Multilevel Monte Carlo Path Simulation

Transcription

1 Mutieve Monte Caro Path Simuation Mike Gies Oxford University Computing Laboratory First IMA Conference on Computationa Finance Mutieve Monte Caro p. 1/34

2 Generic Probem Stochastic differentia equation with genera drift and voatiity terms: ds(t) = a(s, t) dt + b(s, t) dw (t) In many appications, we want to compute the expected vaue of an option dependent on 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. 2/34

3 Standard MC Approach Euer discretisation with timestep h: Ŝ n+1 = Ŝn + a(ŝn, t n ) h + b(ŝn, t n ) W n Simpest 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. 3/34

4 Standard 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 2 (og ε) 2), by combining simuations with different numbers of timesteps same accuracy as finest cacuations, but at a much ower computationa cost. Mutieve Monte Caro p. 4/34

5 Other 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 simiar mutieve use of contro variates, but without an anaysis of its compexity. There are aso cose simiarities to a mutieve technique deveoped by Stefan Heinrich for parametric integration (Journa of Compexity, 1998) Mutieve Monte Caro p. 5/34

6 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. 6/34

7 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. 7/34

8 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. 8/34

9 Mutieve MC Approach For the Euer discretisation and the 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. 9/34

10 Resuts Geometric Brownian motion: ds = r S dt + σ S dw, 0 < t < T, T =1, S(0)=1, r =0.05, σ =0.2 European ca option with discounted payoff exp( rt ) max(s(t ) K, 0) with K =1. Mutieve Monte Caro p. 10/34

11 Resuts GBM: European ca, exp( rt ) max(s(t ) K, 0) og 2 variance 10 og 2 mean P P P P P P Mutieve Monte Caro p. 11/34

12 Resuts GBM: European ca, exp( rt ) max(s(t ) K, 0) ε= ε= ε= ε= ε= N 10 6 ε 2 Cost Std MC MLMC ε Mutieve Monte Caro p. 12/34

13 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. 13/34

14 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 has Mean Square Error MSE E L Ŷ = Ŷ, =0 [ (Ŷ E[P ] ) 2 ] < 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. 14/34

15 Mistein Scheme 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. 15/34

16 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 key idea: within each timestep, mode the behaviour as simpe Brownian motion conditiona on the two end-points anaytic resuts exist for distribution of min/max/average Mutieve Monte Caro p. 16/34

17 Resuts GBM: European ca, exp( rt ) max(s(t ) K, 0) og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 17/34

18 Resuts GBM: European ca, exp( rt ) max(s(t ) K, 0) 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 18/34

19 Resuts GBM: ookback option, exp( rt ) (S(T ) min 0<t<T S(t)) og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 19/34

20 Resuts GBM: ookback option, exp( rt ) (S(T ) min 0<t<T S(t)) 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 20/34

21 Resuts GBM: barrier option, exp( rt ) 1 min S(t)>B max(s(t ) K, 0) og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 21/34

22 Resuts GBM: barrier option, exp( rt ) 1 min S(t)>B max(s(t ) K, 0) 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε Mutieve Monte Caro p. 22/34

23 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. 23/34

24 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. 24/34

25 Mistein Scheme If Ŝc n satisfies and Ŝf n satisfies Ŝ c n+1 = R(Ŝc n), Ŝ f n+1 = R(Ŝf n ) + g n. then if g n 1, putting Ŝf n =Ŝc n+ D n and inearising gives D n+1 = R S D n + g n. Ŝ c n represents cacuation using timestep 2h Ŝ f n represents cacuation using two timesteps of size h Mutieve Monte Caro p. 25/34

26 Mistein Scheme To eading order, error anaysis gives g i,n = h 2 b ij S b k ( Y j,n Z k,n Y k,n Z j,n ). where W f n = 1 2 2h (Yn +Z n ), W f n+ 1 2 = 1 2 2h (Yn Z n ). i.e. Y n is standard N(0, 1) variabe used to construct coarse path, and Z n is N(0, 1) variabe for Brownian Bridge construction of fine path. Note: independence impies that E[g n ] = 0 = E[ D n ] = 0. Mutieve Monte Caro p. 26/34

27 Mistein Scheme Option 1: use contro variate Define Ŷ = N 1 N i=1 ( P (i) P (i) 1 f S ) (i) D, T/2h The contro variate has zero mean and cances out the eading order variation so that [ V P P 1 f ] S D T/2h = O(h 2 ) for twice differentiabe payoffs (and O(h 3/2 ) for usua Lipschitz payoffs?) Mutieve Monte Caro p. 27/34

28 Mistein Scheme Option 2: use antithetic variabes Define Ŷ = N 1 N i=1 ( ( 12 (i) P + ) (i) P ) (i) P 1, P (i) where is based on the same coarse path with Z n repaced by Z n, which eads to canceation of eading order error proportiona to Z n. Very simpe to impement (but sighty more costy?) Mutieve Monte Caro p. 28/34

29 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. 29/34

30 Resuts Heston mode: European ca og 2 variance og 2 mean P P P P 1 P P Mutieve Monte Caro p. 30/34

31 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. 31/34

32 Concusions Resuts so far: (much) improved order of compexity (fairy) easy to impement significant benefits for mode probems However: ots of scope for further improvement need to test ideas on rea finance appications Mutieve Monte Caro p. 32/34

33 Future Work muti-dimensiona SDEs with barrier and digita options quasi-monte Caro integration (F. Kuo, I. Soan UNSW) Greeks and caibration (P. Gasserman Coumbia Business Schoo) numerica anaysis (D. Higham, X. Mao Strathcyde) rea finance appications parae impementation on hyper-core chips (CearSpeed, nvidia cores) Mutieve Monte Caro p. 33/34

34 Working Papers M.B. Gies, Mutieve Monte Caro path simuation, Numerica Anaysis Report NA-06/03 M.B. Gies, Improved mutieve convergence using the Mistein scheme, Numerica Anaysis Report NA-06/22 Emai: Acknowedgements: Pau Gasserman and Mark Broadie for eary feedback Microsoft for current research funding Mutieve Monte Caro p. 34/34