AN Asian option is a kind of financial derivative whose

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1 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 A Class of Control Variates for Pricing Asian Options under Stochastic Volatility Models Kun Du, Guo Liu, and Guiding Gu Abstract In this paper we present a strategy to form a class of control variates for pricing Asian options under the stochastic volatility models by the ris-neutral pricing formula Our idea is employing a deterministic volatility function σt) to replace the stochastic volatility σ t Under the Hull and White model] and the Heston model], the deterministic volatility function σt) can be chosen with the same order moment as that of σ t, and then a control variate can be derived he numerical experiments report that our control variates wor quite well by showing the standard deviation reduction ratio Index erms Asian Options pricing Monte Carlo method control variates I IRODUCIO A Asian option is a ind of financial derivative whose payoff includes a time average of the underlying asset prices he primary purpose for basing an option payoff on an average asset price is to mae it more difficult for anyone to significantly affect the payoff by manipulation of the underlying asset price So Asian options can be used to reduce the ris caused by unusual behaviors of the underlying asset price before expiry, and they are quite popular in ris management According to different sampling types and strie price types, there are eight types of Asian options in this paper, we do not distinguish call and put options), four fixed-strie options and four floating-strie options he payoff functions of four fixed-strie options are: ) fixed-strie continuous sampling arithmetic average Asian call) option caao), V caao t= = S t dt K) ) fixed-strie discrete sampling arithmetic average Asian call) optiondaao), V daao t= = S i K) 3) fixed-strie continuous sampling geometric average Asian call) optioncgao), V cgao t= = e log Stdt K) Manuscript received ov 7th, revised Feb 7th, 3 his wor was supported by the Cultivation Fund of the Key Scientific and echnical Innovation Project, Ministry of Education of Chinao784], the Leading Academic Discipline Program, Project for Shanghai University of Finance and Economicsthe 3rd phase), and Graduate Research and Innovation Fund of Shanghai University of Finance and EconomicsCXJJ--395) Kun Du and Guo Liu are with the School of Finance, Shanghai University of Finance and Economics, PRChina d34597@6com guoliu shufe@yahoocn Guiding Gu is with the Department of Applied Mathematics, Shanghai University of Finance and Economics, SH, 433 PRChina guiding@mailshufeeducn 4) fixed-strie discrete sampling geometric average Asian call) optiondgao), V dgao t= = e logsi K), K and S t are the fixed-strie price and the price of underlying asset at time t, respectively S i S i denotes the price of underlying asset at the ith observation date i with = < < < < =, ] represents the valid period of the option Replacing K with S in V caao t= and V cgao t=, we can derive the payoff functions of the floating-strie continuous sampling arithmetic and geometric average Asian put) options, denoted as caao and cgao, V caao t= = S t dt S, V cgao t= = e log Stdt S Also, replacing K with S in V daao t= and V dgao t=, we have the payoff function of the floating-strie discrete sampling arithmetic and geometric average Asian put) option, denoted as daao and dgao, V daao t= = S i S, V dgao t= = e log Si S he Monte Carlo method is a numerical method based on probability and statistics, and is widely used in many fields, especially in the field of computational finance One of the main advantages of the Monte Carlo method is that its convergence is independent on the number of state variables It is usually used when the number of state variables is greater than three However, the drawbac of Monte Carlo method is that its convergence rate is slow Let V be a random variablerv for short), and we want to calculate µ = EV ] By simulation, we get identically independent distributed iid for short) samples {V i } n of V Law of Larger umber guarantees V n = n n V as i µ Central Limit heorem guarantees that µ asymptotically falls in the confidence interval V n σ Z δ, V n σ ] Z δ n n with probability δ, σ is the standard deviation of V, n is the number of simulation paths, δ is the significance level and Z δ is the quantile of standard normal distribution under δ It is clear that the convergence rate of the Monte Carlo method is On ), and a better way to improve Advance online publication: May 3)

2 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 the accuracy is reducing the standard deviation σ We refer to Glasserman 8] for a summary of various techniques to reduce the variance he method of control variates is one of the most widely used variance reduction techniques Suppose on each replication we can calculate another output X i along with V i, that the pairs {X i, V i )} n are iid and that the expectation EX] of X i is nown We use X, V ) to denote a generic pair of rvs with the same distribution as each X i, V i ) hen for any fixed b R, we can calculate V i b) = V i bx i EX]), i =,, n through the ith replication and compute the sample mean V n b) = V n bx n EX]) his is a control variate estimator It is proved in Glasserman 8] that V n b) is a unbiased and consistent estimator of µ V b) has variance V arv b)) = σ V bσ X σ V ρ XV b σ X ) he minimum point on b is b = σ V σ X ρ XV Substituting b in ), we have V arv b )) V arv ) = ρ XV ) We choose a control variate X for V, if X satisfies two conditions: i) the expectation EX] is nown ii) the correlation ρ XV is close to In practice, b can t be derived exactly as σ V and ρ XV are generally unnown We can use its sample counterpart yields the estimate b = n X i X n )V i V n ) n X i X n ) to approximate b As mentioned in Glasserman 8], we may still get most of the benefit of a control variate using an estimate of b Strictly speaing, to measure the efficiency of the Monte Carlo method, we need not only the variance reduction ratio but also expected computing time per replication But in this paper, the computational effort per replication is roughly the same with and without a control variate, so we focus on the variance reduction ratio see Ma and Xu 3] Kemma and Vorst ] studied the valuation of arithmetic average Asian options by using the counterpart geometric average Asian options as control variates his is one of the most successful applications of control variates in financial engineering In the case of stochastic volatility models, a constant volatility can be chosen to replace the stochastic volatility in some conditions, and then this tractable dynamic process is used as an auxiliary process to form a control variate How to choose this constant volatility is the ey problem of the efficiency of control variates he most intuitive way is to choose the initial value of the stochastic volatility as the constant volatility Both Fouque and Han 5] and Han and Lai 9] use a method named as the Martingale Control Variate method to choose an effective volatility which is dependent on the initial value of the stochastic volatility as the constant volatility his method has many advantages and can be used to other financial derivatives besides Asian options see Fouque and Han 4, 6]) But the martingale control variate method also has a potential drawbac Calculating the effective volatility needs the invariant distribution function of stochastic volatility If the stochastic volatility satisfies Ornstein-Uhlenbec process under which the invariant distribution of stochastic volatility is easy to handle, the martingale control variate method is easy to implement, but if the stochastic volatility satisfies a process, which the invariant distribution of stochastic volatility is hard to handle such as Square-Root Diffusion, or the invariant distribution is unnown, the martingale control variate method is difficult to implement here are many types of stochastic volatility models, such as those in Scott 4], Stein and Stein 6] and Ball and Roma ] We refer to Fouque et al 7] for a summary of various stochastic volatility models In this paper, we present a strategy to form a class of control variates for pricing Asian options under a stochastic volatility model Our idea is employing a deterministic volatility function σt) to replace the stochastic volatility σ t his deterministic volatility σt) is not only dependent on the initial value of the stochastic volatility but also dependent on time t, so that σt) can trac down the stochastic volatility Under the Hull and White model ] and the Heston model ], the deterministic volatility function σt) can be chosen with the same order moment as that of σ t, and then a control variate can be derived he numerical experiments in our paper report that our control variates wor quite well in terms of showing the standard deviation reduction ratio It is worth noting that our control variate is a generalization of the control variate in 3] for pricing variance swap under the Hull and White model ] he rest of this paper is organized as follows We introduce some basic settings for the model used in this paper in Section I and derive the idiographic control variates under the Hull and White model in Section II In Section III we present an algorithm to estimate the standard deviation reduction ratio and then report some numerical results in terms of showing the standard deviation reduction ratios under the Hull and White model and the Heston model Finally we give some conclusions in Section IV A Basic Setting In this section we model the underlying asset price, but we do not give the concrete stochastic differential equation which the volatility satisfies We get some general conclusions which will be useful in the following sections We begin with a probability space Ω, {F t } t, P), here P is the ris-neutral measure In this paper, all expectations are derived under the ris-neutral measure P unless there is a special statement Suppose that the price of underlying asset S t follows the geometric Brownian motion ds t = rs t dt σ t S t dw t, 3) r is the ris-free interest rate which is a constant, W t is the Winner process and σ t is the stochastic volatility which satisfies a diffusion process driving by another Winner process W t W t and W t satisfy covdw t, dw t ) = ρdt, so we have W t = ρw t ρ B t, in which B t is the Winner process and independent with W t Let {F t } t be the filtration generated by the two-dimension Brownian Advance online publication: May 3)

3 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 motion W t, B t ), so S t and σ t are adapted to the filtration {F t } t Suppose that σ t satisfies the square-integrability condition which is E t σ sds < It is nown that by the ris-neutral pricing formula, the prices are V caao t= = Ee r V caao t= )] ) = e r E S t dt K for caao fixed-strie discrete sampling arithmetic average Asian call) option), for daao, V daao t= = Ee r V daao t= )] = e r E S i K V cgao t= = Ee r V cgao t= )] for cgao, and = e r Ee log Stdt K ] V dgao t= = Ee r V dgao t= )] = e r Ee log Si K ] for dgao Also for four floating-strie Asian options, the prices are V caao t= = e r E V daao t= = e r E V cgao t= = e r Ee V dgao t= = e r Ee S t dt S ], S i S ], log Stdt S ], log Si S ] As said in Fouque and Han 5], when the volatility is randomly fluctuating, there is no analytic solution for GAO in general, neither for AAO But if the volatility is a deterministic functionnot necessarily constant), the prices of GAO have analytic solutions In such case, these analytic solutions can be used as control variates for pricing corresponding Asian options with stochastic volatility For GAO with deterministic volatility, we have following theorems ) is the standard normal distribution function in this paper heorem Suppose that the stochastic volatility σ t in 3) is replaced by a deterministic square-integrable volatility σt), there is an analytic solution for the fixed-strie continuous sampling geometric average Asian call) option, X cgao t= = Ee r X cgao t= )] = e r E e log St)dt K = e σ r a d ) Ke r d ), t a = log S r σ s)ds]dt, σ j n = lim n n n j ] σ s)ds, j= and d = a logk, d = d σ σ Proof By 3) and the assumptions, we have t log St) = log S rt σ s)ds σs)dw s at It) 4) t and log St)dt = at)dt It)dt By heorem 449 in Shreve 4], we get It) = It is easy to see a t σs)dw s, t at)dt = log S r σ s)ds) t σ s)dsdt ext, we focus on proving It)dt, σ ) Let = t < t < < t n = t i = t i t i = t = n, i =,,, n t i = i t, i =,,, n Denote Θ It)dt hus, we have Θ = It)dt = lim n It i) t i = lim n n It i) lim Θ n n as Since it holds for any path, we have Θ n Θ as means convergence in almost surely sense) By heorem 53 and as d heorem 55 in 7], we now that Θ n Θ = Θ n Θ d means convergence in distribution sense) Since It ) It ) Σ t t It n ) σ s)ds σ s)ds t σ s)ds t t By setting =,,, ), we have Θ n =, Σ, t σ s)ds σ s)ds t σ s)ds σ s)ds t tn σ s)ds σ s)ds n It i) =,,, ) n, n Σ) =, σ n), It ) It ) It n ) Advance online publication: May 3)

4 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 and σ n = n n j= tj n j ] σ s)ds σ Since Θ n, σ n) for any n, the characteristic function ϕ n u) of Θ n satisfies ϕ n u) = e u σ n e u σ = ϕu) It is easy to prove that in any interval U, U ], ϕ n u) uniformly converges to ϕu) as ϕ n u) and ϕu) are both continuous functions By Levi-Cramer heorem7], heorem 54), we get δ n, σ ) hus as the uniqueness d of limitation, we have Θ, σ ) and ξ logst)dt = a Θ a, σ ) By the ris-neutral pricing formula, it holds that X cgao t= = Ee r X cgao t= )] = e r Ee log St)dt K ] = e r Ee ξ K ] 5) By setting ξ = a σz, Z, ), we have X cgao t= = e r Ee a σz K ] = e r e a σz K e z dz π d = e r e a σz K) e z dz π = e σ r a d ) Ke r d ), d = a logk, d = d σ σ heorem Suppose that the stochastic volatility σ t in 3) is replaced by a deterministic square-integrable volatility σt), there is an analytic solution for the fixed-strie discrete sampling geometric average Asian call) option, X dgao t= = Ee r X dgao t= )] = e r E e log Si) K a = log S r σ = = e σ r a d ) Ke r d ), i j ] j= and d = a log K, d = d σ σ j i σ s)ds, σ s)ds, We omit the proof of heorem since it is similar to that of heorem For the floating-strie Asian options, we also have the following theorems heorem 3 Suppose that the stochastic volatility σ t in 3) is replaced by a deterministic square-integrable volatility σt), there is an analytic solution for the floating-strie continuous sampling geometric average Asian put) option, X cgao t= = E e r X cgao t= ) ] = e r E e log St)dt S ) = S e b a d ) S d ), t a = r σ s)dsdt σ s)ds, b j n = lim n n n j ] σ s)ds j= lim n n j= and d = a b, d = d b j n σ s)ds σ s)ds, 6) Proof Set J ) = e log St)dt By the ris-neutral pricing formula, we have X cgao t= = E e r X cgao t= ) ] = e r E e log St)dt S ) Set Z ) = e = e r E J ) S )] J ) = e r E S ) S ) σs)dws σ s)ds and PA) = A Z )dp, A F By Girsanov s heorem, Ŵ s W s s σu)du is a Winner process under the new probability measure P hen we have ) ] J ) X cgao t= = e r Ê S ) S ) Z ) J ) = S Ê S ) By 5), we have log J ) S ) = a Θ, a = r Θ = t Ît)dt Î ), σ s)dsdt σ s)ds, and Ît) = t σs)dŵs Under the new probability measure P, similar to the proof of heorem, we can prove Θ, b ), and we omit it Set ξ log J ) S ) hen we have ξ a, b ) under the measure P hus it holds that J ) X cgao t= = S Ê S ) = S Ê e ξ ] Also similar to the proof of heorem, we can get the conclusion of heorem 3 heorem 4 Suppose that the stochastic volatility σ t in 3) is replaced by a deterministic square-integrable volatility Advance online publication: May 3)

5 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 σt), there is an analytic solution for the floating-strie discrete sampling geometric average Asian put) option, X dgao t= = E e r X dgao t= ) ] = e r E e log Si) S ) = S e b a d ) S d ), a = r ) ) σ s)ds b = j= j j= i ] j j ] σ s)ds σ s)ds σ s)ds, i σ s)ds and d = a b, d = d b he proof is similar to that of heorem 3 ote that σt) should be chosen such that the limitations in 4) and 6) both exist By the call-put parity formula, for the fixed-strie GAO put option, the price formula is Ke r d ) e σ r a d ), and for the floatingstrie GAO call option, the price formula is S d ) S e b a d ) II COROL VARIAES UDER WO MODELS he analytic solutions for GAO derived in Section I could be employed as control variates for valuing Asian options with stochastic volatility models in Section I For example, we can employ X cgao as a control variate to get V cgao and V caao, and X dgao as a control variate to get V dgao and V daao, et al However, by ), it is important that how to choose the deterministic square integrable volatility σt) to mae ρ XV as large as possible In this section, we show a strategy to choose an appropriate deterministic volatility σt) under the Hull and White model ] and the Heston model ] he idea is that σt) is chosen with the same order moment as that of σ t A Hull and White Model Hull and White ] introduced the concept of stochastic volatility Suppose that square of the stochastic volatility Y t σ t = Y t ) satisfies the following equation dy t = µy t dt σy t dw t, 7) µ, σ are constants It is hold that Y t = σ t = Y e µ σ )tσw t = σ e µ σ )tσw t 8) We choose σt) such that σt) and σ t have the same mth order moment, that is Y t)] m = σt)] m = Eσ m t ] = EY m t ] 9) By 8) and the property of lognormal distribution, we have σt) = σ e amt, ) ], a m = µ 4 m )σ and m is any real number Substituting σt) in heorem 4, we can solve the parameters a, σ) of the analytic solutions in heorem 4 heorem 5 Suppose that σt) is defined by ) hen the parameters a, σ and b in heorem 4 have the expressions i) in heorem, log S r 4 σ, if a m = a = log S σ a m a m e am ) ], if a m { 3 σ, if a m = σ = σ a e am ) σ 3 m a σ m a m, if a m ii) in heorem, a = log S r i σ = log S r i σ i, if a m = σ a m e ami, if a m { σ j= j ] j, if a m = σ a m j= j ]eamj ], if a m iii) in heorem 3, { a = r σ ), if a m = σ a e am ) σ m a m e am r, if a m { b 3 = σ, if a m = σ a m a m )e am eam ) ], if a m a m iv) in heorem 4, r ) i] σ a = ) i], if a m = r ) i] σ a m ) e am ) )], if a eami m σ j ] j σ j σ, if a m = b σ = a m j= j ) ]eamj σ a m j= ) eamj σ a m e am ), if a m he proof of this theorem is computational process and we omit it he only one point is that when solving the limitations in 4) and 6), we should use the aylor expansion e x = x x Ox 3 ) and the concept of the same order infinitesimal hus we can obtain a control variate X to an option V since the expectation of X can be solved analytically by the theorems B Heston Model he Hull and White model is the earliest stochastic volatility model and because of its tractable in mathematics, it s applied very widely But in the long run, it is unreasonable in financial sense If the volatility Y t satisfies 7), by 9) and ), we have Eσ t ] = σ e µ 4 σ )t which illustrates that the volatility mean grows exponentially his is not liely Advance online publication: May 3)

6 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 to be true Heston ] supposed that square of the volatility satisfies the mean-reversion process dy t = α βy t )dt σ Y t dw t, ) α >, β >, σ > he process in ) is a square-root diffusion process, which was first studied by Cox, Ingersoll and Ross 3] his model guarantees that Y t converges to its long run mean α/β and Y t is nonnegative In financial point of view, the Heston model is more reasonable than the Hull-White model, but the Heston model is less tractable in mathematics Unlie 7), ) doesn t have a closed-form solution, but we can easily solve its expectation ],pp4,ex44) Eσ t ] = EY t ] = e βt Y α β e βt ) We now choose σt) such that σt) and σ t have the same order moment Y t) = σ t) = Eσ t ] = e βt Y α β e βt ) ) hus, σt) can be used in heorem 4, and then X can be employed as a control variate to a option V under the Heston model III UMERICAL EXPERIME By ), the efficiency of a control variate X to an option V can be shown by the correlation ρ XV, or by the standard deviation reduction ratio R = ρ 3) XV A larger R means that a control variate X has more efficiency to an option V In this section, we first present a algorithm to estimate R, then perform some numerical experiments to report the efficiency of our control variates by showing the estimation of R Following the way of Ma and Xu ], we present the following numerical algorithm to estimate R for the control variate X dgao to the option V dgao under the Hull-White model Algorithm Estimate R for X dgao to V dgao under the Hull-White model ) Divide, ] into n intervals with mesh size t = /n =, and mae sure that the set of time discrimination points { } n = covers the set of observation dates { i } ) After putting σt) into 3), we can generate S ) from S ) also see 5)) by S ) =S ) exp { r t t } σs)dw s t t σ s)ds As σs)dw t, σ s)ds), we generate standard normal random number Z,j and get { S j ) =S j ) exp r t t σ s)ds t } t σ s)dsz,j, j =,, p) S j t ) = S and p is the number of the replication simulation hus a replication j of the underlying asset price St) is derived 3) By the contract of the option, set the value of control variate X j dgao = e log Sj i) K 4) 4) Similarly, we generate S t from S t by { } S j = S j exp r σj ) ) t σ j t tz,j, with S j t = S, σt i = Y j, and Y from Y t by Y j = Y j exp µ σ ) t σ ] tz,j, 5) Z,j is the standard normal random number with the correlative coefficient ρ with Z,j hus a replication j of the underlying asset prices S t following processes 3) and 3) is simulated 5) By the clause of the option, set the value of the option dgao = e log Sj i K 6) 6) Let X p = p p j= X j, V p = p p j=, then p j= ρ XV = X j X p ) V p ) p j= X p, j X p ) j= V p ) and R = ρ XV ) Remar: ) For other control variate X to other option V, it is only need to modify 4) and 6) ) For the Heston model, it is only need to modify 5) by ) A Hull-White Model Based on the algorithm, we perform some numerical experiments to report the efficiency of our control variates by showing the standard deviation reduction ratio R under the Hull-White model We report our numerical results of with a Matlab 7 implementation of the algorithm Following Ma and Xu 3], we set the parameters =, n =, = 5, r = 5, µ = 5, S =, σ =, Y = σ = 5, p = We test serval groups of the other parameters m, ρ, K ote that if m = 4µ σ, Y t) = σ t) = σ = Y is constant he data in all the tables are the standard deviation reduction ratio R, rather than the variance reduction ratio R Experiment In this experiment, we report the efficiency of the control variate X dgao to the option V dgao by showing the standard deviation reduction ratio R in able I We test serval groups of the parameters m, ρ, K he data in able I show us that: ) when m = 4µ σ at the last column, σt) = σ in ) is a constant, so σt) can t trac down σ t In such Advance online publication: May 3)

7 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 ABLE I X dgao O V dgao m=-5 m= m= m= m= m= 4µ σ K= ρ = K= K= K= ρ = 9 K= K= case, the efficiency of the control variate X dgao to the option V dgao is small For the other m, the difference of the efficiency is not significant ) there is some influence for different ρ he larger ρ is, the larger R is 3) when the option is in-the-money ıe, K < ), the control variate wors better his is because when the option is out-of-the-money ıe, K > ), there are many paths giving zero payoff o overcome this drawbac, we can use the call-put parity formula, V dgao t= =E e r e log Si K =E E It is clear that if e e r K e log Si e r e log Si e r K log Si K is deep) out-of-themoney, K e log Si is deep) in-the-money hus we can use the Monte Carlo method with our control variate to simulate the deep) in-the-money option V dgao t= Experiment In this experiment, we report the efficiency of the control variate X dgao to the option V daao by showing the standard deviation reduction ratio R in able II In such case, we replace dgao in 6) by daao = S j i K We also test the same group of the parameters m, ρ, K as that in the experiment he data in able II show us that: ) the efficiency of the control variate X dgao to the option V daao is much lower than that to the option V dgao his is reasonable since the difference between V daao and X dgao lies not only in the volatility, but also in the payoff structure Even so, the variance reduce ratio is about 45 ), which means the correlation coefficient between V daao and X dgao is about 9998 ) the efficiency of the control variate with the constant σ ie when m = 4µ σ at the last column) is still lower than others m, but that is not much 3) the effect of K is the same as that in the experiment 4) there is some affect for different ρ, but not very clear Experiment 3 In this experiment, we report the efficiency of the control variate X dgao to the options V dgao and V daao by showing R in able III In such cases, we replace X j dgao in 4) by X j dgao = e log Sj i) S j ) also, V dgao in 44) should be replaced by dgao = e log Sj i S j and by daao = S j i Sj respectively We test several groups of the parameters m and ρ he data in able III show us that: ) just lie the results of the experiment and the experiment, the efficiency of the control variate X dgao to the option V daao is much lower than that to the option V dgao ) the efficiency of the control variate with the constant σ ie when m = 4µ σ at the last column) is still lower than others m 3) there is some affect for different ρ, and basically, the smaller ρ is, the smaller R is ext two experiments are about the continuous sampling Asian options Experiment 4 We report the efficiency of the control variate X cgao to the options V cgao and V caao by showing R in able IV In such cases, we replace X j dgao in 4) by X j cgao = e log Sj t)dt K e n log = Sj ) t K) also, V dgao in 6) is replaced by cgao = e log Sj t dt K e n log = Sj t K and by caao = S j t dt K S j t K = Advance online publication: May 3)

8 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 ABLE II X dgao O V daao m=-5 m= m= m= m= m= 4µ σ K= ρ = K= K= K= ρ = 9 K= K= ABLE III X dgao O V dgao AD O V daao to m=-5 m= m= m= m= m= 4µ σ ρ = ρ = V dgao ρ = ρ = ρ = ρ = ρ = V daao ρ = ρ = ρ = ABLE IV X cgao O V cgao AD O V caao to m=-5 m= m= m= m= m= 4µ σ V cgao K= K= V caao K= K= respectively We set the parameter ρ = 9, and test several groups of the parameters m and K Experiment 5 We report the efficiency of the control variate X cgao to the options V cgao and V caao by showing R in able V In such cases, we replace X j dgao in 4) by X j cgao = e log Sj t)dt S j ) e n log = Sj ) t S j )) also, V dgao in 6) should be replaced by cgao = e log Sj t dt S j e n log = Sj t S j and by caao = S j t dt S j S j t S j = respectively We test several groups of the parameters m and ρ he numerical results of two experiments above for the control variates to the continuous sampling Asian options show the similar efficiency lie those to the discrete sampling Asian options B Heston Model Experiment 6 In this experiment, we report the efficiency of the control variate X dgao to the option V dgao under the Heston model by showing R in able VI In such case, we replace 5) by Y j = Y j α βy j ) t σ Y j Z,j We set the parameters by n =, r =, α = 5, β = 5, S =, σ =, =, Y = σ = 4, p =, =, K = We test several parameters ρ and two ind forms of the control variates X dgao one is based on the deterministic volatility function ), and the other is based on the constant volatility Y t) = Y he numerical results show that our control variate also wors well under the Heston model IV COCLUSIO In this paper, we present a strategy to form a class of control variates for pricing Asian options under the stochastic volatility models Our idea is using a deterministic volatility σt) to replace the stochastic volatility σ t by choosing σt) with the same order moment as that of σ t under the Hull- White model and the Heston model umerical experiments report that our control variates wor quite well by showing the standard deviation reduction ratio R and the efficiency is obviously better than one formed by the constant volatility σ, the initial value of the stochastic volatility Our strategy can also be extend to other stochastic volatility models, as long as their order moment can be obtained in the closedform his is much easier than to calculate the distribution Advance online publication: May 3)

9 IAEG International Journal of Applied Mathematics, 43:, IJAM_43 ABLE V X cgao O V cgao AD O V caao to m=-5 m= m= m= m= m= 4µ σ ρ = ρ = V cgao ρ = ρ = ρ = ρ = ρ = V caao ρ = ρ = ρ = ABLE VI X dgao COROL V dgao BASED O WO Y t) Y t) ρ = 9 ρ = 5 ρ = ρ = 5 ρ = 9 e βt Y α β e βt ) Y function of the stochastic volatility such as in the Heston model In addition, our strategy can be extend to pricing other financial derivatives under stochastic volatility models REFERECES ] C Ball and A Roma, Stochastic volatility option pricing, Journal of Financial and Quantitative Analysis: 9, ) ] W Boughamoura and rabelsi F, Variance Reduction with Control Variate for Pricing Asian Options in a Geometric Levy Model, IAEG International Journal of Applied Mathematics 3] J C Cox, E Ingersoll, and SA Ross, A heory of the erm Structure of Interest Rates, Econometrica: 53, ) 4] J P Fouque and CH Han, A Control Variate Method to Evaluate Option Prices under Multi-factor Stochastic Volatility Models, 3 5] J P Fouque and CH Han, Variance Reduction for Monte Carlo methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models, Quantitative Finance, 4 6] J P Fouque and CH Han, A Martingale Control Variate Method for Option Pricing with Stochastic Volatility, Probablity and Statistics, 7 7] J P Fouque, G Papanicolaou, R Sircar, Derivatives in Financial Maret with Stochastic Volatility, Cambridge University Press, 8] P Glasserman, Monte Carlo Methods in Financial Engineering, Springer, ew Yor, 4 9] C H Han and Y Z Lai, Generalized Control Variate Methods for Pricing Asian Options, 9 ] S L Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, he Review of Financial Studies: 6, ) ] J Hull and A White, he pricing of options on assets with stochastic volatilities, Journal of Finance: 4, ) ] A G Kemma and A C Vorst, A Pricing method for options based on average asset values, Journal of Baning and Fianace: 4, ) 3] J M Ma and C L Xu, An Efficient control Variate Method for Pricing Variance Derivatives, Journal of Computational and Applied Mathematic: 35, 8-9 ) 4] L O Scott, Option pricing when the variance change randomly: theory, estimation, and an application, Journal of Financial and Quantitative Analysis:, ) 5] S E Shreve, Stochastic Calculus for Finance II, Springer, 6] E M Stein and J C Stein, Stoc price distributions with stochastic volatility: an analytic approach, Review of Finance Studies: 4, ) 7] S Zhou, S Xie and C Pan, Probability heory and Mathematical Statistics Iin Chinese), People s Education Press, 98 Advance online publication: May 3)