Variance Reduction Through Multilevel Monte Carlo Path Calculations

Transcription

1 Variance Reduction Through Mutieve Monte Caro Path Cacuations Mike Gies Oxford University Computing Laboratory Mutieve Monte Caro p. 1/30

2 Mutigrid A powerfu technique for soving PDE discretisations: Fine grid more accurate more expensive Coarse grid ess accurate ess expensive Mutieve Monte Caro p. 2/30

3 Mutigrid Mutigrid combines cacuations on a nested sequence of grids to get the accuracy of the finest grid at a much ower computationa cost. We wi use a simiar idea to achieve variance reduction in Monte Caro path cacuations, combining simuations with different numbers of timesteps same accuracy as finest cacuations, but at a much ower computationa cost. Mutieve Monte Caro p. 3/30

4 Generic Probem SDE with genera drift and voatiity terms: ds(t) = a(s, t) dt + b(s, t) dw (t) Suppose 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. 4/30

5 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. 5/30

6 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) (In 2005, Ahmed Kebaier pubished a two-eve method which reduces the cost to O ( ε 2.5), equivaent to a singe appication of Richardson extrapoation.) Mutieve Monte Caro p. 6/30

7 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. 7/30

8 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. 8/30

9 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. Mutieve Monte Caro p. 9/30

10 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 Lh ). 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. 10/30

11 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. 11/30

12 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 4 ε 2, β > 1, C c 4 ε 2 (og ε) 2, β = 1, c 4 ε 2 (1 β)/α, 0 < β < 1. Mutieve Monte Caro p. 12/30

13 Resuts Geometric Brownian motion: ds = r S dt + σ S dw, 0 < t < 1, S(0)=1, r =0.05, σ =0.2 Heston mode: ds = r S dt + V S dw 1, 0 < t < 1 dv = λ (σ 2 V ) dt + ξ V dw 2, S(0)=1, V (0)=0.04, r =0.05, σ =0.2, λ=5, ξ =0.25, ρ= 0.5 A cacuations use M =4, more efficient than M =2. Mutieve Monte Caro p. 13/30

14 Resuts GBM: European ca, max(s(1) 1, 0) og M variance 4 6 og M mean P 8 P P P 1 P P Mutieve Monte Caro p. 14/30

15 Resuts GBM: European ca, max(s(1) 1, 0) ε= ε= ε= ε= ε= N 10 6 ε 2 Cost Std MC MLMC ε Mutieve Monte Caro p. 15/30

16 Resuts GBM: ookback option, S(1) min 0<t<1 S(t) og M variance 4 6 og M mean P 8 P P P 1 P P Mutieve Monte Caro p. 16/30

17 Resuts GBM: ookback option, S(1) min 0<t<1 S(t) ε= ε= ε= ε= ε= N 10 6 ε 2 Cost Std MC MLMC ε Mutieve Monte Caro p. 17/30

18 Resuts Heston mode: European ca og M variance 4 6 og M mean P 8 P P P 1 P P Mutieve Monte Caro p. 18/30

19 Resuts Heston mode: European ca N ε= ε= ε= ε= ε=0.001 ε 2 Cost Std MC MLMC ε Mutieve Monte Caro p. 19/30

20 Comments Resuts so far: improved order of compexity easy to impement significant benefits for mode probems Future work: use of Mistein method and a contro variate or antithetic variabes to reduce compexity to O(ε 2 ) adaptive samping to treat discontinuous payoffs and pathwise derivatives for Greeks use of quasi-monte Caro methods additiona variance reduction techniques Mutieve Monte Caro p. 20/30

21 Mistein Scheme Generic 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). Simpest Mistein scheme sets Lévy areas to zero to give Ŝ i,n+1 = Ŝi,n+a i h+b ij W j,n using impied summation convention. b ij S b k ( W j,n W k,n h Ω jk ) Mutieve Monte Caro p. 21/30

22 Mistein Scheme In scaar case: O(h) strong convergence O(ε 2 ) compexity for Lipschitz payoffs O(ε 2 (og ε) 2 ) compexity for digitas and Greeks In vector case: sti ony O(h 1/2 ) strong convergence but Ŝ n E[S W n ] = O(h) Mutieve Monte Caro p. 22/30

23 Mistein Scheme If a coarse path with timestep 2h is constructed using W c n = 2h Y n where the Y n are N(0, 1) random variabes, and the fine path uses a Brownian Bridge construction with W f n = 1 2 2h (Yn +Z n ), W f n+ 1 2 = 1 2 2h (Yn Z n ). where the Z n are aso N(0, 1) random variabes, then perturbation anaysis shows that the O(h 1/2 ) difference between the two paths comes from a sum of terms proportiona to Y j,n Z k,n Y k,n Z j,n. Mutieve Monte Caro p. 23/30

24 Mistein Scheme Using the idea of antithetic variabes, we use the estimator Ŷ = N 1 N i=1 ( ( 12 (i) P + ) (i) P ) (i) P 1, P (i) where is based on the same coarse path Y n, but with Z n repaced by Z n, which eads to the canceation of the eading order error proportiona to Z n. V [Ŷ] = O(h 2 ) for smooth payoffs, O(h 3/2 ) for Lipschitz in both cases, gives O(ε 2 ) compexity for O(ε) accuracy Mutieve Monte Caro p. 24/30

25 Adaptive samping With digita options, the probem is that sma path changes ead to an O(1) change in the payoff For the Euer discretisation, O(h 1/2 ) strong convergence = O(h 1/2 ) paths have an O(1) vaue for Ŷ Hence, V [Ŷ] = O(h 1/2 ). For improved resuts, need more sampes of paths near payoff discontinuities. Mutieve Monte Caro p. 25/30

26 Adaptive samping Two ideas for adaptive samping are both based on Brownian Bridge constructions, using coarse timestep reaisations to decide which paths are interesting (i.e. ikey to produce a arge variance) idea 1: start with ots of paths, and prune those which are not interesting idea 2: start with reativey few paths, and sub-divide those which ook interesting in each case, need to use path weights to ensure estimator remains unbiased no resuts yet, but I think this wi make digita and barrier options as efficient as Lipschitz payoffs Mutieve Monte Caro p. 26/30

27 Quasi-Monte Caro Quasi-Monte Caro methods can offer greaty improved convergence with respect to the number of sampes N: in the best case, O(N 1+δ ) error for arbitrary δ > 0, instead of O(N 1/2 ) depends on knowedge/identification of important dimensions in an appication Brownian Bridge Principa Component Anaysis most theory doesn t appy to financia appications because of ack of payoff smoothness confidence intervas can be obtained by using randomized QMC my pans are to start by using Sobo sequences Mutieve Monte Caro p. 27/30

28 Other Variance Reduction stratified samping probaby not, because QMC has aready done a good job of eading dimensions contro variate probaby not (except perhaps for geometric Asian) mutieve approach can be viewed as using the coarse path vaue as a contro variate importance samping might be usefu for over-samping the tais of the Norma distributions Mutieve Monte Caro p. 28/30

29 Fina words Resuts so far: improved order of compexity easy to impement significant benefits for mode probems Future work: use of Mistein method and a contro variate or antithetic variabes to reduce compexity to O(ε 2 ) adaptive samping to treat discontinuous payoffs and pathwise derivatives for Greeks use of quasi-monte Caro methods additiona variance reduction techniques Mutieve Monte Caro p. 29/30

30 Working Paper M.B. Gies, Muti-eve Monte Caro path simuation Oxford University Computing Laboratory Numerica Anaysis Report NA-06/03 Emai: Mutieve Monte Caro p. 30/30