IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______

Fast Monte Carlo Simulation for Pricing Covariance under Correlated Stochastic Models Junmei Ma, Ping He

Abstract—The modeling and pricing of covariance swap with probability 1 − δ, where σn is the standard deviation derivatives under correlated models are of estimation of V , δ is significance level and Z δ is the 2 studied. The pricing problem of the under the case δ of discrete sampling covariance is mainly discussed, by efficient quantile of standard normal distribution under 2 . The error √σn is Z δ , which just rely on the deviation σ and sample Control Variate Monte Carlo simulation. Based on the closed n 2 form solutions derived for approximate models with correlated storage n.However, although Monte Carlo method is easy deterministic volatility by partial differential equation method, to implement and effective to solve the multi-dimensional two kinds of acceleration methods are therefore proposed when problem, it’s disadvantage is its slow convergence. The the volatility processes obey the Hull-White stochastic processes. − 1 Through analyzing the moments for the underlying processes, convergence rate of Monte Carlo method is O(n 2 ).We the efficient control volatility under the approximate model have to spend 100 times as much as computer time in order is constructed to make sure the high correlation between the to reduce the error by a factor of 10 [8]. control variate and the problem. The numerical results illustrate Many financial practitioners and researchers concentrate the high efficiency of the control variate Monte Carlo method; on the problem of computational efficiency, several ap- the results coincide with the theoretical results. The idea in the paper is also applicable for the valuation of other financial proaches to speed up Monte Carlo simulation, such as derivatives with discrete features under multi-factor models. control variate, antithetic variables, importance sampling and stratification[7] have been proposed over the last few years. Index Terms—Covariance swap, Stochastic volatility, Monte Carlo, Control variate, Variance reduction. These techniques aim to reduce the variance per Monte Carlo simulation so that a given level of accuracy can be achieved with a smaller number of simulations. Control variate is one I.INTRODUCTION of the most widely used variance reduction techniques, main- ly because of the simplicity of its implementations, and the ONTE Carlo method is a numerical method based on fact that it can be accommodated in an existing Monte Carlo M the probability theory. Monte Carlo method becomes calculator with a small effort. This paper mainly studies the more and more popular, as the rapid growth of the financial pricing of covariance derivatives under correlated stochastic derivative markets and the increase of the complexity of the volatility models through fast Monte Carlo simulation. pricing models. The advantage of Monte Carlo method is that Different control variates are proposed to accelerate the the efficiency of the method is independent of the number convergence rate of the simulation errors, based on the of state variables. If the number of state variables is greater closed form solutions of simplified models. The numerical than three, Monte Carlo method is suitably used to solve experiments show the high efficiency of our acceleration the high dimensional pricing problem and often becomes the methods, which can also be extended to the pricing of other only computationally feasible means of derivative pricing. path-dependent derivatives in an extended Black-Scholes Suppose the price of the derivative∑ µ = E[V ] will be framework, such as Asian options, Lookback options and 1 n { }n estimated by mean value V n = n i=1 Vi, where Vi i=1 other variance derivatives. are independent and identical distribution (i.i.d) samples of The covariance swap is a covariance forward contact of the random variable V . Then by the Central Limit Theorem, the underlying assets S1 and S2, and its payoff at the price µ asymptotically falls into the interval is equal to σn σn P = M × (Cov (S ,S ) − K2 ), [V n − √ Z δ , V n + √ Z δ ], R 1 2 cov n 2 n 2 where Kcov is a , M is the notional amount, Manuscript received June 11, 2015; revised Oct 2, 2015. This work and CovR(S1,S2) is the realized covariance between two was supported by NSFC (NO.11226252, NO.11271243 and NO.11271240), assets S1 and S2. Therefore, in the risk-neutral world, the fair the Special Fund for Shanghai Colleges Outstanding Young Teacher- strike covariance can be attained by E[CovR(S1,S2)]. The s Scientific Research Project(No.ZZCD12007), the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Educa- procedure for calculating the realized covariance is usually tion of China(No.708040) and Research Projects of Young Teachers in clearly specified in the contract and includes details about SUFE(No.2011220656). the source and observation frequency of the price of the Junmei Ma is with the School of Mathematics, Shanghai University of Finance and Economics, Shanghai Key Laboratory of Financial Information underlying assets, and the method to calculate the covariance. Technology, 777 Guoding Road, Shanghai 200433, China(phone:+86-21- The covariance swap is the main representation of the 65903296;fax:+86-21-65904320;e-mail:[email protected]) new generation of variance and derivatives, Ping He is with the School of Mathematics, Shanghai Universi- ty of Finance and Economics,777 Guoding Road, Shanghai 200433, and becoming very popular and actively traded in financial China(phone:+86-21-65908857;e-mail:[email protected]) practice in recent years. Variance and volatility derivatives

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IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______

are based on a single underlying asset. Their payoffs just and discretely sampled covariance. For the continuously depend on the realized variance or volatility of the underlying sampled covariance derivatives, a closed form solution is asset or an index. The market for variance and covariance derived directly by the PDE method. For the discretely swaps has been growing, which are now actively quoted and sampled covariance derivatives, two kinds of fast Monte used to speculate on future (co)variance levels, to trade the Carlo simulation methods are developed and studied. The spread between implied and realized (co)variance levels, or discretely sampled covariance derivatives are more applicable to the (co)variance exposures of other positions by in financial practice as the assets ”covariance” cannot be many financial institutions. The objective of this paper is observed directly in the market. The choice of efficient to demonstrate the modeling and the pricing of a covariance control variates is also discussed in the paper. swap. This contract pays the excess of the realized covariance The rest of the paper is arranged as follows. We begin by between two currencies over a constant specified at the outset modeling covariance swaps with the PDE method and ∆- of the contract. Such a contract is growing as a useful hedging principle under the stochastic volatility assumption complement for the variance contracts that trade OTC on in Section 2. In Section 3, the first control variate method of several currencies. By combining variance and covariance Monte Carlo simulation for the valuation of the covariance swaps, the realized variance of return on a portfolio of swap is proposed. The analytical solution for the control currencies can be locked in. variate is attained. Then, with the moment analysis for the Logically, a correlation swap is a correlation forward underlying and auxiliary stochastic processes, an approx- contract of two underlying rates S1 and S2, whose payoff imate “optimal volatility” is provided for developing the at expiration is equal to: highly efficient control variate in this framework. In Section 4, the second improved control variate method is constructed × − P = M (CorrR(S1,S2) Kcorr), based on the algorithm in Section 3 and the choice of

where Kcorr is a strike correlation, M is the notional amount, the optimal control volatility constants is also considered. and CorrR(S1,S2) is the realized correlation of two under- In Section 5, the computational and analytical results are lying assets S1 and S2. Therefore the fair strike correlation illustrated and coincide with the theoretical analysis well. is E[CorrR(S1,S2)]. The correlation CorrR(S1,S2) can be Concluding remarks are given in Section 6. presented by the covariance, II.MODELINGAND PRICINGOF COVARIANCE SWAP CovR(S1,S2) CorrR(S1,S2) = . N this section, the partial differential equation pricing σR(S1)σR(S2) I model for the covariance swap under the correlated Therefore, the modeling and pricing of correlation swap will stochastic volatility assumption is obtained. The variance be attained with the research results of covariance swap. products are all derivatives based on stochastic volatility The research on the pricing of and volatility model. The research on the stochastic volatility starts from swap is very massive and we do not intend to give a list here. the early 1980’s. In 1973, Black and Scholes made a major We will give a brief introduction of the pricing of covariance breakthrough by deriving pricing formulas for vanilla options swap and correlation swap. As these are two-dimensional written on the stock. The Black-Scholes model assumes path-dependent derivatives. Their pricing problems are more that the volatility term is a constant. This assumption is difficult than general variance swap or volatility swap which not always satisfied by real-life options as the probability is only based on single underlying asset. Andrei Badescu distribution of an equity has a fatter left tail and thinner etal [1] discussed the analytical pricing of pseudo-variance, right tail than the lognormal distribution as in Hull [14], pseudo-volatility, pseudo-covariance and pseudo-correlation and the assumption of constant volatility in financial model swaps. Sebastien Bossu [2]and[3] from JPMorgan proposed is incompatible with derivatives prices observed in the mar- a ”toy model” for modeling and pricing correlation swaps ket, verified by volatility smile. The concept of stochastic on the components of an equity index and found that the volatility was introduced by Hull and White (1987) [15], and fair strike of a correlation swap is approximately equal subsequent developments include the work of Scott (1987) to a particular measure of implied correlation. Giovanni [19], Stein and Stein (1987)[16], Ball and Roma (1994) Salvi and Anatoliy Swishchuk[4] researches the modeling [17], and Heston (1993) [18]. They proposed and improved and pricing of covariance and correlation swaps with semi- different stochastic volatility models to satisfy the various Markov volatilities assumption. Da Foneseca et al.[5] solved needs in financial practice. a portfolio optimization problem in a market with risky The stochastic volatility model used in this paper is the assets and volatility derivatives to discuss the influence of Geometric Brownian Motion proposed by Hull and White variance and covariance swap in a market. J. Drissien etal in 1987. Under the martingale measure, the underlying [6] discussed the price of correlation risk for equity options. assets and volatilities are assumed obeying the stochastic The research on the covariance swap and correlation swap differential equations: √ is developed extensively due to its importance and broad dS(1)(t) = rdt + σ(1)dW (1), σ(1) = Y (1)(t), applicability in risk-hedging, and manage the risk S(1)(t) t t t of the fund in the financial transactions. dY (1)(t) In this paper, by PDE method and Monte Carlo variance = µ(1)dt +σ ˆ(1)dW (3). (1) reduction techniques, the modeling and pricing of covariance Y (1)(t) t swap derivatives under the correlated stochastic volatility √ structure are researched. Modeling and pricing of the deriva- dS(2)(t) = rdt + σ(2)dW (2), σ(2) = Y (2)(t), tives are separately discussed under the cases of continuously S(2)(t) t t t

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IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______

(2) dY (t) (2) (2) (4) The solution of the above equation (5) has the following = µ dt +σ ˆ dWt . (2) Y (2)(t) semi-linear expression, √ Where r is deterministic ; µ(i) > 0, (i = 1, 2) is W (Y (1),Y (2), I, t) = A(t; T )I+B(t; T ) Y (1)Y (2)+C(t; T ), the drift of the volatility; σˆ(i) > 0, (i = 1, 2) is the volatility (i) of volatility; Wt , (i = 1, 2, 3, 4) are Wiener processes, and where (i) (j) Cov(dW , dW ) = ρ dt. We further suppose that the −r(T −t) t t ij Mte − − N A(t; T ) = ,C(t; T ) = −MK2 e r(T t), market is no-arbitrage and no transaction costs. There are T cov observation dates 0 = T0 < T1 < T2 <, ··· , < TN = T . (1) (2) −a(T −t) −r(T −t) S = S(1)(T ) S = S(2)(T ) Mρ12[e − e ] i i and i i are the asset price on B(t; T ) = , (1) (1) (2) (2) T (r − a) the ith date, and Yi = Y (Ti) and Yi = Y (Ti) are the instantaneous variance at Ti. The payoff function for the

covariance swap at the maturity T is 1 (1) 2 1 (2) 2 1 (1) (2) 1 (1) 1 (2) a = (ˆσ ) + (ˆσ ) − ρ34σˆ σˆ − µ − µ +r. ∑N 1 S(1) S(2) 8 8 4 2 2 V | = M × [ (ln( i ))(ln( i )) − K2 ] (6) t=T T (1) (2) cov i=1 Si−1 Si−1 Therefore the price formula of this kind of covariance swap is obtained from the above price expression. (1) (2) (1) (2) ··· (1) (2) = h(S0 ,S0 , S1 ,S1 , ,SN ,SN ). (3) Theorem 2.1 The price of the covariance swap for the For the case of continuously sampled covariance, the payoff case of continuously sampled covariance with payoff (4) is √ function denoted by W (.) for the covariance swap at the Mρ Y (1)Y (2)[e−aT − e−rT ] maturity T is | 12 0 0 − 2 −rT ∫ √ √ W t=0 = − KcovMe , 1 T T (r a) W | = M × ( ρ Y (1) Y (2)dt − K2 ), (4) t=T T 12 t t cov (1) (2) 0 where Kcov,Y0 ,Y0 , M, µ, T, r and a are given as in as the realized covariance of the two assets S(1) and S(2) is equations (1), (2), (3)and (6). given by Giovanni S.etal. [4] For the case of discretely sampled covariance, the payoff ∫ T function is expressed by equation (3). Then we will discuss (1) (2) 1 (1) (2) 1 (1) (2) the pricing model for the covariance derivative with PDE CovR(S ,S ) = [ln ST , ln ST ] = ρ12σt σt dt. T T 0 method. There are four sources of randomness about this Then the pricing model for the price of the covariance problem. When constructing a risk-free portfolio, the deriva- swap with continuously sampled covariance will be estab- tives cannot be perfectly hedged with just the underlying lished, and the closed form solution will be deduced. assets. Instead we need another two covariance swaps called 1∗ 2∗ Define a auxiliary state variable It to measure the accu- Vt and Vt on the same underlying assets. Then during an (1) (2) mulated covariance of S and S , arbitrary observation interval (T − ,T ), we construct a risk- ∫ √ √ i 1 i t (1) (1) less portfolio Π, containing the product V , the quantity ∆1 ρ12 Y (s) Y (s)ds I = 0 . of the underlying asset S , the quantity ∆ of the underlying t t 1 2 asset S2, the quantity ∆3 of another traded covariance swap This state variable is known at time t and satisfies the 1∗ Vt and the quantity ∆4 of another traded covariance swap ordinary differential equation, 2∗ − √ √ Vt . By Itoˆ lemma and ∆ hedging principle, choosing (1) (2) dI (ρ Y (t) Y (t) − I ) ∆1, ··· , ∆4 to make Π risk-less during [t, t + dt]. Then the t = 12 t . dt t equation governing V can be written as in Jiang [20] The price process of the derivative is denoted by W = L ∂V (1) ∂V (1) (1) ∂V (2) ∂V (1) (2) V = ∂t + rS ∂S(1) + µ Y ∂Y (1) + rS ∂S(2) + ∑ 2 W (Yt ,Yt ,It, t). During the small time interval (t, t + (2) (2) ∂V − 1 4 ∂ V µ Y (2) rV + ρijaiajZiZj = 0. dt), by Itoˆ lemma, the change of the price of the covariance ∂Y 2 i,j=1 ∂Zi∂Zj √ (7) swap will be (1) Where,√ for convenience with a1 = Y , a2 = ∂W ∂W ∂W 1 ∂2W 2 (2) (1) (2) dW = dt + dY (1) + dI + dY (1) + Y , a3 = σˆ , a4 =σ ˆ and Z1 = S1,Z2 = S2,Z3 = (1) 2 (1) (2) ∂t ∂Y ∂I 2 ∂Y (1) Y ,Z4 = Y in the last part of the above expression. 2 2 As the market is assumed no arbitrage, the price of the 1 ∂ W (2)2 ∂ W (1) (2) ∂W (2) dY +ρ34 dY dY + dY covariance swap should be continuous at observation date 2 ∂Y (2)2 ∂Y (1)∂Y (2) ∂Y (2) Ti. Take V = Vi(i = 1, 2, ··· ,N), t ∈ (Ti−1,Ti), the partial As there is no default risk and based on risk neutral con- differential equation pricing model for the covariance swap ditions, E[dW ] = rW dt will be satisfied. Then the partial (1) (2) is therefore given by differential equation of the price process W (Yt ,Yt ,It, t)  of the covariance swap with payoff (4) is attained as follows,  LV = 0, 0 < S(1),S(2),Y (1),Y (2) < ∞,  N ∂W (1) (1) ∂W 1 (1) 2 (1) 2 ∂2W TN−1 ≤ t < TN = T,  + µ Y + (ˆσ ) (Y ) 2 +   ∂t ∂Y (1) 2 ∂Y (1) (1) (2) (1) (2) (1) (2)  2 V | = h(S ,S , ··· ,S ,S ,S ,S ),  µ(2)Y (2) ∂W + 1 (ˆσ(2))2(Y (2))2 ∂ W + N t=T 0 0 N−1 N−1 ∂Y (2) 2 ∂Y √(2)2 2 (1) (2) (1) (2) (1) (2) ∂ W ∂W ρ12 Y Y −I and  ρ34σˆ σˆ Y Y ∂Y (1)∂Y (2) + ∂I t {  − (1) (2) ∞ ≤ ≤ L (1) (2) (1) (2) ∞ ≤  rW = 0, 0 < I, Y ,Y < , 0 t T, Vi = 0, 0 < S ,S ,Y ,Y < ,Ti−1 t < Ti. | − 2 | | ··· − W t=T = M(IT Kcov). Vi t=Ti = Vi+1 t=Ti , (i = 1, 2, ,N 1). (5) (8)

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IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______

∗ Cov[X,P ] If the analytical solution The optimal coefficient b = Var(X) minimizes the vari- (1) (2) (1) (2) ··· (1) (2) (1) (2) VN (S0 ,S0 , S1 ,S1 ,SN−1,SN−1,S ,S , t) ance, which is given by is obtained from the first problem, then at t = TN−1, ∗ − 2 (1) (1) (2) (2) | Var(P (b )) = (1 ρXP )Var(P ). S = SN−1,S = SN−1, and VN−1 t=TN−1 = (1) (2) (1) (2) ··· (1) (2) VN (S0 ,S0 ,S1 ,S1 ,SN−1,SN−1,TN−1), which is This indicates that a rather high degree of correlation ρXV the terminal value of the recursion second problem with is needed for the control variate to yield better effects of i = N − 1. Therefore, the following solutions VN−1, ··· ,V1 variance reduction. And the error reduction ratio is denoted by √ 1 . The popularity of control variate rests on the can be solved by induction method. − 2 1 ρXP It is very difficulty to derive the analytical expression of ease of implementations, the availability of controls, and on (8). If the finite difference method is used directly to solve the straight intuition of the underlying theory. According to the above path dependent variable problem, there will need the variety of problems, different types of instruments can too much computational cost. So in next two sections, we be chosen as controls, including underlying assets, tractable will use fast Monte Carlo simulation method to discuss the options, bond prices, tractable dynamics, hedges and so on. above multi-dimensional pricing problem. Control variate technique is widely used mainly because of the simplicity of its implementations, and the fact that III.FAST MONTE CARLO SIMULATION FOR PRICINGTHE they can be accommodated in an existing Monte Carlo DERIVATIVE calculator with a small effort. The skill of the control variate method lies in how to choose suitable and efficient HE aim of this section is to develop fast simulation control variate. Examples of successful implementations of T algorithms for pricing covariance swap under the cor- control variate for pricing the derivatives include Hull and related stochastic volatility models. The control variate accel- White(1987)[9], Kemna and Vorst(1990)[10], Turnbull and eration techniques are proposed and used to price covariance Wakeman(1991)[11], Fu, Madan and Wang (1999)[12] and derivatives under this structure. The method of control variate Ma and Xu (2010)[13]. is one of the most widely used variance reduction tech- niques. Its popularity rests on the ease of implementation, A. Analytical Solution for the Covariance Swap with Deterministic the availability of controls, and on the straight intuition Volatility of the underlying theory. The method of control variate In this subsection, we consider another covariance swap exploits information about the errors in estimates of known under the assumption of correlated deterministic volatility quantities to reduce the error in an estimate of an unknown instead of correlated stochastic volatility models (1). The quantity explained in detail in Glasserman[7] and Xu[8]. assets are assumed to follow, These techniques aim to reduce the variance per Monte Carlo observation so that a given level of accuracy can be obtained dS(1)(t) = rdt + σ(1)dW (1), (9) with a smaller number of simulations. We just give a simple S(1)(t) t introduction on this method. (2) The main idea of control variate is through exploiting dS (t) (2) (2) = rdt + σ dWt , (10) information about the errors in estimates of known quantities S(2)(t) to reduce the errors in an estimate of an unknown quantity, as where r is interest rate; σ(1) and σ(2) are the deterministic explained in detail in [7]. Consider the problem of estimating volatility constants which will be chosen in next subsection; the expectation E[P ], where the random variable P is the (1) (2) Wt and Wt are the same Wiener processes as in (1) and discounted payoff of a derivative. Denote P , ··· ,P be (1) (2) 1 m (2), and E(dW dW ) = ρ dt. Let W = W (S, t) denote outputs from m replications of simulations and suppose that t t 12 the price of this auxiliary derivative and the payoff function P , (i = 1, ··· , m) are independent and identical distributed i at maturity is also h(S(1),S(2),S(1),S(2), ··· ,S(1),S(2)) as (i.i.d.). The Monte Carlo estimator of the price is the average 0 0 1 1 N N in (3). Then similar to the derivation in Section 2, the PDE P1 + ·· · + Pm pricing model of the product is given by P = .  m 2  ∂W (1) ∂W 1 (1)2 (1)2 ∂ W  L W := + rS + σ S 2 +  1 ∂t ∂S(1) 2 ∂S(1) Suppose that on each simulation there is another output X  2 2 2 i  rS(2) ∂W + 1 σ(2) S(2) ∂ W +  ∂S(2) 2 ∂S(2)2 along with Pi and the pairs (Xi,Pi) are i.i.d. The expectation  2 ρ σ(1)σ(2)S(1)S(2) ∂ W − rW = 0, E[X] = E[Xi] is assumed known. Then for any fixed 12 ∂S(1)∂S(2)  (1) (2) ∞ ≤ coefficient b we can calculate Pi(b) = Pi − b(Xi − E[X])  0 < S ,S < ,Ti−1 < t Ti,  (1) (2) (1) (2) (1) (2) from the ith simulation. So the control variate estimator of  W | = h(S ,S ,S ,S , ··· ,S ,S ),  t=T 0 0 1 1 N N the derivative price is given by W | + = W | − , (i = 1, 2, ··· ,N). t=Ti−1 t=Ti−1 (11) 1 ∑m P (b) = (P − b(X − E[X])) = P − b(X − E[X]), The closed form solution to the above problem (11) will be m i i i=1 derived by partial differential equation method. We use the recursive method to solve these equations. − here the observed error (X E[X]) serves as a control. When there’s a single observation point, the model is sim- − − The control variate estimate P (b) = P b(X E[X]) is plified as an unbiased estimator of E(P ), and its variance is { (1) (2) √ L1W = 0 0 < S ,S < ∞, 0 < t ≤ T, 2 (1) (2) Var(P (b)) = Var(P )+b Var(X)−2bρXP Var(P )Var(X). W |t=T = h(S ,S ).

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IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______

The analytical solution is given by Theorem 3.1 The price of the covariance swap deriva- (1) (2) W (S0 ,S0 , 0) tive with N discrete observation dates under the stochastic ∫ ∞ ∫ ∞ processes (9) and (10) is given by − (1) − (2) − rT A1 α1 A2 α2 ∑N = e h(S0 e ,S0 e ) (1) (2) M exp(−rT ) (1)2 (2)2 −∞ −∞ − σ − σ W (S0 ,S0 , 0) = T [ (r 2 )(r 2 ) j=1 2 (1) (2) − 2 f(α1, α2)dα1dα2, (12) ∆Tj + ρ12σ σ T KcovT ]. (1) (2) (1)2 (2)2 where M, r, σ , σ , ∆T ,K , T, ρ are given in Sec- where A = (r − σ )T, A = (r − σ )T , and f(α , α ) j cov 12 1 2 2 2 1 2 tion 1 and 2. is the density function of the two dimensional normal distri- Proof Substituting h(S(1),S(2),S(1),S(2), ··· ,S(1),S(2)) bution N(0, 0, σ(1)2T, σ(2)2T, ρ ), which has the expression 0 0 1 1 N N 12 into (12) yields of ∫ ∫ ∫ ∫ 1√ (1) (2) Me−rT ∞ ∞ ∞ ∞ f(α1, α2) = × ·· · (1) (2) − 2 W (S0 ,S0 , 0) = T −∞ −∞ −∞ −∞ 2πσ σ 1 ρ12T f1(α1, α2)f2(β1, β2) ··· fN (γ1, γ2) 2 2 2 (2) − (1) (2) 2 (1) [(A1 − α1)(A2 − α2) + (B1 − β1)(B2 − β2) + ··· α1σ 2ρ12σ σ α1α2 + α2σ exp(− ). +(Z − γ )(Z − γ )]dα dα dβ dβ ··· dγ dγ (1)2 (2)2 2 1 1 2 2 1 2 1 2 1 2 2σ σ (1 − ρ )T −rT 12 − 2 −rT Me − 2 KcovMe = T (J KcovT ). Then substitute the payoff function into Equation (12) and Then by calculating directly, yield the below Lemma 3.1. (1) (2) (1) (2) Lemma 3.1 The price of the covariance swap with single J = (A1A2 + ρ12σ σ ∆T1) + (B1B2 + ρ12σ σ (1) (2) discrete observation date under the stochastic process (9) and ∆T2)+, ··· , +(Z1Z2 + ρ12σ σ ∆TN ). (10) is given by Substitute these 2N expressions A1,A2,B1,B2, ··· ,Z1,Z2 (1)2 (2)2 to J, −rT σ σ 2 (1) (2) W |t=0 = Me [(r − )(r − )T + ρ12σ σ T 2 2 ∑N (1)2 (2)2 − σ − σ 2 (1) (2) J = (r )(r )∆Tj + ρ12σ σ T. − 2 2 2 Kcov], j=1

(1) (2) Therefore where Kcov, σ , σ , M, T, r, ρ12 are given as in equations ∑N (9) and (10). (1) (2) M exp(−rT ) − σ(1)2 − σ(2)2 W (S0 , S0 , 0) = T [ (r 2 )(r 2 ) Then when there are N observation points, the model is j=1 2 (1) (2) − 2 expressed as ∆Tj + ρ12σ σ T KcovT ]. { (1) (2) Remark When N → ∞ and max ∆Ti → 0, the above LWN = 0, 0 < S ,S < ∞,TN−1 ≤ t < TN = T, 1

Where 0 < T1 < ··· < TN = T , Wi(i = 1, 2 ··· N) denotes The explicit algorithm for the valuation of the covariance the value of W during [Ti−1,Ti]. By formula (12) and the swap by control variate Monte Carlo simulation is given as method of recursion, the pricing formula of the derivative follows: with N observation dates can be expressed as (i). Divide [0,T ] into n parts with mesh size ∆t = T/n = − { }n ∫ ∫ ∫ ∫ tk+1 tk, and set time discrimination ponits tk k=1 cover (1) (2) −rT ∞ ∞ ··· ∞ ∞ { }N W (S0 ,S0 , 0) = e −∞ −∞ −∞ −∞ the set of observation dates Ti i=1. (1) − ··· A1 α1 (ii). Generate correlated standard normal random numbers f1(α1, α2)f2(β1, β2) fN (γ1, γ2)h(S0 e , 1,j 2,j 3,j 4,j (2) − (1) − − ··· − Z ,Z ,Z ,Z A2 α2 ··· (A1 α1)+(B1 β1)+ +(Z1 γ1) k k k k , according to Cholesky factor and cor- S0 e , ,S0 e , relation matrix [ρkl], (k, l = 1, 2, 3, 4). Based on diffusions (2) (A2−α2)+(B2−β2)+···+(Z2−γ2) S0 e ) (1) and (2), set dα1dα2dβ1dβ2 ··· dγ1dγ2. √ − 1 (1),j 2 (1),j 1,j (13) (1),j (1),j ((r 2 (σt ) )∆t+σt ∆tZk ) S (tk+1) = S (tk)e k k , − σ(1)2 − σ(1)2 Where A1 = (r 2 )∆T1, A2 = (r 2 )∆T1, (1)2 (2)2 (1),j (1) − σ − σ ·· · St = S0 , B1 = (r 2 )∆T2,B2 = (r 2 )∆T2, ,Z1 = 0 (1)2 (2)2 √ − σ − σ ((r− 1 (σ(2),j )2)∆t+σ(2),j ∆tZ2,j ) (r )∆TN ,Z2 = (r )∆TN , and (2),j (2),j 2 t t k 2 2 S (tk+1) = S (tk)e k k , fi(∗, ∗),(i = 1, 2, ··· ,N) is the density function (2),j (2) of the two dimensional correlated normal distribution St0 = S0 , √ N(0, 0, σ(1)2∆T , σ(2)2∆T , ρ ) (1),j (2),j (1),j (1),j i i 12 . where σ and σ are governed by σ = Y Substitute the payoff function h into Equation (13) and tk √ tk tk k σ(2),j = Y (2),j solve the 2N dimensional integration problem, then Theorem and tk k , then 3.1 is achieved, which will be an important result for fast 1 √ Y (1),j = Y (1),j exp((µ − (ˆσ(1))2)∆t +σ ˆ(1) ∆tZ3,j), Monte Carlo simulation for pricing covariance swap. k+1 k 2 k

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√ (2),j (2),j − 1 (2) 2 (2) 4,j Yk+1 = Yk exp((µ (ˆσ ) )∆t + σˆ ∆tZk ), For the underlying processes (1) and (2), the mean and the 2 variance are separately The replication j of the stock prices S(1),S(2) following ∫ ∫ √ t t T T 1 (1) (1) (1) (1) ( (r− Y (t))dt+ Y (t)dW1t)) processes (1) and (2) are simulated. 0 2 0 E(ST ) = E(S0 e (iii). According to the clause of the covariance swap and replication j in (ii), set the jth price of the product (1) rT = S0 e , ∑N (1),j (2),j (1) 1 S S V ar(ST ) V = M exp(−rT )( ( ln( i ) ln( i ) − K2 )). j (1),j (2),j cov ∫ ∫ √ T T T S − S − (1) (1) i=1 i 1 i 1 (1) 2 2rT − Y (t)dt+2 Y (t)dW1t 0 0 − (iv). Based on the auxiliary processes (8) and (9), set = E[(S0 ) e (e 1)] √ (1c),j 1c,j ((r− 1 (σ(1))2)∆t+σ(1) ∆tZ1,j ) S (tk+1) = S (tk)e 2 k , (1) 2 2rT = (S0 ) e (1),j (1) (2),j (2) ∫ √ ∫ ∫ St = S0 ,St = S0 , T T T 0 0 (1) − (1) (1) √ 2 Y (t)dW1t 2 Y (t)dt+ Y (t)dt 1,j E[e 0 0 0 − 1], (2c),j 2c,j ((r− 1 (σ(2))2)∆t+σ(2) ∆tZ ) S (tk+1) = S (tk)e 2 k , 1,j 2,j and with the same sequences Zk and Zk as in (ii). Then the (1) (2) ∫ ∫ √ T T replication j of the auxiliary stock prices S ,S following − 1 (2) (2) t t (2) (2) ( (r 2 Y (t))dt+ Y (t)dW2t)) processes (9) and (10) are simulated. E(ST ) = E(S0 e 0 0 (v). According to the covariance swap contract and repli- (2) rT cation j in (iv), set the jth value of the control variate = S0 e , (2) V ar(ST ) ∑N 1 S(1c),j S(2c),j − i i − 2 ∫ ∫ √ Xj = M exp( rT )( ( ln( ) ln( ) K )). T T (1c),j (2c),j cov (2) (2) T (2) 2 2rT − Y (t)dt+2 Y (t)dW2t S − S − 0 0 − i=1 i 1 i 1 = E[(S0 ) e (e 1)] (vi). Set the jth control variate estimate = (S(2))2e2rT Vj(b) = Vj − b(Xj − E[X]), 0 (1) (2) ∫ √ ∫ ∫ where E[X] is given by W (S0 ,S0 , 0) in Theorem 3.1. T T T (2) (2) (2) 2 Y (t)dW2t−2 Y (t)dt+ Y (t)dt (vii). The control variate estimate of the price of co- E[e 0 0 0 − 1]. variance swap derivative under stochastic volatility structure ∫ (1),(2) is finally obtained from the mean of m replications, T Y (1)(t)dt ∑m ∫ If the two stochastic integrals 0 and 1 − − T (2) V (b) = Vj(b) = V b(X E[X]). ∫0 Y (t)dt can be approximated∫ by the mean integral m T (1) T (2) j=1 ∗ E(Y (t))dt and E(Y (t))dt, then by the Then the optimal coefficient b which minimizes 0 0 ˆ Exponential Martingale Theorem, the below equations can the variance Vj(b) is estimated by bm = ∑m ∑m be obtained 2 −1 (Xi − X)(Vi − V )( (Xi − X) ) . ∫ T (1) µ(1)t i=1 i=1 (1) (1) 2 2rT Y e dt 0 0 − The important step of the control variate techniques of the V ar(ST ) = (S0 ) e [e 1], above algorithms lies in how to choose suitable and favorable (1) (2) ∫ σ σ T (2) µ(2)t volatility control constants and , to make sure high (2) (2) 2 2rT Y e dt 0 0 − correlation between the control variate X and the problem V . V ar(ST ) = (S0 ) e [e 1]. We researched the first two moments of stochastic processes If the initial values of stock prices (1), (2) and auxiliary (1),(2) and the auxiliary processes (9),(10), and give a (1) (1c) (2) method to choose efficient control volatility constants. The processes (9), (10) are similar, i.e. S0 = S0 , S0 = (2c) results are shown in Theorem 3.2. S0 , and their control volatility expressions satisfy

Theorem 3.2 If the control volatility constants of process- (1) (2) eµ T − 1 eµ T − 1 es (9) and (10) satisfy (σ(1))2 = Y (1) , (σ(2))2 = Y (2) , √ √ 0 µ(1)T 0 µ(2)T eµ(1)T − 1 eµ(2)T − 1 σ(1) = Y (1) , σ(2) = Y (2) , 0 µ(1)T 0 µ(2)T then their mean and variance are equivalent at maturity T , then the mean and the variance of the auxiliary processes (9) (1c) (1) (1c) (1) E(ST ) = E(ST ), V ar(ST ) = V ar(ST ). and (10) approximately equal to the mean and the variance

of underlying processes (1) and (2) at maturity T . (2c) (2) (2c) (2) Proof For the auxiliary processes (9) and (10), the mean E(ST ) = E(ST ), V ar(ST ) = V ar(ST ). and the variance are as follows, respectively, Theorem 3.2 tells us that if σ(1) and σ(2) are chosen as (1c) (1c) rT E(ST ) = S0 e , in above discussion, the processes (9) and (10) approximate (1c) 2 processes (1) and (2) in the sense of ”moments” at the V ar(S(1c)) = (S(1c))2e2rT (e(σ ) T − 1). T 0 maturity T . The control variate and the problem are expected (2c) (2c) rT to have high correlation. Therefore the control variate in this E(ST ) = S0 e , case will be expected to have a well variance reduction effect, (2c) (2c) 2 2rT (σ(2c))2T − V ar(ST ) = (S0 ) e (e 1). which will be testified in next numerical subsection.

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C. Numerical Results N=10 60

Numerical results will be illustrated the efficiency of the 50 acceleration simulation algorithm discussed in section 3.2.

Parameters M = 1000, r = 0.05,T = 1, µ1 = 0.20, µ2 = 40 2 2 0.10,S10 = 10,S20 = 12,Y10 = 0.15 ,Y20 = 0.15 , σˆ1 = 0.01, σˆ2 = 0.02 are chosen, [0,T ] is divided into 100 parts 30 with the mesh size ∆t = T /100. As strike price Kcov is just a constant decrement, having no effect on the numerical 20 effects, Kcov = 0 is supposed without loss of generality. 10 Table 1 and Table 2 show the results of error ratios, the prices of covariance swap and estimate errors with control

The price for the case of stochastic volatility 0 variate of various simulation path numbers for the case of

N = 10 and N = 20 respectively. The control volatility −10 (1) (2) −10 0 10 20 30 40 50 60 70 constant σ = 0.1578 and σ = 0.1538 according to The price for the case of a single volatility conatant Theorem 3.2. In the tables, Price1 and Error1 denote the simulation√ results∑ of crude Monte Carlo simulation, Error1= Fig. 1. Scatter plots of values of covariance swap with the stochastic √1 1 m − ¯ 2 volatility against the values of covariance swap with a single volatility − j=1(Vj V ) . Price2 and Error2 denote the m m 1 constant for N = 10 simulation results√ of∑ control variate Monte Carlo method, √1 1 m − ¯ 2 Error2= − j=1(Vj(b) V (b)) .Variance reduc- N=20 m m 1 35 tion ratio is denoted by Ratio1=Error1/Error2. The results show that the control variate algorithm has obvious variance 30 reduction effects. All the ratios are greater than 22. As the number of simulation paths increases, all the estimated errors 25

have a decrease tendency. The number of simulations without 20 control variate would have to be increased by a factor of Ratio12 in order to achieve the same accuracy as a given 15 number of simulations with control variate. Furthermore, the 10 effects of variance reduction are stable, not influenced by the simulation paths. 5 Table 1: The results with N = 10 Paths Price1 Price2 Error1 Error2 Ratio1 The price for the case of stochastic volatility 0 500 12.0457 11.6786 0.3651 0.0156 23.4738 1000 10.5649 11.6729 0.2648 0.0113 23.3957 −5 −5 0 5 10 15 20 25 30 35 2000 11.4888 11.6750 0.1858 0.0081 22.8342 The price for the case of a single volatility conatant 5000 11.6548 11.6743 0.1160 0.0052 22.7432 8000 1.5963 11.6715 0.0926 0.0041 22.6306 Fig. 2. Scatter plots of values of covariance swap with the stochastic Table 2: The results with N = 20 volatility against the values of covariance swap with a single volatility Paths Price1 Price2 Error1 Error2 Ratio1 constant for N = 20 500 12.5516 11.6056 0.2557 0.0106 24.1978 150 1000 12.0684 11.6022 0.1799 0.0077 23.2207 Num of paths = 2000 2000 11.9264 11.6051 0.1281 0.0056 22.8404 5000 11.5268 11.6055 0.0813 0.0036 22.5414 8000 11.5678 11.6057 0.0649 0.0029 22.7235

The analysis of correlation between the pricing problem 100 and the control variate are also considered. Fig. 1 and 2 shows scatter plots of simulated values with stochastic volatility against the values with deterministic volatility constant for the case of N = 10 and N = 20, respectively. Error reductionError ratio They all show the strong correlations between the two cases 50 and their resulting correlations are 0.9990. Fig. 3 and 4 demonstrates the relationship between Error reduction ratio and control volatility constant square. An un- V ar[Vj ] constrained optimization problem maxσ{ } 0 minb{V ar[Vj (b)]} 0.01 0.015 0.02 0.025 0.03 0.035 is introduced, and a direct search method is used to solve Volatility square this optimization problem. From Fig. 2, we can see that Error Fig. 3. The relationship between Error reduction ratio and control volatility reduction ratios are very sensitive with volatility constants constant square, for N = 10 square σ2 and symmetrically distributed. The searched opti- mal volatility square σ∗2 is very close to σ(1)2 = 0.15782 = 0.0248 and σ(2)2 = 0.15382 = 0.0236. These numerical IV. IMPROVED CONTROL VARIATE FOR MONTE CARLO results testified the high efficiency of the control variate METHOD proposed by Theorem 3.1. A. Improved Control Variate The control variate Monte Carlo simulation proposed in Section 3 has good variance reduction effects. The control (Advance online publication: 26 August 2016)

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150 (1) (2) Num of paths = 2000 where M, r, σi,c , σi,c , ∆Ti,Kvar,T and ρ12 are given in above sections. Then how to choose the list of deterministic volatility constants σi,c, i = 1, ··· ,N in (14) and (15) to make sure 100 the high correlation between the control variate W2 and the derivative V . We are inspired by the idea in above section and consider the first two moments at discrete observation (1) (2) ··· dates Ti, the piecewise constants σi,c , σi,c , i = 1, ,N are

Error reductionError ratio chosen to make the mean and the variance of the process (1) 50 and (2) approximately equal to those of processes (14) and (15). Then the following Theorem 4.2 is achieved. Theorem 4.2 If the list of piecewise volatility constants σi,c, (i = 1, 2 ··· ,N) satisfy the equations, 0 √ 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Volatility square (1) (1) − (1) (1) Y0 (exp(µ Ti) exp(µ Ti−1)) σi,c = , µ(1)(T − T − ) Fig. 4. The relationship between Error reduction ratio and control volatility i i 1 constant square, for N = 20 √ (2) Y (exp(µ(2)T ) − exp(µ(2)T − )) σ(2) = 0 i i 1 . i,c (2) µ (Ti − Ti−1) volatility constants in processes (9) and (10) are determined by making the mean and the variance of the auxiliary then the mean and the variance of the process (13) and (14) processes (9) and (10) approximate those of processes (1) are nearly equal to those of processes (1) and (2) at the and (2). at the maturity T . Considering the characters of the observation date Ti. variance swap derivative, which depends on the stock prices Proof We just need to prove at observations date Ti, their at observation dates Ti, (0 = T0 < T1 < ··· < TN = T ), fist two moments are nearly equal, which is similar to the 1 2 ··· proof of Theorem 3.2 and the proof is omitted. we try to use piecewise constants σi,c, σi,c, (i = 1, 2 N) in stead of σ1, σ2 of the processes (9) and (10). The im- Then how to choose the list of deterministic volatility ··· provement of the control variate algorithm is further studied constants σi,c, i = 1, ,N in (13) and (14) to make sure in this section. the high correlation between the control variate W2 and the The key research point of the algorithm is still on the derivative V . We are inspired by the idea in above section and choice of the control variate. If the approximate underlying consider the first two moments at discrete observation dates ··· auxiliary processes are further studied and assumed as fol- Ti, the piecewise constants σi,c, i = 1, ,N are chosen lows, instead of process (9) and (10), to make the mean and the variance of the process (1) and (2) approximately equal to those of processes (13) and (14). (1) dS (t) (1) (1) Then the following Theorem 4.2 is achieved. = rdt + σ dW , t ∈ [Ti−1,Ti], (14) S(1)(t) i,c t Theorem 4.2 If the list of piecewise volatility constants σ , (i = 1, 2 ··· ,N) satisfy the equations, (2) i,c dS (t) (2) (2) √ = rdt + σ dW , t ∈ [T − ,T ], (15) (2) i,c t i 1 i (1) S (t) Y (exp(µ(1)T ) − exp(µ(1)T − )) σ(1) = 0 i i 1 , i,c (1) − where r is interest rate; σ(1) > 0 and σ(2) > 0, (i = µ (Ti Ti−1) i,c i,c √ 1 ··· N) are the volatility constants during the ith observation (2) (2) (2) (1) (2) (2) Y (exp(µ Ti) − exp(µ Ti−1)) period [Ti−1,Ti]. The denotation Wt and Wt are the σ = 0 . i,c (2) same Wiener processes as in processes (1) and (2), and µ (Ti − Ti−1) (1) (2) E(dWt dWt ) = ρ12dt. Then the price of this auxiliary then the mean and the variance of the process (13) and (14) (1) (2) derivative is denoted by W2 = W2(S ,S , t), and the are nearly equal to those of processes (1) and (2) at the payoff function at maturity is still denoted as observation date Ti. h(S(1),S(2),S(1),S(2), ··· ,S(1),S(2)) Proof We just need to prove at observations date Ti, their 0 0 1 1 N N fist two moments are nearly equal, which is similar to the just as in Equation (3). Then similar to the derivation in proof of Theorem 3.2 and the proof is omitted. Section 3.1, the PDE pricing model of the auxiliary product can be attained and the closed-form solutions of the price B. Numerical Results of the Improved Acceleration Algorith- of the product is achieved by Theorem 4.1, with the proof m omitted. Theorem 4.1 The price of the covariance swap with N The main difference of the improvement algorithm from discrete observation dates under the stochastic process (13) the algorithm in Section 3 is that during the observation and (14) is given by period [Ti−1,Ti], the volatility constants decided by Theorem 4.2 which will be used to simulate the control processes. In − ∑N σ(1)2 σ(2)2 (1) (2) M exp( rT ) − i,c − i,c this case, the processes (14) and (15) seem to have a better W 2(S0 ,S0 , 0) = T [ (r 2 )(r 2 ) i=1 approximation to the underlying stochastic processes (1) and 2 (1) (2) − 2 ∆Ti + ρ12σi,c σi,c T KcovT ], (2), and the better variance reduction effects will be expected.

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In order to make comparison with previous algorithm. N=10 60 Numerical parameters are chosen as the same as in Subsec-

tion 3.3. Table 3 and Table 4 record the simulation results 50 of the improved control variate Monte Carlo simulation. Here Error1 denotes the simulation errors of crude Monte 40 Carlo simulation and Error3 denotes the simulation errors of improved control variate Monte Carlo simulation. Variance 30 reduction ratio is denoted by Ratio2=Error1/Error3. The selected piecewise control volatility constants for the case 20 of N = 10 by Theorem 4.2 is that σ(1) =[0.1508 0.1523 0.1538 0.1553 0.1569 0.1585 0.1601 0.1617 0.1633 0.1650] 10 and σ(2) =[0.1504 0.1511 0.1519 0.1526 0.1534 0.1542

The price for the case of stochastic volatility 0 0.1550 0.1557 0.1565 0.1573]. Then the selected piecewise

control volatility constants for the case of N = 20 is that −10 (1) −10 0 10 20 30 40 50 60 70 σ =[0.1504 0.1511 0.1519 0.1526 0.1534 0.1542 0.1550 The price for the case of piecewise constants volatility 0.1557 0.1565 0.1573 0.1581 0.1589 0.1597 0.1605 0.1613 (2) 0.1621 0.1629 0.1637 0.1645 0.1654] and σ =[0.1502 Fig. 5. Scatter plots of values of covariance swap with the stochastic 0.1506 0.1509 0.1513 0.1517 0.1521 0.1525 0.1528 0.1532 volatility against the values of covariance swap with piecewise volatility 0.1536 0.1540 0.1544 0.1548 0.1551 0.1555 0.1559 0.1563 constants, for N = 10 0.1567 0.1571 0.1575]. Especially, when N = 1, the two algorithms are same. Table 3: The results with N = 10 of the improved algorithm N=20 Paths Price1 Price3 Error1 Error3 Ratio2 35 500 13.5678 11.3672 0.3651 0.0029 124.5770 1000 12.4537 11.6705 0.2648 0.0021 124.3808 30 2000 11.0264 11.6736 0.1858 0.0015 121.9264 5000 11.9865 11.6729 0.1160 0.0010 120.4219 25 Table 4: The results with N = 20 of the improved algorithm 20 Paths Price1 Price3 Error1 Error3 Ratio2 500 13.0568 11.4026 0.2557 0.0024 106.9722 1000 12.0486 11.6641 0.1799 0.0017 105.4123 15 2000 11.7963 11.6394 0.1281 0.0013 102.8288 5000 11.6528 11.6935 0.0813 0.0009 98.9774 10

From the results in Table 3 and Table 4, we can see that 5 the improved control variate algorithm has better obvious variance reduction effects than the control variate determined The price for the case of stochastic volatility 0

by Theorem 3.2. All the ratios are nearly 100, 4 times as −5 −5 0 5 10 15 20 25 30 35 much as Ratio1. As the number of simulation paths increases, The price for the case of piecewise constants volatility the estimated error also has a decrease tendency. The number of simulations without control variate would have to be Fig. 6. Scatter plots of values of covariance swap with the stochastic increased by a factor of Ratio22 in order to achieve the same volatility against the values of covariance swap with piecewise volatility accuracy as a given number of simulations with improved constants, for N = 20 control variate. Furthermore, the effects of variance reduction are stable, not influenced by the simulation paths. The better variance reduction effect of the improved control variate in Section 4 is because the auxiliary stochastic process with approach and Monte Carlo simulation. Control variate Monte piecewise volatility constants can approximate the original Carlo method is applied to the pricing of the covariance process with stochastic volatility process better. swap. Two kinds of control variate techniques are proposed, Then Fig. 5 and 6 shows scatter plots of simulated values based on the closed form solutions in simplified models with stochastic volatility against the values with piecewise with constant or piecewise constant volatility. The variance constant volatility for the case of N = 10 and N = 20, reduction ratios of the improved control variate algorithm respectively, with correlation coefficient bigger than 0.9999. which is based on the piecewise constant volatility are They all show stronger correlations than the first cases of smaller than that of the original control variate based on a control variate I demonstrated by Fig. 1 and 2. The evident single constant volatility. comparison of variance reduction ratios of the two kinds The algorithms in the paper can also be extended to the of control variate acceleration algorithms is listed in Fig. pricing of other financial derivatives, such as Asian options, 7, where ’o’ denotes the variance reduction ratio of the Lookback options with discrete sampling features under improved algorithm in Section 4, and ’*’denotes the variance multi-factor stochastic volatility models. Multiple control reduction ratio of the algorithm in Section 3. variate techniques can be considered based on the high moments of the stochastic volatility. Furthermore, other V. CONCLUSIONAND FURTHER DISCUSSION variance reduction techniques can be applied to the research In this paper, modeling and pricing of covariance swap of the hedging and the risk of the derivatives next. Then the derivative are discussed through the combination of PDE research on the optimal control variate can be considered[21].

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70 puter Science, 41(2), 102-111,2014.

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50 Variancereduction ratio

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20 0 1000 2000 3000 4000 5000 6000 7000 8000 Simulation paths

Fig. 7. Comparison of the effects of two control variate algorithms

REFERENCES

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(Advance online publication: 26 August 2016)