MULTILEVEL MONTE CARLO FOR BASKET OPTIONS. Michael B. Giles

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1 Proceedings of the 29 Winter Simuation Conference M. D. Rossetti, R. R. Hi, B. Johansson, A. Dunkin, and R. G. Ingas, eds. MULTILEVEL MONTE CARLO FOR BASKET OPTIONS Michae B. Gies Oxford-Man Institute of Quantitative Finance Oxford University Mathematica Institute Oxford United Kingdom ABSTRACT The mutieve Monte Caro method has been previousy introduced for the efficient pricing of options based on a singe underying quantity. In this paper we show that the method is easiy extended to basket options based on a weighted average of severa underying quantities. Numerica resuts for Asian, ookback, barrier and digita basket options demonstrate that the computationa cost to achieve a root-mean-square error of is O( 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 Gies (Gies 27, Gies 28 has recenty introduced a mutieve Monte Caro path simuation method which improves the efficiency of financia option pricing by combining resuts using different numbers of timesteps. This can be viewed as a generaisation of the two-eve method of Kebaier (Kebaier 2 and is aso simiar in approach to Heinrich s mutieve method for parametric integration (Heinrich 21. Reference (Gies 28 introduced the mutieve Monte Caro method and proved that it can ower the computationa compexity of path-dependent Monte Caro evauations, whie reference (Gies 27 demonstrated that the computationa cost can be further reduced by using the Mistein discretisation. In this paper we briefy review the key ideas and show that the same approach can be used for basket options in which the financia payoff functions depends on a weighted averaged of a number of underying assets. 2 MULTILEVEL MONTE CARLO METHOD We start by considering a scaar SDE with genera drift and voatiity terms, ds(t=a(s,tdt + b(s,tdw(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 other cases the vauation depends on the entire path S(t, <t < T. Using a simpe Euer-Maruyama discretisation with first order weak convergence, to achieve a r.m.s. error of woud require O( 2 independent paths, each with O( 1 timesteps, giving a computationa compexity which is O( 3. Consider performing Monte Caro path simuations with different numbers of uniform timesteps h = 2 T, =,1,...,L, so 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, etp denote the payoff, and et 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 =1 E[ 1 ]. ( /9/$ IEEE 1283

2 Gies 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 in expectation between simuations using different numbers of timesteps. In the mutieve method we independenty estimate each of the expectations on the right-hand side in a way which minimises the overa variance for a given computationa cost. The simpest estimator for E[ 1 ] for > is a mean of N independent sampes, Ŷ = N 1 N i=1 ( P (i P (i 1. (3 The key point here is that the quantity P (i P (i 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, and with Ŷ being the usua estimate for E[ P ], 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. The Euer-Maruyama discretisation gives O(h 1/2 strong convergence, provided a(s,t and b(s,t satisfy certain conditions (Koeden and Paten From this it foows that V[ 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( 2 Lh, the variance of the combined estimator Ŷ is O( 2. If L is chosen such that L = og 1 /og2 + O(1, as, then h L = 2 L = O(, and so the bias error E[ P L P] is O( due to standard resuts on weak convergence. Consequenty, we obtain a Mean Square Error which is O( 2, with a computationa compexity which is O( 2 L 2 =O( 2 (og 2. This anaysis is generaised in the foowing theorem (Gies 28: Theorem 1. Let P denote a functiona of the soution of stochastic differentia equation (1 for a given Brownian path W(t, and et 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 1 2,,c 1,c 2,c 3 such that i E[ P] c 1 h { E[ P ], = ii E[Ŷ ]= iii V[Ŷ ] c 2 N 1 h E[ 1 ], > 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 <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 E E[P] < 2 with a computationa compexity C with bound c 4 2, > 1, C c 4 2 (og 2, = 1, c 4 2 (1 /, < <

3 Gies 3 MILSTEIN DISCRETISATION The Mistein discretisation of equation (1 is Ŝ n+1 = Ŝ n + a n h + b n W n b ( n S b n (W n 2 h. (4 In the above equation, the subscript n is used to denote the timestep index, and a n, b n and b n /S are evauated at Ŝ n,t n. Provided certain conditions are satisfied (Koeden and Paten 1992, the Mistein scheme gives O(h strong convergence, and for a Lipschitz European payoff this immediatey eads to the resut that V = O(h 2. Reference (Gies 27 addresses the tougher chaenges of Asian, ookback, barrier and digita options. The key to the Asian, ookback and barrier option constructions is a conditiona piecewise Brownian interpoation. Within the time interva [t n,t n+1 ] we approximate the drift and voatiity as being constant with the interva and use a Brownian interpoation conditiona on the two end vaues Ŝ n and Ŝ n+1,giving Ŝ(t=Ŝ n + (t(ŝ n+1 Ŝ n +b n (W(t W n (t(w n+1 W n, ( where (t =(t t n /(t n+1 t n. Standard resuts for the distribution of the extrema and averages of Brownian motions (Gasserman 24 can then be used to construct suitabe mutieve estimators (Gies 27. Simiary, for the digita option which has a discontinuous payoff, one can use a constant coefficient Brownian extrapoation conditiona on the vaue Ŝ N 1, one timestep before the end. Foowing an approach used for payoff smoothing for pathwise sensitivity anaysis (Gasserman 24, the conditiona expectation for the payoff can be evauated anayticay and this is then used to construct the mutieve estimator (Gies BASKET OPTIONS In basket options the vaue is dependent on a weighted average of J underying assets, S(t= J j=1 j S j (t, each of which satisfies an SDE of the form (1 driven by Brownian motions W j (t with correation matrix. In constructing the mutieve estimator, the important observation is that the average of the Brownian interpoations for the J underying assets gives Ŝ(t = Ŝ n + (t(ŝ n+1 Ŝ n + J j=1 j b j,n (W j (t W j,n (t(w j,n+1 W j,n = Ŝ n + (t(ŝ n+1 Ŝ n +b n (W(t W n (t(w n+1 W n, where W(t is another Brownian motion which is a weighted average of the W j (t, andb n is defined by b 2 n = i b i,n i, j j b j,n. i, j Since the Brownian interpoation for the basket average has the same form as the scaar interpoation, the mutieve estimators can be constructed in exacty the same way as in (Gies 27 usingb n. The numerica resuts to be presented are for a basket of five assets, each modeed as a geometric Brownian motion: ds j = rs j dt + j S j dw j (t, < t < T, using a constant risk-free interest rate r =., and five voatiities =.2,.2,.3,.3,.4. The initia asset vaues are S j (=1, the simuation interva is taken to be T =1, and the driving Brownian motions have a correation of.2. In each case, the option price is based on a simpe arithmetic average of the five assets. 128

4 Gies 1 og 2 variance og 2 mean eve eve N =.1 =.2 =. =.1 =.2 2 Cost 1 Std MC MLMC eve accuracy Figure 1: Asian option 4.1 Asian Option ( The Asian basket option has the discounted payoff P = exp( rt max,s K, where S is the time-average of the average of the underying assets, and the strike is K =1. The top eft pot in Figure 1 shows the behaviour of the variance of both and 1. The sope of the atter is approaching a vaue approximatey equa to 2, indicating that V =O(h 2.Oneve =2, which has just 4 timesteps, V is aready amost 1 times smaer than the variance V[ ] of the standard Monte Caro method with the same timestep. The top right pot shows that E[ 1 ] is approximatey O(h, corresponding to first order weak convergence. This is used to determine the number of eves that are required to reduce the bias to an acceptabe eve (Gies 28. The bottom two pots have resuts from five mutieve cacuations for different vaues of. 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 decreases, requiring a ower bias error (Gies 28. The bottom right pot shows the variation with of 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 2 C is amost constant for the mutieve method, as expected, whereas for the standard Monte Caro method it increases with L. For the most accurate case, =.1, the mutieve method is approximatey 1 times more efficient than the standard method. 1286

5 Gies 1 og 2 variance og 2 mean eve eve N =.1 =.2 =. =.1 =.2 2 Cost Std MC MLMC 1 2 eve accuracy Figure 2: Lookback option 4.2 Lookback Option ( The basket ookback option we consider has the discounted payoff P = exp( rt S(T min S(t. <t<t The top eft pot in Figure 2 shows that the variance is O(h 2, whie the top right pot shows that the mean correction is O(h. 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 1 for =.1. The computationa cost of the mutieve method is amost perfecty proportiona to Barrier Option The barrier option which is considered is a down-and-out ca with payoff P = exp( rt (S(T K + 1 >T, where the notation (S(T K + denotes max(,s(t K, 1 >T is an { indicator } function taking vaue 1 if the argument is true, and zero otherwise, and the crossing time is defined as = inf S(t < B. The barrier vaue is taken to be B=8, and the strike t> is again K =1. The top eft pot in Figure 3 shows that the variance is approximatey O(h 3/2. The reason for this is that an O(h 1/2 fraction of the paths have a minimum which ies within O(h 1/2 paths the difference between the coarse and fine path payoff vaues is O(h 1/2 which is O(h 3/2. of the barrier. In (Gies 27 it is argued that for these, giving a contribution to the overa variance 1287

6 Gies 1 og 2 variance og 2 mean eve eve N =.1 =.2 =. =.1 =.2 2 Cost Std MC MLMC 1 2 eve accuracy Figure 3: Barrier option The top right pot shows that the mean correction is O(h, corresponding to first order weak convergence. The bottom right pot shows that the computationa cost of the mutieve method is again amost perfecty proportiona to 2,andfor =.1 it is 1 times more efficient that the standard Monte Caro method. 4.4 Digita Option The digita option has the discounted payoff P = exp( rt K 1 S(T >K with strike K =1. The top eft pot in Figure 4 shows that the variance is approximatey O(h 3/2. 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. The fine path and coarse path trajectories differ by O(h, due to the first order strong convergence of the Mistein scheme and this resuts in an O(h 1/2 difference between the coarse and fine path evauations. One strikingy different feature is that the variance of the eve estimator is zero. This is because the mutieve treatment introduced in (Gies 27 uses a conditiona expectation (based on a simpe Brownian extrapoation for which the expectation is known anayticay evauated one timestep before the end. At eve = where there woud usuay be one timestep, there is no path simuation at a; one simpy uses the anaytic expression for the conditiona expectation. This reduces the cost of the mutieve cacuations even more than usua, giving more than a factor of computationa savings for =

7 Gies 1 og 2 variance og 2 mean eve eve N =.1 =.2 =. =.1 =.2 2 Cost Std MC MLMC eve accuracy Figure 4: Digita option CONCLUSIONS In this paper we have reviewed the mutieve Monte Caro method and have demonstrated that it achieves an O( 2 compexity when computing the vaue of basket options to within a root-mean-square error of. This buit on the singe asset methods introduced in (Gies 27, by noting that the weighted average of a set of simpe correated Brownian motions is itsef a simpe Brownian motion, and so the same techniques can be appied as in the singe asset case. This paper does not present any numerica anaysis of the techniques used. Previous work (Gies, Higham, and Mao 29, Avikainen 29 has anaysed the mutieve method using the Euer-Maruyama discretisation. Current work by Gies, Debrabant and Rößer extending this anaysis to the Mistein discretisation supports the orders of convergence demonstrated in this paper. Future work wi address the use of the mutieve approach for more genera mutivariate cases, in particuar when the payoff function is a genera discontinuous function of the underying asset vaues at a set of discrete times. Other extensions to be considered are the computation of sensitivities (the Greeks in computationa finance, and the appication to genera Lévy processes. ACKNOWLEDGMENTS This research has been supported by the Oxford-Man Institute of Quantitative Finance. 1289

8 Gies REFERENCES Avikainen, R. 29. Convergence rates for approximations of functionas of SDEs. Finance and Stochastics 13 (3: Gies, M. 27. Improved mutieve Monte Caro convergence using the Mistein scheme. In Monte Caro and Quasi-Monte Caro Methods 26, ed. A. Keer, S. Heinrich, and H. Niederreiter, Springer-Verag. Gies, M. 28. Mutieve Monte Caro path simuation. Operations Research 6 (3: Gies, M., D. Higham, and X. Mao. 29. Anaysing mutieve Monte Caro for options with non-gobay Lipschitz payoff. Finance and Stochastics 13 (3: Gasserman, P. 24. Monte Caro methods in financia engineering. Springer, New York. Heinrich, S. 21. Mutieve Monte Caro methods, Voume 2179 of Lecture Notes in Computer Science, Springer- Verag. Kebaier, A. 2. Statistica Romberg extrapoation: a new variance reduction method and appications to options pricing. Annas of Appied Probabiity 14 (4: Koeden, P., and E. Paten Numerica soution of stochastic differentia equations. Springer, Berin. AUTHOR BIOGRAPHY MICHAEL B. GILES is a Professor of Scientific Computing in the Oxford University Mathematica Institute, and a member of the Oxford-Man Institute of Quantitative Finance. He received his Ph.D. in Aeronautics and Astronautics from MIT and worked on computationa fuid dynamics for many years at MIT and in the Oxford University Computing Laboratory before moving into computationa finance. He serves as an Associate Editor for the Journa of Computationa Finance and the SIAM Journa of Financia Mathematics. His emai address is <mike.gies@maths.ox.ac.uk>. 129