Application of Global Sensitivity Analysis and

.

Quasi Monte Carlo methods in financeS i K h k.Sergei Kucherenko

Imperial College London, UK s.kucherenko@imperial.ac.uk

Outline

Application of MC methods to high dimensional path dependent integrals

Comparison of MC and Quasi MC methods. Properties of Low Discrepancy Sequences

.

Why Quasi MC remains superior over MC in high dimensions and for discontinuous payoffs although classical theory does not support this effect?

.

Global Sensitivity Analysis and Sobol’ Sensitivity Indices

Effective dimensions and how to reduce them ( Brownian bridge, PCA, control ( gvariate techniques )

Practical examples ( Asian options; Cox, Ingersoll and Ross interest rate model; martingale control variate approximations )

Derivative based Global Sensitivity Measures and “Global Greeks”

2

MC methods in finance

Many problem in finance can be formulated ashigh dimensional integralshigh-dimensional integrals

[ ] ( )nHI f f x dx= ∫

.Option pricing: the Wiener integral over paths of Brownian motion

.

in the space of functionals [ ( )] :

[ ( )] W

F x t

I F x t d x= , ∫ W

0( ) continuous in 0 , (0) C

x t t T x x− ≤ ≤ =

∫

( [ ( )]), W( ) random WMonte Carl

iener processes (a Brownian motion) to construct many random paths W( ),o approach:

I E F W t tt

= −

3eva alu te functional and average results

Why integrals are high dimensional ?

Main factors to consider:Main factors to consider:

1. The number of risk factors (underlyings) involved:

.

( y g )e.g., valuation of basket option, yield-curve sensitivities of an interest rate option, VaR.

2 The number of time steps required (for each factor):.2. The number of time steps required (for each factor):

e.g., valuation of Asian options, stochastic volatility (path dependence). (p p )

3. The number of options (basket of options)

When all factors are present, the dimension is the product of all factors.

44

Monte Carlo integration methods

D i i i i i h d i hi h di iDeterministic integration methods in high dimensionsare not practical because of the "Curse of Dimensionality"

[ ] [ ( )]I f E f x=

. 1

1Monte Carlo : [ ] ( )N

N ii

I f f zN =

= ∑.

is a sequence of random points in E

niz H−

rror: [ ] [ ]NI f I fε = −E

2 1/21/2

rror: [ ] [ ]

( )= ( ( ))

N

N

I f I f

fEN

ε

σε ε = →

Convergence does not depent on dimensionality but it is slow

N

5

but it is slow

How to improve MC ?

1/ 2

( )Slow convergence: = Nf

Nσε

To improve MC convergence:( ) b l i i d i h if

.

I. Decrease ( ) by applying variance reduction techniques: antithetic variables;

fσ

.

control variates; stratified sampling; importance

1/ 2

sampling.1but the rate of convergence remains ~ Nε 1/ 2

II. Use better ( more uniformly di

g

stributed )

N N

66

sequ es.enc

Quasi random sequences (Low discrepancy sequences):Discrepancy is a measure of deviation from uniformity

n1 2

:Definition: ( ) , ( ) [0, ) [0, ) ... [0, ),

( )

Disc

vol

repancy is a measure of deviation fro

ume of

m uniformity

nQ y H Q y y y ym Q Q

∈ = × × ×

( )

( )

( ) volume of

sup ( )n

Q yN

Q H

m Q QN

D m QN

−

= −

.

( )

1/2 1/2Random sequences: (ln ln ) / ~ 1/

nQ y H N

D N N N

∈

→.Random sequences: (ln ln ) / ~ 1/

(l( )

N

N

D N N N

D c d

→

≤n ) Low discrepancy sequences (LDS)

nNN

−

Convergence: [ ] [ ] ( ) ,

(ln )QMC N N

n

NI f I f V f D

O N

ε = − ≤

Assymptotically ~ (1/ ) much higher than

(ln )

QMC

QMC

O N

O NN

ε

ε =

→

7~ (1/ )

Q

MC O Nε

Sobol’ Sequences vrs Random numbersand regular gridg g

Unlike random numbers, successive Sobol’ points “know" about the position of previously sampled points and fill the gaps between them

Regular Grid/ 64 Points Random Numbers/ 64 Points Sobol Numbers/ 64 Points

..

Sobol Numbers/ 256 PointsRandom Numbers/ 256 PointsRegular Grid/ 256 Points

88

What is the optimal way to arrange N points in two dimensions?

Regular Grid Sobol’ Sequence

..

Low dimensional projections of low discrepancy sequences are better di t ib t d th hi h di i l j ti

9

distributed than higher dimensional projections

Comparison of Sobol’ Sequences and random numbers

Normal probability plots Histograms

..

1010

Evaluation of quantiles I. Low quantile

Distribution dimension n = 5.

are independent standard normal variates

2

1

( ) ,n

ii

f x x=

= ∑(0 1)x N are independent standard normal variates(0,1)ix N∼

0.0100

.w)

"MC"

"QMC"

"LHS"Expon. ("QMC")

Expon. ("MC")

Expon. ("LHS").

0.0010

og(M

eanE

rror

Low

Lo

0.000110 11 12 13 14 15 16 17 18

Log2(N)

11

Low quantile (percentile for the cumulative distribution function) = 0.05A superior convergence of the QMC method

Evaluation of quantiles II. High quantile

Distribution dimension n = 5.

are independent standard normal variates

2

1

( ) ,n

ii

f x x=

= ∑(0 1)x N are independent standard normal variates(0,1)ix N∼

0.0100"MC"

.igh)

"QMC"

"LHS"

Expon. ("MC")

Expon. ("QMC")

Expon. ("LHS").

0.0010

Log(

Mea

nErr

or H

0 0001

Hi h til ( til f th l ti di t ib ti f ti ) 0 95

0.000110 11 12 13 14 15 16 17 18

Log2(N)

12

High quantile (percentile for the cumulative distribution function) = 0.95QMC convergences faster than MC and LHS

Van der Corput sequence:How to construct a low discrepancy sequence ?

4 3 2 1 0Number written in base : (··· )

In the decimal system: , 0 1

p q

bm

jj j

i b i a a a a a

i a b a b

=

= ≤ ≤ −∑1

Reverse the digits and add a radix point to obtain a number within the unit interj=

1 2 3 4

val: (0. ···) by a a a a=

.

1

1In the decimal system: ( ; )

Example:

mj

jj

h i b a b− −

=

= ∑.

2 1 02

4, 24 1 2 0 2 0 2 (10

Example:

0)i b= =

= × + × + × =1 2 3

Holton LDS

0.001 0 2 0 2 1 2 1/ 8

:

− − −→ × + × + × =

1

1

Holton LDS

( ; )

( ( ) ( ) ( )

:m

jj

jh i b a b

h h h b

− −

=

= ∑

13

( ( ;2), ( ;3),..., ( ; ),for e

nh i h i h i bach dimension a Van der Corput sequence but with a different prime number nb

How to construct Sobol’ Sequence ?

4 3 2 1 0 2

A permutation of Van der Corput sequence in base 2Number written in base 2 : (··· )

m

i i a a a a a=

1

0 1 2 3 2

In the decimal system: 2 , 0 1

( ... )

mj

j jj

i L

i a a

y a a a a a=

= ≤ ≤

=

∑

.

0 1 2 3 2( )

Construct vector , permuti L

j ji i

y

g C y C= ation matrix for dimension

-th element for dimension is given by:i

j

x j.

1,

1

In pract

( ) 2

ice:

Lj l

i i ll

x i g − −

=

=∑

1 2 3 2

In pract1. Construct direction numbers:

(0. ... )

ice:

j j j j jbv v v v v= 1 2 3 2( )

The numbers are given by the equatiob

jv n = / 2 , where < 2 is an odd integer, satisfy a recurrence relation

j j j

j j j

v mm v

14

using the coefficients of a primitive polynomial in the Galois field G(2)

2 The Sobol' integer number:jx

How to construct Sobol’ Sequence ?

1 1 2 2

2. The Sobol' integer number:

( ) ...where is an addition modulo 2 operator: 0 0 =0, 1 1=0, 0 1=1, 1 0=1.

jn

j j j jn b

x

x i a v a v v= ⊕ ⊕ ⊕⊕ ⊕ ⊕ ⊕ ⊕p , , ,

can also be seen as bit wise XOR. 3. Convert integer ( ) to a j

nx i⊕

uniform variate:

.

( ) ( ) / 2

Generation of direction numbers:

jbj jn ny i x i=

.

Primitive polynomial (irreducible polynomial with binary coefficients over the Galois filed G(2

Generation of direction nu

):

mbers:

11 1 ... 1, 0,1

Exa

q ql q kP x c x c x a−

−= + + + + ∈2 3 3 2mples of primitive polynomials: 1, 1, 1, 1.x x x x x x x+ + + + + + +

21 1 2 2

A different primitive polynomial is used in each dimension: 2 2 ··· 2

To fully define the direction numbe

qi i i i q i qm c m c m m m− − − −= ⊕ ⊕ ⊕

rs initial numbers

1515

To fully define the direction numbe

1 2

rs initial numbers, ,..., are required.nm m m

2D projections from adjacent dimensions for Sobol’ LDSIn Sobol's algorithm direction numbers is a key component to its efficiencyg y p y

Ref: Peter Jackel, Monte Carlo Methods in Finance, John Wiley & Sons, 2002

..

Sobol’ LDS with inefficient direction numbers

Finance in focus, 201016Sobol’ LDS with efficient direction numbers

Why Sobol’ sequences are so efficient and widely used in finance ?

1

(ln )Convergence: for all LDS

(l )

n

n

O NN

O N

ε

−

= −

1(ln )For Sobol' LDS: , if 2 , integern

kO N N kN

ε = = −

.

Sobol' LDS:1. Best uniformity of distribution as N goes to infinity.

.

2. Good distribution for fairly small initial sets.3. A very fast computational algorithm.Known LDS: Faure, Sobol’, Niederreiter

Many practical studies have proven that the Sobol’ LDS is superior to other LDS:

"Preponderance of the experimental evidence amassed to date points to Sobol' sequences as the most effective quasi-Monte Carlo method for application in financial engineering."

17

Paul Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2003

Sobol LDS. Property A and Property A’

A low-discrepancy sequence is said to satisfy Property A if for any binary segment (not an arbitrary subset) of the n-dimensional sequence of length 2n there is exactly one point in each 2n hyper-octant that results from subdividing the unit hypercube along each of its length extensions into halfhypercube along each of its length extensions into half.

A low-discrepancy sequence is said to satisfy Property A’ if for any binary segment (not an arbitrary subset) of the n-dimensional sequence of length 4n there is exactly one point in each 4n hyper octant that results from subdividing the unit

.

exactly one point in each 4n hyper-octant that results from subdividing the unit hypercube along each of its length extensions into four equal parts.

.

18Property A Property A’

Distributions of 4 points in two dimensions

Property A

NoMC -> No

..

LHS -> No

Sobol’ -> Yes

19

Distributions of 16 points in two dimensions

Property A’

NoMC -> No

..

LHS -> No

Sobol’ -> Yes

20

Comparison of different Sobol’ sequence generatorsg

..

The winner is SobolSeq generator: Sobol' sequences satisfy two additionaluniformity properties: Property A for all dimensions and Property A' foradjacent dimensions. Maximum dimension: 16384

Free version www broda co uk ( including 64-bit version )

21

Free version www.broda.co.uk ( including 64 bit version )

Comparison of different Sobol’ sequence generators. Integration test

( )( )[ ]∫ ∏

=

−+=n

s

iiii dxxcI

1,0 1

5.01

..

22

Full report www.broda.co.uk/doc/HD_SobolGenerator.pdf

DiscrepancyI. Low Dimensions

Discrepancy, n=5

RandomHalton

0.001T

HaltonSobol

.0.0001 .1024 2048 4096 8192

N

0.01

Discrepancy, n=20

Random

0.0001

0.001

T

a doHaltonSobol

1 06

1e-05

23

1e-06128 256 512 1024 2048 4096 8192 16384 32768 65536

N

DiscrepancyII. High Dimensions

0.001

Discrepancy, n=50

Random

1e-07

1e-06

1e-05

0.0001

T

HaltonSobol

.1e-10

1e-09

1e-08

.1e-11

128 256 512 1024 2048 4096 8192 16384 32768

N

1e-16

Discrepancy, n=100

R d

1e 18

1e-17

T

RandomSobol

MC in high-dimensions

1e-19

1e-18T MC in high-dimensions has smaller discrepancy !

24

1e-20128 256 512 1024 2048 4096 8192 16384

N

Are QMC efficient for high dimensional problems ?

(ln )nO N(ln )

Assymptotically ~ (1/ )

QMC

QMC

O NN

O N

ε

ε

=

.21

but increseas with u

not achievable for prac

ntil exp( )

50, 5 10 tical applications

Q

QMC N N n

n N

ε ≈

= ≈ − . p, pp

“For high-dimensional problems (n > 12)For high-dimensional problems (n > 12),

QMC offers no practical advantage over Monte Carlo”

( Bratley Fox and Niederreiter (1992)) ?!( Bratley, Fox, and Niederreiter (1992)) ?!

25

Option pricing. Discretization of the Wiener process

1/ 2

The asset follows geometrical Brownian motion:

, ( ) , ~ (0,1)dS Sdt SdW dW z dt z N= µ + σ =

2

, ( ) , (0,1)Using Ito's lemma

1( ) [( ) ( ))] ( ) Wi th

dS Sdt SdW dW z dt z N

S t S t W t W t

µ + σ

.

20( ) exp[( ) ( ))], ( ) Wiener path

2For time step

S t S t W t W t

t

= µ − σ + σ −

∆.

21( ) ( )exp[( ) ( ( ) ( ))2

S t t S t r t W t t W t+ ∆ = − σ ∆ + σ + ∆ − ]

For the standard discretization algorithm

( ) ( ) / 0 1W t W t t t T i∆ ∆ ≤ ≤1 1( ) ( ) , / , 0 1A terminal asset value:

1

i i iW t W t t z t T n i n+ += + ∆ ∆ = ≤ ≤ −

26

20 1 2

1( ) exp[( ) ( ... )]2 nS T S r T t z z z= − σ + σ ∆ + + +

Approximations of path dependent integralswith Brownian bridge scheme

dS Sdt SdW= µ +σGeometrical Brownian motion:

, ~ (0,1)dW z dt z N=

Brownian bridge algorithm:

SDE:

.

g g

0 1( ) ,W T W T z= +t T.

0 21 1( / 2) ( ( ) ) ,2 21 1( / 4) ( ( / 2) ) / 2

W T W T W T z

W T W T W T

= + +t0 T

T/2

0 3

4

( / 4) ( ( / 2) ) / 2 ,2 21 1(3 / 4) ( ( / 2) ( )) / 2 ,2 2

W T W T W T z

W T W T W T T z

= + +

= + +2 2

1 1(( 1) / ) ( (( 2) / ) ( )) 2 /W n T n W n T n W T T nz− = − + +

27

(( 1) / ) ( (( 2) / ) ( )) 2 / .2 2 nW n T n W n T n W T T nz− = − + +

MC simulation of option pricing

( )The value of European style options

C( ) ( )rT QK T E P S t K− ⎡ ⎤⎣ ⎦( )C( , ) ( ),

The payoff function for an Asian call option

rT QK T e E P S t K⎡ ⎤= ⎣ ⎦

. 1/

=max( - ,0),An

n

P S K

∏. 1/

=1For a geometric average Asian call: S=( ) n

ii

S∏1/n

⎡ ⎤2

0C( , ) max[0,( exp[(2n

rT

H

K T e S r− σ= −∫

1/1

11) ( )]

nn i

i jji

Tt un

−

==

⎡ ⎤+σ Φ⎢ ⎥

⎢ ⎥⎣ ⎦∑∏

nH 11

1 .)] ...And there is a closed form solution

ji

nK du du⎢ ⎥⎣ ⎦

−

28

And there is a closed form solution

MC simulation of option pricing. Discretization

( )( ) ( )

In a general case

1C ( )N

rT i iK T e P S S S K− ⎡ ⎤⎢ ⎥∑

.

( )( ) ( )0 1

1C ( , ) , , , ,

For the case of a European call

N Ti

K T e P S S S KN =

= ⎢ ⎥⎣ ⎦∑

.

( ) ( )

For the case of a European call

1 1( , ) max( ,0) N N

i rT iN TC K T C e S K

N N− ⎡ ⎤

= = −⎢ ⎥∑ ∑1 1

( , ) ( , )N Ti iN N= =

⎢ ⎥⎣ ⎦

∑ ∑

29

MC and QMC methods with standardand Brownian Bridge discretizations

Asian Call (32 observations)

S=100, K=105, r=0.05, s=0.2, T=0.5, C=3.84 (analytical)

4.5

.

4

Valu

e

.

3.5

Opt

ion

V

QMC, Brownian Bridge

QMC Standard Approximation

30 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

QMC, Standard Approximation

MC, Brownian Bridge

Analytical value

Call price vrs the number of paths.

MC l i hi hl ill ti QMC

N_path

30

MC - slow convergences, convergence curve is highly oscillating. QMC convergence – monotonic. Convergence is much faster for Brownian bridge

Asian call. Convergence curves

Asian Call with geometric averaging 252 observationsAsian Call with geometric averaging. 252 observations

S=100, K=105, r=0.05, s=0.2, T=1.0, C=5.56 (analytical)

1

10

1/ 21 K⎛ ⎞

.0.1

1

(RM

SE)

2

1

1 ( )K

kN

k

I IK

ε=

⎛ ⎞= −⎜ ⎟⎝ ⎠∑

.

0.01

Log(

QMC, Brownian BridgeQMC, Standard ApproximationMC Brownian Bridgeα

0.00110 100 1000 10000

MC, Brownian Bridge Trendline -QMC, BB, 1/N 0.82Trendline - QMC, Stand., 1/N 0.56Trendline - MC, 1/N 0.5

~ , 0 1N αε α− < <

Log-log plot of the root mean square error versus the number of paths.

Log(N_path)

31Brownian bridge – much faster convergence with QMC methods: ~1/N0.8 even in high dimensions. Why ?

ANOVA decomposition and Sensitivity Indices

Model f(x)Ω∈x YConsider a model

Model, f(x)Ω∈x Y( )Y f x=

x is a vector of input variablesY is the model output.

ANOVA decomposition:

1 2( , ,..., )0 1

k

i

x x x xx

=≤ ≤

.( ) ( ) ( )0 1,2,..., 1 2

1( ) , ... , , ..., ,

k

i i ij i j k ki i j i

Y f x f f x f x x f x x x>

= = + + + +∑ ∑∑

ANOVA decomposition:

.

1 1

1

1

...0

( , , ..., ) 0, , 1i s is k

i i j i

i i if x x dx k k s

= >

= ∀ ≤ ≤∫0

Variance decomposition: 2 2 2 2, 1,2,...,i iji i j nσ σ σ σ= + +∑ ∑ …

kijlij

k

i SSSS 21...1 ++++= ∑∑∑Sobol’ SI:

32

klji

ijlji

iji

i ,...,2,11

∑∑∑<<<=

ANOVA decomposition and Sensitivity Indices

( ) ( ) ( )0 1 2 1 2( ) , ... , , ..., ,k

i i ij i j k kf x f f x f x x f x x x= + + + +∑ ∑∑( ) ( ) ( )0 1,2,..., 1 21

1

( ) , ... , , ..., ,

( , , ..., ) 0, , 1

i i ij i j k ki i j i

i i i

f x f f x f x x f x x x

f x x dx k k s

= >

+ + + +

= ∀ ≤ ≤

∑ ∑∑

∫

.

1 1...0

1

( , , ..., ) 0, , 1

( ) ( ) 0, .

i s is ki i if x x dx k k s

f x f x dx v u

∀ ≤ ≤

= ∀ ≠

∫

∫ .

( )0

0

( ) ( ) 0, .

,

v v u uf x f x dx v u

f f x dx

∀ ≠

=

∫

∫

( ) ( ) 0 ,

nHn

i i jj i

f x f x dx f= −∏∫

( ) ( ) ( ) ( ) 0, , ...

n

n

j iHn

ij i j k i i j jk i j

f x x f x dx f x f x f

≠

≠

= − − −∏∫

33

,n k i jH ≠

ANOVA decomposition and Sensitivity Indices. Test case

1 2 0 1 1 2 2 12 1 2( , ) ( ) ( ) ( , ),1

f x x f f x f x f x x= + + +

( ) ( )

1 2 1 2 01( , ) ,4

1 1

f x x x x f

f x f x dx f x

= → =

∫

.

( ) ( )

( ) ( )

1 1 2 0 1 ,2 41 1

nH

f x f x dx f x

f x f x dx f x

= − = −

= − = −

∫

∫ .( ) ( )2 2 1 0 2

12 1 2 1 2 1 2

,2 4

1 1 1( )

nH

f x f x dx f x

f x x x x x x

= − = −

= − − +

∫

( )

12 1 2 1 2 1 2

21 1 1

( , ) .2 2 4

3nH

f x x x x x x

f x dx

+

∫1 2

3 ,7

3 1

nHS

S S S

σ= =

= = =

34

2 1 12, .7 7

S S S= = =

Sobol’ Sensitivity Indices (SI)

Definition:1 1

2 2... ... /

s si i i iS σ σ=

- partial variances( )1 1 1 1

12 2... ...

0

,..., ,...,s si i i i i is i isf x x dx xσ = ∫

12

∫.

- variance

Sensitivity indices for subsets of variables:

( )( )220

0

f x f dxσ = −∫( )x y z.Sensitivity indices for subsets of variables: ( ),x y z=

( )1

2 2, ,

1s

m

y i is i i

σ σ= ⟨ ⟨ ∈Κ

=∑ ∑ …

The total variance:( )11 ... ss i i= ⟨ ⟨ ∈Κ

( ) 222z

toty σσσ −=

Corresponding global sensitivity indices:

/ 22 σσS ( ) / 22tottotS35

,/σσ yyS = ( ) ./ 2σσ toty

totyS =

How to use Sobol’ Sensitivity Indices?

0 1toty yS S≤ ≤ ≤

tot SS t f ll i t ti b t d ( )ytoty SS −

iS totiS

accounts for all interactions between y and z, x=(y,z).

The important indices in practice are and

.

( )0totiS f x= → does not depend on ;

only depends on ;ix

( )1iS f x= → ix.

corresponds to the absence of interactions between

and other variables

( )i itotii SS = ix

If then function has additive structure:

Fixing unessential variables∑=

=n

siS

1,1 ( ) ( )0 i i

i

f x f f x= +∑Fixing unessential variables

If does not depend on so it can be fixed( )1totzS f x<< → z

36

complexity reduction, from to variables( ) ( )0,f x f y z≈ → znn −n

Evaluation of Sobol’ Sensitivity Indices

Straightforward use of Anova decomposition requires

2n integral evaluations – not practical !

There are efficient formulas for evaluation of Sobol’ Sensitivity :

.

1

2 0

1 [ ( , )[ ( , ') ( ', ')] ' ',yS f y z f y z f y z dydy dzdzσ

= −∫ .0

1 22 0

1 [ ( , ) ( ', )] ',2

totyS f y z f y z dydzdz

σ

σ= −∫

12 2 200

( , )f y z dydz fσ = −∫

Evaluation is reduced to high-dimensional integration by MC/QMC methods.

37

The number of function evaluations is N(n+2)

Applications of Global Sensitivity Analysis

identify key parameters whose uncertainty most strongly affects the

Global Sensitivity Analysis Global Sensitivity Analysis can be used tocan be used to

.

identify key parameters whose uncertainty most strongly affects the output; rank variables, fix unessential variables and reduce model complexity;

.

select a model structure from a set of known competing models;identify functional dependencies;analyze efficiencies of numerical schemes.y

38

Effective dimensions

The effective dimension of ( ) in superposition sensei th ll t i t h th t

Let u be a cardinality of a set of variables . uf x

d

0

is the smallest integer such that(1 ), 1

S

S

uu d

dS ε ε

< <≥ − <<∑

.

It means that ( ) is almost a sum of f x -dimensional functions.Sd.

is

The effective dimension of ( ) in truncation sense the smallest integer such thatT

f xd

∑

___________________________________________________________

( )

1,2,...,

1Example:

(1 ), 1T

n

i T

uu

S

d

n

S

f x x d d

ε ε⊆

→ = ==

≥ − <<

∑

∑

( )1

1,Examp

does

le:

not depend on t rhe oi

i TS

S

nf x x d d

d=

→∑

can be reducedder in which the input variables are sampled,

- depends on the order by reodering variablesT Td d→

39

can be reduced depends on the order by reodering variables T Td d→

Classification of functions

Type B,C. Variables are Type A. Variables are yp ,equally important

S S d↔

ypnot equally important

T Ty zS S d>> ↔ <<

.

i j TS S nd≈ ↔ ≈y z

y zT n

n nd>> ↔ <<

.

Type B. Dominant low order indices

Type C. Dominanthigher order indices n

11

n

ii

SS nd=

≈ ↔ <<∑1

1n

ii

SS nd=

<< ↔ ≈∑4040

Classification of functions

..

Majority of problems in finance either have low effective dimensions or

ff ti di i b d d b i i l t h i

41

effective dimensions can be reduced by using special techniques

( )f x x x x

Effective dimensions. Test case

( )1 2 1 2

1 1 1

( , ) ,1 1 ,2 4

f x x x x

f x x

=

= −

( )2 2 2

2 41 1 ,2 4

f x x= −

.

12 1 2 1 2 1 21 1 1( , )2 2 4

3 1

f x x x x x x= − − +

.

1 2 123 1, 7 76 0 85 1

S S S

S S

= = =

1 2 0.85 17

effective dimensions:

S S+ = = ≈ →

( ) ( )1 2 0 1 1 2 2

d 2, d 1 second order interactions are not important( , )

T S

f x x f f x f x= = →

≈ + +

42

Global Sensitivity Analysis of standard and Brownian Bridge discretizations

Apply global SA to payoff function i i i(Z )=max( (Z )- ,0), Z , 1,AP S K i n=

1Standard Approximation

Brownian Bridge

.0 01

0.1

otal .

0.001

0.01

S_to

0.00011 6 11 16 21 26 31

time step number

Log of total sensitivity indices versus time step number i.

Standard discretization - Si total slowly decrease with i Brownian bridge -

time step number

43

Standard discretization - Si_total slowly decrease with i. Brownian bridge -Si_total of the first few variables are much larger than those of the subsequent variables.

Global Sensitivity Analysis of the standard discretization

For the standard discretization1/2

10C( , ) max[0,( exp[( ) ( )]

2

nn i

rTi j

TK T e S r t un

− −⎡ ⎤σ

= − +σ Φ⎢ ⎥⎢ ⎥⎣ ⎦

∑∏∫

.

11

1

2

)] ... n jiH

n

n

K du du==⎢ ⎥⎣ ⎦

−

∫

.

1 2( ), ( , ,..., ) uncertaint parameterskY f x x x x x= = −

Apply global SA to a payoff function :1/2

n

n iT⎡ ⎤21

011

( ) max(0, exp[( ) ( )]2

n i

i jji

Tf x S r t xn

−

==

⎡ ⎤σ= − +σ Φ⎢ ⎥

⎢ ⎥⎣ ⎦∑∏

44

Global Sensitivity Analysis of two algorithms at different n (time steps)

..

The effective dimensions

Standard approximation: dT ≈ ndS > 2

Brownian Bridge approximation: dT ≈ 2d 2

45

dS ≈ 2

Option pricing: Why Brownian Bridge is more efficientthan standard discretization in the case of QMC ?

(A) The initial coordinates of LDS are much better distributed than the later high dimensional coordinates.(B) Low dimensional projections of low discrepancy sequences are better distributed than higher dimensional projections

.

The Brownian bridge discretization1) well distributed coordinates are used for important variables

.

and higher not so well distributed coordinates are used for far less important variables ( A )2) the effective dimension d is also reduced – approximation2) the effective dimension ds is also reduced approximation function becomes additive ( B )

The standard construction does not account for the specifics of LDS distribution properties.

Application of QMC with the Brownian bridge discretization results in

46

Application of QMC with the Brownian bridge discretization results in the 102-106 time reduction of CPU time compared with MC !

Cox, Ingersoll and Ross interest rate model

( )where > 0, > 0, > 0

t t t tdy y dt y dW= α +β +σ

α β σ Data: the 9-month Euribor interest rate daily time series using 250 daily

.

observations, starting at 30 Dec 1998The generalised method of moments (GMM) estimation .

is used to obtain , and estimates.

= 1e-005, = 0 1109, = 0 1929

α β σ

→ α β σ 1e 005, 0.1109, 0.1929

Euler discretisation:

→ α β σ

( )Euler discretisation:

t t t t t ty y y t y t+∆ − = α +β ∆ +σ ∆ ε

47

CIR model: 50 day forecasts of the 9-month Euribor rateQMC E l QMC Mil t iQMC Euler

cast

03.

74

QMC Milstein

cast

03.

74

50-p

erio

d fo

rec

3.66

3.70

50-p

erio

d fo

rec

3.66

3.7

.

Number of simulation runs (N)

0 50 100 150 200 250

3.62

Number of simulation runs (N)

0 50 100 150 200 250

3.62

.MC Euler

ast 3.

65

MC Milstein

ast

63.

70

50-p

erio

d fo

reca

13.

63

50-p

erio

d fo

reca

3.62

3.66

MC and QMC estimators with using standard Euler and Milstein schemesNumber of simulation runs (N)

0 50 100 150 200 250

3.61

Number of simulation runs (N)

0 50 100 150 200 250

3.58

48

MC and QMC estimators with using standard Euler and Milstein schemes

QMC produces much smother and much faster convergence than MC

RMSE convergence for 250-days forecasting horizon

01/ 2

1 K⎛ ⎞

6

-4

-2

)

MC Euler (H = 250)QMC Euler (H = 250)Trendline MC (-0.53)Trendline QMC (-0.79)

2

1

1 ( )kN

kI I

Kε

=

⎛ ⎞= −⎜ ⎟⎝ ⎠∑

.-10

-8

-6lo

g2(R

MSE

)

.

-14

-12~ , 0 1N αε α− < <

-166 8 10 12 14 16 18

log2(N)

QMC estimator displays much faster convergence compared to MC regardless of the forecasting horizon (dimension)

49

RMSE convergence for 250-days forecasting horizon

1/ 21 K⎛ ⎞2

1

1 ( )kN

kI I

Kε

=

⎛ ⎞= −⎜ ⎟⎝ ⎠∑

..

~ , 0 1N αε α− < <

There is no notable advantage of Brownian bridge over the standard Euler scheme

50

scheme

Global Sensitivity Analysis of Standard and Brownian Bridge discretizations (CIR model)Bridge discretizations (CIR model)

1 2 n (Z , Z , ... , Z ), is a number of daysTy nApply global SA to a function

1Standard Approximation

Brownian Bridge

.

0.6

0.8_t

otal

.

0.2

0.4

S_

01 6 11 16 21 26 31

time step number

Log of total sensitivity indices versus time step number i.

Standard discretization - Si total are constant Brownian bridge - Si total of the

51

Standard discretization - Si_total are constant. Brownian bridge - Si_total of the first variable is much larger than those of the subsequent variables.

Global Sensitivity Analysis of Standard and Brownian Bridge discretizations (CIR model)

iS∑iS∑n Standard BBii

32 0.99 0.99

64 0.99 0.99

.

128 0.99 0.99

.

Standard approximation: the effective dimension dT ≈ nthe effective dimension dS =1the effective dimension dS =1

Brownian Bridge approximation: the effective dimension dT ≈ 2the effective dimension dS =1the effective dimension dS 1

The low effective dimension dS for both schemes =1, hence QMC efficiency can not be further improved by changing sampling strategy

52

p y g g p g gy

Control Variate Method

We define the control variate by

The control is sampled along with and centered at zero.)( CXX λλ −=

C XThe control is sampled along with and centered at zero,

i.e., is the control parameter so that

C X

0. E C λ=

.

, p

Variance Decomposition

EX(λ) = EX.

2))(( 222XV λλλ.

Where is the correlation between and,2))(( 222

CXCCXXXVar σλρσλσσλ +−=XCρ X C

The optimal control parameter

)( CXCovσ)(

),(*

XVarCXCov

XCC

X == ρσσλ

))(( λXV53

is obtained by minimizing the variance )).(( λXVar

Control Variate Method with Monte Carlo method

The basic Monte Carlo estimator is1 )(∑

NiXSXE

The control variate estimator is defined by

,:1

)(∑=

=≈i

iN X

NSXE

1 N

.The variance reduction ratio is

.)(11

)(*)(*

∑=

−=N

i

iiN CX

NS λλ

.The variance reduction ratio is

*

( ) 1 1, if 0NXC

Var S ρ= > ≠* 2 1, if 01( ) XC

XCNVar Sλρ

ρ> ≠

−

54

Monte Carlo Pricing with MartingaleControl Variate (MCV)Control Variate (MCV)

P i i E ti b MCV th d l d tPricing a European option by a MCV method leads to

P(0,S0,σ 0) ≈ 1 e−rT H(ST(i)) − λM ( i)( ˜ p )[ ]

N

∑ ,

.

where ( , 0, 0)

N( T ) (p)[ ]

i=1∑ ,

*)(~

)~( SSS

T

SSrs dWSSspepM σσ∫∂

= −.

is a martingale with being an approximation of P.0

),,()( SSSSS dWSSsx

epM σσ∫ ∂

: control parameter (empirically chosen as 1)λ

: martingale control.( )PM ~

55

Homogenization Method

Fouque and H. (07) * use the homogenized Black-Scholes )()(i PMprice to construct a martingale control

The effective variance is defined as the averaging of

);,( σtBS StP )()(BS

i PM

2σ

.

The effective variance is defined as the averaging of variance function w.r.t. the invariant distribution of the volatility process.

σ

.

Martingale control variate method is very general for option pricing problems. The control is related to the accumulative value of delta-hedging portfolios.g g pMartingale control variate corresponds to a smoother payoff function so that QMC methods can be effective.

* Fouque, J.P. and Han (2007) A martingale control variate method for option pricing with stochastic volatility. ESAIM Probability & Statistics, 11 (2007), pp. 40-54.

56

European Options Pricing by Variance Reduction

..

blue: Basic MC samples

57

green: MCV samples

One-Factor Stochastic Volatility model

*dS rS dt S dWσ= +

( )0

exp / 2t t t t t

t t

dS rS dt S dWY

σ

σ

= +

=

.

( ) ( )* 2 *1 0 1 11t t t tdY m Y dt dW dWα β ρ ρ= − + + −

.( )

is an Ornstein-Uhlenbeck processtY

1/ , 2 /so that is varying on the time scalesYα ε β ν ε

ε= =

so that is varying on the time scales tY ε

58

European Call Option Prices. Sobol’ LDS

m=‐2.5, =1, ρ=‐0.7; =‐2.5, =110; r=0.05; K=100, T=1ν 0Y 0S

RatioMCVε SE BMC (mean) )(meanSE MCV

r 0.05; K 100, T 1

.

1/50 0.352 (24.115) 0.039 (24.348) 811/10 0 303 (23 340) 0 038 (23 400) 63

ε SE (mean) )(meanSE

.1/10 0.303 (23.340) 0.038 (23.400) 6310 0.268 (20.771) 0.019 (20.964) 19350 0 276 (20 488) 0 017 (20 711) 26750 0.276 (20.488) 0.017 (20.711) 267100 0.279 (20.428) 0.017 (20.660) 277

*BCM – basic MC; RatioMCV – variance reduction ratio

59

Time Discretization – 128 steps; Total number of paths ‐ 8192

Global Sensitivity Analysis of standard and MartingaleControl Variate (MCV) approximationsControl Variate (MCV) approximations

0 0

i i

(0, , ) ( ) ( ).

(Z ), Z , 1,

rTT

T T

P S e H S M P

S S i n

σ −≈ −

= =

Apply global SA to payoff function

i iT T

1Standard approximationMartingale control variate

.0.1_tot

al

Effective dimensions:Standard: dT ≈ n

dS =2.

S_

MCV: dT ≈ 2dS =1

0.011 6 11 16

time step number

Log of total sensitivity indices versus time step number i.

Standard discretization - Si_total slowly decrease with i. MCV - Si_total of the

60

first few variables are much larger than those of the subsequent variables.

Derivative based Global Sensitivity Measuresand “Global Greeks”

Consider the differentiable function defined in the unit hypercube.

2⎛ ⎞

( )1,..., nf x x

2

ni

iH

f dxx

ν⎛ ⎞∂

= ⎜ ⎟∂⎝ ⎠∫

⎛ ⎞

.

Unlike partial derivative which is defined at the nominal point and by

definition is local measure is globali

fx

⎛ ⎞∂⎜ ⎟∂⎝ ⎠

iν .definition is local, measure is global.

How to use in finance:

iν

Static hedge: Consider plain vanilla option . Delta hedging strategy

Is based on a local sensitivity measure . Parametersiν

( , , , )C S r tσ

( , , , ) /S r t C Sσ∆ = ∂ ∂y

are uncertain.

( , , , )

1/22⎛ ⎞

∫

( , , )S rσ

61

Define “global delta” and use it for a static hedge( )2/C S dSd drdtσ⎛ ⎞

∆ = ∂ ∂⎜ ⎟⎝ ⎠∫

Derivative based Global Sensitivity Measuresand “Global Greeks”

Similarly we can define

22

2if dxγ

⎛ ⎞∂= ⎜ ⎟∂∫y

1/2

2n

iiH x

γ ⎜ ⎟∂⎝ ⎠∫

.

And use it to define “global gamma” ( )1/2

22 2/C S dSd drdtσ⎛ ⎞

Γ = ∂ ∂⎜ ⎟⎝ ⎠∫

.Similarly we can define other “global greeks”

There is a link between variance based and derivative based measures

2 2 Theorem: tot ii

vSπ σ

≤

62

Summary

Global Sensitivity Analysis is a general approach for uncertainty, complexity reduction and structure analysis of non-linear models. It can be widely applied in finance.

.

In Sobol's algorithm direction numbers is a key component to its efficiency. The Sobol Sequence generator satisfying uniformity properties A and A' has

.

superior performance over other generators.

Quasi MC methods based on Sobol' sequences outperform MC regardless of i l di i li f bl i h l ff i di inominal dimensionality for problems with low effective dimensions.

Great success of Quasi MC methods in finance is explained by problems t t l ff ti di i hi h b d d b t h i hstructure - low effective dimensions which can be reduced by techniques such

as Brownian bridge, PCA, control variate techniques

63

Acknowledgments

.

Prof. Sobol’

.

Dario Cziraky, Barclays Capital

Sean Han, NTHU, Taiwan

64