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Motivation Theoretical results Features Application conclusion
A general software tool for
constructing rank-1 polynomial lattice rules
Polynomial Lattice Builder
Mohamed Hanini & Pierre L’Ecuyer
Département de mathématiques et de génie industriel
École Polytechnique de Montréal
CORS/INFORMS Montreal June 2015
Mohamed Hanini & Pierre L’Ecuyer Polynomial Lattice Builder 1 / 24

Presentation outline
1 Motivation
2 Theoretical results
3 Features
4 Application
5 conclusion

Monte Carlo (MC)
Multidimensional Integration methods
1 Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)t
f(u) du

Monte Carlo (MC)
Multidimensional Integration methods
1 Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)t
f(u) du
MC Estimator
ˆµn,MC =
1
n
n−1∑
i=0
f(Ui)
U0, . . . , Un−1 : n i.i.d points in (0, 1)t.
Var[ˆµn,MC] = E[(ˆµn,MC − µ)2)] = O(n−1) converges slowly
as a function of nMohamed Hanini & Pierre L’Ecuyer Polynomial Lattice Builder 3 / 24

Quasi-Monte Carlo (QMC)
Multidimensional Integration
µ = E[f(U)] =
∫
[0,1)t
f(u) du

Quasi-Monte Carlo (QMC)
µ = E[f(U)] =
∫
[0,1)t
f(u) du
QMC Estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(Ui)
U0, . . . , Un−1 : a set of n (structured) points that cover the
unit cube more evenly (0, 1)t, thus the Var[ˆµn,QMC] = 0
Solution: square error or RQMC

Randomized Quasi-Monte Carlo (RQMC)
µ = E[f(U)] =
∫
[0,1)t
f(u) du

Randomized Quasi-Monte Carlo (RQMC)
µ = E[f(U)] =
∫
[0,1)t
f(u) du
RQMC Estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(Ui)
U0, . . . , Un−1 : a set of n (structured) points shifted
individually
With m i.i.d estimators, we can construct the unbiased
estimator ˆµn,RQMC = 1
m
∑m
l=1 ˆµn,RQMC,l

Rank-1 Lattice Rules
P0
n =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
number of points: n
generating vector: a ∈ Zs 0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
u1
u2
shifted

P0
n =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
number of points: n
generating vector: a ∈ Zs
Randomly-Shifted Lattice Rule
Pn = {Ui = (ui + U) mod 1 : 0 ≤ i ≤ n − 1}
random shift: U ∼ U[0, 1)
0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
U
u1
u2
shifted

Rank-1 Polynomial Lattice Rules
QMC square error
Can converge with n faster than with Monte Carlo depending
on:
Smoothness of the integrand
Uniformity of the polynomial lattice points (choice of the
parameter g(z))

Rank-1 Polynomial Lattice Rules
QMC square error
Can converge with n faster than with Monte Carlo depending
on:
Smoothness of the integrand
Uniformity of the polynomial lattice points (choice of the
parameter g(z))
polynomial lattice
points for n =
64, P(z) = z6 + z + 1
in 2D
0
1
good lattice
g(z) = (1, z5
+
z3
+ z2
+ z + 1)
0
1
very bad lattice
g(z) = (1, 1)

How to Choose a Generating Vector?
Ideal Approach
1 Consider all possible generating vectors
2 Minimize the error or variance for our problem

Ideal Approach
Practical Approach
1 Narrow down the search space for generating vectors
2 Minimize a figure of merit
representative of:
square error (QMC)
variance (RQMC)
others?

Ideal Approach
Practical Approach
1 Narrow down the search space for generating vectors
2 Minimize a figure of merit
representative of:
square error (QMC)
variance (RQMC)
others?
That is the purpose of Polynomial Lattice Builder!

What Was Needed
Find Good Polynomial Lattice Rules
Figures of merit and weights adapted to one’s problem
Any number of points, any polynomial and any dimension
Various construction methods
Optional normalization and filters

What Was Needed
Find Good Polynomial Lattice Rules
Figures of merit and weights adapted to one’s problem
Any number of points, any polynomial and any dimension
Various construction methods
Optional normalization and filters
Research Purposes
Evaluate a figure of merit for a polynomial lattice
Compare performance of algorithms
Financial application

What Was Already Out There
Constructing
Dirk Nuyens’ Matlab Code
One method construction (Fast CBC)
Very small choice of parameters (ex:primitive polynomials)
Only one type of weight (product weights)
No QMC/RQMC points generated

What Was Already Out There
Constructing
Dirk Nuyens’ Matlab Code
One method construction (Fast CBC)
Very small choice of parameters (ex:primitive polynomials)
Only one type of weight (product weights)
No QMC/RQMC points generated
Tables of Generating Vectors
J. Dick
D. Nuyens

Structure of Polynomial Lattice Builder
Polynomial lattice
rank-1 points
Figure of merits Type of weights
Construction
methods
Error on Walsh
space
Error on Ko-
robov space
Heuristic
Product
weights
Order depen-
dant weights
General
weightsProduct-
and-order-
dependant
(POD) weights
CBC
Fast CBC
Random Ko-
robov
Korobov
Exhaustive

Points of a rank-1 polynomial lattice rule
We want to approximate an integral with rank-1
polynomial lattice rule
Qn(f, Ui
n−1
i=0 ) =
1
n
n−1∑
i=0
f(Ui)

Qn(f, Ui
n−1
i=0 ) =
1
n
n−1∑
i=0
f(Ui)
The point Ui
Ui(z) = φ(q(z)g(z)
P(z) ) ∈ (Zb[z−1])t
Pn = {φ(q(z)g1(z)
P(z) , . . . , q(z)gt(z)
P(z) ) : q(z) ∈ Zb[z]/P(z)}

Qn(f, Ui
n−1
i=0 ) =
1
n
n−1∑
i=0
f(Ui)
The point Ui
Ui(z) = φ(q(z)g(z)
P(z) ) ∈ (Zb[z−1])t
Pn = {φ(q(z)g1(z)
P(z) , . . . , q(z)gt(z)
P(z) ) : q(z) ∈ Zb[z]/P(z)}
The kth coordonate uj,k of the kth point Uk
uj,k =
∑L
l=w xlz−l, where xl ∈ F2

Lattice representation
Lt = {v(z) =
t∑
j=1
hjvj such that hj ∈ Zb[z]}
v1(z), v2(z), v3(z), . . . , vt(z) ∈ Lt
b, and vj(z) = gj(z)/P(z) , Thus
Pn = Lt ∩ [0, 1)t

Lattice representation
Lt = {v(z) =
t∑
j=1
hjvj such that hj ∈ Zb[z]}
v1(z), v2(z), v3(z), . . . , vt(z) ∈ Lt
b, and vj(z) = gj(z)/P(z) , Thus
Pn = Lt ∩ [0, 1)t
Dual lattice representation
L∗⊥
t = {h(z) ∈ Lt
b : h(z)v(z) ∈ Zb[z] for each v(z) ∈ Lt}

Quality measures
Figure of merit used for a practical example
E(f, g(z))1/q
= e(g(z); n; ||.||p,α,γ)
e(g(z); n; ||.||p,α,γ) =
∑
0̸=v⊂1:t
γv
q/2
∏
j∈v
ω(uj)

Quality measures
Figure of merit used for a practical example
E(f, g(z))1/q
= e(g(z); n; ||.||p,α,γ)
e(g(z); n; ||.||p,α,γ) =
∑
0̸=v⊂1:t
γv
q/2
∏
j∈v
ω(uj)
The kernel
ω(uj) =
∞∑
h=1
wal2,h(uj)
2qα⌊log2hj⌋
= 12(
1
6
− 2⌊log)2uj⌋−1
)

Construction methods
CBC construction
et(gt(z); n; ||.||p,α,γ)q
= et−1(g(z)t−1; n; ||.||p,α,γ)q
+ Θt(g(z)t)

CBC construction
et(gt(z); n; ||.||p,α,γ)q
= et−1(g(z)t−1; n; ||.||p,α,γ)q
+ Θt(g(z)t)
CBC construction (fast)
Ωb =
(
ω
(
q(z)g(z)
P(z)
))
g(z)=1,2,...,bm−1

Results – g(z) with Fast CBC
Values of optimal g(z)
P(z) = z10 + z3 + 1 P(z) = z7 + z + 1
t gt(z) Square Error gt(z) Square Error
1 1 1.58 10−8 1 1.01 10−6
2 z9 + z8 + z5 3.6 10−8 z6 + z5 + z3 + z 2.22 10−6
3 z9 + z8 + z6 + z2 + z + 1 6.36 10−8 z6 + z5 + z4 + z2 3.74 10−6
4 z8 + z6 + z5 + z4 + z3 + z2 + z 1.03 10−7 z5 + z4 + z + 1 5.70 10−6
5 z8 + z6 + z1 1.57 10−7 z5 + z3 8.18 10−6
6 z9 + z7 + z4 + z2 + z 2.36 10−7 z5 + z4 + z3 1.16 10−5
7 z8 + z7 + z6 + z3 + z2 3.34 10−7 z6 + z3 + z + 1 1.56 10−5
8 z9 + z8 + z6 + z5 + z4 + z3 + 1 4.55 10−7 z6 + z5 + z3 + z2 + 1 2.03 10−5
9 z9 + z8 + z6 + z4 + z3 + z2 6.25 10−7 z6 + z5 + z4 + z3 + z + 1 2.7 10−5
10 z8 + z6 + z5 + z4 + z2 + 1 8.64 10−7 z5 + z4 + z3 + z 3.43 10−5

QMC square error
This was done in C++
Encouraging results
Uniformity of the polynomial lattice points (optimal
parameter g(z))

QMC square error
This was done in C++
Encouraging results
Uniformity of the polynomial lattice points (optimal
parameter g(z))
polynomial lattice
points for
n = 1023, P(z) =
z10 + z3 + 1 in 2D
0 1
0
1
good lattice
g(z)

Flexibility with Performance
Challenges
Support different:
Figures of merit
Types of weights
Lattice types
Normalizations
Filters

Challenges
Support different:
Figures of merit
Types of weights
Lattice types
Normalizations
Filters
Allow for extensions

Challenges
Support different:
Figures of merit
Types of weights
Lattice types
Normalizations
Filters
Maintain good performance

Challenges
Support different:
Figures of merit
Types of weights
Lattice types
Normalizations
Filters
Maintain good performance
Avoid duplication of code

Financial application–Call Asian option
Stochstic Differential equation (SDE)
dSt = µSt dt + σSt dWt
The solution of SDE is St = S0 exp(r−σ2/2)t+σBt

Financial application–Call Asian option
The solution of SDE is St = S0 exp(r−σ2/2)t+σBt
Asian pricing
The payoff of the option is X = e−rt max(0, 1
s
∑s
j=1 Stj − K),
and the price
c = E[X |Ft]

Financial application–Asian call option
The solution of SDE is St = S0 exp(r−σ2/2)t+σWt

Financial application–Asian call option
The solution of SDE is St = S0 exp(r−σ2/2)t+σWt
Asian call pricing
The payoff f(S1, . . . Sd) = max(0, 1
d
∑d
j=1 Stj − K), and the
price
v(s0, T) = E[e−rt
f(S1 . . . St) |Ft]

Call Asian option– Numerical results
v(s0, T) = e−rt
∫
(0,1)d
max(0,
1
d
d∑
i=1
S0 exp [(r − σ2
)ti
+σ
√
ti − ti−1
i∑
j=1
Φ−1
(uj)] − K)du1 . . . dud
Reduction variance techniquess
control variate CV1 = e−rt max(0, Y − K) where
Y =
∏t
j=1(Stj )1/d
control variate CV2 = e−rt max(0, Y − K) where
Y =
∑t
j=1(Stj )1/d
Antithetic variate (AV)

Call Asian option– Numerical results (n=1023)
Results with MC points
Method r = 0.08, σ = 0.2 Average Variance Ratio
Strike price K=80
No CV 20.83 49.87 1
CV1 20.90 1.00 10−2 4961
CV2 20.90 9.25 10−4 53913
CV1 +CV2 20.90 8.45 10−4 59000
CV1+CV2+AV 20.91 3.23 10−4 154284
Strike price K=90
No CV 11.23 4.68 1
CV1 11.28 8.79 10−3 5326
CV2 11.29 4.61 10−1 101
CV1 +CV2 11.28 8.52 10−3 5492
CV1+CV2+AV 11.30 3.36 10−3 13925
Strike price K=100
No CV 3.54 22. 58 1
CV1 3.55 5.66 10−3 3984
CV2 3.57 4.27 5
CV1 +CV2 3.55 5.5710−3 4051
CV1+CV2+AV 3.56 1.4010−3 16066

Call Asian option– Numerical results
Comparison with QMC points
MC RQMC
Strike price K= 100
Option value 3.54 3.33
Standard deviation 4.52 0.015
Variance Ratio 90802
Efficiency ratio 1804200

Conclusion
Search for good polynomial lattice rules with specific input
parameters for RQMC integration.
Analyze and compare algorithms and figures of merit for
RQMC research.
What else are you interested in?

Functional ANOVA Decomposition
ANOVA Decomposition
f(u) = µ +
∑
q
fq(uq) where


∑
q
→
∑
∅̸=q⊆{1,...,s}


q: coordinate set
uq: projection of u onto q
fq(uq): integrates to zero

ANOVA Components of the RQMC Variance
ANOVA Decomposition
Var[ˆµ(f)] =
∑
q
Var[ˆµ(fq)]

ANOVA Components of the RQMC Variance
ANOVA Decomposition
Var[ˆµ(f)] =
∑
q
Var[ˆµ(fq)]
Weighted Figure of Merit
D2
(Pn) =
∑
q
γqD2
q(Pn)
D2
q(Pn) depends only on the
projection of Pn onto q
γq: weight (free to be chosen)

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