An analysis of faster convergence in certain finance applications for quasi-monte Carlo

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1 An analysis of faster convergence in certain finance applications for quasi-monte Carlo a,b a School of Mathematics and Statistics, University of NSW, Australia b Department of Computer Science, K.U.Leuven, Belgium 15th International Conference on Computing in Economics and Finance University of Technology, Sydney, Australia July 15 17, 2009

2 New things have happened Motivation A paper by Boyle, Lai and Tan (2005) discusses the usage of lattice rules (a quasi-monte Carlo technique) for derivative pricing. But: Shows very impressive results using quasi-monte Carlo methods. (Their paper is a good introduction to QMC for finance.) Lattice rules perform dramatically better than Monte Carlo and even Sobol (a favorite quasi-monte Carlo method in finance). Nevertheless: lattice rules are not the magic bullet. Fact: the periodization used is a very delicate process in practice. Since then, there is new technology and theory: Lattice sequences (instead of fixed lattice rules); Higher order digital nets. This talk tries to address these new developments.

3 The problem What is QMC? The underlying numerical problem we are solving is approximating I( f ) := f (x) dx, [0,1] s by an equal-weight quadrature/cubature rule Q( f, {x k } N 1 k=0 ) := 1 N 1 f (x k ). N k=0 If the points x k ( samples paths) are taken 1 (pseudo) randomly, then this the Monte Carlo method; 2 deterministically, then this is the Quasi-Monte Carlo method, using low-discrepancy points. E.g., using Sobol & Niederreiter sequences or lattice sequences.

4 The problem Derivative pricing... Lets assume a stock to adhere to S(t) S(t 0 ) e (µ σ2 /2) t + σ W(t), with W(t) BM(0, 1), and thus W(t) N(0, t), then for a discretization on 0 = t 0 < t 1 <... < t n = T we have with W := (W(t 1 ), W(t 2 ),..., W(t n )) N n (0, C), C i,j = min(t i, t j ).

5 The problem... as a multi-dimensional integral Since the price of a derivative contract is defined as the expected payoff; this expectation can be written as an integral over all possible underlier paths. Since W N n (0, C), the payoff is given by E(payoff) = for some factorization C = AA. Rn g(w) exp( 1 2 w C 1 w) dw (2π) n det(c) Rn = g(a z) exp( 1 2 z z) dz (2π) n = g(a Φ 1 (x)) dx, [0,1] n

6 Low-discrepancy point sets What sample points to use? Figure: Three point sets with each 64 samples in the unit square. 1 The product left-rectangle rule. Classical product rule Note: grids don t work for high dimensions since, e.g., taking the minimum 2 points per dimensions in 100 dimensions requires a total of = points , a quintillion 2 Pseudo-random numbers. Monte Carlo 3 Low-discrepancy points. Quasi-Monte Carlo

7 Low-discrepancy point sets What sample points to use? Figure: Three point sets with each 64 samples in the unit square. 1 The product left-rectangle rule. Classical product rule 2 Pseudo-random numbers. Monte Carlo (Fig: mt19937, the Mersenne Twister, with default initial state.) 3 Low-discrepancy points. Quasi-Monte Carlo (Fig: A good lattice sequence in base 3.)

8 Low-discrepancy point sets Lattice rules and sequences An N-point lattice rule with generating vector z Z s : x k = k z mod 1, for k = 0, 1,..., N 1. N A lattice sequence generates points x k = ϕ b (k) z mod 1, for k = 0, 1,..., with ϕ b a good permutation. E.g. radical inverse in base b. Figure: A lattice sequence in base 3 with maximal 729 = 3 6 points and z = [1, 140,...] stopped at 81, 100, 600, 700 and 729 points.

9 Low-discrepancy point sets Digital sequences (e.g., Sobol and Niederreiter sequence) A digital sequence in base b (for efficiency, think b = 2) generates points {x k } bm 1 k=0 by taking for the j-th dimension x k,j = C j k, i.e., C j,1,1 C j,1,m. =..... C j,m,1 C j,m,m x k,j,1 x k,j,m k 0. k m 1 over F b, where the x k,j,i, for i = 1,..., m, and k i, for i = 0,..., m 1, are the base b digits of x j and k, m m 1 x k,j = x k,j,i b i [0, 1), k = k i b i Z b m. i=1 i=0 (In reality this can be a little bit more complicated.)

10 Low-discrepancy point sets Lattice sequences Note: lattice here has *nothing* to do with binomial pricing. A lattice sequence in base b (again, think b = 2) generates points {x k } bm 1 k=0 by setting x k = ϕ b (k) z mod 1, where typically ϕ b is the radical inverse in base b (or its Gray code variant). E.g., the radical inverse φ b just reverses the digits: m 1 m 1 φ b (k) := k i b m 1, given k = k i b i. i=0 Typically for a digital sequence C 1 = I, the identity matrix, and for a lattice sequence z 1 = 1. Then the first dimensions coincide. (And form the van der Corput sequence.) i=0

11 Digital sequences versus lattice sequences Digital versus lattice sequences: Practical differences For a digital sequence one needs good generating matrices C 1, C 2,..., C s F m m b ; preferably base 2, then each of the columns of the matrices can be represented by an integer and adding up columns is done by xor ing; preferably the k s are taken in Gray code ordering such that only one column has to be xor d to the previous result when generating sequence elements. For a lattice sequence one needs a good generating vector z Z s b m ; multiplication modulo 1 is no problem at all; preferably base 2, then radical inverse can be done by bit masks in O(log 2 (m)); or Gray code variant by 2 assembly instructions.

12 Some examples of usage Pricing of an Asian option std. err asian-64-random asian-64-sobol asian-64-lattice asian-64-sobol-bb asian-64-lattice-bb asian-64-sobol-pca asian-64-lattice-pca n Figure: 64 dimensions, 10 random shifts

13 Some examples of usage Pricing of an Asian option Monte Carlo std. err Order-2 lattice n Figure: 100 dimensions, 10 random shifts

14 The need for speed A reprise on the multivariate integration We are interested in approximating I( f ) := f (x) dx [0,1] s by some quadrature / cubature rule of the form N 1 Q N ( f ; {x k } k, {w k } k ) := w k f (x k ). For moderate to high dimensions: equal weights w k = N 1 are nice, with random points (Monte Carlo, error O(N 1/2 )); or low-discrepancy points (quasi-monte Carlo, error O(N 1 )). Here: not so interested in high dimensions and wish error O(N α ). k=0

15 Smooth functions Smooth functions Consider the class H α s v 1,...,v s f (x) x v 1 1 xvs s of s-dimensional functions for which all are of BV for all 0 v i α 1. (Bounded variation (BV) in the sense of Hardy and Krause: s-dimensional.) E.g., if for a 1-dimensional function f we have (sufficient, not necessary) 1 0 f (x) dx <, then it is of bounded variation (BV). (In the sense of Vitali: 1-dimensional.)

16 Smooth functions Smooth functions Consider the class H α s v 1,...,v s f (x) x v 1 1 xvs s of s-dimensional functions for which all are of BV for all 0 v i α 1. (Bounded variation (BV) in the sense of Hardy and Krause: s-dimensional.) It follows that if (sufficient, not necessary) αs f (x) x α 1 xα s exists and is continuous on [0, 1] s, then f H α s. If this holds than we might expect the error I( f ) Q N ( f ) of a good cubature rule Q N to decrease like almost O(N α ).

17 Smooth functions Smooth periodic functions Consider a similar class E α s of s-dimensional functions for which all v 1,...,v s f (x) x v 1 1 xvs s are of BV for all 0 v i α 1 and additionally the multivariate analog of f (v) (1) = f (v) (0), 0 v α 2, holds: v 1,...,v s f (x) x v 1 1 = v 1,...,v s f (x) xvs s x v 1 1 xvs s for all j and 0 v i α 2. xj =1 xj =0

18 Smooth functions Then by a theorem of Zaremba (1968, 1972): the Fourier series of f Es α converges uniformly and the Fourier coefficients are bounded by ˆf (h) 3V α( f ) (2π) α r(h) α c r(h) α, (denote f E α s (c) for a fixed c) where r(h) := s r(h j ), r(h j ) := max(1, h j ), j=1 and V α ( f ) is a bound on the variation of (α 1)s f / x α 1 1 x α 1 s ; or if αs f / x α 1 xα s is continuous the largest maximum of over all choices of 0 v i α. v 1+ +v s f x v 1 1 xvs s

19 Lattice rules and periodization In come lattice rules... Suppose f Es α (c), then, using a (rank-1) lattice rule Q N ( f ; z) := 1 N 1 ( ) k z mod N f, N N k=0 the error is given in terms of the Fourier coefficients of f by Q N ( f ; z) I( f ) = ˆf (h) c 0 h Z s h z 0 (mod N) For a good lattice rule the last sum can be estimated as P α (z) := 0 h Z s h z 0 (mod N) 0 h Z s h z 0 (mod N) 1 r(h) α = O(N α (log N) α(s 1) ). 1 r(h) α.

20 Lattice rules and periodization... and lattice sequences Good lattice rules and good lattice sequences can be constructed by the fast component-by-component algorithm (Nuyens and Cools, 2005, 2006; and Cools, Kuo and Nuyens, 2006). With a good lattice sequence One can get successive approximations of lattice rules. By stopping at powers of the base b m. One can stop at any point (like with Monte Carlo). But then one can never do better than O(N 1 ), whatever the point set, see Hickernell, Kritzer, Kuo and Nuyens (2009). Using a weighted QMC algorithm (WQMC), one can stop at any point and have higher order of convergence O(N α ). New algorithm from Hickernell, Kritzer, Kuo and Nuyens (2009). Note: There is a published lattice sequence generator in Cools, Kuo and Nuyens, SIAM Journal on Scientific Computing, 28(6): , 2006.

21 Lattice rules and periodization Periodization Given f H α s, i.e., all v 1+ +v s f (x) x v 1 1 xvs s are of BV for all 0 v i α 1, then we want to replace f by F Es α for which 1 F(x) dx = [0,1] s f (x) dx. [0,1] s 2 3 v 1+ +v s F(x) x v 1 1 xvs s v 1,...,v s F(x) x v 1 1 xvs s xj =1 are of BV for all 0 v i α 1; = v 1,...,v s F(x) x v 1 1 xvs s xj =0 for all j and 0 v i α 2.

22 Lattice rules and periodization Variable transformation Given a smooth increasing function ϕ : [0, 1] [0, 1] for which ϕ(0) = 0 ϕ(1) = 1 ϕ (v) (0) = ϕ (v) (1) for v = 1,..., α 1, then set Periodizers by F(x) = f (ϕ(x 1 ),..., ϕ(x s )) ϕ (x 1 ) ϕ (x s ). Korobov (1963): polynomial forms; Sag & Szekeres (1964): tanh transform; Iri (1970): IMT transform; Mori (1978): double exponential transform; Sidi (1993): sin m forms; Laurie (1996): polynomial forms.

23 Lattice rules and periodization A weighted cubature formula As expliticized by Hickernell (2002), periodizing a function f gives a new weighted cubature rule Q N (F; g) = 1 N 1 f (ϕ(x k,1 ),..., ϕ(x k,s )) ϕ (x k,1 ) ϕ (x k,s ) N = k=0 N 1 k=0 s j=1 ϕ (x j ) f (ϕ(x k,1 ),..., ϕ(x k,s )) N s = Q N (f ; {ϕ(x k )} k, {N 1 ϕ (x k,j )} k ). j=1

24 Lattice rules and periodization A change of point set The stronger the periodization the more the points are pushed to the edges.

25 Lattice rules and periodization And non-equal weights equal weights! = 3! = 4! = The stronger the periodization the more the weights deviate from their original equal weight situation.

26 Lattice rules and periodization Some problems with this approach Korobov (1963) and Zaremba (1972) already show problems for increasing α and s. Recent work by Kuo, Sloan & Woźniakowski (2007) adds to this picture. However: Periodization can only work in low dimensions (say s 7). Periodization needs high precision. At least for calculating weights and points. Periodization costs time! But can be pre-calculated and traded for memory. Very impressive results by Boyle, Lai and Tan (2005) on derivative pricing. Nice and effective quadrature packages by Robinson et al.

27 Lattice rules and periodization Exit periodization, enter higher order nets? The paper by Kuo, Sloan and Woźniakwoski (2007), titled Periodization strategy may fail in high dimensions, has the following footnote: Since this paper was finished, we became aware of a result by Dick and Pillichshammer (2007): the optimal convergence rate of O(N α ), i.e., a rate matching the smoothness of the integrands, can be nearly achieved by digitally shifted polynomial lattice rules; moreover, the implied constant can be bounded independently of s under appropriate conditions on the weights. This further undermines the motivation for periodizing the integrands.

28 Higher order nets An explicit construction for higher order d Dick (2007, 2008,... ) gave an explicit construction based on the generating matrices of existing nets and sequences to get higher order. Given a (t, m, ds)-net or (t, ds)-sequence over F b with points in ds dimensions x k = ( x k,1 x k,2 x k,ds ), k = 0, 1,..., where x k,j = x k,j,1 b 1 + x k,j,2 b 2 + = i=1 x k,j,i b i, and all x k,j,i Z b. Then define new points in only s dimensions for which y k = ( y k,1 y k,2 y k,s ), y k,j = y k,j,1 b 1 + y k,j,2 b 2 + = i=1 v=1 It follows that y k,j,(i 1)d+v = x k,(j 1)d+v,i. d x k,(j 1)d+v,i b ((i 1)d+v).

29 Higher order nets Convergence What to expect when using such a higher order net / sequence up to N = b m points? For a function f Hs α one can construct a net of order d from a (t, m, ds)-net for which the error behaves like b min(α, d) max(0, m (t + s(d 1)/2 )). Then we need to have s(d 1) m > t +, 2 to see the higher order convergence of O(N d ).

30 Higher order nets Need for precision Assuming one picks basis b = 2, then a higher order net with N = 2 m points needs md bits. Using double precision with 53 bits of precision: d maximum N = 2 m possible On the other hand we need m > t sd + (s(d 1))/2. Using Niederreiter-Xing points, m min is s\d

31 Periodization versus higher order nets Periodization versus higher order nets Periodization by variable transformation blows up variation: possible exponential (in α and s) constant for convergence. Periodization needs high precision; mainly for calculating the weights and points. Periodization takes time, but can be pre-computed. Higher order nets are still equal-weight! Amazing! Higher order nets also need high precision. High precision is also directly needed in evaluating f ; unless f is periodic (see Dick (2007)); this is more problematic.

32 A lookback option Lookback option The payoff for a lookback call option with s discrete monitoring steps is given by P(S, T) = max(max(s(t 1 ),..., S(t n )) K, 0). In the paper by Boyle, Lai and Tan (2005) lattice rules are shown to do dramatically better than the Sobol sequence. Two reasons: 1 Black-Scholes kind of analytic solution in function of multivariate normal distribution. Genz (1992) did some work on approximating this integral. (Hard for high dimensions.) 2 For this function periodization works nice. The analytic solution works so well because there the integrand function is smooth!

33 A lookback option The solution The value of this option can be expressed as n V(S 0, K, r, σ, T) = exp( rt) S 0 exp(rt j ) H j I n j (1 L n ) K, where H j, I n j and L n are all multivariate normal probabilities j=1 1 b1 bs... exp( 1 Σ (2π) n 2 x Σ 1 x) dx, with different dimensions and covariance matrices.

34 Multivariate normal integrals Multivariate normal integrals Consider calculating 1 b1 bn exp( 1 Σ (2π) n 2 x Σ 1 x) dx, then following Genz (1992) this can be written as an in effect (n 1)-dimensional integral with 1 e 1 e 2 (e 1, w 1 ) e s (e 1:n 1, w 1:n 1 ) 0 dw, ( bi i 1 j=1 e i (e 1:i 1, w 1:i 1 ) = Φ c ) ijφ 1 (w j e j ) c ii and C the lower triangular Cholesky factorization of Σ.

35 Multivariate normal integrals Using WQMC For the numerical tests it is nice to see the whole convergence graph, not just powers of 2. Furthermore: that is the usual practice for Monte Carlo simulations. However, it can be shown that only O(N 1 ) can be obtained for all N. Therefore the graphs are made using a reweighted sum from Hickernell, Kritzer, Kuo & Nuyens (2009). For powers of 2 the approximations obtained by the WQMC algorithm are exactly the same as those for a standard QMC algorithm.

36 Multivariate normal integrals Mvn in 3 dimensions (2 dimensional integral) s= stderr (10 shifts) MC wlatseq baker wlatseq sin3 sobol ho3 sobol nx ho3 nx N

37 Multivariate normal integrals Mvn in 5 dimensions (4 dimensional integral) s= stderr (10 shifts) MC wlatseq baker wlatseq sin3 sobol ho3 sobol nx ho3 nx N

38 Multivariate normal integrals Mvn in 7 dimensions (6 dimensional integral) s= stderr (10 shifts) MC wlatseq baker wlatseq sin3 sobol ho3 sobol nx ho3 nx N

39 New things will happen Conclusion For the derivatives in Boyle, Lai and Tan (2005) lattice sequences work very well! Not so surprising. One can recover O(N α ), but α 1 should be carefully chosen such that periodization does not backfire. One should pre-calculate weights and points in high precision if pushing the limits. The new higher order nets are not yet up to the task, but look promising and are easy to use. The higher order nets perform better than plain Monte Carlo anyway and are very useful by that fact alone already. Anyone interested in software for lattice sequences or other quasi-monte Carlo techniques: please contact me! dirk.nuyens@cs.kuleuven.be