Multilevel quasi-monte Carlo path simulation Michael B. Giles and Ben J. Waterhouse Lluís Antoni Jiménez Rugama January 22, 2014
Index 1 Introduction to MLMC Stochastic model Multilevel Monte Carlo Milstein discretisation 2 Quasi-Monte Carlo method Transforming the integration Choice of parameters 3 Definition of the algorithm Assumptions Algorithm 4 Example Financial example
Our model The SDE we shall work with along this work is, ds(t) = a(s, t)dt + b(s, t)dw (t), 0 < t < T
Principles For a time step h l = 2 l T where l = 0, 1,..., L, let P denote the payoff and ˆP l its approximation using a numerical discretisation with time step h l. Then, E[ ˆP L ] = E[ ˆP 0 ] + and the estimators considered are, L E[ ˆP l ˆP l 1 ] l=1 Ŷ 0 = N 1 0 N 0 i=1 ˆP (i) 0, Ŷ l = N 1 l N l ( i=1 ˆP (i) l ˆP (i) l 1 )
Main theorem Under strong assumptions on the estimators and the bias, it is stated that ɛ < e 1, L, N l where Ŷ = L l=0 Ŷ l has a mean square error with bound E[(Ŷ E[P ])2 ] < ɛ 2
Improving the complexity According to one of the theorem s conclusions, the complexity of Ŷ is optimal with the Milstein discretisation, Ŝ n+1 = Ŝn + a n h + b n W n + 1 b n 2 S b n(( W n ) 2 h)
Index 1 Introduction to MLMC Stochastic model Multilevel Monte Carlo Milstein discretisation 2 Quasi-Monte Carlo method Transforming the integration Choice of parameters 3 Definition of the algorithm Assumptions Algorithm 4 Example Financial example
How to compute it? For a scalar SDE with n T time steps, the dimensionality needed is d = n T. With two transformations we can obtain: Rd P (x) exp( 1 2 xt Σ 1 x) (2π) d/2 det Σ dx = Rd P (Ay) exp( 1 2 yt y) (2π) d/2 dy = [0,1] d P (AΦ 1 (z))dz
How to choose the matrix A? While any choice of A such that AA T = Σ is suitable, there are three established ways:
How to choose the matrix A? While any choice of A such that AA T = Σ is suitable, there are three established ways: Cholesky, W n = hφ 1 (x i,n )
How to choose the matrix A? While any choice of A such that AA T = Σ is suitable, there are three established ways: Cholesky, W n = hφ 1 (x i,n ) Brownian Bridge construction, B t B u N(t/uB u, t t 2 /u)
How to choose the matrix A? While any choice of A such that AA T = Σ is suitable, there are three established ways: Cholesky, W n = hφ 1 (x i,n ) Brownian Bridge construction, B t B u N(t/uB u, t t 2 /u) PCA, λ n v n
Index 1 Introduction to MLMC Stochastic model Multilevel Monte Carlo Milstein discretisation 2 Quasi-Monte Carlo method Transforming the integration Choice of parameters 3 Definition of the algorithm Assumptions Algorithm 4 Example Financial example
Important assumptions We assume that there is first order weak convergence. Then we have that, E[P ˆP L ] ŶL Being cautious with the sign, we estimate the magnitude of the bias using, { 1 } max ŶL 1, ŶL 2
Final algorithm The mean square error is the sum of the combined variance L l=0 V l plus the square of the bias E[P ˆP L ]. We choose to make both of them smaller than ɛ 2 /2 to achieve the RMS accuracy of ɛ: Set L = 0. Initial estimate of V L (using 32 random shifts and N L = 1). While L l=0 V l > ɛ 2 /2, double N l on the level with largest V l /(2 l N l ). If L < 2 or the bias estimate is greater than ɛ/ 2, set L := L + 1 and go to step 2.
Index 1 Introduction to MLMC Stochastic model Multilevel Monte Carlo Milstein discretisation 2 Quasi-Monte Carlo method Transforming the integration Choice of parameters 3 Definition of the algorithm Assumptions Algorithm 4 Example Financial example
Lookback call option 0 0 5 10 5 log 2 variance 15 20 25 30 35 1 16 256 4096 40 0 2 4 6 8 l log 2 mean 10 15 P l P l P l 1 20 0 2 4 6 8 l N l 10 5 10 4 10 3 10 2 ε=0.0001 ε=0.0002 ε=0.0005 ε=0.001 ε=0.002 ε 2 Cost 10 0 10 1 10 2 Std QMC MLQMC 10 1 10 0 0 2 4 6 8 l 10 3 10 4 10 3 ε