The parareal algorithm for American options

Contents lists available atScienceDirect

C. R.

Acad.

Sci.

Paris,

Ser. I

www.sciencedirect.com

Numerical analysis

The

parareal

algorithm

for

American

options

La

méthode

pararéelle

pour

les

options

américaines

Gilles Pagès

a

,

Olivier Pironneau

b

,

Guillaume Sall

a

,

b

a_{Laboratoiredeprobabilitésetmodèlesaléatoires,UMR7599,case188,4,placeJussieu,75252Pariscedex 05,France} b_{LaboratoireJacques-Louis-Lions,UMR 7598,case187,4,placeJussieu,75252Pariscedex 05,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

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a

c

t

Articlehistory: Received 23 May 2016

Accepted after revision 27 September 2016 Available online 5 October 2016 Presented by Alain Bensoussan

ThisnoteprovidesadescriptionofthepararealmethodforAmericancontracts,anumerical sectiontoassessitsperformance.Thescalarcaseisinvestigated.Least-SquareMonteCarlo (LSMC)andpararealtimedecompositionwithtwoormorelevelsareused,leadingtoan efficientparallelimplementation.Itcontainsalsoaconvergenceargumentforthetwo-level pararealMonteCarlomethodwhenthetimestepusedfortheEulerschemeateachlevel isappropriate.Thisargumentprovidesalsoatoolforanalyzingthemultilevelcase.

©2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Danscettenote,laméthodepararéelleestintroduitepourlecalculd’optionsaméricaines. L’algorithmeLSMC(Least-SquareMonteCarlo)deLongstaff–Schartz estparalléliségrâceà unedécompositionentempsmulti-niveaux.Dansunesectionnumérique,lesperformances delaméthodesont donnéesdans deuxcasscalaires.Unrésultatpartieldeconvergence est énoncé lorsque la méthode d’Euler explicite est utilisée avec des pas de temps appropriés sur chaque niveau. Une estimation est obtenue, qui permet d’analyser la méthodepararéellemulti-niveaux.

©2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

In quantitative finance, risk assessmentis computer intensive and expensive, and there is a market for cheaper and fastermethods,asseenfromthelargeliteratureonparallelismandGPUimplementationofnumericalmethodsforoption pricing[1,6,7,11,12,14–16,21].

Americancontractsarenoteasytocomputeonaparallelcomputer;evenifalargenumberofthemhavetobecomputed at once, an embarrassingly parallel problem, still the cost ofthe transfer of data makes parallelism at the level of one contractattractive.Butthetaskisnoteasy,especiallywhenthenumberofunderlyingassetsislarge[3,5,22],rulingoutthe PDE approach[2].Furthermorethemostpopularsequentialalgorithmisthe Least-SquareMonteCarlo(LSMC)methodof

E-mailaddresses:[email protected](G. Pagès), [email protected](O. Pironneau), [email protected](G. Sall). http://dx.doi.org/10.1016/j.crma.2016.09.010

1631-073X/©2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

LongstaffandSchwartz[19].Exploiting parallelismbyallocatingblocksofMonteCarlopathstodifferentprocessorsisnot convincinglyefficient[7],becausethebackwardregressionisessentiallysequentialandneedsallMonteCarlopathsinthe sameprocessor.

Inthisnote,weinvestigatethepararealmethod,introducedin[18],forthetask.AnearlierstudybyBalandMaday[4] haspavedthewaybutitisrestrictedtoStochasticDifferentialEquations(SDE)withoutLSMC.Yetitcontainsaconvergence proofforthetwo-levelmethodintherestrictedcasewherethesolutioniscomputedexactlyatthelowestlevel[4].

ThisnoteprovidesadescriptionofthepararealmethodforAmericancontracts,anumericalsectiontoassessits perfor-mance.Thescalarcaseisinvestigated.Least-SquareMonteCarlo(LSMC)andpararealtimedecompositionwithtwoormore levelsare used,leading toanefficientparallel implementation.Itcontains alsoaconvergenceargumentforthetwo-level pararealMonteCarlomethodwhenthetimestepusedfortheEulerschemeateachlevelisappropriate.

Convergence ofLSMC forAmericancontractshasbeenproved by Clément,LambertonandProtter [9]; itisnot unrea-sonabletoexpectanextensionoftheirestimatesforthepararealmethod,butthisnotedoesnotcontainsucharesult,only anumericalassessment.

2. Theproblem

Withtheusualnotations[17],considera probabilityspace

(,

A,

P)

,andfunctionsb

,

σ

,

f

: [

0

,

T

]

× R

→ R

,uniformly Lipschitzcontinuousinx

,

t.

Let W

= (

Wt

)

t∈[0,T] be astandard Brownian motionon

(,

A,

P)

. Let X

= (

Xt

)

t∈[0,T]

,

Xt

∈ R

,be a diffusionprocess, strongsolutiontotheSDE

dXt

=

b

(

t

,

Xt

)

dt

+

σ

(

t

,

Xt

)

dWt

,

X

(

0

)

=

X0

∈ R.

(1) A(vanilla)Europeancontracton X isdefinedbyitsmaturityT anditspayoff

E[

f

(

T

,

XT

)

]

,typically f

(

t

,

x

)

=

er(T−t)

(

κ

−

x

)

+ inthecaseofaputofstrikeprice

κ

andinterestrater.AnAmericanstylecontractallows theownertoclaimthepayoff f

(

t

,

Xt

)

atanytime

∈ [

0

,

T

]

.So arationalstrategytomaximize theaverageprofit

V

attime t istofindthe

[

t

,

T

]

-valued

F

-stoppingtimesolutiontotheSnellenvelope problem:

V

(

t

,

Xt

)

:= E[

e−r(τt−t)f

(

τ

t

,

Xτt

)

|

F

t

] = P

-ess supτ∈TtF

E[

e

−r(τ−t)_f

₍

_τ

_,

_X

τ

)

|

F

t

]

where

F = (F

t

)

t∈(0,T) isthe(augmented)filtration ofW and

T

tF denotesthesetof

[

t

,

T

]

-valued

F

-stoppingtimes.Such anoptimalstoppingtimeexists(see[8,20]).Wedonotspecifyb,

σ

or f tostayinageneralOptimalStoppingframework. Inpractice,Americanstyleoptionsarereplacedbytheso-calledBermuda optionswheretheexerciseinstantsarerestricted toa timegridtk

=

kh, k

=

0

,

. . . ,

K ,whereh

=

KT (K

∈ N

∗).Owing totheMarkovpropertyof

{

Xtk

}

K

k=0,thecorresponding Snellenvelopereads

(

V

(

tk

,

Xtk

))

k=0,...,K andsatisfiesaBackwardDynamicProgrammingrecursiononk:

V

(

T

,

XT

)

=

f

(

T

,

XT

),

V

(

tk

,

Xtk

)

=

max

{

f

(

tk

,

Xtk

),

e−

rh

_E[

_V

₍

_t

k+1,Xtk+1

)|

Xtk

]},

k

=

K

−

1

, . . . ,

0

.

(2)

Theoptimalstoppingtimes

τ

k(fromtimetk)aregivenbyasimilarbackwardrecursion:

τ

K

=

T

,

τ

k

=

tkif f

(

tk

,

Xtk

) >

e−

rh

_E[

_V

₍

_t

k+1

,

Xtk+1

)

|

Xtk

],

τ

k

=

τ

k+1otherwise

,

k

=

K

−

1

, . . . ,

0

.

(3)

When

(

Xtk

)

k=0,...,K cannotbe simulatedata reasonable computationalcost,it canbe approximatedby theEulerscheme withsteph,denoted

( ¯

Xh

tk

)

k=0,...,K,whichisasimulableMarkovchainrecursivelydefinedby

¯

X_th k+1

= ¯

X h tk

+

b

(

tk

, ¯

X h tk

)

h

+

σ

(

tk

, ¯

X h tk

)

Wk

, ¯

X h 0

=

X0,k

=

0

, . . . ,

K

−

1

,

(4) where



Wk

:=

Wtk+1

−

Wtk

=

√

h Zksothat

{

Zk}_kK₌−₀1arei.i.d.

N (

0

,

1

)

-distributedrandomvariables.Fromnowonweswitch totheEulerscheme,itsSnellenvelope,etc.

InLSMC, foreach k, theconditional expectation

E[

V

(

tk+1

,

X

¯

htk+1

)

| ¯

X h

tk

]

asa function ofx

= ¯

X

h

tk,is approximatedby its

L2-projection onthelinearspacespannedby themonomials

{

xp

}

P_p=0 fromthevalues

{

e−rhV

(

tk+1

,

X

¯

thk,(+1m)

)

}

M

m=1 generated by M MonteCarlopathsusing(4);then eachpathhasits ownoptimalstoppingtimeateachk

∈ {

0

,

. . . ,

K

−

1

}

givenby (forthestoppingproblemstartingat k)

τ

_K(m)

=

T

,

τ

_k(m)

=

tkif f

(

tk

, ¯

Xthk,(m)

) >

e −rh P



p=0

¯

a_kp

( ¯

X_th,(m) k

)

p

_,

_τ

(m) k

=

τ

(m) k+1otherwise where

{¯

ao_k

, . . . ,

a

¯

_kP

} =

arg min {(a_k0,...,aP_k)∈RP+1_} M



m=1



V



tk+1

, ¯

Xhtk,(+m1)



−

P



p=0 a_kp

 ¯

Xh_t,(m) k



p

2

.

FinallythepriceoftheAmericancontractiscomputedby

V

(

0

,

X0)

=

max

{

f

(

0

,

X0), 1 M M



m=1 e−rτ1(m)_f

₍

τ

(m) 1

, ¯

X h,(m) τ₁(m)

)

}.

Note that

P

₀a

¯

p_k

( ¯

Xh_tk

)

p _{is the best approximation of}

_E[

_V

₍

_t

k+1

,

X

¯

htk+1

)

| ¯

X h

tk

]

in the least-square sense (e.g., Longstaff and

Schwartz’spaper[19])inthevectorsubspace

( ¯

X_th_k

)

p

,

p

=

0

:

P



ofL2

(

P)

. 3. Atwo-levelpararealalgorithm

3.1. Thepararealmethod ConsideranODE

˙

x

=

f

(

x

,

t

),

x

(

0

)

=

x0,t

∈ [

t0,tK

] = ∪

₀K−1

[

tk

,

tk+1].

AssumethatGδ

(

xk

,

tk

)

isahigh-precisionintegratorthatcomputesx attk+1 fromxkattk.AssumeGisasimilarintegrator

but oflow precision. The parareal algorithm isan iterativeprocess withn

=

0

,

. . . ,

N

−

1 above a forwardloop in time, k

=

0

,

. . . ,

K

−

1

xn_k+₊₁1

=

G

(

xn_k+1

,

tk

)

+

Gδ

(

xnk

,

tk

)

−

G

(

xnk

,

tk

).

(5)

Sothecoarse-gridsolutioniscorrectedbythedifferencebetweenthefine-gridpredictioncomputedfromtheoldvalueon thatintervalandthecoarse-gridoldsolution.Inthisanalysis,Gδ andGareEulerexplicitschemeswithtimesteps

δ

t

< 

t respectively.

The samemethodcanbe appliedtoan SDEinthecontext ofthe MonteCarlomethodprovided therandomvariables

{

Zm_k,j

}

m_j:==₁1_,...,,...,M_{J−1,k:=1,...,K} defining



Wk in(4)aresampledonceandforallintheinitialphaseofthealgorithmandreused foralln (seetheinitializationstepinAlgorithm 3.2forthenotations).

Theiterativeprocess(5)isappliedoneachsamplepathwithGasinglestepof(4)withh

= 

t andGδ theresultof J

stepsof(4)withh

= δ

t.Anerroranalysisisavailablein[4]forthestochasticcaseinthelimitcase

δ

t

=

0,i.e.whenthefine integratoristheexactsolution.ForfurtherresultsofthepararealmethodappliedtodeterministicODEsandPDEs,see[18] and[13].Inthisnote,wealsoextendtheresultof[4]tothecase0

< δ

t

< 

t.

3.2. Algorithm

We denoteby Vk a realizationof V

(

tk

,

Xtk

),

k

=

0

,

. . . ,

K

=

T

t; considera refinement ofeach interval

(

tk

,

ttk+1

)

by a

uniformsub-partitionoftimestep

δ

t

=

t

J ,forsomeinteger J

>

1.Then

[

tk

,

tk+1] = ∪Jj=−01

[

tk,j

,

tk,j+1]with tk,j+1

=

tk,j

+ δ

t

,

j

=

0

, . . . ,

J

−

1

,

so that tk

=

tk,0

=

tk−1,J

.

Denoteby

P

f theL2_{-projectionof f onthemonomials1}

_,

_x

_,

_{. . . ,}

_xP_.

Letn

=

0

,

. . . ,

N

−

1 betheiterationindexofthepararealalgorithm. Initialization Generate

{

Zm

k,j

}

m=1,...,M

k=1,...,K,j=1,...,J fortheM pathsoftheMonteCarlomethodwiththecoarseandfinemesh. Computerecursively forwardall Monte Carlopaths

{ ˆ

X0

tk

(

ω

m

₎

_}

M

m=1 from X

ˆ

00

=

X0 by using(4)withh

= 

t and then recursivelybackwardV

ˆ

_k0

=

max

{

f

(

tk

,

X

ˆ

0tk

),

e

−rt

_P

_{E[ ˆ}

_V0 k+1

| ˆ

X 0 tk

]},

k

=

K

−

1

,

. . . ,

0 fromV

ˆ

0 K

(

ω

m

₎

₌

_e−rT_f

₍

_T

_,

_X

ˆ

0 T

(

ω

m

₎₎

_, m

=

1

. . . ,

M. for n

=

0

,

. . . ,

N

−

1

forall M paths,

for k

=

0

,

. . . ,

K

−

1 (forwardloop): (i) computethefine-gridsolution

{ ˜

Xtδ,k, jn

}

J

j=0 of(4)withrefinedsteph

= δ

t

=

 t

J ,startedattk,0

=

tkfrom X

ˆ

n tk.

(ii) computethecoarse-gridsolutionattk+1: X

¯

tk+1

= ˆ

X n+1 tk

+

b

(

tk

,

X

ˆ

n+1 tk

)

t

+

σ

(

tk

,

X

ˆ

n+1 tk

)

Wk; (iii) set X

ˆ

tnk++11

= ¯

Xtk+1

+ ˜

X δ,n tk, J

− ˆ

X n tk+1. endk-loop endM-loop. initialization:ComputeV

¯

_Kn+1

= ˆ

V_Kn+1

=

f

(

T

,

X

ˆ

n_T+1

)

for k

=

K

−

1

,

. . . ,

0 (backwardloop):

(i) oneach

(

tk

,

tk+1

)

,fromV

˜

_kδ,_,n_J

= P E( ˆ

V_kn₊₁

| ˜

X_kδ,_,n_J

)

,computebyabackwardloopin j

˜

V_kδ,_,n_j

=

max



f

(

tk,j

, ˜

Xtδ,k,nj

),

e −rδt

_P

_{E[ ˜}

_Vδ,n k,j+1

| ˜

X δ,n tk,j

]

,

j

=

J

−

1

, . . . ,

0

;

(ii) computeV

¯

_kn+1

=

max



f

(

tk

,

X

ˆ

tnk+1

),

e −rt

_P

_{E[ ¯}

_V k+1

| ˆ

Xtnk+1

]

; (iii) setV

ˆ

_kn+1

= ¯

V_kn+1

+ ˜

V_kδ,_,0n

− ˆ

V_kn.

endbackwardk-loop endn-loop

Remark1.Notethatallfine-gridcomputationsarelocalandcanbeallocatedtoaseparateprocessorforeachk,for paral-lelization.

The followingpartialresults canbe established forAlgorithm 3.2bisobtainedfrom3.2by changingthe firststep into

˜

V_kδ,_,n_J

= P E( ¯

V_kn₊₁

| ˜

X_kδ,_,n_J

)

andthelaststepinto: V

ˆ

_kn+1

= ¯

V_kn+1

+ ˜

V_kδ,_,0n

− ¯

V_kn.

Theorem3.1. Assumeb

,

σ

: [

0

,

T

]

×R

continuous,C2inx withspatialderivativesuniformlyLipschitzint

∈ [

0

,

T

]

.Thenthereexist C , independentofk

,



t andn,suchthatfork

=

0

,

. . . ,

K ,n

=

0

,

. . . ,

N:

ˆ

X_tn k

− ¯

X δ tk

L2(P)

≤ (

C



t

)

n



k n



¯

X_t k

− ¯

X δ tk

L2(P)

≤ (

C



t

)

n



k n

√



t

.

(6) Furthermore,X

ˆ

ntk

= ¯

X δ

tkforalln

≥

k (e.g.,definition(iii)).

Corollary3.2. Forafixed

δ

t andn pararealiterations,thefinalanduniformerrorssatisfy

ˆ

Xn_T

− ¯

Xδ_T

_L2_(P)

≤ (

C



t

)

n 2



t n

!

and





max k=0,...,K

| ˆ

X n tk

− ¯

X δ tk

|





L2_(P)

≤

(

C



t

)

n2

√

(

n

+

1

)!

(7)

respectivelywhereC onlydependsontheLipschitzconstantsandnormsofb

,

b

,

b

,

σ

,

σ

,

σ

andonT.

Thisestimateshowsthatwhen



t

<<

C ,themethodconvergesexponentiallyinn andgeometricallyin



t.

Remark2.Theestimate (6) indicatesthat a recursive useofparareal witheach sub-intervalredivided into J

=

O

(

t−1

)

smallerintervals, theso-calledmultilevelsparareal,isbetter thanmanyiterations atthesecondlevelonly. Indeed,asthe errordecreasesproportionallyto

(

t

)

n2 _{ateachlevelandas}

_

_{t becomes}

_

_t2 _{atthenextgridlevel,theerrorafterL levels}

isdecreasedby

(

t

)

nL2. Proposition3.3.

(

a

)

Let

˜

V_t,n k

= P

-ess supτ∈T_tkF

E[

e −r(τ−tk)_f

₍

_τ

_{, ˆ}

_Xn τ

)|

F

tk

], ¯

V ,δ tk

= P

-ess supτ∈T_tkF

E[

e −r(τ−tk)_f

₍

_τ

_{, ¯}

_Xδ τ

)|

F

tk

]

where

T

_tkFdenotesthesetof

{

tk

,

tk+1

,

. . . ,

tK}-valued

F

-stoppingtimes.Then,forsomeconstantC ,





max k=0,...,K

 ˜

V ,n tk

− ¯

V ,δ tk



_L2_(P)

≤ [

f

]Lip

(

C



t

)

n2

√

(

n

+

1

)!

.

(Notethat

( ¯

V_t,δ

k

)

k=0,...,KisbutthecoarseSnellenvelopeoftherefinedEulerscheme.)Atafixedtimetk,wehavethebetterestimate

 ˜

V_t,n k

− ¯

V ,δ tk



2

≤ [

f

]Lip

(

C



t

)

n+1 2



K

+

1 n

+

1



−



k n

+

1



.

(8)

(

b

)

Let

( ¯

Vδ

tk

)

k=0,Kdenotethe“fine”SnellenvelopeoftherefinedEulerschemeattimestkdefinedby

˜

V_tδ

k

= P

-ess supτ∈T_tkF

E[

e

−r(τ−tk)_f

₍

_τ

_{, ¯}

_Xδ

τ

)|

F

tk

].

Then,forsomeconstantC ,

 ˜

V_tδ,n k,0

− ¯

V δ tk



L2_(P)

≤

C

√



t

.

Remark3.A result similar to

(

a

)

can be obtained for

( ¯

V_tn_k

)

k=0,...,K, i.e.when

F

tk is replaced by X

ˆ

n

tk in the expectation

defining V_tn_k atthecost oflosinga

√



t inthe errorestimate. Bothquantities V

˜

_t,_kn andV

¯

_tn_k donot coincideas X

ˆ

n isnot Markovian(italsodependson X

ˆ

n−1).

Table 1

Absolute error from the American payoff computed on the fine grid by a sequential LSMC standard algorithm and the same computed using the parareal iterative Algorithms 3.2 and 3.2bis. The coarse grid has K intervals;

the coarse time

step is t/K ;

the fine grid has a fixed number of points, hence each interval

(tk,ttk+1)has J sub-intervals.

The top 4

lines of numbers correspond to Algorithm 3.2, while the last 4 lines correspond to Algorithm 3.2bis, for which a partial convergence estimate can be obtained, but which does not work as well numerically.

K J t n=1 n=2 n=3 n=4 2 16 0.666667 0.60338 0.152339 0.0171122 0.000833293 4 8 0.4 0.237451 0.0437726 0.00217885 0.000725382 8 4 0.222222 0.0854814 0.0156243 0.000735309 0.000515332 16 2 0.117647 0.0257407 0.00120513 0.000439038 0.000262921 2 16 0.666667 0.5912463 0.1434691 0.0418341 0.0414722 4 8 0.4 0.2245711 0.0743709 0.0225051 0.0224303 8 4 0.222222 0.0740923 0.0205441 0.0072178 0.0072066 16 2 0.117647 0.0194701 0.0021758 0.0021592 0.0021509

Fig. 1. Black–Scholes

case: errors on the payoff versus

t on

the left for several values of

n and

versus

n on

the right for several values of

t.

Both graphs

are for Algorithm 3.2in log–log scales and indicate a general behavior of the error not incompatible with (3.1).

4. Numericaltests

Thepayoffis f

(

t

,

x

)

=

er(T−t)

(

κ

−

x

)

+withX0

=

36,r

=

0

.

05,

κ

=

40,T

=

2.WehavechosenM

=

100

.

000 asinLongstaff andSchwartz’swork[19].TheinterpolationusedintheLSMCisonthebasis

{

1

,

x

,

x2

_}

_,i.e.P

₌

_{2.TheAmericanpayoffis} then4

.

478 atanearlyexercise

τ

=

0

.

634.

4.1. Convergenceofthepararealalgorithm 4.1.1. TheBlack–Scholescase

Here theunderlyingasset isgivenby theBlack–Scholes SDE,

σ

(

x

,

t

)

=

σ0

x,b

(

x

,

t

)

=

rx. Inthe test,

σ0

=

0

.

2. Wehave chosenafinegridwith

δ

t

=

T

/

32.Thefreeparametersare



t,whichgovernsthenumberofpointsonthecoarsegridand

n thenumberofpararealalgorithms. TheerrorbetweentheAmericanpayoff computedonthefinegridbyLSMCandthe

samecomputedbythepararealalgorithmisdisplayedonTable 1forbothAlgorithms 3.2 and3.2bis.

ThesameinformationaboutconvergenceisnowdisplayedinthetwographsinFig. 1fortheerrorsversus



t andthe errorsversus n.Wewerenotabletodecrease



t tosmallervaluesbecausethecomputingtimebecomestoolarge. 4.1.2. Theconstantelasticitycase

Thevolatilityisnowafunctionofprice[10]:

σ

(

x

,

t

)

=

σ0

x0.7_{.Allparametershavethesamevaluesasabove.Theresults} areshowninFig. 2.

4.2. Multilevelpararealalgorithm

Thepreviousconstructionbeingrecursive,onecanagainapplythetwo-levelpararealalgorithmtoLSMConeachinterval

[

tk

,

ttk+1

]

.Theproblemoffindingtheoptimalstrategyforparallelismandcomputingtimeiscomplex,becausethereareso

manyparameters;inwhatfollows,thenumberoflevelsisL

=

4;furthermore,whenanintervalwith J

+

1 pointsisdivided intosub-intervals,each oneisendowedwitha partitionusing J

+

1 pointsaswell.So,ifthecoarsegridhas K intervals,

Fig. 2. Constant elasticity case: same legend as inFig. 1.

Table 2

Absolute error between the computed payoff with the multilevels parareal method and the reference value of Longstaff– Schwartz. The number of levels is L=4, each level is subdivided into K intervals; K4_{is the number of intervals at the}

deepest level. K K4 _{n=1 n=2 n=3 n=4} 2 16 0.353319 0.118118 0.0477764 0.0276106 3 81 0.208997 0.0390674 0.0225218 0.0186619 4 256 0.141692 0.0259727 0.0192504 0.0136437 5 625 0.105196 0.0220218 0.0179706 0.0129702 Table 3

Absolute error between the computed payoff with the multilevel parareal method and the reference value of Longstaff– Schwartz. There are L=4 levels; at level l−1, to obtain level l each

interval is divided into

Jlintervals. The errors are

given versus the number of parareal iterations n=1,2,3,4. Note that all subdivisions give more or less the same precision; computationally and for parallelism the last one is the best.

Time-step Total Absolute-error

J1 J2 J3 J4 n=1 n=2 n=3 n=4 6 5 4 3 360 0.108593 0.0305688 0.0202016 0.0142071 3 4 5 6 360 0.35365 0.0316707 0.0167488 0.0135151 20 2 2 2 160 0.0231221 0.0163731 0.0155624 0.013314 2 20 2 2 160 0.354477 0.0835047 0.0231243 0.0121775 2 2 20 2 160 0.351854 0.115285 0.015826 0.0137373 2 2 2 20 160 0.355166 0.119577 0.0444797 0.0110232

Fig. 3. Comparison

between a standard LSMC solution and the parareal solution for the same number of time intervals at the finest level. The 4 points have

respectively 1,2,3,4 levels; the first data point has one level and 4 intervals, the second has 2 levels and 16 intervals, the third one 3 levels and 64 intervals, the fourth one 4 levels and 256 intervals. The total number of time steps is on the horizontal axis, in log scale, and the error at n=2 is on the vertical axis, in log scale as well.

Fig. 4. Speed-up

versus the number of processors, i.e. the parareal CPU time on a parallel machine divided by the parareal CPU time on the same machine

but running on one processor. There are two levels only; the parameters are K=1,2..,32, n=2 and J=100 so as to keep each processor fully busy. the 4thgrid has K4 _{intervals. Theresults arecompared withthereferencevalue of Longstaff–Schwartz,4.478, shownin} Table 2andinFig. 3.

The numberofpararealiterations is4andthe errorisdisplayed ateach n.We havealsocarriedout sometestswith sub-partitionsusing J

=

K .Thuseachlevelhasitsownnumberofpointsperinterval, Jl.ErrorsshownonTable 3arealso shown inFig. 3 forn

=

2. Itseems to be O

(

K4

₎

_{for K small and O}

₍

_K2

₎

_{for K bigger. The methodwas implemented in} parallel;eachintervalisallocatedtoaprocessor,ateachlevelinaround-robinorder.Almostperfectparallelismisobtained inourtestsonamachinewith32processors,asshowninFig. 4andTable 3.

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