_𝑡,𝑆_𝑡=−𝐽^𝑡_𝑡𝑆_𝑡^{𝑇−𝑡1/𝑇}𝑒^{(𝑟∗−𝑟)(𝑇−𝑡)+(𝜎∗2/2)(𝑇2𝐻−𝑡2𝐻)}𝑁−𝑑₁+𝐾𝑒^{−𝑟(𝑇−𝑡)}𝑁−𝑑₂, (33) where 𝑟^∗=𝑟()−𝜎2𝑇𝑇−𝑡²𝑇^2𝐻−𝑡^2𝐻2()+𝑇−𝑡𝐻𝜎²𝑇^2𝐻+1−𝑡^2𝐻+1𝑇()+12𝐻+1)(𝑇−𝑡2𝐿𝑒(𝐻)𝜎²𝑇−𝑡𝑇−12𝜎²𝑏𝑆_𝑡(𝛿𝑡)^2𝐻−1𝑇−𝑡𝑇 (34) and the remaining symbols accord with Theorem 3.

4. 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: 𝑆_𝑡=80,𝑡=0,𝑇=1,𝑟=0.05,𝜎=0.4,𝐾=80,𝐻=0.5,𝑎=0.009,𝑏=0.002,𝛿𝑡=0.02. (35) 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 𝑎 and parameter 𝑏. We can draw such a conclusion: the option value increases with the parameter 𝑏 increasing and decreases with the parameter 𝑎 increasing. This is mainly because transaction cost rate is a decreasing function of 𝑏 and an increasing function of 𝑎.

Figure 1 

Asian call option pricing corresponding to the different 𝐻.

Figure 2 

Asian put option pricing corresponding to the different 𝐻.

Figure 3 

Asian put option pricing corresponding to the different 𝑎.

Figure 4 

Asian put option pricing corresponding to the different 𝑏.

5. 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.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (2013XK03).