Application of Quasi Monte Carlomethods in finance

.Sergei Kucherenko

Imperial College London UKImperial College London, UK

BRODA Ltd.S.Kucherenko@broda.co.uk

Outline

Application of MC methods to high dimensional path dependent integralsApplication 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 ?

Global Sensitivity Analysis and Sobol’ Sensitivity Indices

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

Practical examples ( Asian options; martingale control variate approximations )

2

MC methods in finance

Many problem in finance can be formulated ashigh-dimensional integrals

[ ] ( )nHI f f x dx= ∫

.Option pricing: the Wiener integral over paths of Brownian motionin the space of functionals [ ( )] :F x t

[ ( )] WI F x t d x=

0

,

( ) continuous in 0 (0)C

x t t T x x− ≤ ≤ =

∫

0( ) continuous in 0 , (0)

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

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

x t t T x x

I E F W t tt

≤ ≤

= −

3

y p ( ),ev

ppa alu te functional and average results

Why integrals are high dimensional ?

Main factors to consider:

1. The number of risk factors (underlyings) involved: e.g., valuation of basket option, yield-curve sensitivities of an interest rate option, VaR.

.

p2. The number of time steps required (for each factor): e.g.,

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

3. The number of options (basket of options)

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

44

Monte Carlo integration methods

Deterministic integration methods in high dimensionsDeterministic integration methods in high dimensionsare not practical because of the "Curse of Dimensionality"

[ ] [ ( )]1M t C l [ ] ( )

N

I f E f x

I f f

=

∑. 1

Monte Carlo : [ ] ( )

is a sequence of random points in

N ii

ni

I f f zN

z H=

=

−

∑

E

2 1/2

rror: [ ] [ ]

( )= ( ( ))

NI f I f

fE

ε

σε ε

= −

→1/2

Convergence does not depe

= ( (

nt on dimensionality

))N EN

ε ε = →

5

but it is slow

How to improve MC ?

( )fσ1/ 2

( )Slow convergence: = Nf

Nσε

To improve MC convergence:I. Decrease ( ) by applying variance reduction techniques:fσ

.

antithetic variables; control variates;

t tifi d li stratified sampling; importance sampling.

11/ 2

II. Use better

1but the rate of co

( more uniformly di

nverg

strib

ence remains

uted )

~ N Nε

66

sequ es.enc

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

n1 2Definition: ( ) , ( ) [0, ) [0, ) ... [0, ),

( ) vol

p y

ume of

y

nQ y H Q y y y ym Q Q

∈ = × × ×

−

( )

( )sup ( )

n

Q yN

Q y H

ND m Q

N∈= −

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

(lND N N N→

n )nN(l( )ND c d≤n ) Low discrepancy sequences (LDS)

Convergence: [ ] [ ] ( ) ,QMC N N

NN

I f I f V f Dε

−

= − ≤

Assymptotically ~ (1/ ) much higher than

(ln )

Q C

QMC

n

O N

O NN

ε

ε =

→

7

Assymptotically (1/ ) much higher than

~ (1/ )QMC

MC

O N

O N

ε

ε

→

Sobol’ Sequences vrs Random numbersand regular grid

U lik d b i S b l’ i t “k " b t th iti f i l l dUnlike 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

Comparison between Sobol sequencesand random numbers

.

9

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

10

p j p y qdistributed than higher dimensional projections

Evaluation of quantiles. Low 5% quantilen

Distribution dimension n = 5.

are independent standard normal variates

2

1

( ) ,ii

f x x=

= ∑(0,1)ix N∼

0.0100"MC"

"QMC"

"LHS"

.

0.0010nErr

or L

ow)

LHSExpon. ("QMC")

Expon. ("MC")

Expon. ("LHS")

Log(

Mea

n

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

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

4 3 2 1 0

1

Number written in base : (··· )

In the decimal system: , 0 1

b

mj

j jj

i b i a a a a a

i a b a b=

=

= ≤ ≤ −∑Reverse the digits and add a radix point to obtain a number within the unit inter

1 2 3 4

1

val: (0. ···)

In the decimal system: ( ; )

b

mj

j

y a a a a

h i b a b− −

=

=∑.

1

In the decimal system: ( ; )

4, 2Example:

jj

h i b a b

i b

=

= =

∑

2 1 02

1 2 3

4 1 2 0 2 0 2 (100)0.001 0 2 0 2 1 2 1/ 8− − −

= × + × + × =

→ × + × + × =

1

1

Holton LDS

( ; )

:m

jj

j

h i b a b− −

=

= ∑

12

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

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

A i f V d C i b 2

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 : (··· )

In the decimal system: 2 0 1m

j

i i a a a a a

i a a

=

= ≤ ≤∑1

0 1 2 3 2

In the decimal system: 2 , 0 1

( ... )

Construct vector , permut

j jj

i L

j ji i

i a a

y a a a a a

g C y C

=

= ≤ ≤

=

=

∑

ation matrix for dimension j

.

Construct vector , permuti ig C y C

1,

ation matrix for dimension

-th element for dimension is given by:

( ) 2

iL

j li i l

j

x j

x i g − −=∑ ,1

In pract1. Construct direction numbers:

ice:

i i ll=∑

1 2 3 2 (0. ... )

The numbers are given by the equatio

j j j j jb

j

v v v v v

v

=

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

j j j

j j j

v mm v

13

where 2 is an odd integer, satisfy a recurrence relationusing the coefficients of a primitive polynomial in the Galois field G(2)

m v

2. The Sobol' integer number: jnx

How to construct Sobol’ Sequence ?

1 1 2 2

g

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

can also be seen as bit wise XOR

nj j j j

n bx i a v a v v= ⊕ ⊕ ⊕⊕ ⊕ ⊕ ⊕ ⊕

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

nx i⊕

uniform variate:

( ) ( ) / 2 jbj jn ny i x i=

.

Primitive polynomial (irreducible polynomial with binary coefficients over Generation of direction numbers:

11 1

the Galois filed G(2): ... 1, 0,1

Exa

q ql q kP x c x c x a−

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

21 1 2 2

mples of primitive polynomials: 1, 1, 1, 1.A different primitive polynomial is used in each dimension:

2 2 ··· 2qi i i i q i q

x x x x x x x

m c m c m m m− − − −

+ + + + + + +

= ⊕ ⊕ ⊕

1414

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 efficiency

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

.

Sobol’ LDS with inefficient direction numbers

Finance in focus, 201015Sobol’ LDS with efficient direction numbers

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

(l )nO N

1

(ln )Convergence: for all LDS

(ln )For Sobol' LDS: if 2 integer

n

nk

O NN

O N N k

ε

ε−

= −

= = −For Sobol LDS: , if 2 , integer

Sobol' LDS:

N kN

ε = =

.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 datepoints to Sobol' sequences as the most effective quasi-Monte Carlomethod for application in financial engineering "

16

method for application in financial engineering.

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 segmentA 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 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 hypercube along each of its length extensions into four equal parts.

.

yp g g q p

17Property A Property A’

Distributions of 4 points in two dimensions

MC ->

Property A

No

.

LHS -> No

Sobol’ -> Yes

18

Comparison of different Sobol’ sequence generators

.

The winner is SobolSeq generator: Sobol' sequences satisfy two additionaluniformity properties: Property A for all dimensions and Property A' foradjacent dimensions. Maximum dimension: 32000

F i b d k

19

Free version www.broda.co.uk

Discrepancy

0.001

0.01

Discrepancy, n=20

RandomHaltonSobol

1e-05

0.0001T

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

N

1e-17

1e-16

Discrepancy, n=100

RandomSobol

1e-19

1e-18T QMC in high-dimensions doesn’t show smaller discrepancy than MC !

20

1e-20128 256 512 1024 2048 4096 8192 16384

N

Are QMC efficient for high dimensional problems ?

(ln )n

QMCO Nε =

Assymptotically ~ (1/ )

but increseas with until exp( )

QMC

QMC

QMC

NO N

N N n

ε

ε ≈

.21 not achievable for prac

p( )

50, 5 10 tical applicationsQMC

n N= ≈ −

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

QMC offers no practical advantage over Monte Carlo”

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

21

Option pricing. Discretization of the Wiener process

1/ 2

The asset follows geometrical Brownian motion:

, ( ) , ~ (0,1)i ' l

dS Sdt SdW dW z dt z N= µ + σ =

20

Using Ito's lemma1( ) exp[( ) ( ))], ( ) Wiener path2

S t S t W t W t= µ − σ + σ −

.

2

For time step 1( ) ( )exp[( ) ( ( ) ( ))2

t

S t t S t r t W t t W t

∆

+ ∆ = − σ ∆ + σ + ∆ − ]2

For the standard discretization algorithm

1 1( ) ( ) , / , 0 1A terminal asset value:

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

22

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

~ (0 1)dW z dt z N=SDE:

dS Sdt SdW= µ + σGeometrical Brownian motion:

, (0,1)dW z dt z N

Brownian bridge algorithm:

SDE:

.0 1

0 2

( ) ,1 1( / 2) ( ( ) ) ,2 2

W T W T z

W T W T W T z

= +

= + +t0 T

T/2

0 3

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

W T W T W T z= + +

41 1(3 / 4) ( ( / 2) ( )) / 2 ,2 2

1 1

W T W T W T T z= + +

23

1 1(( 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 E P S t K− ⎡ ⎤= ⎣ ⎦( )The payoff function for an Asian call option

=max( - ,0),AP S K

⎣ ⎦

. 1/

=1

( , ),

For a geometric average Asian call: S=( )

An

ni

iS∏

=1

2

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

i

rTK T e S r− σ= −∫

1/1) ( )]

nn i

i jTt u−

⎡ ⎤+σ Φ⎢ ⎥

⎢ ⎥∑∏ 0 2nH

∫11

1 .)] ...

i jji

n

n

K du du==⎢ ⎥⎣ ⎦

−

∑∏

24

And there is a closed form solution

MC simulation of option pricing. Discretization

In a general case

( )( ) ( )0 1

1

g

1C ( , ) , , , ,N

rT i iN T

iK T e P S S S K

N− ⎡ ⎤

= ⎢ ⎥⎣ ⎦∑

.

1

For the case of a European call

1 1

i

N N

N =⎣ ⎦

⎡ ⎤( ) ( )

1 1

1 1( , ) max( ,0) i rT iN T

i iC K T C e S K

N N−

= =

⎡ ⎤= = −⎢ ⎥

⎣ ⎦∑ ∑

25

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

.

3 5

4

Opt

ion

Valu

e

3

3.5QMC, Brownian Bridge

QMC, Standard Approximation

MC, Brownian Bridge

Analytical value

Call price vrs the number of paths.

30 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

N_path

26

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 observations

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

10

11/ 2

2

1

1 ( )K

kN

kI I

Kε ⎛ ⎞= −⎜ ⎟⎝ ⎠∑

.0.1

Log(

RMS

E)

QMC B i B id

1kK =⎝ ⎠

0.001

0.01QMC, Brownian BridgeQMC, Standard ApproximationMC, 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

10 100 1000 10000

Log(N_path)

27

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

Brownian 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 modelx is a vector of input variables

, f( ) Y

1 2

( )( )k

Y f xx x x x==

Y is the model output.

k

ANOVA decomposition:

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

k

i

x x x xx≤ ≤

.

( ) ( ) ( )0 1,2,..., 1 21

1

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

( ) 0 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

f d k k

= >

= = + + + +

∀

∑ ∑∑

∫ 1 1...0

( , , ..., ) 0, , 1i s is ki i if x x dx k k s= ∀ ≤ ≤∫

Variance decomposition: 2 2 2 21 2i i jσ σ σ σ= + +∑ ∑ …

k

SSSS1 ++++= ∑∑∑

p

Sobol’ SI:

, 1,2,...,i iji i j nσ σ σ σ+ +∑ ∑ …

28

klji

ijlji

iji

i SSSS ,...,2,11

...1 ++++= ∑∑∑<<<=

Sobol SI:

Sobol’ Sensitivity Indices (SI)

Definition:1 1

2 2... ... /

s si i i iS σ σ=

( )1

2 2∫ - partial variances

- variance

( )1 1 1 1

2 2... ...

0

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

( )( )1

220f x f dxσ = −∫

.Sensitivity indices for subsets of variables:

( )( )0∫

( ),x y z=2 2

m

∑ ∑

The total variance:( )

1

1

2 2, ,

1 ...s

s

y i is i i

σ σ= ⟨ ⟨ ∈Κ

= ∑ ∑ …

( ) 222z

toty σσσ −=

Corresponding global sensitivity indices:

( ) zy

( )229

,/ 22 σσ yyS = ( ) ./ 22σσ tot

ytotyS =

How to use Sobol’ Sensitivity Indices?

0 1toty yS S≤ ≤ ≤

ytoty SS − accounts for all interactions between y and z, x=(y,z).yy

iS totiS

( )0totiS f x= →

The important indices in practice are and

does not depend on ;ix.

( )i fonly depends on ;

corresponds to the absence of interactions between

i

( )1iS f x= → ixtotii SS = ix

and other variables

If then function has additive structure: ∑ =n

iS ,1 ( ) ( )0 i ii

f x f f x= +∑Fixing unessential variables

If does not depend on so it can be fixed

=s 1 i

( )1totzS f x<< → z

30

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

ANOVA decomposition and Sensitivity Indices

( ) ( ) ( )0 1,2,..., 1 21

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

i i ij i j k ki i j i

f x f f x f x x f x x x= >

= + + + +∑ ∑∑

1 1

1

1

...0

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

i i j i

i i if x x dx k k s

>

= ∀ ≤ ≤∫

.

( )

1

0

( ) ( ) 0, .v v u uf x f x dx v u= ∀ ≠∫

∫ ( )

( ) ( )

0 ,nH

n

f f x dx

f x f x dx f

=

= −

∫

∏∫( ) ( )

( ) ( ) ( ) ( )

0

0

,

, , ...

ni i j

j iHn

ij i j k i i j j

f x f x dx f

f x x f x dx f x f x f

≠

= −

= − − −

∏∫

∏∫

31

( ) ( ) ( ) ( ),n

j j j jk i jH ≠∏∫

ANOVA decomposition and Sensitivity Indices. Test case

1 2 0 1 1 2 2 12 1 2

1 2 1 2 0

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

f x x f f x f x f x x

f x x x x f

= + + +

= → =

( ) ( )

1 2 1 2 0

1 1 2 0 1

( , ) ,4

1 1 ,2 4nH

f f

f x f x dx f x= − = −∫

.( ) ( )2 2 1 0 21 1 ,2 4n

H

H

f x f x dx f x= − = −∫

( )

12 1 2 1 2 1 2

21 1 1

1 1 1( , ) .2 2 4

f x x x x x x

f x dx

= − − +

∫ ( )1 1 1

1 2

3 ,7

3 1

nH

f x dxS

σ= =∫

32

2 1 123 1, .7 7

S S S= = =

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 :There are efficient formulas for evaluation of Sobol Sensitivity :

.

1

2 0

1 2

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

1 [ ( ) ( ' )] '

y

tot

S f y z f y z f y z dydy dzdz

S f y z f y z dydzdz

σ= −

= −

∫

∫2 0

12 2 200

[ ( , ) ( , )] ,2

( , )

yS f y z f y z dydzdz

f y z dydz f

σ

σ

= −

= −

∫

∫

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

33

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

Global Sensitivity Analysis

Modern models contain a large number of parameters and are oftenModern models contain a large number of parameters and are oftenHighly non-linearHave large uncertainty ranges for the parameters Computationally expensive to runTraditional methods for uncertainty and sensitivity analysis not suitable due to their computational expense, local nature and the difficulty in interpreting the results

.

Global Sensitivity Analysisevaluates the effect of an uncertain input while all other inputs are varied as well; accounts for interactions between variables; the results do not depend on the stipulation of a nominal point.

It can be used toIt can be used toidentify key parameters whose uncertainty most strongly affects the output; rank variables in order of importance, fix unessential variables and reduce model complexity;

34

identify functional dependencies;analyze efficiencies of numerical schemes.

Effective dimensions

The effective dimension of ( ) in superposition senseis the smallest integer such that

Let u be a cardinality of a set of variables .

S

uf x

d

0(1

h ( ) i l f

), 1S

uu d

f

S ε ε< <

≥ − <<∑

di i l f id.

It means that ( ) is almost a sum of f x

is The effective dimension of ( ) in truncation sense

-dimensional functions.S

f x

d___________________________________________________________

1,2,...,

the smallest integer such that(1 ), 1

T

n

uu

T

d

dS ε ε

⊆≥ − <<∑

( )1

1,Examp

does

le:

not depend on t rhe o

n

ii TS

S

nf x x d d

d=

→ = ==∑

b d dder in which the input variables are sampled,

d d th d b d i i bld d→

35

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

Classification of functions

Type B,C. Variables are equally important

Type A. Variables are not equally important equally important

i j TS S nd≈ ↔ ≈

not equally importantT Ty z

TS S nn n

d>> ↔ <<

.

y zn n

Type B. Dominant low order indices

Type C. Dominanthigher order indicesDominant low order indices

11

n

ii

SS nd≈ ↔ <<∑

higher order indices

11

n

ii

SS nd=

<< ↔ ≈∑3636

1i= 1i=

Classification of functions

.

Majority of problems in finance either have low effective dimensions or

37

effective dimensions can be reduced by using special techniques

1 2 1 2( , ) ,f x x x x=

Effective dimensions. Test case

( )1 2 1 2

1 1 1

( , ) ,1 1 ,2 41 1

f

f x x= −

( )2 2 21 1 ,2 4

1 1 1( )

f x x

f x x x x x x

= −

= − − +

.

12 1 2 1 2 1 2

1 2 12

( , )2 2 4

3 1, 7 7

f x x x x x x

S S S

= +

= = =

1 2

7 76 0.85 17

effective dimensions:

S S+ = = ≈ →

( ) ( )1 2 0 1 1 2 2

effective dimensions: d 2, d 1 second order interactions are not important

( , )T S

f x x f f x f x= = →

≈ + +

38

( ) ( )1 2 0 1 1 2 2( )f f f f

Global Sensitivity Analysis of the standard discretization

1/2

For the standard discretizationn

i⎡ ⎤21

011

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

n irT

i jjiH

TK T e S r t un

− −

==

⎡ ⎤σ= − +σ Φ⎢ ⎥

⎢ ⎥⎣ ⎦∑∏∫

.1)] ... nK du du−

1 2( ), ( , ,..., ) uncertaint parametersApply global SA to a payoff function :

kY f x x x x x= = −

1/

1/2

10

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

2

nn i

i jji

Tf x S r t xn

−

==

⎡ ⎤σ= − +σ Φ⎢ ⎥

⎢ ⎥⎣ ⎦∑∏

39

⎣ ⎦

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=

0.1

1Standard Approximation

Brownian Bridge

.0.01

S_to

tal

0.0001

0.001

1 6 11 16 21 26 31

Log of total sensitivity indices versus time step number i.

1 6 11 16 21 26 31

time step number

40

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 two algorithms at different n (time steps)

.

The effective dimensions

St d d i ti dStandard approximation: dT ≈ ndS > 2

Brownian Bridge approximation: dT ≈ 2

41

g pp TdS ≈ 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 ( ) p j p y qare better distributed than higher dimensional projections

The Brownian bridge discretization

.

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 )less important variables ( A )2) 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.

42

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

Control Variate Method

We define the control variate byWe define the control variate by

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

C X

i.e., is the control parameter so that 0. E C λ=EX(λ) = EX

.Variance Decomposition

Where is the correlation between and

EX(λ) = EX.

,2))(( 222CXCCXXXVar σλρσλσσλ +−=

ρ X CWhere is the correlation between and

The optimal control parameter

XCρ X C

)(),(*

XVarCXCov

XCC

X == ρσσλ

43

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

Control Variate Method with Monte Carlo method

The basic Monte Carlo estimator is

,1: )(∑=≈N

iN X

NSXE

The control variate estimator is defined by1

∑=iN

.)(1 )(*)(*

∑ −=N

iiN CXS λλ

.The variance reduction ratio is

)(1∑=i

N N

( ) 1V S* 2

( ) 1 1, if 01( )

NXC

XCN

Var SVar S λ

ρρ

= > ≠−

44

Monte Carlo Pricing with MartingaleControl Variate (MCV)

Pricing a European option by a MCV method leads to

whereP(0,S0,σ 0) ≈ 1

Ne−rT H(ST

(i)) − λM ( i)( ˜ p )[ ]i=1

N

∑ ,

.

where

i ti l ith b i i ti f P

*

0),,(

~)~( SSS

T

SSrs dWSSs

xpepM σσ∫ ∂∂

= −

is a martingale with being an approximation of P.

: control parameter (empirically chosen as 1)λ : control parameter (empirically chosen as 1)

: martingale control.( )PM ~

λ

45

Homogenization Method

Fouque and H. (07) * use the homogenized Black-Scholes price to construct a martingale control);,( σtBS StP )()(

BSi PMp g

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

);,( tBS

2σ

.

variance function w.r.t. the invariant distribution of the volatility process.

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

* 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.

46

European Options Pricing by Variance Reduction

.

blue: Basic MC samples

47

green: MCV samples

One-Factor Stochastic Volatility model

*0t t t t tdS rS dt S dWσ= +

( )

( ) ( )* 2 *

exp / 2

1

t tY

d d d d

σ

β

=

.

( ) ( )* 2 *1 0 1 11

is an Ornstein-Uhlenbeck process

t t t tdY m Y dt dW dW

Y

α β ρ ρ= − + + −

is an Ornstein Uhlenbeck process

1/ , 2 /tY

α ε β ν ε= =so that is varying on the time scales tY ε

48

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

RatioMCV

1/50 0 352 (24 115) 0 039 (24 348) 81

ε SE BMC (mean) )(meanSE MCV

.

1/50 0.352 (24.115) 0.039 (24.348) 811/10 0.303 (23.340) 0.038 (23.400) 6310 0 268 (20 771) 0 019 (20 964) 19310 0.268 (20.771) 0.019 (20.964) 19350 0.276 (20.488) 0.017 (20.711) 267100 0.279 (20.428) 0.017 (20.660) 277100 0.279 (20.428) 0.017 (20.660) 277

*BCM – basic MC; RatioMCV – variance reduction ratio

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

Global Sensitivity Analysis of standard and MartingaleControl 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 two

payoff functions 1) standard; 2) MCV:

1Standard approximation

Martingale control variate

Effective dimensions:Standard: d ≈ n

.0.1

S_to

tal

Standard: dT ≈ ndS =2

MCV: dT ≈ 2

0.011 6 11 16

dS =1

Log of total sensitivity indices versus time step number i.

Standard discretization - Si total slowly decrease with i MCV - Si total of the

time step number

50

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

2f d

⎛ ⎞∂⎜ ⎟∫

( )1,..., nf x x

ni

iH

f dxx

ν = ⎜ ⎟∂⎝ ⎠∫

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

⎛ ⎞∂⎜ ⎟∂⎝ ⎠

.

p p y

definition is local, measure is global.

How to use in finance:

ix∂⎝ ⎠

iν

How to use in finance:

Static hedge: Consider plain vanilla option . Delta hedging strategy ( , , , )C S r tσ

Is based on a local sensitivity measure . Parameters

are uncertain.

iν ( , , , ) /S r t C Sσ∆ = ∂ ∂

1/2⎛ ⎞

( , , )S rσ

Define “global delta” and use it for a static hedge( )1/2

2/C S dSd drdtσ⎛ ⎞

∆ = ∂ ∂⎜ ⎟⎝ ⎠∫

Derivative based Global Sensitivity Measuresand “Global Greeks”

Similarly we can define

22

2n

iiH

f dxx

γ⎛ ⎞∂

= ⎜ ⎟∂⎝ ⎠∫

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

22 2/C S dSd drdtσ⎛ ⎞

Γ ∂ ∂⎜ ⎟∫

iH ⎝ ⎠

.

And use it to define “global gamma”

Similarly we can define other “global greeks”

( )/C S dSd drdtσΓ = ∂ ∂⎜ ⎟⎝ ⎠∫

There is a link between variance based and derivative based measures

2 2 Theorem: tot ii

vSπ σ

≤

Summary

Global Sensitivity Analysis is a general approach for uncertainty, complexity y y g pp y, p yreduction and structure. 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

.

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 gnominal dimensionality for problems with low effective dimensions.

Great success of Quasi MC methods in finance is explained by problems structure low effective dimensions which can be reduced by techniques suchstructure - low effective dimensions which can be reduced by techniques such as Brownian bridge, PCA, control variate techniques

53

References

1. Sobol’ I., Kucherenko S. On global sensitivity analysis of quasi-Monte Carlo algorithms. Monte Carlo Methods and Simulation, 11, 1, 1-9, 2005 http://www.broda.co.uk/gsa/brownian_bridge.pdf2 Sobol’ I Kucherenko S Global Sensitivity Indices for Nonlinear Mathematical Models Review Wilmott 56-61 12. Sobol I., Kucherenko S. Global Sensitivity Indices for Nonlinear Mathematical Models. Review, Wilmott, 56 61, 1, 2005. http://www.broda.co.uk/gsa/sobol_global_sa.pdf3. Kucherenko S., Shah N. The Importance of being Global.Application of Global Sensitivity Analysis in Monte Carlo option Pricing Wilmott, 82-91, July 2007 http://www.broda.co.uk/gsa/wilmott_GSA_SK.pdf4. Sobol’ I.M., Kucherenko S., Derivative based Global Sensitivity Measures and their link with global sensitivity indices, Mathematics and Computers in Simulation, V 79, Issue 10, pp. 3009-3017, June 2009.

.

http://www.broda.co.uk/sobol/DGSA_MATCOM_2009.pdf5. Temnov G., Kucherenko S. An approach to actuarial modelling with Quasi-Monte Carlo: simulation of random sums depending on stochastic factors, J. of Mathematical Sciences, Informatics and its Applications, V.3, N 3, pp 39-45, 2009. http://www.broda.co.uk/gsa/temnov_kucherenko_loss_aggregation.pdf6. Feil B., Kucherenko S., Shah N., Volatility Calibration using Spline and High Dimensional Model Representation Models Wilmott V 1 N 2 pp 179 195 2009 http://www broda co uk/doc/fsk volatility calibration pdfModels, Wilmott, V 1, N 2, pp 179-195, 2009 http://www.broda.co.uk/doc/fsk_volatility calibration.pdf7. Sobol’ I.M., Kucherenko S. A new derivative based importance criterion for groups of variables and its link with the global sensitivity index Computer Physics Communications, V. 181, Issue 7, p. 1212-1217., 2010 http://www.broda.co.uk/sobol/Computer_Phys_Comm_paper.pdf8. Kucherenko S., Feil B., Shah N., Mauntz W. The identification of model effective dimensions using global sensitivity analysis Reliability Engineering and System Safety 96 (2011) 440–449analysis Reliability Engineering and System Safety 96 (2011) 440 449 http://www.broda.co.uk/doc/ApplicationOfGSI_Kucherenko_RESS.pdf9. I. Sobol’, D. Asotsky, A. Kreinin, S. Kucherenko. Construction and Comparison of High-Dimensional Sobol’Generators, 2011, Wilmott Journal, Nov, pp. 64-79, 2012 http://www.broda.co.uk/doc/HD_SobolGenerator.pdf10. S. Kucherenko, S. Tarantola, P. Annoni. Estimation of global sensitivity indices for models with dependent variables, Computer Physics Communications, V. 183 (2012) 937–946 http://www.broda.co.uk/gsa/COMPHY4624.pdf

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Grazie per la vostra attenzione!Grazie per la vostra attenzione!

Acknowledgments

.

Prof. Sobol’

Dario Cziraky, Barclays Capital

Sean Han, NTHU, Taiwan

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