Improved multilevel Monte Carlo convergence using the Milstein scheme

Transcription

1 Improved mutieve Monte Caro convergence using the Mistein scheme M.B. Gies Oxford University Computing Laboratory, Parks Road, Oxford, U.K. Summary. In this paper we show that the Mistein scheme can be used to improve the convergence of the mutieve Monte Caro method for scaar stochastic differentia equations. Numerica resuts for Asian, ookback, barrier and digita options demonstrate that the computationa cost to achieve a root-mean-square error of ɛ is reduced to O(ɛ 2 ). This is achieved through a carefu construction of the mutieve estimator which computes the difference in expected payoff when using different numbers of timesteps. 1 Introduction In many financia engineering appications, one is interested in the expected vaue of a financia option whose payoff depends upon the soution of a stochastic differentia equation. To be specific, we consider an SDE with genera drift and voatiity terms, ds(t) = a(s, t) dt + b(s, t) dw (t), < t < T, (1) with given initia data S. In the case of European and digita options, we are interested in the expected vaue of a function of the termina state, f(s(t )), but in the case of Asian, ookback and barrier options the vauation depends on the entire path S(t), <t<t. Using a simpe Monte Caro method with a numerica discretisation with first order weak convergence, to achieve a root-mean-square error of O(ɛ) woud require O(ɛ 2 ) independent paths, each with O(ɛ 1 ) timesteps, giving a computationa compexity which is O(ɛ 3 ). We have recenty introduced a new mutieve approach [Gi6] which reduces the cost to O(ɛ 2 (og ɛ) 2 ) when using an Euer path discretisation for a European option with a payoff with a uniform Lipschitz bound. This mutieve approach is reated to the two-eve method of Kebaier [Keb5], and is simiar to the muti-eve method proposed

2 2 M.B. Gies by Speight [Spe5] based on the quasi contro variate method of Emsermann and Simon [ES2]. There are aso strong simiarities to Heinrich s mutieve approach for parametric integration [Hei1]. In the previous work, it was aso proved that the computationa cost can be further reduced to O(ɛ 2 ) for numerica discretisations with certain mutieve convergence properties. The objective of this paper is to demonstrate that this improved compexity is attainabe for scaar SDEs with a variety of exotic options through using the Mistein path discretisation. For European options with a Lipschitz continuous payoff, it can be proved that this an immediate consequence of the improved strong order of convergence of the Mistein discretisation compared to the simper Euer discretisation. However, for Asian, ookback, barrier and digita options, specia numerica treatments have to be introduced, and that is the focus of the paper. Furthermore, no a priori convergence proofs have yet been constructed for these cases and so the paper reies on numerica demonstration of the effectiveness of the agorithms that have been deveoped. The paper begins by reviewing the mutieve approach, and the theorem which describes its computationa cost given certain properties of the numerica discretisation. The next section discusses the Mistein discretisation and the chaenges of achieving higher order variance convergence within the mutieve method. Asian, ookback, barrier and digita options are a considered, and O(ɛ 2 ) computationa cost is demonstrated for each through the use of Brownian interpoation to approximate the behaviour of paths within each timestep. The fina section indicates the direction of future research, incuding the need for a priori convergence anaysis, the chaenges of extending this work to muti-dimensiona SDEs, and the use of quasi-monte Caro methods for further reduction of the computationa compexity. 2 Mutieve Monte Caro method Consider Monte Caro path simuations with different timesteps h = 2 T, =, 1,..., L. Thus on the coarsest eve, =, the simuations use just 1 timestep, whie on the finest eve, = L, the simuations use 2 L timesteps. For a given Brownian path W (t), et P denote the payoff, and et P denote its approximation using a numerica discretisation with timestep h. Because of the inearity of the expectation operator, it is ceary true that E[ P L ] = E[ P L ] + E[ P P 1 ]. (2) This expresses the expectation on the finest eve as being equa to the expectation on the coarsest eve pus a sum of corrections which give the difference =1

3 Improved mutieve Monte Caro convergence using the Mistein scheme 3 in expectation between simuations using different numbers of timesteps. The idea behind the mutieve method is to independenty estimate each of the expectations on the right-hand side in a way which minimises the overa variance for a given computationa cost. Let Ŷ be an estimator for E[ P ] using N sampes, and et Ŷ for > be an estimator for E[ P P 1 ] using N paths. The simpest estimator is a mean of N independent sampes, which for > is Ŷ = N 1 N i=1 ( ) (i) (i) P P 1. (3) The key point here is that the quantity 1 comes from two discrete approximations with different timesteps but the same Brownian path. The variance of this simpe estimator is V [Ŷ] = N 1 V where V is the variance of a singe sampe. Combining this with independent estimators for each of the other eves, the variance of the combined estimator Ŷ = L = Ŷ is V [Ŷ ] = L = N 1 V, whie its computationa cost is proportiona to L = N h 1. Treating the N as continuous variabes, the variance is minimised for a fixed computationa cost by choosing N to be proportiona to V h. In the particuar case of an Euer discretisation, provided a(s, t) and b(s, t) satisfy certain conditions [BT95, KP92, TT9] there is O(h 1/2 ) strong convergence. From this it foows that V [ P P ] = O(h ) for a European option with a Lipschitz continuous payoff. Hence for the simpe estimator (3), the singe sampe variance V is O(h ), and the optima choice for N is asymptoticay proportiona to h. Setting N = O(ɛ 2 L h ), the variance of the combined estimator Ŷ is O(ɛ2 ). If L is chosen such that L = og ɛ 1 / og 2 + O(1), as ɛ, then h L = 2 L = O(ɛ), and so the bias error E[ P L P ] is O(ɛ) due to standard resuts on weak convergence. Consequenty, we obtain a Mean Square Error which is O(ɛ 2 ), with a computationa compexity which is O(ɛ 2 L 2 ) = O(ɛ 2 (og ɛ) 2 ). P (i) P (i) This anaysis is generaised in the foowing theorem: Theorem 1. Let P denote a functiona of the soution of stochastic differentia equation (1) for a given Brownian path W (t), and et P denote the corresponding approximation using a numerica discretisation with timestep h = M T. If there exist independent estimators Ŷ based on N Monte Caro sampes, and positive constants α 1 2, β, c 1, c 2, c 3 such that i) E[ P P ] c 1 h α E[ P ], = ii) E[Ŷ] = E[ P P 1 ], >

4 4 M.B. Gies iii) V [Ŷ] c 2 N 1 h β iv) C, the computationa compexity of Ŷ, is bounded by C c 3 N h 1, then there exists a positive constant c 4 such that for any ɛ < e 1 there are vaues L and N for which the mutieve estimator Ŷ = L Ŷ, = has a mean-square-error with bound [ (Ŷ ) ] 2 MSE E E[P ] < ɛ 2 with a computationa compexity C with bound c 4 ɛ 2, β > 1, C c 4 ɛ 2 (og ɛ) 2, β = 1, c 4 ɛ 2 (1 β)/α, < β < 1. Proof. See [Gi6]. The remainder of this paper addresses the use of the Mistein scheme [Ga4, KP92] to construct estimators with variance convergence rates β > 1, resuting in an O(ɛ 2 ) compexity bound. Provided certain conditions are satisfied [KP92], the Mistein scheme gives O(h) strong convergence. In the case of a Lipschitz continuous European payoff, this immediatey eads to the resut that V = O(h 2 ), corresponding to β = 2. Numerica resuts which are not presented here demonstrate this convergence rate, and the associated O(ɛ 2 ) compexity This paper addresses the tougher chaenges of Asian, ookback, barrier and digita options. These cases require some ingenuity to construct estimators for which β > 1. Unfortunatey, there is no accompanying theoretica anaysis as yet, and so the paper reies on numerica demonstration of their effectiveness. 3 Mistein discretisation For a scaar SDE, the Mistein discretisation of equation (1) is Ŝ n+1 = Ŝn + a h + b W n b S b ( W n) 2. (4)

5 Improved mutieve Monte Caro convergence using the Mistein scheme 5 In the above equation, the subscript n is used to denote the timestep index, and a, b and b/ S are evauated at Ŝn, t n. A of the numerica resuts to be presented are for the case of geometric Brownian motion for which the SDE is ds(t) = r S dt + σ S dw (t), < t < T. By switching to the new variabe X = og S, it is possibe to construct numerica approximations which are exact, but here we directy simuate the geometric Brownian motion using the Mistein method as an indication of the behaviour with more compicated modes, for exampe those with a oca voatiity function σ(s, t). 3.1 Estimator construction In a of the cases to be presented, we simuate the paths using the Mistein method. The refinement factor is M = 2, so each eve has twice as many timesteps as the previous eve. The difference between the appications is in how we use the computed discrete path data to estimate E[ P P 1 ]. In each case, the estimator for E[ P P 1 ] is an average of vaues from N independent path simuations. For each Brownian input, the vaue which is computed is of the form P f P 1 c f. Here P is a fine-path estimate using timestep h = 2 T, and P 1 c is the corresponding coarse-path estimate using timestep h = 2 ( 1) T. To ensure that the identity (2) is correcty respected, to avoid the introduction of an undesired bias, we require that E[ P f c ] = E[ P ]. (5) This means that the definitions of P when estimating E[ P P 1 ] and E[ P +1 P ] must have the same expectation. In the simpest case of a European option, this can be achieved very simpy by defining P f and P c to be the same; this is the approach which was used for a appications in the previous work using the Euer discretisation [Gi6]. However, for more chaenging appications such as Asian, ookback, barrier c and digita options, the definition of P wi invove information from the discrete simuation of P f f +1, which is not avaiabe in computing P. The reason for doing this is to reduce the variance of the estimator, but it must be shown that equaity (5) is satisfied. This wi be achieved in each case through a construction based on a simpe Brownian motion approximation.

6 6 M.B. Gies 3.2 Asian option The Asian option we consider has the discounted payoff P = exp( rt ) max (, S K ), where T S = T 1 S(t) dt. is The simpest approximation of S, which was used in previous work [Gi6], n T 1 Ŝ = T h (Ŝn+Ŝn+1), where n T = T/h is the number of timesteps. This corresponds to a piecewise inear approximation to S(t) but improved accuracy can be achieved by approximating the behaviour within a timestep as simpe Brownian motion, with constant drift and voatiity, conditiona on the computed vaues Ŝn. Taking b n to be the constant voatiity within the time interva [t n, t n+1 ], standard Brownian Bridge resuts (see section 3.1 in [Ga4]) give tn+1 t n S(t) dt = 1 2 h(s(t n) + S(t n+1 )) + b n I n, where I n, defined as I n = tn+1 t n (W (t) W (t n )) dt 1 2 h W, is a N(, h 3 /12) Norma random variabe, independent of W. Using b n = b(ŝn, t n ), this gives the fine-path approximation n T 1 S = T 1 ( 1 2 h (Ŝn+Ŝn+1) + b n I n ). The coarse path approximation is the same except that the vaues for I n are derived from the fine path vaues, noting that tn+2h (W (t) W (t n )) dt h(w (t n +2h) W (t n )) t n = + tn+h t n (W (t) W (t n )) dt 1 2 h (W (t n+h) W (t n )) tn+2h t n+h (W (t) W (t n +h)) dt 1 2 h (W (t n+2h) W (t n +h)) h (W (t n+h) W (t n )) 1 2 h (W (t n+2h) W (t n +h)),

7 Improved mutieve Monte Caro convergence using the Mistein scheme 7 og 2 variance P P P og 2 mean P P P N ε=.5 ε=.1 ε=.2 ε=.5 ε=.1 ε 2 Cost Std MC MLMC Fig. 1. Asian option ε and hence I c = I f1 + I f h( W f1 W f2 ), where I c is the vaue for the coarse timestep, and I f1 and W f1 are the vaues for the first fine timestep, and I f2 and W f2 are the vaues for the second fine timestep. Figure 1 shows the numerica resuts for parameters S()=1, K =1, T =1, r =.5, σ =.2. The top eft pot shows the behaviour of the variance of both P and P P 1. The sope of the atter is approaching a vaue approximatey equa to 2, indicating that V = O(h 2 ), corresponding to β = 2. On eve = 2, which has just 4 timesteps, V is aready more than 1 times smaer than the variance V [ P ] of the standard Monte Caro method with the same

8 8 M.B. Gies timestep. The top right pot shows that E[ P P 1 ] is approximatey O(h ), corresponding to first order weak convergence, α=1. This is used to determine the number of eves that are required to reduce the bias to an acceptabe eve [Gi6]. The bottom two pots have resuts from five mutieve cacuations for different vaues of ɛ. Each ine in the bottom eft pot shows the vaues for N, =,..., L, with the vaues decreasing with because of the decrease in both V and h. It can aso be seen that the vaue for L, the maximum eve of timestep refinement, increases as the vaue for ɛ decreases, requiring a ower bias error. The bottom right pot shows the variation with ɛ of ɛ 2 C where the computationa compexity C is defined as C = 2 N, which is the tota number of fine grid timesteps on a eves. One ine shows the resuts for the mutieve cacuation and the other shows the corresponding cost of a standard Monte Caro simuation of the same accuracy, i.e. the same bias error corresponding to the same vaue for L, and the same variance. It can be seen that ɛ 2 C is amost constant for the mutieve method, as expected, whereas for the standard Monte Caro method it is approximatey proportiona to ɛ 1. For the most accurate case, ɛ=5 1 5, the mutieve method is more than 1 times more efficient than the standard method. 3.3 Lookback option The ookback option we consider has the discounted payoff ( ) P = exp( rt ) S(T ) min S(t). <t<t In previous work [Gi6], the minimum vaue of S(t) over the path was approximated numericay by ) Ŝ min = min (Ŝn β b n h. n Here b n is the voatiity in the n th timestep, and β.5826 is a constant which corrects the O(h 1/2 ) eading order error due to the discrete samping of the path, and thereby restores O(h) weak convergence [BGK97]. However, using this approximation, the difference between the computed minimum vaues and fine and coarse paths is O(h 1/2 ), and hence the variance V is O(h ), corresponding to β = 1. In the previous work, this was acceptabe because β = 1 is the best that can be achieved in genera with the Euer path discretisation which was used, but in this work we aim to achieve an improved convergence rate using the Mistein scheme.

9 Improved mutieve Monte Caro convergence using the Mistein scheme 9 To achieve this, we again approximate the behaviour within a timestep as simpe Brownian motion, with constant drift and voatiity, conditiona on the computed vaues Ŝn. For the time interva [t n, t n+1 ], standard Brownian Interpoation resuts (see section 6.4 in [Ga4]) give the minimum of Brownian motion, conditiona on the end vaues, as ( ) ) 2 Ŝ n,min = 1 2 Ŝ n + Ŝn+1 (Ŝn+1 Ŝn 2 b 2 n h og U n, (6) where b n is the constant voatiity and U n is a uniform random variabe on [, 1]. The fine-path vaue P f is defined in this way using b n = b(ŝn, t n ), and then taking the minimum over a timesteps to obtain the goba minimum. However, for the coarse-path vaue P 1 c, we do something different. Again assuming simpe Brownian motion conditiona on the end-points, the vaue at the midpoint of the time interva [t n, t n+1 ] is given by ) Ŝ n+1/2 = 1 2 (Ŝn + Ŝn+1 b n D n, (7) where D n = W n+1 2W n+1/2 + W n = ( W n+1 W n+1/2 ) ( Wn+1/2 W n ), is a N(, h) random variabe which corresponds to a difference in the consecutive Brownian increments of a finer path with timestep h/2. Given this midpoint vaue, the minimum vaue over the fu timestep is the smaer of the minima for each of the two haf-timesteps, { ( ) ) 2 Ŝ n,min = min Ŝ n + Ŝn+1/2 (Ŝn+1/2 Ŝn b 2 n h og U 1,n, ( Ŝ n+1/2 + Ŝn+1 (Ŝn+1 Ŝn+1/2 ) 2 b 2 n h og U 2,n )} In computing P 1 c, we use the vaues for D n, U 1,n and U 2,n that come from the fine-path simuation for P f. D n is the difference of the Brownian increments for the two fine-path timesteps, and U 1,n and U 2,n are the uniform random variabes used to compute the minima for the two fine-path timesteps. Since these a have the correct probabiity distribution, it foows that the expected vaues of (6) and (8) are identica, and therefore equaity (5) is satisfied. Figure 2 shows the numerica resuts for parameters S() = 1, T = 1, r =.5, σ =.2. The top eft pot shows that the variance is O(h 2 ), corresponding to β = 2, whie the top right pot shows that the mean correction is O(h ),. (8)

10 1 M.B. Gies og 2 variance P P P og 2 mean P P P N ε=.5 ε=.1 ε=.2 ε=.5 ε=.1 ε 2 Cost Std MC MLMC Fig. 2. Lookback option ε corresponding to first order weak convergence, α = 1. The bottom eft pot shows that more eves are required to reduce the discretisation bias to the required eve. Consequenty, the savings reative to the standard Monte Caro treatment are greater, up to a factor of approximatey 2 for ɛ= The computationa cost of the mutieve method is amost perfecty proportiona to ɛ Barrier option The barrier option which is considered is a down-and-out ca for which the discounted payoff is

11 Improved mutieve Monte Caro convergence using the Mistein scheme 11 P = exp( rt ) (S(T ) K) + 1{τ >T }, where the notation (S(T ) K) + denotes max(, S(T ) K), 1(τ > T ) is an indicator function taking vaue 1 if the argument is true, and zero otherwise, and the crossing time τ is defined as τ = inf {S(t) < B}. t> Foowing a standard approach for continuousy monitored barrier crossings (see section 6.4 in [Ga4]), for a particuar Brownian path input samped discretey at uniform intervas h, the conditiona expectation of the payoff can be expressed as n T 1 exp( rt ) (Ŝn T K) + p n, where n T = T/h is again the number of timesteps, and p n represents the probabiity that the path did not cross the barrier during the n th timestep. If we again approximate the motion within each timestep as simpe Brownian motion conditiona on the endpoint vaues, then ( 2 (Sn B) + (S n+1 B) + ) p n = 1 exp b 2. (9) n h This is the expression used to define the payoff P f for the fine-path cacuation, with b n set equa to b(ŝn, t n ), as in the ookback cacuation. For the coarse path cacuation, in which each timestep corresponds to two fine-path timesteps, we again use equation (7) to construct a midpoint vaue Ŝ n+1/2. Given this vaue, the probabiity that the simpe Brownian path does not cross the barrier is { ( 2 (Sn B) + (S n+1/2 B) + )} p n = 1 exp { 1 exp n= b 2 n h ( 2 (Sn+1/2 B) + (S n+1 B) + b 2 n h )}. (1) The conditiona expectation of (1) is equa to (9) and so equaity (5) is satisfied. Figure 3 shows the numerica resuts for parameters S() = 1, K = 1, B =.85, T = 1, r =.5, σ =.2. The top eft pot shows that the variance is approximatey O(h β ) for a vaue of β sighty ess than 2. An expanation for this is that a sma O(h 1/2 ) fraction of the paths have a minimum which ies within O(h 1/2 ) of the barrier, for which the product p n is neither cose to zero nor cose to unity. The fine path and coarse path trajectories differ by O(h ), due to the first order strong convergence of the Mistein scheme.

12 12 M.B. Gies og 2 variance P P P og 2 mean P P P N ε=.5 ε=.1 ε=.2 ε=.5 ε=.1 ε 2 Cost Std MC MLMC Fig. 3. Barrier option ε Since the p n have an O(h 1/2 ) derivative, this resuts in the difference between pn for this sma subset of coarse and fine paths being O(h 1/2 ), giving a contribution to the variance which is O(h 3/2 ). The top right pot shows that the mean correction is O(h ), corresponding to first order weak convergence, α = 1. The bottom right pot shows that the computationa cost of the mutieve method is again amost perfecty proportiona to ɛ 2, and for ɛ=5 1 5 it is over 1 times more efficient that the standard Monte Caro method.

13 Improved mutieve Monte Caro convergence using the Mistein scheme Digita option The digita option which is considered has the discounted payoff P = exp( rt ) 1{S(T ) > K}. The standard numerica discretisation woud be to simuate the path of S(t) right up to the fina time T. This is the approach adopted previousy for mutieve cacuations using the Euer discretisation [Gi6]. In that case, the variance V was O(h 1/2 ), because O(h 1/2 ) of the paths terminate within O(h 1/2 ) of the strike K, and for these paths there is an O(1) probabiity that the coarse and fine paths wi terminate on opposite sides of the strike, giving an O(1) vaue for P P 1. Using the same approach with the Mistein method, there woud be O(h ) of the paths terminating within O(h ) of the strike K, for which there woud be an O(1) probabiity that the coarse and fine paths woud terminate on opposite sides of the strike. This woud resut in V being O(h ). This corresponds to β =1 and woud give a computationa cost which is O(ɛ 2 (og ɛ) 2 ). To achieve a better mutieve variance convergence rate, we instead smooth the payoff using the technique of conditiona expectation (see section in [Ga4]), terminating the path cacuations one timestep before reaching the termina time T. If Ŝn T 1 denotes the vaue at this time, then if we approximate the motion thereafter as a simpe Brownian motion with constant drift a nt 1 and voatiity b nt 1, the probabiity that Ŝn T > K after one further timestep is ) (ŜnT 1+a nt 1h K p = Φ, (11) b nt 1 h where Φ is the cumuative Norma distribution. For the fine-path payoff P f = exp( rt ) p, with a nt 1 = a(ŝn T 1, T h) and b nt 1 = b(ŝn T 1, T h). For the coarse-path we therefore use P f payoff, we note that given the Brownian increment W for the first haf of the N th timestep, then the probabiity that Ŝn T > K is p = Φ (ŜnT 1+a nt 1h+b nt 1 W K ). (12) b nt 1 h/2 The vaue for W is taken from the fina timestep of the fine-path cacuation, which corresponds to the first haf of the N th timestep in the coarse-path cacuation. The conditiona expectation of (12) is equa to (11), and so again equaity (5) is satisfied. Figure 4 shows the numerica resuts for parameters S()=1, K =1, T =1, r =.5, σ =.2. The top eft pot shows that the variance is approximatey

14 14 M.B. Gies og 2 variance P P P og 2 mean P P P N ε=.1 ε=.2 ε=.5 ε=.1 ε=.2 ε 2 Cost Std MC MLMC Fig. 4. Digita option ε O(h 3/2 ), corresponding to β = 1.5. The reason for this is simiar to the argument for the barrier option. O(h 1/2 ) of the paths have a minimum which ies within O(h 1/2 ) of the strike, for which the p is neither cose to zero nor cose to unity. The fine path and coarse path trajectories differ by O(h ), due to the first order strong convergence of the Mistein scheme. Since p has an O(h 1/2 ) derivative, this resuts in the difference between p for the coarse and fine paths being O(h 1/2 ), and that resuts in the variance being O(h 3/2 ). One strikingy different feature is that the variance of the eve estimator, V, is zero. This is because at eve = there woud usuay be ony one timestep, and so here it is not simuated at a; one simpy uses equation (11)

15 Improved mutieve Monte Caro convergence using the Mistein scheme 15 to evauate the payoff. This reduces the cost of the mutieve cacuations even more than usua, eading to a factor 1 computationa savings for ɛ= Concusions and future work In this paper we have demonstrated numericay the abiity of mutieve Monte Caro path simuation using the Mistein discretisation to achieve an ɛ RMS error for a range of financia options at a computationa cost which is O(ɛ 2 ). This requires the use of Brownian interpoation within each timestep for Asian, ookback and barrier options, and the use of conditiona expectation to smooth the payoff of digita options. There are three major directions for future research. The first is the theoretica anaysis of the agorithms presented in this paper, to prove that they do indeed have variance convergence rates with β > 1. The anaysis of earier agorithms for ookback, barrier and digita options based on the Euer discretisation [Gi6] is currenty being deveoped; it is hoped this can then be extended to the Mistein discretisation for scaar SDEs. The second is the extension of the agorithms to muti-dimensiona SDEs, for which the Mistein discretisation usuay requires the simuation of Lévy areas [GL94, Ga4]. Current investigations indicate that this can be avoided for European options with a Lipschitz payoff through the use of antithetic variabes. However, the extension to more difficut payoffs, such as the Asian, ookback, barrier and digita options considered in this paper, ooks more chaenging. The third direction for future research is the use of quasi-monte Caro methods. The anaysis in section 2 showed that the optima number of sampes on eve is proportiona to V h. If V = O(h β ), then this number is proportiona to h (β+1)/2. Since the cost of an individua sampe is proportiona to the number of timesteps, and hence inversey proportiona to h, the computationa cost on eve is proportiona to h (β 1)/2. For β > 1, this shows that the computationa effort decreases geometricay as one moves to finer eves of discretisation. Thus, when using the Mistein discretisation most of the computationa effort is expended on the coarsest eves of the mutieve computation. For these ow dimensiona eves it is reasonabe to expect that quasi-monte Caro methods [KS5, Ecu4, Nie92] wi be very much more effective than the standard Monte Caro methods used in this paper.

16 16 M.B. Gies Acknowedgements This research was funded in part by a research grant from Microsoft Corporation, and in part by a Springboard Feowship from the UK Engineering and Physica Sciences Research Counci. References [BGK97] M. Broadie, P. Gasserman, and S. Kou. A continuity correction for discrete barrier options. Mathematica Finance, 7(4): , [BT95] V. Bay and D. Taay. The aw of the Euer scheme for stochastic differentia equations, I: convergence rate of the distribution function. Probabiity Theory and Reated Fieds, 14(1):43 6, [Ecu4] P. L Ecuyer. Quasi-Monte Caro methods in finance. In R.G. Ingas, M.D. Rossetti, J.S. Smith, and B.A. Peters, editors, Proceedings of the [ES2] 24 Winter Simuation Conference, pages IEEE Press, 24. M. Emsermann and B. Simon. Improving simuation efficiency with quasi contro variates. Stochastic Modes, 18(3): , 22. [Gi6] M.B. Gies. Mutieve Monte Caro path simuation. Technica Report NA6/3, Oxford University Computing Laboratory, 26 (to appear in Operations Research). [GL94] [Ga4] [Hei1] [Keb5] [KP92] [KS5] J.G. Gaines and T.J. Lyons. Random generation of stochastic integras. SIAM J. App. Math., 54(4): , P. Gasserman. Monte Caro Methods in Financia Engineering. Springer- Verag, New York, 24. S. Heinrich. Mutieve Monte Caro Methods, voume 2179 of Lecture Notes in Computer Science, pages Springer-Verag, 21. A. Kebaier. Statistica Romberg extrapoation: a new variance reduction method and appications to options pricing. Annas of Appied Probabiity, 14(4): , 25. P.E. Koeden and E. Paten. Numerica Soution of Stochastic Differentia Equations. Springer-Verag, Berin, F.Y. Kuo and I.H. Soan. Lifting the curse of dimensionaity. Notices of the AMS, 52(11): , 25. [Nie92] H. Niederreiter. Random Number Generation and Quasi-Monte Caro Methods. SIAM, [Spe5] A. Speight. A mutieve approach to contro variates. Working paper, Georgia State University, 25. [TT9] D. Taay and L. Tubaro. Expansion of the goba error for numerica schemes soving stochastic differentia equations. Stochastic Anaysis and Appications, 8:483 59, 199.