Multilevel Monte Carlo Path Simulation

Transcription

1 Mutieve Monte Caro p. 1/32 Mutieve Monte Caro Path Simuation Mike Gies Oxford University Mathematica Institute Oxford-Man Institute of Quantitative Finance Workshop on Stochastic Computationa Methods with Appications in Finance, Insurance and the Life Sciences RICAM, November 17-21, 2008

2 Mutieve Monte Caro p. 2/32 Generic Probem Stochastic differentia equation with genera drift and voatiity terms: ds(t) = a(s,t) dt + b(s,t) dw(t) 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.

3 Mutieve Monte Caro p. 3/32 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

4 Mutieve Monte Caro p. 4/32 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.

5 Mutieve Monte Caro p. 5/32 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

6 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. 6/32

7 Mutieve Monte Caro p. 7/32 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

8 Mutieve Monte Caro p. 8/32 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 ).

9 Mutieve Monte Caro p. 9/32 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 ).

10 Mutieve Monte Caro p. 10/32 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.

11 Mutieve Monte Caro p. 11/32 MLMC Resuts GBM: European ca, exp( rt) max(s(t) K, 0) og 2 variance 10 og 2 mean P P P P P P

12 Mutieve Monte Caro p. 12/32 MLMC 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

13 Mutieve Monte Caro p. 13/32 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

14 Mutieve Monte Caro p. 14/32 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.

15 Mutieve Monte Caro p. 15/32 Mistein Scheme The theorem suggests use of Mistein approximation 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 + ah + b W n b b ( ) ( W n ) 2 h.

16 Mutieve Monte Caro p. 16/32 Mistein Scheme In scaar case: O(h) strong convergence O(ε 2 ) compexity for Lipschitz payoffs trivia O(ε 2 ) compexity for more compex cases using carefuy constructed estimators based on Brownian interpoation or extrapoation digita, with discontinuous payoff Asian, based on average ookback and barrier, based on min/max

17 Mutieve Monte Caro p. 17/32 MLMC Resuts GBM: European ca, exp( rt) max(s(t) K, 0) og 2 variance og 2 mean P P P P 1 P P

18 Mutieve Monte Caro p. 18/32 MLMC 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

19 Mutieve Monte Caro p. 19/32 MLMC Resuts GBM: Asian option, exp( rt) max(t 1 T S(t) dt 1, 0) og 2 variance og 2 mean P P P P 1 P P

20 Mutieve Monte Caro p. 20/32 MLMC Resuts GBM: Asian option, exp( rt) max(t 1 T 0 S(t) dt 1, 0) 10 8 ε= ε= ε= ε= ε= Std MC MLMC N 10 6 ε 2 Cost ε

21 Mutieve Monte Caro p. 21/32 MLMC 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

22 Mutieve Monte Caro p. 22/32 MLMC 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

23 Mutieve Monte Caro p. 23/32 Extensions 1) Mistein scheme for vector SDEs significanty more difficut because it invoves 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. 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

24 Mutieve Monte Caro p. 24/32 Extensions 2) Quasi-Monte Caro standard Monte Caro has a random samping error proportiona to N 1/2 Quasi-Monte Caro uses a deterministic choice of sampe points to achieve an error which is neary O(N 1 ) in the best cases Not much appicabe theory because financia payoffs don t have required smoothness In practice, get great resuts using rank-1 attice rues deveoped by Ian Soan s group at UNSW Haven t yet tried Sobo sequences

25 Mutieve Monte Caro p. 25/32 Extensions 3) Numerica Anaysis Finance & Stochastics paper with Des Higham and Xeurong Mao (Strathcyde) on anaysis of Euer discretisation with compex options Rainer Avikainen (Jyväskyä) has a paper in the same issue with a tighter bound for the digita option Kaus Ritter (Darmstadt) and Thomas Müer-Gronbach (Magdeburg) have generaised anaysis of Euer discretisation to path dependent options with a Lipschitz property more work needed to anayse Mistein approximation

26 Mutieve Monte Caro p. 26/32 Extensions 4) Greeks this is the name given to derivatives such as E[P] S 0 [ ] P under certain circumstance, this is equa to E S 0 this eads to the pathwise differentiation approach the mutieve approach shoud again work we but not tried yet can aso incorporate the adjoint approach deveoped with Pau Gasserman more efficient when many Greeks are wanted for one payoff function

27 Mutieve Monte Caro p. 27/32 Extensions 5) vibrato Monte Caro probem with discontinuous payoffs is that sma changes in path can ead to a big change in the payoff so far, have treated digita options using a trick in Pau Gasserman s book, taking the conditiona expectation one timestep before maturity, which effectivey smooths the payoff the vibrato Monte Caro idea generaises this to cases in which the conditiona expectation is not known in cosed form

28 Mutieve Monte Caro p. 28/32 Extensions 6) American options with European options, the buyer can ony exercise the option at maturity, the fina time T with American options, the buyer can exercise at any time, eading to an optima contro probem in PDE approaches, this is soved using a inear compementarity approach which marches backwards in time modifying Monte Caro methods is much harder an active research topic I have some ideas on how to incorporate the mutieve approach hope to start a project on this soon

29 Mutieve Monte Caro p. 29/32 Extensions 7) Adaptive time integration Rau Tempone and Anders Szepessy (KTH) are combining their adaptive timestepping methods (based on adjoint-weighted error indicators) with the mutieve method Have appied it very successfuy to first exit time appications Shoud aso be very good for CIR and Heston modes which invove a v singuarity

30 Mutieve Monte Caro p. 30/32 Extensions 8) SPDEs (stochastic PDEs) working with a coeague Christoph Reisinger on a financia SPDE which is a convection-diffusion PDE with a stochastic convection veocity : dv = µ v x dt v x 2 dt ρ v x dw working with Rob Scheich (Bath) on an eiptic SPDE where the diffusivity is a og-norma stochastic fied: (κ(x) p ) = 0. Kaus Ritter and others are aso using mutieve approach for SPDE s

31 Mutieve Monte Caro p. 31/32 Extensions 9) CUDA impementation on NVIDIA graphics cards advances in computer hardware/software are important as we as advances in mathematics graphics cards are very powerfu parae processors, with up to 240 cores per graphics chip (GPU) 2 years ago, NVIDIA introduced the CUDA deveopment environment which uses minor extension to C/C++ with a visiting student, Xiaoke Su, achieved 100 speedup on a Monte Caro appication using 128 cores (more recenty, achieved 50 speedup for simpe PDE appications, incuding impicit ADI time-marching)

32 Mutieve Monte Caro p. 32/32 Concusions Mutieve Monte Caro method has aready achieved improved order of compexity significant benefits for mode probems but much more research is needed, both theoretica and appied. Acknowedgements: Pau Gasserman, Mark Broadie, Terry Lyons for eary discussions Microsoft, EPSRC, Oxford-Man Institute for funding