Accelerated Computing Through Quasi-Monte Carlo Methods

Motivation Theoretical approach C++ practicing examples Asian Option Pricing Asian Option Valuation p
Accelerated computing through
Quasi-Monte Carlo (QMC) constructions
Polynomial lattice builder, C++ software
Mohamed Hanini,
CEO & Chief Scientist at Koïos Intelligence
Research & development
Montreal, January 28th 2020,
mohamed.hanini@koiosintelligence.ca
Mohamed Hanini, CEO & Chief Scientist at Koïos IntelligenceACCELERATED COMPUTING 1 / 42

Summary
1 Motivation
2 Theoretical approach
3 C++ practicing examples
4 Asian Option Pricing
5 Asian Option Valuation powered by Polylatbuilder

Why you should care?
For many applications, using QMC structured points is a very
efficient way to cut infrastructure and development costs.
Applications
1 Accelerate your training algorithms (Initializing
parameters),
2 Bayesian Deep Learning (reducing the variance of the
gradient estimator),
3 Parallel computing (optimize your GPU),
4 Evaluate complex financial applications,

Why you should care?
For many applications, using QMC structured points is a very
efficient way to cut infrastructure and development costs.
Applications
1 Accelerate your training algorithms (Initializing
parameters),
2 Bayesian Deep Learning (reducing the variance of the
gradient estimator),
3 Parallel computing (optimize your GPU),
4 Evaluate complex financial applications,
We need an efficient software!

Multidimensional Integration Methods
Methods
1 Monte Carlo (MC),
2 Quasi-Monte Carlo (QMC) and Randomized QMC
(RQMC).

Methods
1 Monte Carlo (MC),
2 Quasi-Monte Carlo (QMC) and Randomized QMC
(RQMC).
Practical approach QMC-RQMC
1 Start a search for a given construction method to find a
generator vector. How to find this vector?
2 Build a Low Discrepancy sequence (grid points)
Different representations:
Worst case error, square error (QMC),
Variance (RQMC),
Others?

Monte Carlo (MC)
Monte Carlo (MC), Quasi-Monte Carlo (QMC) et Randomized
QMC (RQMC)
Let X be a system (output) with real values
X = f(u1, u2, . . . , ud).
µ = E[X] = E[f(u1, u2, . . . , ud)] =
∫
[0,1)d
f(u) du
Inversion technique Example: X ∼ exp(λ)
F(X) = 1 − exp(−λX) = U ∼ U[0, 1)
⇒ X = − log(U)/λ.

Monte Carlo (MC)
Monte Carlo (MC), Quasi-Monte Carlo (QMC) and
Randomized QMC (RQMC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
standard MC: generate n random values i.i.d of U(0, 1)d, let
U0, . . . , Un−1;

Monte Carlo (MC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
U0, . . . , Un−1; estimate µ with ˆµn,MC = 1
n
∑n−1
i=0 f(Ui),
En = ˆµn,MC − µ converges with Op(n−1/2).

Monte Carlo (MC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
n
∑n−1
i=0 f(Ui),
Var[ˆµn,MC] = E[(ˆµn,MC − µ)2)] = σ2/n = O(n−1) converges
slowly as a function of n, where σ2 =
∫
[0,1)d f(u) du − µ2.

Monte Carlo (MC)
µ = E[X] = E[f(U)] =
∫
[0,1)d
f(u) du
n
∑n−1
i=0 f(Ui),
Var[ˆµn,MC] = E[(ˆµn,MC − µ)2)] = σ2/n = O(n−1) converges
slowly as a function of n, where σ2 =
∫
[0,1)d f(u) du − µ2.
central limit theorem
√
n(ˆµn,MC − µ)/σ ⇒ N(0, 1) when n → ∞.

Quasi-Monte Carlo (QMC)
µ = E[f(U)] =
∫
[0,1)d
f(u) du
QMC estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(ui).
We replace independent points MC by u0, . . . , un−1 a set of n
structured points () which covers the hypercube (0, 1)d more
uniformly than MC points.

µ = E[f(U)] =
∫
[0,1)d
f(u) du
QMC estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(ui).
Integration methods: Polynomial lattice rules and digital nets.
Construction methods: example Korobov, CBC etc...

µ = E[f(U)] =
∫
[0,1)d
f(u) du
QMC estimator
ˆµn,QMC =
1
n
n−1∑
i=0
f(ui).
Integration methods: Polynomial lattice rules and digital nets.
Construction methods: example Korobov, CBC etc...
Worst case error dans O(n−α+ϵ) where α ≥ 1

Randomized Quasi-Monte Carlo (RQMC)
RQMC estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(ui).
To create variance, we need to randomize the n structurted
points u0, . . . , un−1 : by shifting them individually. Each point
Ui ∼ U[0, 1)d.

RQMC estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(ui).
Ui ∼ U[0, 1)d. With m i.i.d estimators, we can construct an
unbiased estimator ˆµn,RQMC = 1
m
∑m
l=1 ˆµn,RQMC,l.

RQMC estimator
ˆµn,RQMC,l =
1
n
n−1∑
i=0
f(ui).
Ui ∼ U[0, 1)d. With m i.i.d estimators, we can construct an
unbiased estimator ˆµn,RQMC = 1
m
∑m
l=1 ˆµn,RQMC,l.
Var[ˆµn,RQMC] = Var[f(ui)]
n + 2
n2
∑
i<j Cov[f(ui), f(uj)].

Integration lattice
Primal lattice
Ld =



v =
d∑
j=1
hjvj such that each hj ∈ Z



,
Where v1(z), v2(z), . . . , vd(z) ∈ Rd are linearly independant
points over R and Ld contains Zd, Pn = Ld ∩ [0, 1)d.
.

Integration lattice
Primal lattice
Ld =



v =
d∑
j=1



,
A rank 1 lattice: ui = iv1 mod 1 for i = 0, 1, . . . , n − 1.
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
.

Integration lattice
Primal lattice
Ld =



v =
d∑
j=1



,
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
Example: Korobov construction a = (1, a, a2 mod n, . . . , ad−1
mod n) = O(n), because 0 ≤ a ≤ n − 1.

Integration lattice
Primal lattice
Ld =



v =
d∑
j=1



,
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
Dual lattice
L⊥
d =
{
h ∈ Rd
: hT
v ∈ Z for each v ∈ Ld
}
⊆ Zd
.

Integration lattice
Primal lattice
Ld =



v =
d∑
j=1



,
nv1 = a = (a1, a2, . . . , ad) ∈ Zd
n.
Dual lattice
L⊥
d =
{
h ∈ Rd
: hT
v ∈ Z for each v ∈ Ld
}
⊆ Zd
.
A dual representation

How to choose the generator vector?
Naive approach
1 Consider all the possibilities of finding a generator vector,
2 minimize the worst case error or the variance

Naive approach
Practical approach
1 Refine the search space g(z)
2 minimize an error function (cost function)
With a construction method
Component by component(CBC),
Fast component by component (Fast-CBC),
Korobov
Others?

Naive approach
Practical approach
1 Refine the search space g(z)
2 minimize an error function (cost function)
With a construction method
Component by component(CBC),
Fast component by component (Fast-CBC),
Korobov
Others?
We clearly need a software (Polylatbuilder)!

Example of a rank 1 lattice rule
Rank1 lattice
Pn =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
,
Rank1 lattice rule = 1
n
∑n−1
i=0 f(ui),
Number of points: n,
Generator vector: a ∈ Zd.
0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
u1
u2
shifted

Example of a rank 1 lattice rule
Rank1 lattice
Pn =
{
ui =
ia
n
mod 1 : 0 ≤ i ≤ n − 1
}
,
Rank1 lattice rule = 1
n
∑n−1
i=0 f(ui),
Number of points: n,
Generator vector: a ∈ Zd.
Randomized lattice rule
Pn,∆ = {Ui = (ui + U) mod 1 : 0 ≤ i ≤ n − 1} ,
Random point : U ∼ U[0, 1)d.
0 1
0
1
a/n
u1
u2
original
a = (1, 3)
0 1
0
1
U
u1
u2
shifted

Integration lattice
Projection of the lattice points
⇒ We ideally want n distinct values for each one-dimensional
projection Pn({j}) of Pn so the components of a are found in :
Un = {a ∈ Z : 1 ≤ a ≤ n − 1 et pgcd(a, n) = 1}.
For n prime:
There are n − 1 choices for each component of a,
there are (n − 1)d possibilities to generate the vector a.
For each subset ν ⊆ {1, 2, . . . , d}, the projection of Ld(ν) under
Ld is also a lattice, having a set points Pn(ν).

Integration polynomial lattice
Primal lattice
Ld =



v(z) =
d∑
j=1
hjvj as each hj ∈ Zb[z]



,
where v1(z), v2(z), . . . , vd(z) ∈ Ld
b (a set of Laurent series) where
vj(z) = gj(z)/P(z), which P(z) ∈ Zb[z] a polynomial with degree
m. Pn = φ(Ld) ∩ [0, 1)d.

Primal lattice
Ld =



v(z) =
d∑
j=1



,
where v1(z), v2(z), . . . , vd(z) ∈ Ld
m. Pn = φ(Ld) ∩ [0, 1)d.
Dual lattice
L∗⊥
d =



h ∈ Nd
0 :
d∑
j=1
hj(z)vj(z) ∈ Zb[z] for each v(z) ∈ Ld



,

Primal lattice
Ld =



v(z) =
d∑
j=1



,
where v1(z), v2(z), . . . , vd(z) ∈ Ld
m. Pn = φ(Ld) ∩ [0, 1)d.
Dual lattice
L∗⊥
d =



h ∈ Nd
0 :
d∑
j=1
hj(z)vj(z) ∈ Zb[z] for each v(z) ∈ Ld



,
Dual representation

Construction of points
Structure of the points for a rank 1 polynomial lattice rule
ui,j = φ
(
qi(z)gj(z) mod P(z)
P(z)
)
such asφ : Lb → R.
=⇒ φ
( ∞∑
l=w
xlz−l
)
=
∞∑
l=w
(xl)b−l
,

ui,j = φ
(
qi(z)gj(z) mod P(z)
P(z)
)
=⇒ φ
( ∞∑
l=w
xlz−l
)
=
∞∑
l=w
(xl)b−l
,
Practical approach
1 Truncate Laurent’s series to L ui,j =
∑L
l=w xlb−l,
2 Find the xl ∈ {0, 1} for a base b = 2.

ui,j = φ
(
qi(z)gj(z) mod P(z)
P(z)
)
=⇒ φ
( ∞∑
l=w
xlz−l
)
=
∞∑
l=w
(xl)b−l
,
Practical approach
1 Truncate Laurent’s series to L ui,j =
∑L
l=w xlb−l,
2 Find the xl ∈ {0, 1} for a base b = 2.
ui = (u1,i, u2,i, . . . , uj,i, . . . , ui,d) the i-th point

Example: rank 1 polynomial lattice rules
Rank 1 polynomial lattice rules
Pn =
{
ui =
qi(z)g(z) mod P(z)
P(z)
: q(z) ∈ Zb[z]
}
rank 1 polynomial lattice rules = 1
n
∑n−1
i=0 f(ui)
prime polynomial : P(z) with degree m.
generator vector : g(z) ∈ Zb[z]d
Example (particular case): Korobov rules
g(z) = (1, g(z), g(z)2 mod P(z), . . . g(z)d−1 mod P(z))
Ṟandomized Rank 1 polynomial lattice rules
Pn,∆ = {ui = ui + ∆ : 0 ≤ i ≤ n − 1}
random point: ∆ ∼ U[0, 1)d, ˆui,j = (ui,j + ∆i,j) mod b

Example: Rank 1 polynomial lattice rules
Quadratic error QMC
Could converge more fast the MC depending on:
the regularity of the function to be integrated
the uniformity of points in the polynomial lattice (choice of
parameter g(z))

Example: Rank 1 polynomial lattice rules
Quadratic error QMC
Could converge more fast the MC depending on:
the regularity of the function to be integrated
the uniformity of points in the polynomial lattice (choice of
parameter g(z))
Polynomial lattice of
n = 64 points
P(z) = z6 + z + 1 in
2D
0 1
0
1
Good lattice
g(z) = (1, z5
+
z3
+ z2
+ z + 1)
0 1
0
1
Very bad lattice
g(z) = (1, 1)

Réseaux digitaux
Définition
Let a prime number b ≥ 2, called base, a digital net in base b
formed by n = bm points in d dimensions is defined by selecting
d generating matrices C1, C2, . . . , Cd, where each Cj is an
∞ × m. The matrix whose elements belong to the finite field
Fb = 0, 1, 2, . . . , b − 1. ⇒ The matrix Cj determines the
coordinate j of all the points.
We define ui pour i = 0, 1, 2, . . . bm − 1,
we write the binary representation of i in base b,
the vector of the binary coefficient is multiplied by Cj
mod b to obtain the binary coefficients of ui,j.

Réseaux digitaux
Définition
Let a prime number b ≥ 2, called base, a digital net in base b
formed by n = bm points in d dimensions is defined by selecting
d generating matrices C1, C2, . . . , Cd, where each Cj is an
∞ × m. The matrix whose elements belong to the finite field
Fb = 0, 1, 2, . . . , b − 1. ⇒ The matrix Cj determines the
coordinate j of all the points.
⇒ Similar to the component gj(z) of the vector g(z)
We define ui pour i = 0, 1, 2, . . . bm − 1,
we write the binary representation of i in base b,
the vector of the binary coefficient is multiplied by Cj
mod b to obtain the binary coefficients of ui,j.

Digital Nets
Let
i =
k−1∑
l=0
ai,lbl
,
and let



ui,j,1
ui,j,2
...


 = Cj





ai,0
ai,1
...
ai,k−1





mod b.
ui,j =
∞∑
l=1
ui,j,lb−l
et ui =
(
ui,1, ui,2, . . . , ui,d
)
.
⇒ The resulting set of points is a basic digital net in base b. In
practice we truncate the expansion ui,j to L. For b = 2, the Cj
are the L × m binary matrcies.

Walsh transform
f(u) =
∑
h∈Nd
0
ˆf(h) walb,h(u),
where
ˆf(h) =
∫
[0,1)d f(u)walb,h(u)du et walb,h(u) = exp (2πih · u)/b.

Walsh transform
f(u) =
∑
h∈Nd
0
ˆf(h) walb,h(u),
where
ˆf(h) =
∫
[0,1)d f(u)walb,h(u)du et walb,h(u) = exp (2πih · u)/b.
Integration error (Sloan’s property)
En =
1
n
n−1∑
i=0
f(ui) −
∫
[0,1)d
f(u) du =
∑
0̸=h∈L∗⊥
d
ˆf(h).

Expression of variance
For a randomized lattice, the variance can be expressed:
Var[ˆµn,RQMC] =
∑
0̸=h∈L∗⊥
d
|ˆf(h)|2
.
From a variance reduction perspective, an optimal network
for f minimize Df(Pn)2 = Var[ˆµn,RQMC].
Not widely used as criterion solution ?
⇒ Worst case error
e(HK; Pn) = sup
f∈HK,||f||K≤1
|Qn(f) − I(f)|.

Practical criteria
Consider the following class of functions:
Eν,d(c) = {f : [0, 1]d
→ R : |ˆf(h)| ≤ c b−νµ1(h)
pour tout h ∈ Nd
0},
où ν > 1 et c > 0. We thus obtain:
|Qn(f) − I(f)| ≤
∑
0̸=h∈L∗⊥
d
c b−νµ1(h)
, ν > 1,
The quantity
Tν =
∑
0̸=h∈L∗⊥
d
c b−νµ1(h)
, ν > 1,
Depends only on the function class Eν,d(1).
To implement!

What do we need?
To find good digital and polynomial lattice
Figures of merit and weights adapted to a problem,
for a number of points n, for a polynomial and a dimension
d,
several construction methods,
Options: like normalization etc.
Our research work
Establish merit figures for digital networks and
polynomial,
compare (application) the performance of our software
integration methods compared to other integration
methods,
define new criteria for non-functional classes,
considered (explored) ⇒ unbounded functions!?

What already exists
Dirk Nuyens’ Code Matlab
a construction method (Fast CBC)
Limited choice of parameters (ex: only ten primitive
polynomials)
only one type of weight (product weight)
No QMC / RQMC points generated and tested! no
application!!

What already exists
Dirk Nuyens’ Code Matlab
a construction method (Fast CBC)
Limited choice of parameters (ex: only ten primitive
polynomials)
only one type of weight (product weight)
No QMC / RQMC points generated and tested! no
application!!
Generator vector tables
J. Dick
D. Nuyens

Structure of Polylatbuilder
Points of the
polynomial and
digital nets
Figures of merit Weight type
Construction
methods
Error in a
Walsh space
t-value
Spectral test
other heuris-
tic??
Product weight
Order depend-
ing weights
General weight
Product and
order depen-
dent weights
(POD)
CBC
Fast CBC
Random Ko-
robov
Korobov
Exhaustive

Quality measures (Discrepancies)
Implemented figure of merit
E(f, g(z))1/q
) = e(g(z); n; ||.||p,α,γ)
The worst function:
χ(g(z); n; ||.||p,α,γ) =
∑
0̸=v⊂1:t
γv
q/2
∏
j∈v
ω(uj)

Quality measures (Discrepancies)
Implemented figure of merit
E(f, g(z))1/q
) = e(g(z); n; ||.||p,α,γ)
The worst function:
χ(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 Constructions
Knowing n, we construct a generator vector
g(z) = (g1(z), g2(z), . . . , gd(z)).
We set g1(z) = 1,
g1(z) remains fixed, we choose g2(z) from the subset of the
components of generator vectors to minimize the desired
error criterion in 2 dimensions,
With g1(z), g2(z) remain fixed, we choose g3(z) from the
subset of the components of generator vectors to
minimize the desired error criterion in 3 dimensions,
ed(gd(z); Pn; ||.||p,α,γ)q
= ed−1(g(z)d−1; n; ||.||p,α,γ)q
+ Θd(g(z)d)

Setup intger to bit
#include "PolyType.h"
namespace PolyLatBuilder {
// choose the type of lattice
std::ostream& operator<<(std::ostream&
os, PolyLatType polylatType)
{ return os << (polylatType ==
PolyLatType::EMBEDDED ? "embedded" :
"ordinary"); }
}
}

C++code
boost::dynamic_bitset<>
PolyLatBuilder::FastCBC::getbin(int
integer){
unsigned long long dec;
dec = integer;
std::string str;
for(unsigned long long d = dec; d>0;
d/=2)
str.insert(str.begin(),
boost::lexical_cast<char>(d&1));
boost::dynamic_bitset<>
binaryNumber(str);
std::cout << "The number " << dec << "
in binary is: " << binaryNumber<<'n';
return binaryNumber;
}

Setup Bit vector to polynomial
std::string PolyLatBuilder::FastCBC::
pretty(const GF2X& p) {
std::ostringstream os;
if ( deg(p)<0 ) os << "0" ;
for (int i = deg(p); i >= 0; i--) {
if (rep(coeff(p, i))) {
if (i < deg(p)) os << " + ";
if (i >= 2)
os << "x^" << i;
else if (i == 1)
os << "x";
else
os << "1";
}
}
return os.str();
}

Perform the mapping function Vm(g(z)q(z)modP(z)
GF2X ww(INIT_SIZE,1);
ww.SetLength(1);
std::vector <double> w(rr.length());
std::vector <double> w1(rr.length());
for ( long j= 0; j <rr.length(); j++) {
w[j]=0;
double b = 1.0;
for( int k=0; k< m; k++){
ww[k] = rr[j][m-k-1];
for ( int l =0; l <k; l++){
ww[k] = ww[k] - ww[l] *
P2[m+(l-k)];
}
b *= 0.5;
if (IsOne(ww[k]))
w[j] += b;
}
std::cout << w[j] << std::endl;

Perform the vector of real values by applying the
Kernel
fftw<double>::real_vector usob(n-1);
for (long i = 0 ; i < w.size(); i++){
usob[i] = 1.0/6 - pow(2,
floor(log2(w[i]))-1);
}
float usob_0=1.0/6;
fftw<double>::complex_vector fft_psi =
fftw<double>::fft(usob);
}

Results – g(z) with Fast CBC
Values of 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
Built in C++
Encouraging results
Good uniformity of the polynomial lattice rules (good
parameter g(z))

QMC square error
Built in C++
Encouraging results
Good uniformity of the polynomial lattice rules (good
parameter g(z))
Polynomial lattice
rules points
n = 1023, P(z) =
z10 + z3 + 1 en 2D
0 1
0
1
good lattice
g(z)

Financial application: asian option
Stochastic differential equation(SDE)
dSt = µSt dt + σSt dWt
The SDE solution St = S0 exp(r−σ2/2)t+σBt

Financial application: asian option
Stochastic differential equation(SDE)
dSt = µSt dt + σSt dWt
The SDE solution St = S0 exp(r−σ2/2)t+σBt
Option pricing
The payoff of the option X = e−rt max(0, 1
d
∑s
j=1 Stj − K),
and the price
c = E[X |Ft]
v(s0, T) = E[e−rt
f(S1 . . . St) |Ft]

Asian purchase 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
Variance reduction techniques for MC
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 variable (AV)

Call Asian option– Numerical results (n=1023)
Risk free rate r = 0.08, volatility σ = 0.2, and the strike price K
MC points with variance reduction techniques
Approach PD LGD EAD
No CV 20.83 49.87
CV1 20.90 1.00 10−2
Data CV2 20.90 9.25 10−4
CV1 +CV2 20.90 8.45 10−4
CV1+CV2+AV 20.91 3.23 10−4
No CV 11.23 4.68
1
CV1 11.28 8.79 10−3
Key Features CV2 11.29 4.61 10−1
CV1 +CV2 11.28 8.52 10−3
CV1+CV2+AV 11.30 3.36 10−3
No CV 3.54 22. 58
1
CV1 3.55 5.66 10−3
100 CV2 3.57 4.27
CV1 +CV2 3.55 5.5710−3
CV1+CV2+AV 3.56 1.4010−3

Call Asian option– Numerical results
Comparison with LatticeBuilder software
Table: Value of the Asian option with a low exercise based on the
following configuration:(j)/10, for j = 1, 2, . . . , d = 10, number of
n = 210
points, and an exercise price K = 100
MC RQMC RQMC
Polylatbuilder LatticeBuilder
Option value 3.37 3.33 3.354
Standard deviation 4.52 0.015 0.025
Variance Ratio 95.17 33.06

Java Asian Option Valuation (using QMC points)
import umontreal.iro.lecuyer.rng.*;
import umontreal.iro.lecuyer.hups.*;
import umontreal.iro.lecuyer.stat.Tally;
import umontreal.iro.lecuyer.util.Chrono;
public class AsianQMCSSJ extends AsianSSJ {
public AsianQMCSSJ (double r, double
sigma, double strike, double s0, int
s, double[] zeta) {
super (r, sigma, strike, s0, s, zeta);
}

Java code
public void simulateQMC (int m, PointSet p,
RandomStream noise, Tally statQMC) {
Tally statValue = new Tally ("stat on
value of Asian option");
PointSetIterator stream = p.iterator ();
for (int j=0; j<m; j++) {
simulateRuns (p.getNumPoints(), stream,
statValue);
statQMC.add (statValue.average());
p.addRandomShift (0, 1, noise);
stream.resetStartStream();
}
}

Java code
public static void main (String[] args) {
try {
int s = 9;
double[] zeta = new double[s+1];
for (int j=0; j<=s; j++)
zeta[j] = 12.0*j/365;
AsianQMCSSJ process = new AsianQMCSSJ
(0.08, 0.2, 100.0, 100.0, s, zeta);
Tally statValue = new Tally ("value of
Asian option");
Tally statQMC = new Tally ("QMC averages
for Asian option");

Java code
timer.init();
// QMC Polynomial points
PointSet ps2 = new
PointSetFromFile("/Users/QuasiMonteAsian/src/options/pointsl
int m;
m =10;
// Number of QMC randomizations
process.simulateQMC ( m, ps2, new
MRG32k3a(), statQMC);
System.out.println ("QMC point set with
" + n +
" points and affine matrix
scramble:n");

Java code
System.out.printf ("Vlaue of Asian with
QMC: %9.4g%n", valueQMC);
double varQMC = ps2.getNumPoints() *
statQMC.variance();
double cpuQMC = timer.getSeconds() / (m *
n);
System.out.printf ("Variance ratio:
%9.4g%n", varMC/varQMC);
System.out.printf ("Efficiency ratio:
%9.4g%n",
(varMC * cpuMC) / (varQMC * cpuQMC));
System.out.printf ("Efficiency ratio 22:
%9.4g%n",
(cpuMC) / (cpuQMC));
System.out.println ("Test CPU time: "+
cpuMC);
System.out.println ("Test CPUQMC time: "+

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