Fast orthogonal transforms and generation of Brownian paths

Transcription

1 Fast orthogonal transforms and generation of Brownian paths G. Leobacher partially joint work with C. Irrgeher MCQMC 2012 G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

2 Outline 1 Discrete Brownian paths 2 Linear construction Examples Why use BB/PCA? Cost of generation 3 Dependence on payoff 4 Orthogonal constructions 5 Fast completion of orthogonal systems Problem formulation Givens rotations Hoeseholder Reflections 6 References G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

3 Discrete Brownian paths What - and why Many financial derivatives can be priced (approximately) using ( ) price = E f(b1,...,bn ) n n where B is a standard Brownian motion, B = (B t ) t [0,1] and f is some payoff function (Black-Scholes formula) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

4 Discrete Brownian paths What - and why We therefore want efficient ways to generate a random vector where B1,B2 n n B = (B1,...,Bn ) n n B1,...,Bn n n Bn 1 n are independent normal variables with mean 0 and variance 1 n, from X 1,...,X n...i.i.d.n(0,1) Discrete Brownian path G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

5 Linear construction Three classical methods: Forward construction Brownian bridge construction (BB) First application in financial/qmc context: Moskowitz & Caflisch (1996) Principal component analysis (PCA) First application in financial context: Acworth, Broadie & Glasserman (1996) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

6 Linear construction All constructions are of the form where B = AX B = (B1,...,Bn ) n n (X 1,...,X n ) indep. std. normal variables A an n n matrix G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

7 Linear construction Necessary and sufficient for B being a (discrete) Brownian path: AA = =: Σ n n G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

8 Linear construction Examples Forward method: A = 1 n =: S SS = Σ... Cholesky decomposition of Σ Brownian bridge construction: A = something else PCA construction: take A = VD where VD 2 V = Σ is the singular value decomposition of Σ. G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

9 Linear construction Why use BB/PCA? BB/PCA probabilistically equivalent to forward construction, but Influence of X k on overall behavior of B is decreasing with k makes methods useful for MC with stratified sampling makes methods useful for quasi-monte Carlo (low effective dimension / decreasing weights) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

10 Linear construction Cost of generation Forward method: O(n) Brownian bridge: O(n) PCA: Originally thought to be O(n 2 )!! Scheicher (2007): PCA generation costs O(n log(n)) by using the fast sine transform in dimension 2n By fast we mean O(nlog(n)) (or faster) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

11 Dependence on payoff (Some people are never satisfied) Papageorgiou (2002), Sloan & Wang (2011) Integration error depends on payoff So there is no best generation method Suggests to look for optimal A (in some sense) for given payoff But multiplication with A should be fast! G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

12 Orthogonal constructions We can write any matrix A with AA = Σ as where U is an orthogonal matrix and A = SU S is scaled summation (multiplication with S is fast) suggests we look for good/optimal U for given payoff but multiplication with U should be fast! PCA/BB provide good and fast U for Asian options (and many other types) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

13 Orthogonal constructions Makes method generic: every finite-dimensional (quasi-)mc problem can by written as E(f(X)) where X is a standard normal vector. E(f(X)) = E(f(UX)) for any orthogonal matrix U. Choose U such that variance/variation is concentrated on few variables (low effective dimension / decreasing weights) Sample application: BB/PCA-type generation for Lévy paths (L (2006)) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

14 Orthogonal constructions Examples Σ = AA with A = SU Forward method: U = Id R n. Brownian Bridge: U = H 1 where H is the Haar transform PCA: U = S 1 VD (trivially) G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

15 Orthogonal constructions One advantage of orthogonal formulation: there are many fast generation methods for BM besides forward, BB and PCA method. (L 2011) reviews transforms that need at most O(n log n) operations Discrete Sine/Cosine transform Hartley-, Hilbert-, W- transform Walsh transform Haar transform, general wavelet transforms Direct sums and Kronecker products of the above and more G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

16 Orthogonal constructions At least one of these fast constructions is actually useful in practice Theorem (L 2011) Let Σ = VD 2 V be the PCA of Σ C the discrete cosine transform of type IV in dimension n d n (P,Q) 2 := n n l=1 k=1 (P Q)2 lk for n n matrices P,Q Then for all n N we have d n (SC,VD) < 1 and lim supd n (SC,VD) 2 2( 48 π 2) n (π 2 24) 2 = G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

17 Orthogonal constructions Comparison PCA/DCT-IV, n = 64 G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

18 Orthogonal constructions Comparison PCA/DCT-IV, n = 256 G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

19 Orthogonal constructions Thus we have found a construction method that gives essentially same result as PCA is very easy to implement uses roughly half the time for path generation However it is not easy to find a payoff depending on a single Brownian path for which one of our constructions performs better than BB/PCA G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

20 Fast completion of orthogonal systems Problem formulation Imai & Tan (2007): Find (approximate) U that works best for Asian basket and Heston model Sloan & Wang (2011): Provide more theoretical insight into efficient orthogonal transforms Both papers above identify most important components, i.e. the first k rows of U. U is subsequently orthogonalized, i.e. filled up with orthogonal rows. G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

21 Fast completion of orthogonal systems Problem formulation U =? k n k G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

22 Fast completion of orthogonal systems We aim to: Find good U that admits fast matrix-vector multiplication Find it fast: Gram-Schmidt orthogonalization takes O(n 3 ) operations! Solution (Irrgeher & L): Givens rotations! G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

23 Fast completion of orthogonal systems Givens rotations Recall G =... 1 cos(α) sin(α) sin(α) cos(α) 1... G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

24 Fast completion of orthogonal systems Givens rotations Recall Givens rotations: n n-matrix G, G is a rotation in a 2-dimensional subspace spanned by two different canonical basis vectors e j, e k one Givens rotation needs 4 multiplications and two additions the angle can be chosen such that for a given matrix A the (j,k)-th is made zero, i.e. (AG) j,k = 0 Therefore any orthogonal n n-matrix can be written as the product of at most n(n 1)/2 Givens rotations and n reflections along the canonical basis vectors G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

25 Fast completion of orthogonal systems Idea: if U is of the following form U = 0 k n k then U is the product of at most k(k 1) 2 +k(n k) kn Givens rotations G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

26 Fast completion of orthogonal systems Givens rotations Therefore multiplication by U can be done using O(kn) operations! Remaining question: can we compute the Givens rotations efficiently? Answer: Yes, can be done using O(k 2 n) operations G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

27 Fast completion of orthogonal systems Hoeseholder Reflections Altenatively, we may Householder reflections (L & Irrgeher). Gives same complexities. G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28

28 References Leobacher (2006) Stratified sampling and quasi-monte Carlo simulation of Lévy processes, Monte-Carlo Methods Appl. Leobacher (2011) Fast orthogonal transforms and generation of Brownian paths, J. Complexity. Irrgeher & Leobacher (2012) Fast completion of orthogonal systems and the LT method, preprint. Imai & Tan (2007) A general dimension reduction technique for derivative pricing, J. Comput. Finance. Sloan & Wang (2011) Quasi-Monte Carlo methods in financial engineering: An equivalence principle and dimension reduction, Oper. Res. G. Leobacher (JKU Linz, Austria) Orthogonal transforms/brownian paths MCQMC / 28