Multilevel Monte Carlo path simulation

Transcription

1 Mutieve Monte Caro path simuation Mike Gies Oxford University Mathematica Institute Oxford-Man Institute of Quantitative Finance Acknowedgments: research funding from Microsoft and EPSRC, and coaboration with Pau Gasserman (Coumbia) and Ian Soan, Frances Kuo (UNSW) Mutieve Monte Caro p. 1/41

2 Outine Long-term objective is faster Monte Caro simuation of path dependent options to estimate vaues and Greeks. Severa ingredients, not yet a combined: mutieve method quasi-monte Caro adjoint pathwise Greeks parae computing on NVIDIA graphics cards Emphasis in this presentation is on mutieve method Mutieve Monte Caro p. 2/41

3 Generic Probem Stochastic differentia equation with genera drift and voatiity terms: ds(t) = a(s, t) dt + b(s, t) dw (t) We want to compute the expected vaue of an option dependent on S(t). In the simpest case of European options, it is a function of the termina state P = f(s(t )) with a uniform Lipschitz bound, f(u) f(v ) c U V, U, V. Mutieve Monte Caro p. 3/41

4 Simpest MC Approach Euer discretisation with timestep h: Ŝ n+1 = Ŝn + a(ŝn, t n ) h + b(ŝn, t n ) W n Estimator for expected payoff is an average of N independent path simuations: Ŷ = N 1 N i=1 f(ŝ(i) T/h ) weak convergence O(h) error in expected payoff strong convergence O(h 1/2 ) error in individua path Mutieve Monte Caro p. 4/41

5 Simpest MC Approach Mean Square Error is O ( N 1 + h 2) first term comes from variance of estimator second term comes from bias due to weak convergence To make this O(ε 2 ) requires N = O(ε 2 ), h = O(ε) = cost = O(N h 1 ) = O(ε 3 ) Aim is to improve this cost to O ( ε p), with p as sma as possibe, ideay cose to 1. Note: for a reative error of ε = 0.001, the difference between ε 3 and ε 1 is huge. Mutieve Monte Caro p. 5/41

6 Standard MC Improvements variance reduction techniques (e.g. contro variates, stratified samping) improve the constant factor in front of ε 3, sometimes spectacuary improved second order weak convergence (e.g. through Richardson extrapoation) eads to h = O( ε), giving p=2.5 quasi-monte Caro reduces the number of sampes required, at best eading to N O(ε 1 ), giving p 2 with first order weak methods Mutieve method gives p=2 without QMC, and at best p 1 with QMC. Mutieve Monte Caro p. 6/41

7 Other Reated Research In Dec. 2005, Ahmed Kebaier pubished an artice in Annas of Appied Probabiity describing a two-eve method which reduces the cost to O ( ε 2.5). Aso in Dec. 2005, Adam Speight wrote a working paper describing a very simiar mutieve use of contro variates. There are aso cose simiarities to a mutieve technique deveoped by Stefan Heinrich for parametric integration (Journa of Compexity, 1998) Mutieve Monte Caro p. 7/41

8 Mutieve MC Approach Consider mutipe sets of simuations with different timesteps h = 2 T, = 0, 1,..., L, and payoff P E[ P L ] = E[ P 0 ] + L =1 E[ P P 1 ] Expected vaue is same aim is to reduce variance of estimator for a fixed computationa cost. Key point: approximate E[ P P 1 ] using N simuations with P and P 1 obtained using same Brownian path. Ŷ = N 1 N i=1 ( (i) P ) (i) P 1 Mutieve Monte Caro p. 8/41

9 Mutieve MC Approach Discrete Brownian path at different eves P 0 P P 2 P 3 P 4 P 5 P P Mutieve Monte Caro p. 9/41

10 Mutieve MC Approach Using independent paths for each eve, the variance of the combined estimator is V [ L =0 Ŷ ] = L =0 N 1 V, V V[ P P 1 ], and the computationa cost is proportiona to L =0 N h 1. Hence, the variance is minimised for a fixed computationa cost by choosing N to be proportiona to V h. The constant of proportionaity can be chosen so that the combined variance is O(ε 2 ). Mutieve Monte Caro p. 10/41

11 Mutieve MC Approach For the Euer discretisation and a Lipschitz payoff function V[ P P ] = O(h ) = V[ P P 1 ] = O(h ) and the optima N is asymptoticay proportiona to h. To make the combined variance O(ε 2 ) requires N = O(ε 2 L h ). To make the bias O(ε) requires L = og 2 ε 1 + O(1) = h L = O(ε). Hence, we obtain an O(ε 2 ) MSE for a computationa cost which is O(ε 2 L 2 ) = O(ε 2 (og ε) 2 ). Mutieve Monte Caro p. 11/41

12 Mutieve MC Approach Theorem: Let P be a functiona of the soution of a stochastic o.d.e., and P the discrete approximation using a 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 0 ], = 0 ii) E[Ŷ] = E[ P P 1 ], > 0 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 Mutieve Monte Caro p. 12/41

13 Mutieve MC Approach then there exists a positive constant c 4 such that for any ε<e 1 there are vaues L and N for which the muti-eve estimator L Ŷ = Ŷ, =0 [ (Ŷ ) ] 2 has Mean Square Error MSE E E[P ] < ε 2 with a computationa compexity C with bound C c 4 ε 2, β > 1, c 4 ε 2 (og ε) 2, β = 1, c 4 ε 2 (1 β)/α, 0 < β < 1. Mutieve Monte Caro p. 13/41

14 Mistein Scheme The theorem suggests use of Mistein scheme better strong convergence, same weak convergence Generic scaar SDE: ds(t) = a(s, t) dt + b(s, t) dw (t), 0<t<T. Mistein scheme: Ŝ n+1 = Ŝn + a h + b W n b b ( ) ( W n ) 2 h. Mutieve Monte Caro p. 14/41

15 Mistein Scheme In scaar case: O(h) strong convergence O(ε 2 ) compexity for Lipschitz payoffs trivia O(ε 2 ) compexity for Asian, ookback, barrier and digita options using carefuy constructed estimators based on Brownian interpoation or extrapoation Mutieve Monte Caro p. 15/41

16 Mistein Scheme Key idea: within each timestep, mode the behaviour as simpe Brownian motion conditiona on the two end-points Ŝ(t) = Ŝn + λ(t)(ŝn+1 Ŝn) ) + b n (W (t) W n λ(t)(w n+1 W n ), where λ(t) = t t n t n+1 t n There then exist anaytic resuts for the distribution of the min/max/average over each timestep. Mutieve Monte Caro p. 16/41

17 Resuts Geometric Brownian motion: ds = r S dt + σ S dw, 0 < t < T, with parameters T =1, S(0)=1, r =0.05, σ =0.2 European ca option: exp( rt ) max(s(t ) 1, 0) ( ) T Asian option: exp( rt ) max T 1 S(t) dt 1, 0 Lookback option: exp( rt ) 0 ( ) S(T ) min S(t) 0<t<T Down-and-out barrier option: same as ca provided S(t) stays above B =0.9 Mutieve Monte Caro p. 17/41

18 MLMC Resuts GBM: European ca og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 18/41

19 MLMC Resuts GBM: European ca 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 19/41

20 MLMC Resuts GBM: Asian option og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 20/41

21 MLMC Resuts GBM: Asian option 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 21/41

22 MLMC Resuts GBM: ookback option og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 22/41

23 MLMC Resuts GBM: ookback option 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 23/41

24 MLMC Resuts GBM: barrier option og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 24/41

25 MLMC Resuts GBM: barrier option 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 25/41

26 Mistein Scheme Generic vector SDE: ds(t) = a(s, t) dt + b(s, t) dw (t), 0<t<T, with correation matrix Ω(S, t) between eements of dw (t). Mistein scheme: Ŝ i,n+1 = Ŝi,n + a i h + b ij W j,n + 1 b ( ) ij 2 b k W j,n W k,n h Ω jk A jk,n S with impied summation, and Lévy areas defined as A jk,n = tn+1 t n (W j (t) W j (t n )) dw k (W k (t) W k (t n )) dw j. Mutieve Monte Caro p. 26/41

27 Mistein Scheme In vector case: O(h) strong convergence if Lévy areas are simuated correcty expensive O(h 1/2 ) strong convergence in genera if Lévy areas are omitted, except if a certain commutativity condition is satisfied (usefu for a number of rea cases) Lipschitz payoffs can be handed we using antithetic variabes Other cases may require approximate simuation of Lévy areas Mutieve Monte Caro p. 27/41

28 Resuts Heston mode: ds = r S dt + V S dw 1, 0 < t < T dv = λ (σ 2 V ) dt + ξ V dw 2, T =1, S(0)=1, V (0)=0.04, r =0.05, σ =0.2, λ=5, ξ =0.25, ρ= 0.5 Mutieve Monte Caro p. 28/41

29 MLMC Resuts Heston mode: European ca og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 29/41

30 MLMC Resuts Heston mode: European ca 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 30/41

31 Quasi-Monte Caro we-estabished technique for approximating high-dimensiona integras for finance appications see papers by Ecuyer and book by Gasserman Sobo sequences are perhaps most popuar; we use attice rues (Soan & Kuo) two important ingredients for success: randomized QMC for confidence intervas good identification of dominant dimensions (Brownian Bridge and/or PCA) Mutieve Monte Caro p. 31/41

32 Quasi-Monte Caro Approximate high-dimensiona hypercube integra [0,1] d f(x) dx by where 1 N N 1 i=0 x (i) = f(x (i) ) [ ] i N z and z is a d-dimensiona generating vector. Mutieve Monte Caro p. 32/41

33 Quasi-Monte Caro In the best cases, error is O(N 1 ) instead of O(N 1/2 ) but without a confidence interva. To get a confidence interva, et [ ] i x (i) = N z + x 0. where x 0 is a random offset vector. Using 32 different random offsets gives a confidence interva in the usua way. Mutieve Monte Caro p. 33/41

34 Quasi-Monte Caro For the path discretisation we can use W n = h Φ 1 (x n ), where Φ 1 is the inverse cumuative Norma distribution. Much better to use a Brownian Bridge construction: x 1 W (T ) x 2 W (T/2) x 3, x 4 W (T/4), W (3T/4)... and so on by recursive bisection Mutieve Monte Caro p. 34/41

35 Mutieve QMC rank-1 attice rue deveoped by Soan, Kuo & Waterhouse at UNSW 32 randomy-shifted sets of QMC points number of points in each set increased as needed to achieved desired accuracy, based on confidence interva estimate resuts show QMC to be particuary effective on owest eves with ow dimensionaity Mutieve Monte Caro p. 35/41

36 MLQMC Resuts GBM: European ca og 2 variance og 2 mean P P P Mutieve Monte Caro p. 36/41

37 MLQMC Resuts GBM: European ca N ε= ε= ε= ε= ε=0.001 ε 2 Cost Std QMC MLQMC ε Mutieve Monte Caro p. 37/41

38 MLQMC Resuts GBM: barrier option og 2 variance og 2 mean P P P Mutieve Monte Caro p. 38/41

39 MLQMC Resuts GBM: barrier option ε= ε= ε= ε= ε= Std QMC MLQMC N 10 4 ε 2 Cost ε Mutieve Monte Caro p. 39/41

40 Concusions Resuts so far: much improved order of compexity fairy easy to impement significant benefits for mode probems However: ots of scope for further deveopment muti-dimensiona SDEs needing Lévy areas adjoint Greeks and vibrato Monte Caro numerica anaysis of agorithms execution on NVIDIA graphics cards (128 cores) need to test ideas on rea finance appications Mutieve Monte Caro p. 40/41

41 Papers M.B. Gies, Mutieve Monte Caro path simuation, to appear in Operations Research, M.B. Gies, Improved mutieve convergence using the Mistein scheme, to appear in MCQMC06 proceedings, Springer-Verag, M.B. Gies, Mutieve quasi-monte Caro path simuation, submitted to Journa of Computationa Finance, Emai: Mutieve Monte Caro p. 41/41