# Asian Option Pricing with Monotonous Transaction Costs under Fractional Brownian Motion

Authors: Di Pan; Shengwu Zhou; Yan Zhang; Miao Han
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 Section2, 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 {BH(t),t≥0} with Hurst exponent H(0<H<1), that is, a continuous, centered Gaussian process with covariance function [12]
(1)Cov(BH(t),BH(s))=12(|t|2H+|s|2H-|t-s|2H),s,t∈R.IfH=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 priceSt of the underlying stock satisfies
(2)dSt=rtStdt+σStdBH(t),
where rt is the expected yields, σ is the stock price volatility, and BH(t) is the fractional Brownian motion with Hurst exponent.(ii)
The risk-free interest ratert is a deterministic function of time t.(iii)
Supposevt shares of the underlying stock are bought (vt>0) or sold (vt<0) at the price St; then the transaction cost is given by h(vt)|vt|St in either buying or selling; here h(vt)=a-bvt(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 everyδt, where δt is a finite and fixed small time step.LetV=V(t,Jt,St) be the value of the geometric-average Asian call option on the time t, where Jt=e(1/t)∫0tlnSτdτ is the geometric average of the underlying asset price in the time period of [0,t].Construct a portfolioΠt: long position of a Geometric-average Asian call option and short position of Δt shares of underlying asset. Then the value of the portfolio at current time t is
(3)Πt=Vt-ΔtSt.In the period[t,t+δt], the change in the value of the portfolio is
(4)δΠt=δVt-ΔtδSt-[(a-b|νt|)|νt|St]=(∂V∂t+Hσ2St2t2H-1∂2V∂St2)δt+(∂V∂St-Δt)δSt+∂V∂JtδJt-[(a-b|νt|)|νt|St],
where
(5)vt=∂V∂St+δt-∂V∂St=∂2V∂St2δSt+∂2V∂St∂JtδJt+O(δt)=∂2V∂St2σStδBH(t)+O(δt).
Owing to
(6)E(a|νt|St)=2πaσSt2(δt)H|∂2V∂St2|+O(δt),
we have
(7)E[b|νt|2St]=bσ2St3(∂2V∂St2)2{Var(δBH(t))-[E(δBH(t))]2}+O(δt2)=bσ2St3(∂2V∂St2)2(δt)2H+O(δt2);
hence we have
(8)E[(a-b|νt|)|νt|St]=E[a|νt|St]-E[b|νt|2St]=2πaσSt2(δt)H|∂2V∂St2|-bσ2St3(∂2V∂St2)2(δt)2H.
By the assumption (iv) we have
(9)E(dΠt)=rΠtdt.
And then
(10)r(V-ΔtSt)δt=(∂V∂t+Hσ2St2t2H-1∂2V∂St2)δt+∂V∂JtδJt-2πaσSt2(δt)H|∂2V∂St2|+bσ2St3(∂2V∂St2)2(δt)2H.
Owing to
(11)Jt=e(1/t)∫0tlnSτdτ,
then
(12)δJtδt=Jtln(St/Jt)t.
Choosing  Δt=∂V/∂St and substituting it into (10), the following formula can be obtained:
(13)∂V∂t+Hσ2St2t2H-1∂2V∂St2+rSt∂V∂St-2πaσSt2(δt)H-1|∂2V∂St2|+bσ2St3(∂2V∂St2)2(δt)2H-1+∂V∂JtJtln(St/Jt)t-rV=0.
Let
(14)Le(H)=2πaσ(δt)H-1,σ~2=2σ2[Ht2H-1-Le(H)sign(VSS)+bStVSS(δt)2H-1].
Substituting these into (13), we get the following conclusions.Theorem 1.
If the underlying asset priceSt 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:
(15)∂V∂t+12σ~2St2∂2V∂St2+rSt∂V∂St+∂V∂JtJtln(St/Jt)t-rV=0,V(T,JT,ST)=(JT-K)+.Remark 2.
For a single European option with long position, the yield at expiration date is(JT-K)+ or (K-JT)+. Because they are convex functions, VJJ>0. Due to Jt=e(1/t)∫0tlnSτdτ, thus VSS>0. Then (14) can be represented as
(16)σ~2=2σ2(Ht2H-1-2πaσ(δt)H-1+bSt(δt)2H-1).


## 3. Option Pricing Formulas

Theorem 3.
Supposing the underlying asset pricesSt  satisfy  (2), then at time t the value V(t,Jt,St) of Geometric-average Asian call option with transaction costs with expiration date T and exercise price K is as follows:
(17)V(t,Jt,St)=(JttStT-t)(1/T)er*(T-t)-∫tTrθdθ+(σ*2/2)(T2H-t2H)×N(d1)-Ke-∫tTrθdθN(d2),
where
(18)d1=ln[(JttStT-t)1/T/K]+r*(T-t)+σ*2(T2H-t2H)σ*T2H-t2H,d2=d1-σ*T2H-t2H,r*=∫tTrθ((T-θ)/T)dθT-t-σ2(T2H-t2H)2(T-t)+Hσ2(T2H+1-t2H+1)T(2H+1)(T-t)+12Le(H)σ2T-tT-12σ2bSt(δt)2H-1T-tT,σ*=σ(1-4H(T2H+1-t2H+1)T(2H+1)(T2H-t2H)+H(T2H+2-t2H+2)T2(H+1)(T2H-t2H)-2Le(H)×(T-t)33T2(T2H-t2H)+2bSt(δt)2H-1(T-t)33T2(T2H-t2H))1/2Le(H)=2πaσ(δt)H-1,N(x)=∫-∞x12πe-(t2/2)dt.Proof.
Let(19)ξt=1T[tlnJt+(T-t)lnSt],V(t,Jt,St)=U(t,ξt);
then
(20)∂V∂t=ln(Jt/St)T∂U∂ξt+∂U∂t,∂V∂St=T-tTSt∂U∂ξt,∂2V∂St2=(T-tTSt)2∂2U∂ξt2-T-tTSt2∂U∂ξt,∂V∂Jt=tTJt∂U∂ξt.
Hence model (15) can be transformed into
(21)∂U∂t+(rt-σ~22)T-tT∂U∂ξt+σ~22(T-tT)2∂2U∂ξt2-rtU=0,U(ξT,T)=(eξT-K)+.
Let
(22)τ=γ(t),ητ=ξt+α(t),W(τ,ητ)=U(t,ξt)eβ(t),
which satisfies the conditions α(T)=β(T)=γ(T)=0; then we have
(23)∂U∂t=e-β(t)(∂W∂τγ′(t)-β′(t)W+∂W∂ητα′(t)),∂U∂ξt=e-β(t)∂W∂ητ,∂2U∂ξt2=e-β(t)∂2W∂ητ2.
Substituting (23) into (21), we have
(24)γ′(t)∂W∂τ+σ~22(T-tT)2∂2W∂ητ2+[(rt-σ~22)T-tT+α′(t)]×∂W∂ητ-(rt+β′(t))W=0.
Let
(25)(rt-σ~22)T-tT+α′(t)=0,rt+β′(t)=0,γ′(t)=-σ~22(T-tT)2,
and combined with terminal conditions α(T)=β(T)=γ(T)=0,  we can get
(26)α(t)=∫tTrθ(T-θT)dθ-∫tT12σ~2(T-θT)dθ=∫tTrθ(T-θT)dθ+Hσ2(T2H+1-t2H+1)T(2H+1)-σ2(T2H-t2H)2+12Le(H)σ2(T-t)2T-12σ2bSt(δt)2H-1(T-t)2Tγ(t)=∫tT12σ~2(T-θT)2dθ=HT2σ2(T2H-t2H2HT2-2TT2H+1-t2H+12H+1+T2H+2-t2H+22H+2)-Le(H)σ2(T-t)33T2+bStσ2(δt)2H-1(T-t)33T2β(t)=∫tTrθdθ.
Thus (21) becomes
(27)∂W∂τ=∂2W∂ητ2,W(η0,0)=(eη0-K)+.
According to the theory of classical heat conduction equation solution, we have
(28)W(ητ,τ)=12πτ∫lnK+∞(ey-K)e-((y-ητ)2/4τ)dy=eητ+τN(2τ+ητ-lnK2τ)-KN(ητ-lnK2τ).
Let
(29)r*=∫tTrθ((T-θ)/T)dθT-t-σ2(T2H-t2H)2(T-t)+Hσ2(T2H+1-t2H+1)T(2H+1)(T-t)+12Le(H)σ2T-tT-12σ2bSt(δt)2H-1T-tT,σ*=σ(1-4H(T2H+1-t2H+1)T(2H+1)(T2H-t2H)+H(T2H+2-t2H+2)T2(H+1)(T2H-t2H)-2Le(H)(T-t)33T2(T2H-t2H)+2bSt(δt)2H-1(T-t)33T2(T2H-t2H))1/2;
then
(30)2τ+ητ-lnK2τ=ln[(JttStT-t)1/T/K]+r*(T-t)+σ*2(T2H-t2H)σ*T2H-t2H≜d1,ητ-lnK2τ=ln[(JttStT-t)1/T/K]+r*(T-t)σ*T2H-t2H=d1-σ*T2H-t2H≜d2.
By variable restored, we obtain that
(31)W(ητ,τ)=(JttStT-t)1/Ter*(T-t)+(σ*2/2)(T2H-t2H)N(d1)-KN(d2).
So the value of geometric-average Asian call option on the time t is
(32)V(t,Jt,St)=U(ξt,t)=We-β(t)=(JttStT-t)1/Ter*(T-t)-∫tTrθdθ+(σ*2/2)(T2H-t2H)N(d1)-Ke-∫tTrθdθN(d2).Corollary 4.
If the riskless interestr and the volatility σ are constant, then the  values  VC and Vp 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
(33)VC(t,Jt,St)=(JttStT-t)1/Te(r*-r)(T-t)+(σ*2/2)(T2H-t2H)N(d1)-Ke-r(T-t)N(d2),VP(t,Jt,St)=-(JttStT-t)1/Te(r*-r)(T-t)+(σ*2/2)(T2H-t2H)N(-d1)+Ke-r(T-t)N(-d2),
where
(34)r*=r2T(T-t)-σ2(T2H-t2H)2(T-t)+Hσ2(T2H+1-t2H+1)T(2H+1)(T-t)+12Le(H)σ2T-tT-12σ2bSt(δt)2H-1T-tT
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:(35)St=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.
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 differentH.Figure 2
Asian put option pricing corresponding to the differentH.Figure 3
Asian put option pricing corresponding to the differenta.Figure 4
Asian put option pricing corresponding to the differentb.


## 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.



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