2009 International Conference on Business Intelligence and Financial Engineering

Modeling of Variance and Improved Control Variate for Monte Carlo method

Junmei Ma Chenglong Xu Department of Mathematics Department of Mathematics Tongji University Tongji University Shanghai, 200092, China Shanghai E-Institute of Scientific Computing E-mail:[email protected] Shanghai Normal University Shanghai, 200092, China E-mail: [email protected]

Abstract—This study proposes the partial differential equation In this study we discuss the modeling and pricing of the pricing model for the Variance Swap derivatives under the variance swap under the stochastic structure. We structure. Control variate technique is first build the PDE pricing model for the variance swap. applied to valuation of the derivatives, based on the closed Then a high efficient Monte Carlo (MC) method for the form solutions in a simpler model. Then with the analysis of valuation of the Variance Swap is provided, based on the the moments for the underlying processes, a method to choose control variate technique and the analysis of the moments high efficient control variate for Monte Carlo simulation is for the underlying processes. The computation results provided. The computation results show that our method can confirm the high efficiency of the selected control variate. reduce variance efficiently, and are in line with the theoretical analysis. The method in the paper can also be extended to the II. MODELING OF VARIANCE SWAP valuation of other types of Variance Swaps, such as Corridor Variance Swap, Gamma Variance Swap, Conditional Variance The study on the stochastic volatility starts from the early Swap and other products with multi-factor models. 1970's. In 1973, Black and Scholes made a major break- through by deriving pricing formulas for vanilla options Keywords-Variance Swap; stochastic volatility; Monte Carlo written on the stock. The model assumes that the volatility method; Control Variate term is a constant. This assumption is not always satisfied by real-life options, since the probability distribution of I. INTRODUCTION equity has a fatter left tail and thinner right tail than the lognormal distribution as in [3] and it is also incompatible Over the last few years, many market participants have with derivatives prices observed in the market, verified by started trading in variance swap. This is a . The concept of stochastic volatility was in which one counterparty agrees to pay the other a notional introduced by Hull and White (1987) [7], and subsequent amount M , times the difference between a fixed level and a developments include the work of Scott (1987) [8], Stein realized level of variance. Its payoff at is equal to (1987) [9], Ball and Roma (1994) [10], Heston (1993) [11]. =×σ 22 − Payoff M (Rvar K ). The stochastic volatility model used in this study is the 2 Geometric Brownian Motion proposed by Hull and White in Where Kvar is called the variance strike for variance swap, 1987. Under the martingale measure, the underlying asset and σ 2 is realized variance which is determined by the R St()and volatility σ are assumed obeying the stochastic variance of the asset’s return over the life of the swap. t Recently, a new generation of variance swaps are born and differential equations: dS becoming popular, such as Corridor Variance Swap, Gamma t =+σσ=rdt dW , Y , (1) S t1tt t Variance Swap, and Conditional Variance Swap. t dY In some recent work of Derman and Kamal (1999) [1], t =μ dt +σˆ dW . (2) Y 2t Detemple and Osakwe (1999) [2], the authors have t examined the properties of variance swap, and they showed Where r is deterministic ; μ > 0 is a reversion σˆ > 0 that variance swap can be replicated theoretically by a static speed, is a volatility of volatility;W1t andW2t are Winner position in European call and put options of all strikes and a processes, Cov(, dW dW )= ȡdt .We suppose the market is no- dynamic trading strategy in the underlying asset, based on 1t 2t arbitrage and no transaction costs. There are N observation the Black Scholes (1973) pricing theory as in [3]. Brockhaus dates ttttT=<<<=0,ˈ " SS is the asset price on the ith and Long (2000) [4] provided an analytical approximation 001 N iti date, and YY is the instantaneous variance at t . The payoff for the valuation of volatility swaps. Little and Pant (2001) iti i [5] developed a finite difference method for the valuation of function for the variance swap at the maturity T is variance swaps. Carr, Geman and Madan (2005) [6] priced N V=× M ( (ln(S S ))22 − K )" h(S ,S , ,S ) (3) tT= ¦ ii1− var 01 N variance options by directly modeling the quadratic = i1 variation of underlying process using a Levy process.

978-0-7695-3705-4/09 $25.00 © 2009 IEEE 735 DOI 10.1109/BIFE.2009.170

Authorized licensed use limited to: East China Univ of Science and Tech. Downloaded on January 27, 2010 at 01:27 from IEEE Xplore. Restrictions apply. During an observation interval (,),ttii−1 we construct a risk- i.i.d. as the variableV . The natural estimator of the price is Π the average V(V=++" V)/m. Suppose that on each less portfolio , containing the productV , the quantity Δ1 of 1m simulation there is anther output X along with V , and the the underlying asset S , and the quantity Δ2 of another traded i i ∗ expectation E[]X of the X i is known and the pairs (,)XYiiare Vt with different maturity and strike but same underlying. By Ito lemma and Δ− hedging principle, i.i.d. Then for any fixed b we can calculate =− − Π + V(b)iii V b(X E[X])from the ith simulation. So the control choosing Δ1 and Δ2 to make risk-less during [,tt dt ]. Then variate estimator of the price is given by the equation governing V can be written as [7] 1 m 223 2 =− − = − − ∂∂ ∂ ∂∂∂ V(b) V b(X E[X]) (Vii b(X E[X])). V1222 V1 V2 V V V ¦ LV ++σ+ YSˆˆ Yρσ++ SY rSμ Y−= rV 0. m i1= ∂∂t2 S22 2 ∂ Y ∂∂∂∂ SY S Y The observed error ([])X − EX serves as a control. Obviously As there is no transaction friction and the market is assumed no arbitrage, the price of the variance swap should be it is an unbiased estimator and the variance is =+2 − ρ == ∈ Var(V(b))iXV Var(V) b Var(X) 2b Var(V) Var(X) . continuous at ti . Let VVi12Ni ,( , ," , ), ttt(,)ii−1 , then the − PDE model for the variance swap is given by The optimal coefficient b*1= Cov[X,V]Var(X) minimizes the = < <+∞ < <+∞ < ≤ = Var(V (b*2 ))=− (1ρ )Var(V ). °­LVNN1N 0 0 S , 0 Y , t− t t T, variance, which is given by iXYi ® = () °VN01N1= h S ,S ," ,S− ,S , The literature in the theory and applications of control ¯ tT = < <+∞ < <+∞ < ≤ variates is quite extensive, and we do not intend to provide °­LVii1i 0 0 S , 0 Y , t− t t , ® ==− (4) an exhaustive list here. The paper by Nelson (1990) [13] VVii1==+ (i1,2,,N1)." ¯° ttii tt contains a very complete list of relevant references.

If the analytical solutionVSNN(,11" , S− ,,) St is obtained from With PDE approach, the closed form solution is easily = = derived when the volatility is equal to a list of piecewise the first problem, then at ttN −1 , SSN −1 , = constants. Then this solution is used as a suitable control to VVSSStNNNNN−−−−11111= (,,,,)" , which is the terminal value of ttN −1 resolve the evaluation of the variance swap under the the second problem with iN1=−. Therefore, the following stochastic volatility structure. Furthermore, we analyze the

solutions VVN −11,," can be solved one by one. Hull and moments of the underlying processes, and give a favorable White (1987) [7] discussed the special case N ==1,ρ 0 of method to choose the high efficient control variate. the above problem, through the introduction of the mean A. PDE Approach to Solve the Control Variate variance. Using the similar method, the formal solution of Here, we consider another product, like standard variance ρ = 0 the problem (4) with can be derived =<< < = swap, with the same observation dates 0,tt01" tN T +∞ +∞ NN −N/2 2 2 2 V(S, Y, 0)=πΔ+Δθ++θ"" q(Y) q(Y )(2 ) exp{ Į t1/2ȕ Ytˉ "Ν  } 2 ³³−∞ −∞ 1N¦¦ iiiii1 notional value M , strike variance K , and the asset price Si i1== i1 var ⋅Δ−βΔΔθ−βΔ h(S exp( Y t Y t ),"" ,6ˇ exp( Y t Y t on the ith observation date. But the asset is assumed to obey 0111110111111 the process (6), with piecewise volatility constants: +Δθ−βΔθθ YNNN t N Y NN t ))d 1"" d N dY 1 dY N . (5) dSt t =+σrdt dW (6) =−i α=−− − 2 β= − ic 1t Where YYdt/(tt),itii1− iiir1/Y(rY/2),¦ ii1/2 r/Y, St ³ti1− σ >= Where r is interest rate; ic 0i() 1" N is the volatility and qY()i denotes the density of Yi satisfying the equation: constant during the ith observation period; is a Winner ∂∂qq1qqYY− ∂∂ W1t ++ii σμˆ 22YY0. += ∂−∂ii ∂ ∂ = tTtY2iii Y Y process as in (1). WWSt(,)denotes the price of this

As there is no closed form solution to qY()i , it is difficult to derivative and the payoff function at maturity is also " derive the analytical expression of (5). If the finite hS(,,,01 S SN ). Then similar to the derivation in Section II, difference method is used directly to solve this path the PDE pricing model of the product is given by dependent variable problem (4), it’s really an enormous ­ ∂∂∂2 =+σW122 W + W −=<<+∞<≤ °LW1ic S2 rS rW 0 0 S ,t i1i− t t costing job. For example, if only 20 parts for S direction, the ° ∂∂∂t2 S S 57 ® N (7) number of computer variables will reach 20≈× 5 10 , which S 22 °WM((ln)K)(i1,2,,N)=⋅i −ˈ = " ° tT= ¦ var means too much computation cost. So, in next section, we ¯ i1= Si1− will discuss this valuation problem using Monte Carlo We can derive the closed form solution to the problem (7), simulation with control variate technique. by means of no arbitrage in market and continuity ofW at ti . III. CONTROL VARIATE TECHNIQUES FOR MC METHOD Firstly, when there’s a single observation point, the model is =<<+∞<≤ °­LW1 0 0 S ,0 t T We start by considering the problem of estimating by MC ® Wh(S,S).= simulation the expectation of a random variable V , the ¯° tT= 0 V The analytical solution is given by discounted payoff of a derivative as in [12]. Let i be an +∞ W(S ,0)=π−−θσθ−βσθ 1 2 exp[ rT22 2]h(S exp( T T))d (8) output of the ith iteration of the simulation, done in a way so 00cc³−∞ 111

that the replications VV1,," m obtained after n iterations are Secondly, when there are N observation points, the model is

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Authorized licensed use limited to: East China Univ of Science and Tech. Downloaded on January 27, 2010 at 01:27 from IEEE Xplore. Restrictions apply. =<<+∞≤<= =<<+∞<≤ °°­LW1N 0,0 S ,t N1− t t N T, ­LW1i 0,0 S ,t i1− t t, i are standard normal random variables with coefficient ®®= ==−+ ρ Wh(S,S,,S,S),N01N1= " − WW(S,t),(i1,2,,N1).ii1ii= + " which can be generated by two independent variables, ¯° ttN ¯° tti 2,j=ρ 1,j + −ρ 2 j << < = = ZZ1Ukk k. Then the replication j of the stock Where 0,(1,2,,)ttTWiN1 ""Ni denotes the value of

prices St following processes (1) and (2) is simulated; W during [,]ttii−1 . By applications of (8) and the method of induction, the value of the derivative with N observation 4. According to the clause of variance swap and replication N dates can be expressed as =−−rT 2,j 2, j 2 2 j in 3, set the price VMe((ln(SS))K)jii1var¦ − ˗ +∞ +∞ i1= W(S ,0)=π−−(θ++θ2⋅"" (2 )−N2 exp[ rT 2 2 ) ] h(S , 01N0³³−∞ −∞ =− − 5. Set V(b)jjj V b(X E[X]) , where E[]X is given by σΔθ−σΔ22 σΔθ−σΔ S0 exp( 1c t 1 1ȕ 1 1ct), 1" ,Sexp( 0 1c t 1 1ȕ 1 1ct 1 WS(,)0 0in Theorem 3.1; +,"" ,+σ Δ t θ −β σ2 Δ t ))d θ d θ (9) Nc N NΝ Nc N 1 N 6. The control variate estimate of the price of variance swap α=−− σ−222 −σ β= − σ 2 Where iiciciicr 1/ 2 (r / 2) , 1/ 2 r / . (10) under stochastic volatility structure is finally obtained from 1 m Since hS(,1 " SN )is given by (3), the Theorem 3.1 is then ==−− the mean of m replications, V(b)¦ Vj (b) V b(X E[X]) . m = obtained. j1 σ Theorem 3.1 The price of variance swap with N observation With a fixed list of piecewise volatility constants ic , and *1=σ ρ σ− = [][]−1 dates under the stochastic process (6) is given by the optimal coefficient b VVXX Cov X,V Var X NN =−rT ⋅ σΔ+ 2 βσΔ− 2 4 2 2 which minimizes the variance Var[()] V b . However, in W(S0iciiicivar ,0) e M (¦¦ t t K ). j i1== i1 practice, Cov[,] X V is unknown, we may use an estimate of Where M ,,rKTσ , , being seen in Section I and IIˈand ic var b* which is given by β = i (1,2,,)iN" being equation (10). mm  =−− −21− bm ¦¦(Vjj V)(X X)( (X j X) ) . Proof: Substituting hS(,1 " SN )into (9) yields j1== j1

N +∞ +∞ − The key step of the control variate in this algorithm lies in W(S , 0)= M"" (2 π )N2 exp[ − rT − ( θ 2 + +θ 2 ) 2]( σ Δ t θ −βσ 2 Δ t ) 2 d θ " d θ 01Niciiiici1N¦³³−∞ −∞ σ = i1= choosing suitable volatility constants ic (1,,)iN" . We hope +∞ +∞ −π−−θ++θθθ=− M""" (2 )−N2 exp[ rT ( 2 2 ) 2]K 2 d d M(I I ). σ ³³−∞ −∞ 1Nvar1N12 to choose a representative group of ic to make sure high Then by calculating directly, correlation between the control variate X and V . With the N +∞ help of moments relations between the processes (1) (2) and I=π K2N/2 (2 )∏ exp[ −−θθ=− rT 2 / 2]d K 2 exp( rT). 2var³−∞ i ivar i1= (6), we obtain Theorem 3.2. N +∞ +∞ Theorem 3.2 When the list of piecewise volatility constants I=−θ++θσΔθ−"" exp[ (2222332242 ) 2]( t 2β σΔ t θ+β σΔ t )d θ " d θ 1¦³³−∞ −∞ 1 N icii iicii iici1 N σ = σ=2 μ − μ μ − i1= ic (,,)i1" Nsatisfy icY 0 (exp( t i ) exp( t i−− 1 )) (t i t i 1 ) ˈthe N ⋅π−N2 − = 1 + 2 + 3 − (2 ) exp( rT)¦ (Qii Q Q i )exp( rT). first and second moments of the process (6) are nearly equal i1= +∞ +∞ to those of processes (1) (2) at observation date ti . Where Qexp(()2)tddt,122222="" −θ+ +θ σΔθθ " θ=σΔ i³³−∞ −∞ 1 N icii1 N ici Proof: At the observation date t1 , the first two moments of +∞ +∞ Q22233242=−θ++θσβΔΔθθθ==βσΔ"" 2exp( ( ) 2) t t d " d 0, Q t . i1Niciiii1Niiii³³−∞ −∞ process (6) are N (1)==+ (1) 2 2σ 2 22422 E(S ) S exp(rt ),E((S ) ) S exp(2rt t ). = − σΔ+βσΔ − t01t11 0 11c1 So W(S0iciiicivar ,0) M exp( rT)(¦ ( t t ) K ). i1= For the processes (1) and (2), we have tt E(S(2) )=−+= E(S exp(11 (r 1 2 Y(t))dt Y(t)dW )) S exp(rt ), B. Control Variate for Valuation of the Variance Swap t01 ³³00 1t01 ttt The algorithm for the valuation of variance swap with E((S(2) ) 2 )=+ S 2 E[exp(2rt 2111 Y(t)dW −+ 2 Y(t)dt Y(t)dt)]. control variate technique is given as follows: t01 1³³³000 1t t If stochastic integral 1 Y(t)dt is approximated by the mean Δ=tTnt = − t ³0 1. Divide [0,T ] into n parts with mesh size k1+ k , t1 and make sure time discretization points {12,," , n} include all integral E(Y(t))dt , then by the exponential martingale ³0 the observation dates {12,," , N} . Based on diffusion (6), theorem, the second moment is expressed as t generate standard normal random number Z 1, j and set E((S(2) ) 2 )=+ S 2 exp(2rt1 E(Y(t))dt) =+μ−μ S 2 exp(2rt Y (exp( t ) 1) ). k t011 ³0 0101 1,j=−σΔ+σΔ= 1,j 2 1,j 1,j S(t)k1+ S(t)exp((r1/2 k ic )t ic tZ),S(t) k 0 S 0 . Then a If the initial values between the two processes are same, and σ 2 μ − μ replication j of the stock price St is attained; 1c is equal to Y(exp(t)01 1) t 1, then their corresponding

2. According to the contract and replication j in , and set moments are equivalent at t1 . N −rT 1,j 1,j 2 2 Then at the observation date t2 , for process (6), we have the price of control variate XMe((ln(SS))K)=−− ; jii1var¦ (1) (1) 2 2 2 2 i1= E(S )= S exp(rt ), E((S ) ) = S exp(2rt +σ t +σ (t − t )). t02t22 0 21c12c21 3. Set S(t)S,S(t)2,j== 2,j S(t)exp((r1/2((t)))t 2,j −σΔ+σΔ j 2 j (t) tZ), 1,j 00 k1k+ k k k For processes (1) and (2), we have 1, j j tt Z σ ()t (2) 22 with the same sequence k in 1, where k is governed by E(S )=−+= E(S exp( (r 1/ 2Y(t))dt Y(t)dW )) S exp(rt ), t02 ³³00 1t02 jjjj2 2,j σ=(t ) Y (t ),Y = Y exp(( μ−σΔ+σΔ 1/ 2ˆˆ ) t tZ ), 2, j 1, j ttt kk+ kZ and Z 222 k1 k k k E((S(2) ) 2 )=⋅ S 2 E[exp(2rt + 2 Y(t)dW − 2 Y(t)dt + Y(t)dt)]. t02 2³³³000 1t

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Authorized licensed use limited to: East China Univ of Science and Tech. Downloaded on January 27, 2010 at 01:27 from IEEE Xplore. Restrictions apply. t t Similarly, the integral 2 Y(t)dt is replaced by 2 E(Y(t))dt, then 01519. given by Theorem 3.2. However, we are glad to see ³0 ³0 that their corresponding variance reduction ratios are almost t E((S(2) ) 2 )=+ S 2 exp(2rt2 E(Y(t))dt) =+μ−μ S 2 exp(2rt Y (exp( t ) 1) ). similar and their errors are also equal. So, they still confirm t022 ³0 0202 In order to make their corresponding moments same at t , the greatest efficiency of the controls given by Theorem 3.2. 2 σ Fig.1 and Fig.2 show that when the searched c is equal we need σ−=22(t t ) Y (exp( μ−μ−σ t ) 1) t . And so on, when 2c 2 1 0 2 1c 1 to 01500. , the ratio of variance reduction is maximal, which σ−2 (t t ) = Y (exp( μ−μ t ) exp( t )) μ= ,(i 1," , N), ic i i−− 1 0 i i 1 is in line with the result in Theorem 3.2. They also show the first two moments of the processes (6) and (1) (2) that the variance reduction ratio is symmetrical distributed at t are approximately equal. σ * = i with the centre c 01500. . That is to say the process (6) is equal to processes (1) (2) σ = in the sense of “moments”, if ic (,i12N" ) satisfies the TABLE I. SIMULATION RESULTS relation in Theorem 3.2. The control variate in this case is The results with N1,0N viewed as the “optimal control” and regarded to have the Paths σ (1) σ (2) R Price Error greatest efficiency of variance reduction. 1000 0.1500 0.1500 159.0614 21.39 0.0061 IV. COMPUTATIONAL RESULTS AND ANALYSIS 2000 0.1500 0.1500 147.3817 21.39 0.0047

In this section, we will present some computational 5000 0.1500 0.1500 146.2758 21.40 0.0031 results based on the algorithm in Section III and the list of σ = 8000 0.1500 0.1500 144.3252 21.39 0.0026 control volatility constants ic (,i12N" )given by Theorem ==σ= ==2 10000 0.1500 0.1500 142.7051 21.39 0.0022 3.2. Parameters M 1000,., r 0 05 0.01, T1Y015,.0 are chosen and time period of[,0T ]is divided into100 parts with the mesh size %tT100 , according to data on the clause. As

strike volatility Kvar is just a constant decrement, having no = effect on the numerical results, Kvar 0 is supposed. The results with parameters μ = 0005010015,. ,. ,. and observed frequency N1N5==, are recorded separately. In order to confirm the optimality of the control variate governed by Theorem 3.2, an unconstrained optimization

problem max{ Var[Vjj ] min{Var[V (b)]}} is also discussed. Direct σ b c Figure 1. The searched results with N1,0N search method is used to solve this optimal problem as in σ * [14]. Then the searched optimal volatility ic is compared with the volatility determined by Theorem 3.2. Where b* is TABLE II. SIMULATION RESULTS −1 equal to Cov[,] X Y Var () X ˈ with their sample counterparts The results with N1,0.05N

 (1) (2) yielding the estimate bm . Paths σ (Reduction σ (Reduction In Table I and Table II, the variance reduction ratio, ratio) ratio) Price Error estimate error with control variate and the searched 1000 0.1519(85.0311) 0.1520(85.0644) 23.33 0.0118 σ * N1==,μ 0 N1==,.μ 005 result c with parameters and are 2000 0.1519(81.9947) 0.1520(82.0698) 23.33 0.0087 listed respectively. Here σ (1) is used to denote volatility 5000 0.1519(82.4042) 0.1520(82.4074) 23.34 0.0056 constant derived from Theorem 3.2, σ (2) to denote the searched volatility of the optimization problem and R to 8000 0.1519(80.5270) 0.1519(80.5270) 23.34 0.0045 denote variance reduction ratio. Fig.1 and Fig.2 present 10000 0.1519(82.2752) 0.1519(82.2752) 23.34 0.0040 some searched results of their optimal volatility constant σ * . c From Table I we see that the searched resultσ ()2 = 01500. equals to the result given by Theorem 3.2˗The algorithm has obvious variance reduction effects, with variance reduction ratios greater than 140. With this method, the satisfied errors can be obtained without need of simulating so many paths. As the number of paths increases, the error with control variate decreases gradually and the simulated price is stable every time. In Table II, the results have the same conclusions as those in Table I, except that the searched 01520. results of the optimal volatility is (when the number of paths m is equal to 1000 5000 ), a little different from Figure 2. The searched results N1,0.05N

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Authorized licensed use limited to: East China Univ of Science and Tech. Downloaded on January 27, 2010 at 01:27 from IEEE Xplore. Restrictions apply. The below Table III-V record separately the outcomes V. CONCLUSIONS with parameters N5==,.,μ 005N5==,.μ 001and N5==,.μ 015. In this paper, we build the pricing model for the variance In addition, a contrast is made to the results (R2) with only a swap with PDE method under stochastic volatility structure. single control volatility constant over all the time range of Then based on the closed form solutions in a simpler model, [,0T ]. Variance reduction ratio with piecewise volatility control variate technique is applied to the valuation of the

controls is denoted by R1. According to Th. 3.2, the piece- swap with more complicated processes. Then with the wise control volatility constants of the case N5==,.μ 005 are analysis of the moments for the underlying processes, we provide a favorable method to determine how to choose an 0.,.,.,.,.. 1504 0 1511 0 1519 0 1526 0 1534 . For the case of N5= , efficient control variate for Monte Carlo simulation. The idea μ = 010. , the piecewise control volatility constants are in this paper can also be extended to other variance swaps, 0.,.,.,.,. 1508 0 1523 0 1538 0 1553 0 1569 . For the case of N5==,.μ 015, such as Corridor variance swap, Gamma variance swap and the piecewise controls are 0.,.,.,.,.. 1511 0 1534 0 1557 0 1581 0 1605 Conditional variance swap, even the derivatives with other The simulation results with control variate show that the stochastic volatility models or multi-factor models. algorithm has obvious variance reduction effects. And the ACKNOWLEDGMENT comparisons show that the variance reduction ratios with piecewise control volatility constants are evidently greater Thank for the support of National Basic Research Program than those with single control volatility constant, especially of China (973 Program) 2007CB814903. And we also for bigger μ . Moreover, the price of the swap is increasing appreciate Professor Lishang Jiang’s instructive guide and along with increasing μ . The reason is that if the volatility suggestions.

has an increase tendency, the asset will have a wild fluctuation. High risk means high return, and therefore the REFERENCES price of the variance swap grows with increase of μ . [1] E. Derman, M. Kamal, J. Zou, and K. Demeterfi, “A guide to volatility and variance swaps,” The Journal of Derivatives, vol. 6, pp. TABLE III. SIMULATION RESULTS 9–32, Summer 1999. [2] J. Detemple and C. Osakwe, “The valuation of volatility options,” The results with N5,0.05N European Finance Review, vol. 4, pp. 21–50, Jan. 2000. Paths R1 R2* Price Error [3] J. Hull, Options Futures and Other Derivatives, 4th ed, Prentice Hall, 1000 86.6001 53.624 22.22 0.0049 New Jersey, 2000. [4] O. Brockhaus and D. Long, “Volatility swaps made simple,” Risk, 2000 87.7722 55.9692 22.22 0.0036 vol. 2, pp. 92–95, Jan. 1999. 5000 86.6895 55.3777 22.22 0.0023 [5] T. Little DQGV. Pant, “A finite difference method for the valuation of variance swaps, ”Journal of Computational Finance, vol. 5, pp. 81– 8000 86.2136 54.9926 22.22 0.0018 103, Fall 2001. [6] J. Carr, P. Geman, H. Madan, and D.Yor, “Pricing options on realized 10000 86.2484 55.0507 22.22 0.0016 variance, ” Finance and Stochastic, vol. 9, pp. 453–475, 2005. [7] J. Hull and A.White, “The pricing of options on assets with stochastic TABLE IV. SIMULATION RESULTS volatilities,” Journal of Finance, vol. 42, pp. 281–300, June 1987. The results with N5,0.05N [8] L.O. Scott, “ pricing when the variance changes randomly: Paths theory, estimation, and an application,” Journal of Financial and R1* R2* Price Error Quantitative Analysis, vol. 22, pp. 419–438, Dec. 1987. 1000 82.0851 32.0454 22.78 0.0053 [9] E.M. Stein, and J.C. Stein, “Stock price distributions with stochastic volatility: an analytic approach,” Review of Financial Studies, vol. 4, 2000 83.4307 33.2450 22.77 0.0039 pp. 727–752, Winter 1991. 5000 82.1819 32.9354 22.77 0.0025 [10] C.Ball and A.Roma, “Stochastic volatility option pricing,” Journal of Financial and Quantitative Analysis, vol. 29, pp. 589–607, Dec. 1994. 8000 81.8193 32.7475 22.77 0.0020 [11] S.L. Heston and L.Steven, “A closed-form solution for options with 10000 81.8470 32.7740 22.77 0.0016 stochastic volatility, ” Review of Financial Studies, vol. 6, pp. 327– 343, 1993.

TABLE V. SIMULATION RESULTS [12] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004, pp.185–195. The results with N5,0.05N [13] B. L. Nelson, “Control variate remedies,” Operations Research, Vol. Paths R1* R2* Price Error 38, pp. 974–992, Nov. 1990. 1000 76.0471 22.2537 23.35 0.0059 [14] A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Science Press, Beijing, 2006, pp. 300–305. 2000 77.4806 23.0953 23.34 0.0043

5000 76.3413 22.8921 23.35 0.0028 8000 76.0562 22.7557 23.34 0.0022 10000 76.1191 22.7560 23.34 0.0019

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