Multilevel Monte Carlo Path Simulation

Transcription

1 Mutieve Monte Caro Path Simuation Mike Gies Oxford University Computing Laboratory 15th Scottish Computationa Mathematics Symposium Mutieve Monte Caro p. 1/34

2 SDEs in Finance In computationa finance, stochastic differentia equations are used to mode the behaviour of stocks interest rates exchange rates weather eectricity/gas demand crude oi prices... The stochastic term accounts for the uncertainty of unpredictabe day-to-day events. Mutieve Monte Caro p. 2/34

3 SDEs in Finance These modes are then used to cacuate fair prices for a huge range of financia options: an option to se a stock portfoio at a specific price in 2 years time an option to buy aviation fue at a specific price in 6 months time an option to se US doars at a specific exchange rate in 3 years time In most cases, the buyer of the financia option is trying to reduce their risk. Mutieve Monte Caro p. 3/34

4 SDEs in Finance Exampes: Geometric Brownian motion (Back-Schoes mode for stock prices) ds = r S dt + σ S dw Cox-Ingerso-Ross mode (interest rates) dr = α(b r) dt + σ r dw Heston stochastic voatiity mode (stock prices) ds = r S dt + V S dw 1 dv = λ (σ 2 V ) dt + ξ V dw 2 with correation ρ between dw 1 and dw 2 Mutieve Monte Caro p. 4/34

5 Generic Probem Stochastic differentia equation with genera drift and voatiity terms: SDE with genera drift and voatiity terms: ds(t) = a(s, t) dt + b(s, t) dw (t) W (t) is a Wiener variabe with the properties that for any q <r <s<t, W (t) W (s) is Normay distributed with mean 0 and variance t s, independent of W (r) W (q). In many finance 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. 5/34

6 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 ) Mutieve Monte Caro p. 6/34

7 Standard MC Approach Two kinds of errors: statistica error, due to finite number of paths so r.m.s. error = O(N 1/2 ). V [Ŷ ] = N 1 V [f(ŝt/h)] discretisation bias, due to finite number of timesteps weak convergence O(h) error in expected payoff strong convergence O(h 1/2 ) error in individua path Mutieve Monte Caro p. 7/34

8 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) Mutieve Monte Caro p. 8/34

9 Mutigrid A powerfu technique for soving PDE discretisations: Fine grid more accurate more expensive Coarse grid ess accurate ess expensive Mutieve Monte Caro p. 9/34

10 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. 10/34

11 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. 11/34

12 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. 12/34

13 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. 13/34

14 Mutieve MC Approach each eve adds more detai to Brownian path E[ P P 1 ] refects impact of that extra detai on the payoff different timescaes handed by different eves simiar to different waveengths being handed by different grids in mutigrid Mutieve Monte Caro p. 14/34

15 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. 15/34

16 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. 16/34

17 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. 17/34

18 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. 18/34

19 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. 19/34

20 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. 20/34

21 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. 21/34

22 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. 22/34

23 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. 23/34

24 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. 24/34

25 Resuts Heston mode: European ca N ε= ε= ε= ε= ε=0.001 ε 2 Cost Std MC MLMC ε Mutieve Monte Caro p. 25/34

26 Concusions Resuts so far: improved order of compexity easy to impement significant benefits for mode probems Current research: use of Mistein method (and antithetic variabes in muti-dimensiona case) to reduce compexity to O(ε 2 ) adaptive samping to treat discontinuous payoffs and pathwise derivatives for Greeks use of quasi-monte Caro methods, to reduce compexity towards O(ε 1 ) Mutieve Monte Caro p. 26/34

27 Working Paper M.B. Gies, Muti-eve Monte Caro path simuation Oxford University Computing Laboratory Numerica Anaysis Report NA-06/03 Emai: Acknowedgements: Pau Gasserman and Mark Broadie for eary feedback Microsoft for current research funding Mutieve Monte Caro p. 27/34

28 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. 28/34

29 Mistein Scheme In scaar case: O(h) strong convergence O(ε 2 ) compexity for Lipschitz payoffs O(ε 2 ) compexity for ookback, barrier and digita options using carefuy constructed estimators In muti-dimensiona case: sti ony O(h 1/2 ) strong convergence but Ŝ n E[S W n ] = O(h) Mutieve Monte Caro p. 29/34

30 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. 30/34

31 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. 31/34

32 Adaptive samping With digita options, the probem is that sma path changes can 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 P P 1 Hence, V = O(h 1/2 ). For improved resuts, need more sampes of paths near payoff discontinuities. Mutieve Monte Caro p. 32/34

33 Adaptive samping Two ideas for adaptive samping are both based on Brownian Bridge constructions, using coarse timestep reaisations to decide which paths are important idea 1: start with reativey few paths, and sub-divide those which ook interesting (spitting) idea 2: start with ots of paths, and prune those which are unimportant (Russian rouette) use path weights to ensure estimator remains unbiased initia resuts (combining 2 ideas to keep a fixed number of paths) ook good for a digita option, and it shoud aso hande barrier options Mutieve Monte Caro p. 33/34

34 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 confidence intervas can be obtained by using randomized QMC working with Soan, Kuo and Waterhouse, wi try both rank-1 attice rues and Sobo sequences Mutieve Monte Caro p. 34/34