Antithetic multilevel Monte Carlo estimation for multidimensional SDES

Transcription

1 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES Michae B. Gies and Lukasz Szpruch Abstract In this paper we deveop antithetic mutieve Monte Caro MLMC estimators for mutidimensiona SDEs driven by Brownian motion. Gies has previousy shown that if we combine a numerica approximation with strong order of convergence O t with MLMC we can reduce the computationa compexity to estimate expected vaues of Lipschitz functionas of SDE soutions with a root-mean-square error of ε from Oε 3 to Oε 2. However, in genera, to obtain a rate of strong convergence higher than O t 1/2 requires simuation, or approximation, of Lévy areas. Recenty, Gies and Szpruch [5] constructed an antithetic mutieve estimator that avoids the simuation of Lévy areas and sti achieves an MLMC correction variance which is O t 2 for smooth payoffs and amost O t 3/2 for piecewise smooth payoffs, even though there is ony O t 1/2 strong convergence. This resuts in an Oε 2 compexity for estimating the vaue of financia European and Asian put and ca options. In this paper, we extend these resuts to more compex payoffs based on the path minimum. To achieve this, an approximation of the Lévy areas is needed, resuting in O t 3/4 strong convergence. By modifying the antithetic MLMC estimator we are abe to obtain Oε 2 ogε 2 compexity for estimating financia barrier and ookback options. 1 Introduction In his origina MLMC paper [4], Gies showed that one coud obtain a good MLMC variance for smooth payoffs by using a numerica approximation with good strong convergence properties. This is in contrast to the standard Monte Caro approach to simuations of SDEs, where ony a good weak order of convergence is required. Michae B. Gies Mathematica Institute, University of Oxford, e-mai: mike.gies@maths.ox.ac.uk Lukasz Szpruch Mathematica Institute, University of Oxford, e-mai: szpruch@maths.ox.ac.uk 1

2 2 Michae B. Gies and Lukasz Szpruch For mutidimensiona SDEs, to obtain good strong convergence, simuation of the Lévy areas is required. Indeed, Cark & Cameron [1] proved for a particuar SDE that it is impossibe to achieve a better order of strong convergence than the Euer- Maruyama discretisation when using just the discrete increments of the underying Brownian motion. The anaysis was extended by Müer-Gronbach [8] to genera SDEs. As a consequence, if we use the standard MLMC method with the Mistein scheme without simuating the Lévy areas the compexity wi remain the same as for Euer-Maruyama. Recenty, Gies and Szpruch [5] constructed an antithetic MLMC estimator, enabing one to negect the Lévy areas and sti obtain a mutieve correction estimator with a variance which decays at the same rate as the scaar Mistein estimator. They achieved an O t 2 MLMC variance for smooth payoffs and amost an O t 3/2 variance for piecewise smooth payoffs, even though there is ony O t 1/2 strong convergence. This resuts in an Oε 2 compexity for estimating the vaue of European and Asian put and ca options. The question remains whether the approach can be extended to more compex payoffs such as those based on the minimum of the path over the simuation interva. For scaar SDEs with the Mistein discretisation, Gies [4] obtained Oε 2 compexity for such payoffs by combining MLMC with conditiona Monte Caro methods. In this paper, we extend these resuts to the mutidimensiona case. Unike the previous mutidimensiona work, we find that a suitabe approximation to the Lévy areas is required. By a suitabe modification of the antithetic MLMC estimator we are abe to obtain Oε 2 ogε 2 compexity for payoffs corresponding to financia ookback and barrier options. We focus on simuations of Cark and Cameron s SDE since it captures the essence of simuations requiring Lévy area simuation to obtain higher that O t 1/2 strong convergence property. Our resuts are supported by numerica experiments. 2 MLMC Mutieve Monte Caro simuation uses a number of eves of resoution, = 0,1,...,L, with = 0 being the coarsest, and = L being the finest. In the context of an SDE simuation, eve 0 may have just one timestep for the whoe time interva [0,T ], whereas eve L might have 2 L uniform timesteps t L = 2 L T. If P denotes the payoff or other output functiona of interest, and P denote its approximation on eve, then the expected vaue E[P L ] on the finest eve is equa to the expected vaue E[P 0 ] on the coarsest eve pus a sum of corrections which give the difference in expectation between simuations on successive eves, E[P L ] = E[P 0 ] + L =1 E[P P 1 ]. 1 Let Y 0 be an estimator for E[P 0 ] using N 0 sampes, and et Y, > 0, be an estimator for E[P P 1 ] using N sampes. The simpest estimator is a mean of N

3 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 3 independent sampes, which for > 0 is Y = N 1 N i=1 P i Pi 1. 2 The key point is that P i Pi 1 shoud come from two discrete approximations for the same underying stochastic sampe. We reca the Theorem from [5]: Theorem 1. Let P denote a functiona of the soution of a stochastic differentia equation, and et P denote the corresponding eve numerica approximation. If there exist independent estimators Y based on N Monte Caro sampes, and positive constants α,β,γ,c 1,c 2,c 3 such that α 1 2 minβ,γ and i E[P P] { c 1 2 α E[P0 ], = 0 ii E[Y ] = E[P P 1 ], > 0 iiiv[y ] c 2 N 1 2 β ivc c 3 N 2 γ, where C is the computationa compexity of Y then there exists a positive constant c 4 such that for any ε < e 1 there are vaues L and N for which the [ mutieve estimator Y = L =0 Y, has a mean-square-error with bound MSE E Y E[P] 2] < ε 2 with a computationa compexity C with bound c 4 ε 2, β > γ, C c 4 ε 2 ogε 2, β = γ, c 4 ε 2 γ β/α, 0 < β < γ. In 2 we have used the same estimator for the payoff P on every eve, and therefore 1 is a trivia identity due to the teescoping summation. However, in [3] Gies expained that it can be better to use different estimators for the finer and coarser of the two eves being considered, P f when eve is the finer eve, and Pc when eve is the coarser eve. In this case, we require that E[P f ] = E[Pc ] for = 0,...,L 1, 3 so that E[P f L ] = E[P f 0 ] + L =1 E[P f Pc 1 ]. The MLMC Theorem is sti appicabe to this modified estimator. The advantage is that it gives the fexibiity to construct approximations for which P f Pc 1 is much smaer than the origina P P 1, giving a arger vaue for β, the rate of variance convergence in condition iii in the theorem. In the next sections we demonstrate how suitabe choice of P f and Pc can dramaticay increase the convergence of the variance of the MLMC estimator.

4 4 Michae B. Gies and Lukasz Szpruch 2.1 Mistein Scheme Let Ω,F,{F t } t 0,P be a compete probabiity space with a fitration {F t } t 0 satisfying the usua conditions, and et wt be a m-dimensiona Brownian motion defined on the probabiity space. We consider the numerica approximation of SDEs of the form dxt = f xt dt + gxt dwt, 4 where xt R d for each t 0, f C 2 R d,r d, g C 2 R d,r d m have bounded first and second derivatives, and for simpicity we assume a fixed initia vaue x 0 R d. For Lipschitz continuous payoffs that depend on finite number of times t n = n t, the MLMC variance can be estimated from the strong convergence of the numerica scheme, that is E [ sup xtn Xn p] 1/p = O t ξ for p 2. 0 n 2 For partition P t := {n t : n = 0,1,2,...,2 = N}, where t = T /N, we consider the Mistein approximation X n with i th component of the form Xi,n+1 =Xi,n + f i Xn t + + m j,k=1 m g i j Xn w j,n j=1 5 h i jk Xn w j,n w k,n δ j,k t [A jk ]+1 where h i jk x = 1 2 d =1 g kx g i j x x, δ j,k is a Kronecker deta, w n = wn+ 1 t wn t and [A jk ]+1 is the Lévy area defined as t [A n+1 jk ]+1 = w j t w j tn dw k t tn t n+1 t n w k t w k tn dw j t. 6 For the Mistein scheme ξ = 1 and therefore β = 2 for smooth payoffs, and hence MLMC has compexity Oε 2. However, there is no method for simuating Lévy areas with a cost per timestep simiar to that of Brownian increments, apart from in dimension 2 [2, 9, 10]. Furthermore, within computationa finance, options are often based on the continuousy-monitored minimum or maximum or the path. The Mistein scheme gives an improved rate of convergence at the simuation times, but to maintain the strong order of convergence for such path-dependent options we use Brownian Bridge interpoation within each timestep [t n,t n+1 ] X t = Xn + λ Xn+1 X n + gxn wt wtn λ w n where λ t t n/ t. Using this interpoant, we have the resut [7] 7

5 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 5 E [ sup xt X t p] = O t og t p. 0 t T 3 Antithetic MLMC estimator The idea for the antithetic estimator is to expoit the fexibiity of the more genera MLMC estimator by defining P 1 c to be the usua payoff PX c coming from a eve 1 coarse simuation X c, and define P f to be the average of the payoffs PX f,px a coming from an antithetic pair of eve simuations, X f and X a. X f wi be defined in a way which corresponds naturay to the construction of X c. Its antithetic twin X a wi be defined so that it has exacty the same distribution as X f, conditiona on X c, which ensures that E[PX f ] = E[PX a ] and hence 3 is satisfied, but at the same time X f X c X a X c and therefore PX f PX c PX a PX c, so that 2 1 PX f + PX a PX c. This eads to 2 1 PX f + PX a PX c having a much smaer variance than the standard estimator PX f PX c. It was proved in [5], that if P x L 1, 2 P x 2 L2. then for p 2, [ E 12 PX f + PX a PX c ] p 2 p 1 L p 1 E [ 1 2 X f +X a X c p] + 2 p 1 L p 2 E [ X f X a 2p]. In the mutidimensiona SDE we wi show that the Mistein approximation with the Lévy areas set to zero, combined with the antithetic construction, eads to X f X a = O t 1/2 but 2 1X f +X a X c = O t. Hence, the variance V[ 1 2 P f +Pa Pc 1 ] is O t 2 for smooth payoffs, which is the same order obtained for scaar SDEs using the Mistein discretisation with its first order strong convergence. 4 Cark-Cameron exampe The Cark and Cameron mode probem [1] is dx 1 t = dw 1 t, dx 2 t = x 1 tdw 2 t, 8 with x 1 0 = x 2 0 = 0, and zero correation between the two Brownian motions w 1 t and w 2 t. These equations can be integrated exacty over a time interva [,+1 ], where = n t, to give x 1 +1 = x 1 + w 1,n x 2 +1 = x 2 + x 1 w 2,n w 1,n w 2,n [A 12] +1 9

6 6 Michae B. Gies and Lukasz Szpruch W W f W c W a Fig. 1 Brownian path w, its piecewise inear interpoations w c and w f, and the antithetic w a, for a singe coarse timestep. The circes denote the points at which the Brownian path is samped. where w i,n w i +1 w i, and [A 12 ] +1 is the Lévy area defined in 6. This corresponds exacty to the Mistein discretisation presented in 5, so for this simpe mode probem the Mistein discretisation is exact. The point of Cark and Cameron s paper is that for any numerica approximation XT based soey on the set of discrete Brownian increments w, E[x 2 T X 2 T 2 ] 4 1 T t. Since in this section we use superscript f,a,c for fine X f, antithetic X a and coarse X c approximations, respectivey, we drop the superscript for the carity of notation. We define a coarse path approximation X c with timestep t, and times n t, by negecting the Lévy area terms to give X c 1,n+1 = X c 1,n + w 1 1,n X c 2,n+1 = X c 2,n + X c 1,n w 1 2,n w 1 1,n w 1 2,n 10 This is equivaent to repacing the true Brownian path by a piecewise inear approximation as iustrated in Figure 1. Simiary, we define the corresponding two haf-timesteps of the first fine path approximation X f. Using w 1 n+1 w+1 w = w+1 w+1/2 + w+1/2 w w n+1/2 + w n, we can combine two haf-timestep approximations to obtain an equation for the increment over the coarse timestep,

7 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 7 X f 1,n+1 = X f 1,n + w 1 1,n X f 2,n+1 = X f 2,n + X f ,n w 1 2,n w 1 1,n w 1 w 1,n w 2,n+1/2 w 2,n w 1,n+1/2 2,n 11. The antithetic approximation Xn a is defined by exacty the same discretisation except that the Brownian increments w n+1/2 and w n+1 are swapped, as iustrated in Figure 1. This gives X a 1,n+1 = X a 1,n + w 1 1,n, X2,n+1 a = X 2,n a + X 1,n a w 1 2,n w 1 1,n w 1 2,n 12 w 1,n w 2,n+1/2 w 2,n w 1,n+1/ Swapping w n and w n+1/2 does not change the distribution of the driving Brownian increments, and hence X a has exacty the same distribution as X f. Note aso the change in sign in the ast term in 11 compared to the corresponding term in 12. This is important because these two terms cance when the two equations are averaged. In [5] Gies and Szpruch proved the foowing resut: Lemma 1. If Xn f, Xn a and Xn c are as defined above, then X f 1,n = X 1,n a = X 1,n c, 1 2 X f 2,n + X 2,n a = X2,n c, n = 1,2,...,N 2 1. and [ X f E 2,N X 2,N a p ] = O t p/2 for p 2. This aows us to prove that for payoffs which are a smooth function of the fina state the MLMC variance [ ] V 12 PX f N + PX N a PXN c has an O t 2 upper bound and therefore the compexity of the MLMC estimator is Oε 2. This matches the convergence rate and compexity for the mutieve method for scaar SDEs using the standard first order Mistein discretisation, and is much better than the O t MLMC convergence obtained with the Euer-Maruyama discretisation. Very few financia payoff functions are twice differentiabe on the entire domain, but Gies and Szpruch have proved that for piecewise smooth put and ca options the variance converges with rate O t 3/2, assuming oca boundedness of the density of the SDE soution 4 near the strike [5]. To perform numerica experiments we cosey foow the agorithm prescribed in [4, Section 5] with predefined root-mean-square errors ε = [1,2,4,8,16] 10 4 : 1. start with eve = 0 2. estimate variance using initia 10 4 sampes

8 8 Michae B. Gies and Lukasz Szpruch og 2 variance 15 P P P 1 ref eve og 2 mean 15 P P P 1 ref eve ε 2 Cost Std MC MLMC accuracy ε og 2 variance X 12 X 1 ref eve Fig. 2 Cark/ Cameron mode probem with payoff max0,x evauate optima number of sampes on each eve as in [4, Section 5] 4. if L 2, test for convergence [4, Section 5] 5. if L < 2 or not converged, set := + 1 and go to 2. In addition, we used 10 6 sampes to generate the pots where we estimate the rate of the strong and weak errors. Figure 2 presents resuts for the payoff function P = max0,x appied to Cark and Camerson mode probem with initia conditions x 1 0 = x 2 0 = 1. The top eft pot with the superimposed reference sope with rate 1.5 shows that the variance P P 1 is O t 1.5. The top right pot shows that E[P P 1 ] = O t. The bottom eft pot shows ε 2 C where C is the computationa compexity as defined in Theorem 1. The pot is versus ε, and the neary horizonta ine confirms that the MLMC compexity is Oε 2, whereas the standard Monte Caro approach has compexity Oε 3. For accuracy ε = 10 4, the antithetic MLMC is approximatey 500 times more efficient than standard Monte Caro. The bottom right pot shows that V[X 2,N X 1 2,N ] = O t, corresponding to the standard strong convergence of order 0.5.

9 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 9 5 Subsamping of Levy areas Consider now a Brownian path wt on the interva [0,T ] with N sub-intervas of size t =T /N. We define w n wn t and w n w n+1 w n. Lemma 2. The Lévy area for wt can be expressed as [A jk ] T N 1 0 = w j,n w j,0 w k,n w k,n w k,0 w j,n + [A s jk ]n+1 t where [A s jk ]n+1 t is the Lévy area for the sub-interva [n t,n+1 t]. Proof. This foows from the definition of the Lévy area by expressing the integra over [0,T ] as the sum of integras over each of the sub-intervas, and using the identity wt w0 = w n w 0 + wt w n to evauate the integra on the n th sub-interva. Ignoring the sub-interva Lévy areas [A s jk ]n+1 t, which corresponds to using the expected vaue of [A jk ] T 0 conditiona on {wn t} 0 n N, gives the Lévy area approximation: [L jk ] T 0 = N 1 w j,n w j,0 w k,n w k,n w k,0 w j,n. We denote by [L a jk ]T 0 the corresponding antithetic quantity generated by reversing the order of the Brownian increments w N, w N 1,..., w 1. The antithetic abe is due to the foowing emma: Lemma 3. [L a jk ]T 0 = [L jk ] T 0. Proof. [L a jk ]T 0 = = N 1 N 1 m =0 n 1 w j,n 1 m w k,n 1 n w k,n 1 m w j,n 1 n m=0 N 1 w j,n w k,m w k,n w j,m n =m +1 N 1 n 1 = w j,m w k,n w k,m w j,n n =0 m =0 = [L jk ] T 0 The second ine in the proof uses the substitutions m =N 1 n, n =N 1 m, and the third ine simpy switches the order of summation.

10 10 Michae B. Gies and Lukasz Szpruch 5.1 Antithetic subsamping In Section 4 we showed that by setting the Lévy area to zero and using a suitabe antithetic treatment we obtained an MLMC variance with the same order as the Mistein scheme for scaar SDEs. However, to obtain simiary good resuts for payoffs which depend on the path minimum or maximum we are not abe to competey negect the Lévy areas. Instead, for reasons which woud require a engthy expanation and wi be addressed in future work, we need to improve the rate of strong convergence from 1/2 to 3/4 by approximating the Lévy areas by sub-samping the driving Brownian path. Let M f denote the number of subsampes required to approximate the Lévy area on the fine timestep. The subsamping timestep is given by δ = 2 T /M f. Since we want to obtain E [L jk ] +1/2 [A jk ] +1/2 2 = O2 3/2, we need to take M f 2 /2 sub-sampes in each fine timestep. By the same reasoning we take M c 2 1/2, with sub-samping timestep δ 1 = 2 1 T /M c. For impementation we round the exponents, using M f = 2 /2 and M c = 2 1/2. W W f W c W a Fig. 3 Brownian path w, its piecewise inear interpoations w c and w f, and the antithetic w a, for a singe coarse timestep. The circes denote the points at which the Brownian path is samped. Figure 3 iustrates a case in which w c has M c =4 sub-samping intervas within each coarse timestep, and w f has M f = 8 sub-samping intervas within each fine timestep this corresponds to eve = 5. With sub-samping, the piecewise inear antithetic fine path w a is defined by a time-reversa of the Brownian increments within each of the coarse sub-samping intervas. In the case iustrated, the first

11 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 11 coarse sub-samping interva contains 4 fine sub-samping intervas, so these 4 increments w f,1, w f,2, w f,3, w f,4 are re-ordered as w f,4, w f,3, w f,2, w f,1 to give the increments for w a. First we represent the Lévy area approximation on the coarse time interva as a sum of two approximations each with M c /2 subsampes [L c jk ]+1 = [L c jk ]+1/2 + [L c jk ]+1 +1/2 + w 1,n+1/2 w 2,n+1 w 2,n+1/2 w 1,n We can represent the Lévy area approximation for the first fine timestep within a coarse timestep as where [L f jk ]+s+1δ 1 +sδ 1 [L f jk ]+1/2 M c /2 1 = [L c jk ]+1/2 + s=0 are Lévy area approximatiosn with 2M f M c [L f jk ]+s+1δ 1 +sδ 1, 14 subsampes. Notice that M c 2 δ 1 = 2. In the same way, we represent the Lévy area approximation for the antithetic path as [L a jk ]+1/2 Due to Proposition 3, [L a jk ]+s+1δ 1 +sδ 1 [L f jk ]+1/2 M c /2 1 = [L c jk ]+1/2 + [L c jk ]+1/2 = s=0 [L a jk ]+s+1δ 1 +sδ = [L f jk ]+s+1δ 1 +sδ. Hence 1 [L a jk ]+1/2 [L c jk ]+1/2, which is the key antithetic property required for higher order MLMC variance convergence. We derive the anaogous approximation for the second fine timestep within a coarse timestep. Returning to the Cark-Cameron exampe, where we focus ony on the equation for x 2, with Lévy area approximation using M c and M f subsampes respectivey, we obtain X c 2,n+1 = X c 2,n + X c 1,n w 1 2,n w 1 1,n w 1 2,n [Lc jk ]+1 16 X f 2,n+1 = X f 2,n + X f 1,n w 1 2,n w 1 1,n w 1 2,n [L f jk ]+1/ [L f jk ]+1 +1/2 w 1,n w 2,n+1/2 w 2,n w 1,n+1/2, X2,n+1 a = X 2,n a + X 1,n a w 1 2,n w 1 1,n w 1 2,n [La jk ]+1/ [La jk ]+1 +1/2 w 1,n w 2,n+1/2 w 2,n w 1,n+1/2, where we use Lévy areas approximations 13 and 14. We present a emma that can be proved in a simiar way to Lemma 3.1 in [5]:

12 12 Michae B. Gies and Lukasz Szpruch Lemma 4. If Xn f, Xn a and Xn c are as defined above, and N = 2 1, then X f 1,n = X 1,n a = X 1,n c, 1 2 X f 2,n + X 2,n a = X2,n c, n = 1,2,...,N and [ X f sup E 2,n X 2,n a p ] = O t 3 4 p 0 n N Our numerica experiments show that for ookback and barrier options the MLMC variance V [ 1 2 PX f + PX a PX c ] has an O t 3/2 upper bound. Since we use subsamping to approximate the Lévy areas, the computationa cost corresponds to γ = 3/2 in Theorem 1, and as a consequence the compexity of the MLMC estimator is Oε 2 ogε 2, whereas the standard Monte Caro simuation compexity is Oε 3. 6 Lookback and barrier options 6.1 Lookback options Lookback options are based on the minimum or maximum of the simuated path. As a specific exampe, we consider the payoff P = x 2 T min 0<t<T x 2 t, based on the second component x 2 of the Cark and Cameron mode probem. To improve the convergence we use X t defined in 7. We have min 0 t<t X 2t = min 0 n<2 1 X 2,n,min, where the minimum of the fine approximation over the fine timestep [tn 1 n+1/2 ] is given by [6] 2 X2,n,min = 2 1 X2,n + X2,n+1/2 X2,n+1/2 X 2,n 2g2 Xn 2 t ogu n,,t 1 19 where U n is a uniform random variabe on the unit interva. The minima for the antithetic path are defined simiary, using the same uniform random numbers U n. For the coarse path, we do something sighty different. Using the same Brownian interpoation, we use equation 7 to define X 1 n+1/2 X 1 n+ 2 1 t 1. Given this interpoated vaue, the minimum vaue over the coarse interva can then be taken to be the smaer of the minima for the two fine intervas

13 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 13 X2,n,min 1 = 1 2 X2,n 1 + X 1 2,n+1/2 X ,n+1/2 X2,n 2g2 Xn 1 2 t ogu n, X 1 2,n+1/2,min = 1 2 X 1 1 2,n+1/2 +X2,n+1 X2,n+1 1 X 1 2,n+1/2 2 2g2 Xn 1 2 t ogu n+1/2 Note that we use g 2 Xn 1 for both fine timesteps, because we have used the Brownian Bridge with diffusion term g 2 Xn 1 to derive both minima. If we changed g 2 Xn 1 to g 2 X 1 1 n+1/2 in X2,n+1/2,min, this woud mean that we used different Brownian Bridge on the first and second haf of the coarse timestep and as a consequence we woud vioate 3. Note aso the re-use of the same uniform random numbers U n and U n+1/2 used to compute the fine path minima. To perform numerica experiments we cosey foow the agorithm prescribed in [4]. The resuts in Figure 4 are for the Cark and Cameron mode probem with this ookback payoff. The top eft eft pot shows the behaviour of the variance of both P and P P 1. The superimposed reference sope with rate 1.5 indicates that the variance V = V[P P 1 ] = O t 1.5, corresponding to Oε 2 ogε 2 computationa compexity for the antithetic MLMC estimator. The top right pot shows that E[P P 1 ] = O t. The bottom eft pot shows computationa compexity C as defined in Theorem 1 with desired accuracy ε. The pot is of ε 2 C versus ε, because we expect to see that ε 2 C is ony weaky dependent on ε for MLMC. For standard Monte Caro without subsamping of the Lévy areas, theory predicts that ε 2 C shoud be proportiona to the number of timesteps on the finest eve, which in turn is roughy proportiona to ε 1 due to the weak convergence order. For accuracy ε = 10 4, the antithetic MLMC is over 100 times more efficient than standard Monte Caro. The bottom right pot shows that V[X2 X 2 1 ] = O t 3/2. This corresponds to the standard strong convergence of order 3/ Barrier options The barrier option which is considered is a down-and-out option for which the payoff is a Lipschitz function of the vaue of the underying at maturity, provided the underying has never dropped beow a vaue B, i.e. P = f x 2 T 1 {τ>t }, where the crossing time τ is defined as τ = inf{t : x 2 t < B}. Using the Brownian Bridge interpoation, we can approximate 1 {τ>t } by 2 1 1/2 1 {X 2,n,min B}, where X 2,n,min is defined in equation 19. This suggests foowing the ookback approximation in computing the minimum of both the fine and coarse paths. However, the variance woud be arger in this case because the payoff is a discontinuous function of the minimum. A better treatment, which is the one used in [3], is to use the conditiona Monte Caro approach to further smooth the payoff. Since the process X n is Markovian, we have

14 14 Michae B. Gies and Lukasz Szpruch [ ] [ [ ]] N 1/2 N 1/2 E f X2,N 1 {X n,min B} = E f X2,N E 1 {X 2,n,min B} X 0,...,X N = E = E [ [ N 1/2 f X2,N N 1/2 f X2,N E [ 1 {X 2,n,min B} X n,x n+1/2 ] ] 1 p n where 2X p n = P inf <t<t 2t n+1/2 X < B Xn,X n+1/2 2,n B + X 2,n+1/2 = exp B+ g 2 Xn 2 t The antithetic path is treated simiary. For the payoff for the coarse path we subsampe X 1 n+1/2, as we did for the ookback option, to obtain ], og 2 variance 15 P P P 1 ref eve og 2 mean 15 P P P 1 ref eve ε 2 Cost Std MC MLMC accuracy ε og 2 variance X X 1 ref eve Fig. 4 Cark / Cameron mode with payoff x 2 1 minx 2 t.

15 Antithetic mutieve Monte Caro estimation for mutidimensiona SDES 15 [ ] [ ] N 1/2 N 1/2 E f X2,N 1 1 {X 1 2,n,min B} = E f X2,N 1 1 p 1 n, where, for integer n, p 1 n = exp p 1 n+1/2 = exp 2X 1 2,n B+ X 1 g 2 Xn 1 2 t n2,+1/2 B+ 2 X 1 2,n+1/2 B+ X 1 g 2 Xn 1 2 t 2,n+1 B+ Note that the same g 2 Xn 1 is used to cacuate both probabiities for the same reason as for the ookback option. The resuts in Figure 5 are for barrier option with barrier B = 0.1. The top eft eft pot shows the behaviour of the variance of both P and P P 1. The superimpose reference sope with rate 1.5 indicates that the variance V = V[P P 1 ] = O t 1.5. This corresponds to an Oε 2 ogε 2 computationa compexity for the antithetic MLMC, due to the additiona cost of the sub-samping to approximate the Lévy areas. The top right pot shows that E[P P 1 ] = O t. The bottom eft pot, og 2 variance 15 P P P 1 ref eve og 2 mean 15 P P P 1 ref eve ε 2 Cost 10 2 og 2 variance 8 12 Std MC MLMC accuracy ε 14 X X 1 ref eve Fig. 5 Cark / Cameron mode with payoff minx 2 1 1,0 1 minx2 t>0.1.

16 16 Michae B. Gies and Lukasz Szpruch shows the variation of the computationa compexity C with desired accuracy ε. For standard Monte Caro without subsamping of the Lévy areas, theory predicts that ε 2 C shoud be proportiona to the number of timesteps on the finest eve, which in turn is roughy proportiona to ε 1 due to the weak convergence order. For accuracy ε = 10 4, antithetic MLMC is amost 10 times more efficient than standard Monte Caro. The bottom right pot shows that V[X2 1 X2 ] = O t 3/2. This corresponds to standard strong convergence of order 3/4. 7 Concusions In this paper we extended resuts from [3] and [5] to ookback and barrier options for mutidimensiona SDEs. By suitabe modification of the antithetic MLMC estimator, using sub-samping of the driving Brownian path to approximate the Lévy areas, we obtained Oε 2 ogε 2 compexity for barrier and ookback options. Simiar resuts have aso been obtained for digita options which are a discontinuous function of the fina state, but they have been omitted here due to ack of space. References 1. J.M.C. Cark and R.J. Cameron. The maximum rate of convergence of discrete approximations for stochastic differentia equations. In B. Grigeionis, editor, Stochastic Differentia Equations, No. 25 in Lecture Notes in Contro and Information Sciences. Springer-Verag, J.G. Gaines and T.J. Lyons. Random generation of stochastic integras. SIAM Journa of Appied Mathematics, 544: , M.B. Gies. Improved mutieve Monte Caro convergence using the Mistein scheme. In A. Keer, S. Heinrich, and H. Niederreiter, editors, Monte Caro and Quasi-Monte Caro Methods 2006, pages Springer-Verag, M.B. Gies. Mutieve Monte Caro path simuation. Operations Research, 563: , M.B. Gies and L. Szpruch. Antithetic mutieve Monte Caro estimation for mutidimensiona SDEs without Lévy area simuation. Arxiv preprint arxiv: , P. Gasserman. Monte Caro Methods in Financia Engineering. Springer, New York, T. Müer-Gronbach. The optima uniform approximation of systems of stochastic differentia equations. The Annas of Appied Probabiity, 122: , T. Müer-Gronbach. Strong approximation of systems of stochastic differentia equations. Habiitation thesis, TU Darmstadt, T. Rydén and M. Wiktorsson. On the simuation of iterated Itô integras. Stochastic Processes and their Appications, 911: , M. Wiktorsson. Joint characteristic function and simutaneous simuation of iterated Itô integras for mutipe independent Brownian motions. Annas of Appied Probabiity, 112: , 2001.