1. Efficient Monte CarloAre low-discrepancy sequencesbetter than pseudo random numbersfor Monte-Carlo simulation ? Ashwin Rao

2. Random numbers for Monte Carlo Say you have 10,000 paths with quarterly time steps for 30 years Say you have a 4 factor model (say 4 uncorrelated dWs at every time step) We pick 10,000 points in a uniform distribution over [0,1]480 This gives us 480 uncorrelated dWs on every path (using inverse gaussian) On each path: Generate diffusion, Calculate payoffs, Discount => Path PV Path price function is a f: [0,1]480 R We approximate Price E[f(x)] by averaging f(x) over the 10,000 480-D pts To get an accurate price, we need to pick good points in the [0,1]480 space The rest of the talk is on picking good [0,1]s points using PRN: Pseudo random number generation QRN: Quasi random number generation (i.e., low-discrepancy sequences) 3. Whats the key objective ? Generate points in the s-dimensional cube [0,1]s Ideally, you want uniformity and independence of the points PRN: Tries to mimic a truly random u.i.i.d deterministically PRN: Moderate success in passing uniformity tests PRN: Moderate success in passing independence tests QRN: Arranges well-spaced points uniformly on the cube QRN: Aims to minimize clusters and holes QRN: Doesnt attempt to make points independent 4. Fundamental Methodology PRN: Cycles over large-sized finite fields PRN: Powers of a primitive root of Zpgenerates Zp PRN: Primitive polynomial over Zp generates Zp extension QRN: Van der Corput sequences QRN: In multi-D, generate permutations of VdC sequences QRN: (t,m,s) nets and (t,s) sequences 5. Multi-dimension PRN: Generated points are in one dimension PRN: Successive 1-D points pieced together to form multi-D points QRN: Generated points are in a specific multi-D QRN: Arranging points uniformly gets harder as dimension increases QRN: Curse of dimensionality 6. Big picture view PRN: Passes independence tests reasonably well PRN: Does okay on uniformity in lower dimensions PRN: In high dimensions, points end up lying in lower dim planes PRN: Mersenne Twister is an exception, doing quite well in high dim QRN: Uniformity is its USP QRN: Correlation between points & between neighboring coordinates QRN: This correlation causes difficulties in high dimensions QRN: Sobol is an exception, doing quite well in high dim 7. Theoretical Error Convergence PRN: O(1/sqrt(n)) PRN: Doesnt depend on s PRN: sqrt(n) convergence is fundamentally slow QRN: O( (log n)^s / n ) 8. Practical performance QRN is much better in lower dimensions But QRN performance deteriorates very quickly as dim goes up Basic QRN methods do worse than basic PRN methods in high dim However, Sobol and Generalized Faure do very well in high dim 9. Abstract Algebra Basics Group: Set with closed operator(+), Associativity, Identity(0), Inverse Monoid: Take away existence of inverse from a GroupSemigroup: Take away existence of identity from a Monoid Ring: Group under +, Monoid under *, Distributivity of + and * Field: Group under +, Group under *, Distributivity of + and * Z is a Ring, Q and R are Fields Zm is a group under + modulo m for any m > 1 Zp is a field under + and * modulo p for any prime p 10. Finite Fields Multiplying Zp by any a Zp simply permutes Zp Hence, every element in Zp has an inverse Fermats Little Theorem: ap-1 1 (mod p ) Primitive Root of Zp : Any a Zp whose powers generate Zp 11. Finite Field Extensions Polynomials over Zp modulo an irreducible polynomial f(x) This set is clearly a ring Multiplying this ring with any element of the ring permutes it Hence, each such polynomial has an * inverse modulo f(x) Hence, this set of pr polynomials form a field K extending Zp For any K s.t. f()=0, attaching to Zp generates K Essentially, polynomial-symbolic x represents If powers of generate K, f(x) is called a primitive polynomial Then powers of x modulo f(x) generate all polynomials of deg < r 12. Simple PRN: Linear Congruential Generator sn = a * sn-1 + b (mod p ) a 1, b 0 Seed 0 s0 p Identify a and b s.t. we cycle through all elements of Zp This is called a generator with a full period For case of b = 0, choose for a => any primitive root 13. Higher order Linear Recursive Generators sn = cr-1 sn-1 + cr-2 sn-2 + + c0 sn-r b (mod p ) Define f(x) = xr cr-1 xr-1 cr-2xr-2 - - c0 Vector vn = < sn+r-1, sn+r-2, , sn > vn is associated with the polynomial gv{n}(x)= sn+r-1 xr-1 + sn+r-2 xr-2 + . + sn Multiplying gv{n}(x) by x modulo f(x) gives gv{n+1}(x) If f(x) is a primitive polynomial,this generation gives a full period of pr 14. QRN - Definition of Discrepancy Discrepancy: Deviation from uniformity Given a point set {x1,, xn}, each xi [0,1]s Given a collection S of Lebesgue measurable subsets of [0,1]s D(x1,, xn ; S) = supAS | (#{xi A}) / n volume(A) | 15. Van der Corput Sequence For each n = 1,2, in sequence, write n in base p Flip the base p representation across decimal point Sequence of resulting [0,1] numbers is low-discrepancy At each incremental n = pr, we fill [0,1] at a finer level Period of cycling over [0,1] is p Hence, lower values of p are desirable Note the connection with finite field extensions 16. Van der Corput Sequence in base 300 0.0011/27 0.0022/270.11/3 0.101 10/27 0.102 11/270.22/3 0.201 19/27 0.202 20/270.01 1/9 0.0114/27 0.0125/270.11 4/9 0.111 13/27 0.112 14/270.21 7/9 0.211 22/27 0.212 23/270.02 2/9 0.0217/27 0.0228/270.12 5/9 0.121 16/27 0.122 17/270.22 8/9 0.221 25/27 0.222 26/27 17. Van der Corput Sequence in base 10 0 0.9 0.81 0.1 0.010.91 0.2 0.110.02 0.3 0.210.12 0.4 0.310.22 0.5 0.410.32 0.6 0.510.42 0.7 0.610.52 0.8 0.710.62 18. (t,m,s) nets and (t,s) sequences in base b b-ary box in base b: i=1..s [ ai / bji, (ai + 1) / bji ]ji {0,1,} and ai {0,1, , bji -1 }. Volume of elementary interval is 1 / bj1++js A (t,m,s)-net in base b is a set of bm points in [0,1]s such that exactly bt points fall in each b-ary box of volume bt-m The net correctly estimates the volume of each b-ary box A sequence of points x1,x2, in [0,1]s is a (t,s) sequence in base bif for all m > t each segment {xi : jbm i < (j+1)bm}, j = 0,1. is a (t,m,s) net in base b Smaller t is better and smaller b is better 19. Multi-dimensional QRN Halton sequences:- Each dimension is a Van der Corput sequence- Each dimension has a different prime base- So the highest base is the s-th prime number (high base is bad!)- Successive dimensions highly collinear- Halton sequences are not (t,s) sequences Faure sequences:- Common base (>s) across dimensions- Permute VdC sequence across dimensions- Permutation at every level of granularity- Think of permutation as multiplying by a finite field element- Faure sequences are (0,s) sequences in a base > s 20. Sobol sequences Base 2 Van der Corput sequences in the 1st dimension Other dimensions are permuted VdC sequences Permutation at every level of granularity Think of permutation as multiplying by a finite field element Permutation matrix consists of direction numbers Primitive polynomials used to construct direction numbers Different primitive polynomial in every dimension Gray code representation makes algorithm very efficient Sobol sequences are (t,s) sequences in base 2 But t is a function of s