Fast orthogonal transforms and generation of Brownian paths G. - - PowerPoint PPT Presentation

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 2012 1 / 28

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 2012 2 / 28

Discrete Brownian paths

What - and why

Many financial derivatives can be priced (approximately) using price = E

• f (B 1

n , . . . , B n n )

• where B is a standard Brownian motion, B = (Bt)t∈[0,1] and f is some

payoff function (Black-Scholes formula)

• G. Leobacher (JKU Linz, Austria)

Orthogonal transforms/Brownian paths MCQMC 2012 3 / 28

Discrete Brownian paths

What - and why

We therefore want efficient ways to generate a random vector B = (B 1

n , . . . , B n n )

where B 1

n , B 2 n − B 1 n , . . . , B n n − B n−1 n

are independent normal variables with mean 0 and variance 1

n, from

X1, . . . , Xn . . . i.i.d.N(0, 1) “Discrete Brownian path”

• G. Leobacher (JKU Linz, Austria)

Orthogonal transforms/Brownian paths MCQMC 2012 4 / 28

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)

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Linear construction

All constructions are of the form B = AX where B = (B 1

n , . . . , B n n )

(X1, . . . , Xn) indep. std. normal variables A an n × n matrix

• G. Leobacher (JKU Linz, Austria)

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Linear construction

Necessary and sufficient for B being a (discrete) Brownian path: AA⊤ = 1 n        1 1 1 . . . 1 1 2 2 . . . 2 1 2 3 . . . 3 . . . . . . . . . ... . . . 1 2 3 . . . n        =: Σ

• G. Leobacher (JKU Linz, Austria)

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Linear construction

Examples

Forward method: A = 1 √n        1 . . . 1 1 . . . 1 1 1 . . . . . . . . . . . . ... . . . 1 1 1 . . . 1        =: S SS⊤ = Σ . . . Cholesky decomposition of Σ Brownian bridge construction: A = something else PCA construction: take A = VD where VD2V ⊤ = Σ is the singular value decomposition of Σ.

• G. Leobacher (JKU Linz, Austria)

Orthogonal transforms/Brownian paths MCQMC 2012 8 / 28

Linear construction

Why use BB/PCA?

BB/PCA probabilistically equivalent to forward construction, but Influence of Xk 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)

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Linear construction

Cost of generation

Forward method: O(n) Brownian bridge: O(n) PCA: Originally thought to be O(n2)!! Scheicher (2007): PCA generation costs O(n log(n)) by using the fast sine transform in dimension 2n By “fast” we mean O(n log(n)) (or faster)

• G. Leobacher (JKU Linz, Austria)

Orthogonal transforms/Brownian paths MCQMC 2012 10 / 28

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)

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Orthogonal constructions

We can write any matrix A with AA⊤ = Σ as A = SU where U is an orthogonal matrix and 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 2012 12 / 28

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 L´ evy paths (L (2006))

• G. Leobacher (JKU Linz, Austria)

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Orthogonal constructions

Examples

Σ = AA⊤ with A = SU Forward method: U = IdRn. Brownian Bridge: U = H−1 where H is the Haar transform PCA: U = S−1VD (trivially)

• G. Leobacher (JKU Linz, Austria)

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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 2012 15 / 28

Orthogonal constructions

At least one of these fast constructions is actually useful in practice

Theorem (L 2011)

Let Σ = VD2V ⊤ be the PCA of Σ C the discrete cosine transform of type IV in dimension n dn(P, Q)2 := n

l=1

n

k=1(P − Q)2 lk for n × n matrices P, Q

Then for all n ∈ N we have dn(SC, VD) < 1 and lim sup

n→∞ dn(SC, VD)2 ≤ 2

• 48 − π2

(π2 − 24)2 = 0.381 . . . .

• G. Leobacher (JKU Linz, Austria)

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Orthogonal constructions

10 20 30 40 50 60 0.6 0.4 0.2 0.2 0.4 0.6

Comparison PCA/DCT-IV, n = 64

• G. Leobacher (JKU Linz, Austria)

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Orthogonal constructions

50 100 150 200 250 1.5 1.0 0.5 0.5

Comparison PCA/DCT-IV, n = 256

• G. Leobacher (JKU Linz, Austria)

Orthogonal transforms/Brownian paths MCQMC 2012 18 / 28

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)

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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

• rthogonal 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)

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Fast completion of orthogonal systems

Problem formulation

U = k n − k

?

• G. Leobacher (JKU Linz, Austria)

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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(n3) operations! Solution (Irrgeher & L): Givens rotations!

• G. Leobacher (JKU Linz, Austria)

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Fast completion of orthogonal systems

Givens rotations

Recall G =                  ... 1 cos(α) − sin(α) 1 ... 1 sin(α) cos(α) 1 ...                 

• G. Leobacher (JKU Linz, Austria)

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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 ej, ek

• ne 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

• f 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 2012 24 / 28

Fast completion of orthogonal systems

Idea: if U is of the following form

U = 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)

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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(k2n) operations

• G. Leobacher (JKU Linz, Austria)

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Fast completion of orthogonal systems

Hoeseholder Reflections

Altenatively, we may Householder reflections (L & Irrgeher). Gives same complexities.

• G. Leobacher (JKU Linz, Austria)

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References

Leobacher (2006) Stratified sampling and quasi-Monte Carlo simulation of L´ evy 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)

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