Recursive lower and dual upper bounds for Bermudan-style options

European Journal of Operational Research xxx (xxxx) xxx

Contents lists available at ScienceDirect

European

Journal

of

Operational

Research

journal homepage: www.elsevier.com/locate/ejor

Interfaces with Other Disciplines

Recursive

lower

and

dual

upper

bounds

for

Bermudan-style

options

R

Alfredo

Ibáñez

a ,∗

_,

_Carlos

_Velasco

b

a_{UniversidadPontificiaComillas,ICADE.c/AlbertoAguilera23,28015Madrid,Spain}

b_{DepartmentofEconomics,UniversidadCarlosIIIdeMadrid.CalleMadrid,126.,Getafe,E-28903Madrid,Spain}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received4August2018 Accepted13July2019 Availableonlinexxx

Keywords:

Finance

Bermudan/Americanoptions Optimal-stoppingtimes Recursivelower/upperbounds Simulationandlocalleastsquares

a

b

s

t

r

a

c

t

AlthoughBermudanoptionsareroutinelypricedbysimulation andleast-squaresmethods usinglower

and dual upper bounds, the latter arehardly optimized. In thispaper, we optimize recursive upper

bounds,whicharemoretractablethantheoriginal/nonrecursiveones,andderivetwonewresults:(1)

Anupperboundbasedon(amartingalethat dependson)stopping timesisindependent ofthe

next-stageexercisedecisionandhencecannotbeoptimized.Instead,weoptimizetherecursivelowerbound,

anduseitsoptimalrecursivepolicytoevaluatetheupperboundaswell.(2)Lesstime-intensiveupper

boundsthatarebasedonacontinuation-valuefunctiononlyneedthisfunctioninthecontinuation

re-gion,wherethiscontinuationvalueislessnonlinearandeasiertofit(thanintheentiresupport).Inthe

numericalexercise,bothupperboundsimproveoverstate-of-the-artmethods(includingstandard

least-squaresandpathwise optimization).Specifically,theverysmallgapbetweenthelower andthe upper

boundsderivedin(1) implies therecursivepolicy andthe associated martingalearenearoptimal,so

thatthesetwospecificlower/upperboundsarehardtoimprove,yettheupperboundistighterthanthe

lowerbound.

© 2019ElsevierB.V.Allrightsreserved.

1. Introduction

Pricing Bermudan options in high dimensionsrequires Monte

Carlo methods, and two simulation-based prices have been

de-veloped:lower anddualupperbounds. Specifically,Longstaff and Schwartz (2001) use a standard least-squares Monte Carlo (LSM)

approach to compute lower bounds. Likewise, upper bounds are

also based onleast squares andsimulation (Andersen & Broadie, 2004 ). Although both boundsare widely used,upper boundsare hardly optimized, which is importantbecause simulation is time

consuming,demandingasmartapproach.

Inthispaper,weoptimize(themoretractable)recursiveupper

boundsandprovidetwo newresults.Lower/upperbounds

gener-atedbysimulationdependonanexercisepolicy,wherebythe up-perboundisderivedfromamartingalebasedonthispolicy.First, weshowarecursiveupperboundisindependentofthenext-stage

R _{A previous version ofthis paper was titled“Pricing Bermudan Options by} Simulation:WhenOptimalExerciseMatters.” WethankPeterCarrforcomments. This paper has been presentedatMorgan Stanley(NYC) and Oxford University (Mathematicsdepartment).Weareespeciallygratefultoallrefereesfortheir con-structivecommentsthatledtoabetterpaper.ResearchfundedbyPlanNacional deI+D+i(ECO2017-86009-P,MDM2014-0431,andPEP-BS-INV/GRF-12002_01)and Comunidad de Madrid ,MadEco-CM(S2015/HUM-34 4 4 ).

∗ _{Correspondingauthor.}

E-mailaddresses:[email protected] (A.Ibáñez),[email protected] (C. Velasco).

exercise decision and hence cannot be optimized. Therefore, we optimizetherecursivelower bound,followingIbáñez and Velasco (2018) local LSM approach, and use its optimal recursive policy

toevaluate the upperbound aswell. We findthesetwo bounds,

whichhaveasimilarcosttothereciprocalboundsbasedona pol-icyestimatedbythestandardLSMmethod,areverytight.Second, westudyseparatelyanupperboundgeneratedfromamartingale based on continuation-valuefunctions, a bound that is lesstime intensiveyetmoreupperbiased,andshowhowtoreduce itsbias aswell.

In our first approach, we consider a given family of exercise policies—orstoppingtimes.IbáñezandVelascomaximizea recur-sivelowerbound—orBermudanprice, L ,withregardtothisfamily ateach exercise stage. Anopen questionis which exercise strat-egy minimizes the upper bound, U . We show the exercise

strat-egy that maximizesa recursive lower bound also minimizesnot

therecursiveupperbounditself,butratherthegapbetweenthem,

U −L .Weprovidearecursiveexpressionforthegap(Theorem 1 ), andshowa recursiveupperbound U isindependentofthe next-stage exercise policy (Proposition 1 ). Therefore, minimizing the gap, U −L,isequivalenttomaximizingtheBermudanprice, L , re-cursively.

Inthesecond approach,we considerafamilyof

continuation-value functions. We show (i) a recursive upper bound is

inde-pendent of the next-stage continuation-value function as well

https://doi.org/10.1016/j.ejor.2019.07.031 0377-2217/© 2019 Elsevier B.V. All rights reserved.

(Proposition 1 ). (ii) By factorizing the two martingales that are basedon either stoppingtimes orcontinuation values,the latter martingaleincludesa thirderrorterm, whichensurestheprocess isactuallyamartingaleyetimpliesmorebiasedupperbounds.The othertwotermsofthemartingalearethoseofthestandard factor-izationoftheAmericanoptionintoanearly-exercisepremiumand theEuropeancounterpart(Carr, Jarrow, & Myneni, 1992 ).Thisthird term,however,dependsonly onthe optioncontinuation value in thewaiting/continuationregion.

The latterconstraint is criticalbecause Bermudan options are highly nonlinear near the exercise boundary but less so in the waitingregion, and fitting a continuation-value function only in thisregion is easier. This new upper bound, based on a contin-uation value estimated onlyin the waitingregion, is asaccurate asan upperbound basedon an exercise policyestimated by the LSMmethod(Andersen & Broadie, 2004 ),butina fractionofthe time. The new bound is especially accurate forat-/in-the-money options,whichdependmostlyonsamplepathsthatcrossthe exer-ciseregionandthatdonotcontributetothemartingale’sthird er-rorterm.Thisdualwaiting-regionconstraintisthereciprocal con-straintofusingin-the-moneypaths,andhence,theexerciseregion, toestimatethecontinuationvalueintheLSM/primalmethod.

Inthenumericalexercise,wepriceup-and-outBermudan max-options(Desai, Farias, & Moallemi, 2012 ). The up-and-out barrier makes this option very sensitive to suboptimal exercise, provid-ing a good test. From the local LSM method (Ibáñez & Velasco, 2018 ), we derive the optimal recursive exercise policy and com-putethetwoboundsassociatedwiththispolicy:thelowerbound

improvesupon thereciprocalboundsbasedon thestandard LSM

andpathwiseoptimization(Desai et al., 2012 )bymorethan100– 200cents;theupperboundyieldsaone-digitgap.Thissmallgap impliestherecursivepolicyandtheassociatedmartingalearenear optimalandthetwoboundsareclosetothetrueprice.The local policyissogoodthatreducingthenumberofsubsimulationpaths by20 (i.e.,5% of the original subsimulations)decreases the time effortbya factorof 10,yetthe upperbound increasesonly bya fewcents.

Notably, the upper bound basedon the local-LSMpolicy only changesmarginallywiththenumberofsubsimulationsandis ro-bustto all refinements, implying the upperbound is tighter and closertothe truepricethan the lower bound.With other meth-ods(e.g.,thestandard LSM)thatyieldanontrivialgap,thisclaim cannotbemade.Thisresultagreeswiththetwo-periodBermudan upperbound, which is independent of the (one-period) exercise policy.Atighter upperboundimpliesamidpointbetweenlower andupperboundsislowerbiased.

The dualityapproachinoptionpricing(Haugh & Kogan, 2004; Rogers, 2001 ) has been extended in many ways. Chen and Glasserman (2007) andRogers (2010) studyoptimaldual bounds;

Belomestny, Schoenmakers, and Dickmann (2013) usea multilevel approach;Glasserman (2004) studiesdualboundsbasedon regres-sionmethods; Christensen (2014) andBhim and Kawai (2018)

de-rive upper bounds using linear and semidefinite programming.

Desai et al. (2012) usea pathwise-optimization approach that is less time consuming. In a novel extension, Brown, Smith, and Sun (2010) uses an informationrelaxationthat neststhe perfect-informationassumptionof themartingaleapproach, andalso ap-pliestoother problemsinoperationsresearch (Balseiro & Brown, 2019 ).

We tailor Haugh–Kogan and Andersen–Broadie results to our

optimalrecursivesetting,whichyields suchtight bounds. Specifi-cally,intheformercasebasedoncontinuationvalues,weimprove theupperboundbycomputingthecontinuationvalueonlyinthe waitingregion.Inthelattercasebasedonstoppingtimes,the up-perboundisbothtightandefficientifweusefewersubsimulation paths,butanear-optimalexercisestrategyasinthelocalLSM

ap-proach.Inboth ways,we bringthe overallcost ofthemartingale approachinlinewithpathwiseoptimizationorinformation relax-ation.Moreover,thefactorizationofthedualmartingalesinterms

of the components ofthe Bermudan process is mostlynew. Our

tightboundsareanusefulbenchmarkfornewmethodsthattryto improveupperboundsintermsofaccuracyortimeeffort.

Section 2 reviewsthelocalLSMmethodandexplainsthe exer-cisepoliciesandcontinuationvaluesneededlaterfordual bound-ing; Section 3 shows the independence of the recursive upper bound onthe next-stageexercise policy; Section 4 showsan up-perboundbasedonacontinuation-valuefunctiononlyneedsthis functioninthe waitingregion; Section 5 providesthecomplexity analysisandexamples;Section 6 concludes.Proofs areleft tothe Appendix.

2. Bermudanoptions:alocalLeast-SquaresSetting

Inthissection,weexplainthedifferencebetweenthestandard andthelocalLSMmethods. Becausethelocalapproach yields an optimalrecursiveexercisepolicy,itisourmethodofchoiceto de-riveboththelowerboundandthemartingaleassociatedwiththe

upperbound.Wenext discussprecisely howwecompute various

exercisepoliciesandcontinuation-valuefunctionsneededlaterfor dualbounding.Lastly,wedefinethelowerandtheupperbound.

Consider a Bermudan option that can be exercised at t ∈

{

1,2,...,T

}

,where t =0istoday.Wedenoteby I t≥0theintrinsic

value (oroption payoff)iftheBermudan optionis exercisedat t . Consider a vector of N stockprices S t. Interest rates are

stochas-tic, R t>0isabank-accountprocess, R 0=1,and R j,t=R t/R j.Ifthe

interest rate r isconstant, R t= e rtt, R t,t+1=e rt,and

t isthe

timebetween t and t +1.

We introduce two binary auxiliary processes, Y and b ; b 0=1

and

Yt∈

{

0,1

}

and bt=bt−1×Yt,t=1,2,...,T. (1)

Bothprocessesare usedinthecaseofbarrieroptions (e.g., secu-rities subjecttodefaultrisk). Inthe numericalexercise, inwhich we studyan up-and-out max-option, Y t=1_{max_{S_{t }}<B_} where B is

theup-and-outbarrier(andthenobarriercaseisequivalentto as-sumethat Y t=1forall t or B →∞).Inthiscase,theintrinsicvalue

isgivenby I t×b t, t ≤1,2,...,T .

FromtheBellmanprinciple,thecontinuationvalue V ∗(t , S t)ofa

Bermudanoptionsatisfies

V∗

(

t,St

)

=EtQ

1

Rt,t+1×

max

{

It+1×bt+1,V∗

(

t+1,St+1

)

}

,

t=0,1,...,T−1, (2)

andV ∗

(

T ,S T

)

=0. E tQ[]istheexpectationoperatorunderthe

risk-neutral measure Q conditional to the information at time t . We refer to V ∗ as the “first-best” Bermudan price. Although the re-sultsbelowcan bederived interms ofabank account(in which

R t=1, t =0,1,...,T ),we workinnominalterms;thus, all

equa-tionscarrydirectlytothecomputer.

Remark. In Eq. (2) ,we assume b t=1, andhence V ∗(t , S t) isthe

continuationvalueseensince t ,thatis,conditionalonnoprevious barrier(otherwise,V ∗

(

t,S t

)

=0if b t=0).

2.1. The local LSM algorithm

Let n max≥1 bethenumberofiterations.Wespecifythefinal

periodt = T ,andrecursivelysolvethecontinuationvaluefor T − 1, T −2, until t =1, where V LSM

T−1 is a standard LSM estimator of

thecontinuation value atthe initialstage T −1.Specifically, y t is

A.IbáñezandC.Velasco/EuropeanJournalofOperationalResearchxxx(xxxx)xxx 3

between t and T ),V n

t isalocalLSMestimatorofthecontinuation

value at t stage and n iteration,and K (z , h ) isa Gaussian kernel (which is evaluated at z , and h is the bandwith). The algorithm providesV _t∗,whichisthefinallocalLSMestimatorofthe continu-ationvalueat t ,andV co

t ,whichistheestimatorofthecontinuation

valueinthecontinuationregionat t (t =1,2,...,T −1).

We assume a specific set of regressors x t, which depend on

prices orstate variables S t (or earliervaluesif necessary),where

f t(x t)isanaffinefunctiontobeestimatedbyleastsquares.

The local LSM algorithm: Consider a set of simulated paths,

ϖ∈

.

0.Atmaturity,set t ₌ T .Define y T+1=0,V T∗=0,andV Tco=0. 1.Updatingpaths,ϖ∈

yt=Yt×

It, i f It≥Vt∗

R−1

t,t+1×yt+1, otherwise.

Ifa barrierexists, Y t=0cancels(set toa 0value)a paththat

hitsthebarrier. Set t =t −1.

2.The newContinuationvalue. Set n =1andV 0

t =V t∗+1.If t =

T −1,setV 0

T−1=V TLSM−1.

2.1.Localizingtheexerciseboundary:alocalregression

Vn

t = argmin

ft∈F

∈

ft

(

xt

)

−Rt−,t1+1×yt+1

2

×K

Vn−1

t

(

xt

)

−It,h

×1_{It>0}1{Yt=1},

Vtn←

Vn t +Vtn−1

2 (ifnecessary,toavoidpotential loops) (3)

Set n = n +1.Gobacktostep2.1until n =n max.

SetV _t∗=V nmax

t .

3.ThenewContinuationvalueinthewaitingregion

Vtco=argmin ft∈F

∈

ft

(

xt

)

−Rt−,t1+1×yt+1

2

×1

_{

_V∗

t≥It

}

. (4)

Gobacktostep1until t =1.

End of the local LSM algorithm

At t =1,wehaveestimatedallcontinuationvalues:V _t∗andV co t ,

t =1,2,...,T −1.

Four remarks . First, from the value-matching condition (i.e.,

V ∗

(

t,S

)

= I t

(

S

)

),thefunction K

(

V _tn−1

(

x t

)

−I t,h

)

isaGaussian

ker-nel that underweights paths that arefar away fromthe optimal-exercise boundaryatthe t stage and n −1 iteration.InEq. (3) ,in the case ofa barrier, if 1_{Y_t =1_}=0,this term excludesthe paths

thathitthebarrierfromtheregression.Second,thestandardLSM method is given by no iterations and no kernel (n max=1 and

K

(

)

=1),where onlyin-the-money paths are used (i.e.,the term 1_{I_t >0}).Third,V tcoisan estimatorofthecontinuationvalue inthe

waitingregion. At time t , we compute many samplesof the dis-countedrealizedpayoff, R −1

t,t+1×y t+1.Then,thepayoffsinthe

wait-ing region (i.e.,ifV _t∗_≥I t) are approximatedusing standard

least-squaresbythefamily F .(Withoutthebinaryfunction1

_{

_V∗ t ≥I_t

}

,V tco

isestimatedbyastandardregression,onlythattheexercisepolicy isbasedonalocalapproach.)

Finally,fourth,step2.1inthelocalLSMalgorithmisa Newton-typeiteration(inamulti-dimensionalsetting),whichconvergesin afew(fromonetothree)iterationsbecausetheintrinsicvalue—I t

isa linearfunctioninthein-the-money regionandthe continua-tionvalue—V ∗isasmooth andmonotonicfunction,inthecaseof standardput/callpayoffs. However,inthecaseofabarrier,which ifhitcancelstheoption, V ∗isnotmonotonic.Becauseofthislack

of monotonicity, whichmakes harder toprice theBermudan

op-tion, we include the stepV n

t ←

V_t n +Vn −1

t

2 , to avoidpotential loops

(intheNewtoniterations).

Thestoppingtime—orexercisestrategy

Let

τ

(

t

)

∈

{

t,t +1,...,T

}

, and hence

τ

(

t

)

≥t, be a stopping time indexed in t , for t ∈

{

1,2,...,T

}

, and

τ

(

T

)

= T . If

τ

is not indexedin t ,

τ

₌

τ

(

1

)

.First,fromthecontinuationvalueobtained fromthelocalLSMmethod(V ∗),the stoppingtime

τ

(

t

)

is recur-sivelydefinedasfollows:

τ

(

t

)

=t if It≥Vt∗;

τ

(

t

)

=

τ

(

t+1

)

otherwise. (5)

The stopping time

τ

is used to compute both the lower bound

(V low

0 )andthemartingale(M /R ) associatedwiththeupperbound,

whicharedefinedbelow.Thisstoppingtime

τ

provides the exer-cisestrategy that optimizesa recursive lowerbound. Second, the other continuation-value function V co _{is used to estimate a new}

martingaleandsecondupperbound.

Henceforth,no otherleast-squares estimatorsarenecessary. In addition,forsimplicity, inthe following lower/upper bounds, the intrinsicvalueiswrittenas I t(insteadof I t×b t).

2.2. Lower and dual upper bounds

WerewriteEq. (2) foraMonte-Carlosetting.LetT bethesetof stopping-times,

τ

∈

{

1,2,...,T

}

.Foragiven

τ

∈T,alowerbound

V low

0 isdefinedasfollows:

Vlow 0 :=E

Q 0

_I τ

R_τ

≤sup

τ∈TE

Q 0

_I τ

R_τ

=E₀Q

_I τ

R_τ

τ=τ∗

:=V₀∗, (6)

where V ₀∗₌V ∗

(

0_,S 0

)

is theBermudanpriceand

τ

∗ isthe

associ-atedfirst-beststoppingtime.

AdualupperboundisanestimatoroftheBermudanpriceand allowsustobuildamidpointandtoassessalower bound. How-ever,adualupperbounddependsonamartingalethatisnot spec-ified.Fora martingale M_Rt

t , t ∈

{

0,1,...,T

}

,upperbounds V

up 0 are

basedonthefollowingresult:

V₀up:=M0+E0Q

max

1≤t≤T

_It−Mt

Rt

≥M0+E0Q

_Iτ

R_τ − Mτ

R_τ

τ=τ∗

=V₀∗. (7)

Thelastequalityfollowsfromtheoptionalsamplingtheoremand theinequalityfollowsfrom

max

1≤t≤T

_I

t−Mt

Rt

≥ Iτ−Mτ

Rτ

τ=τ∗.

The upper bound is binding (i.e., V ₀up=V ₀∗) for the process as-sociated with the first-best Bermudan price, M ∗ (Andersen & Broadie, 2004; Rogers, 2001 ; ourProposition 1 ).Todefine M ∗ (in

Section 3.1 ),weusethestandardfactorizationoftheAmerican op-tion into the early-exercise premium and the European counter-part.

V ₀upisindependentoftheinitialvalue M 0 (seeAppendix A ).In

particular,ifthe initialvalue ofthe process M isset(not tozero but)to M 0=V 0low,itfollows,alongwith

Vlow

0 =M0≤V0∗≤V up 0 ,

thatthefollowingexpectationisapropergap:

EQ 0

max

1≤t≤T

_It−Mt

Rt

=V₀up−Vlow 0 ≥0;

that is, the difference between the upper and the lower bound

isnonnegative.Then,conditional on V low

0 ,weapproximate V up 0 by

simulationintwo waysthatcorrespond totwo differenttypesof martingales.

3. Recursiveupperboundsbasedonstoppingtimes

Becauseoptimizing thedual upperbound is nottractable, we studyarecursiveversion.Ibáñez and Velasco (2018) maximize re-cursivelytheBermudanpricewithrespecttoa familyofstopping

timesat each exercise period; we refer to this price asa recur-sive Bermudan price, which is the objective function of the pri-malproblem.Thedualproblemistominimizetherecursiveupper boundandtodeterminewhetherthesolutiontothesetwo (recur-sive)primalanddualproblemsarelinked.Ifweconsiderafamily ofstoppingtimesthat arespecifiedin arecursive way,a martin-galebasedontheexercisestrategythat maximizestheBermudan pricealsominimizesnottheupperbounditself,butratherthegap betweenthelowerandtheupperbound.

We derive a simple recursive expression for this gap

(Theorem 1 ), from which we prove all results. A recursive

upper bound is independent of the next-stage stopping time

(Proposition 1 ). Therefore, for a martingale based on stopping times,minimizing thegapis equivalentto maximizing thelower bound ina recursive way(Proposition 3 ). InSection 4 ,we show

Theorem 1 andProposition 1 alsoholdforan upperbound based oncontinuationvalues.

FromEq. (6) ,whichisbasedonstoppingtimes,webuilda mar-tingalebyusingtheexercisepolicyinEq. (5) .WedefineZ t as

fol-lows:

Zt=1_{t<τ (t)}Vt+1_{t=τ (t)}It,t=1,2,...,T. (8)

TheprocessZ issimilartothevalueprocessofaBermudanoption (eithertowait ortoexercise)associated with

τ

, inwhichV T=0

and

Vt−1=EtQ−1

Zt

Rt−1,t

,t=1,2,...,T. (9)

WethendefinetheprocessM asfollows;thatis,M 0=V 0and

Mt=Mt−1Rt−1,t+Zt−Vt−1×Rt−1,t, t=1,2,...,T, (10)

sothat M t/R t is a martingale (i.e., E _tQ₋₁[M t/R t−1,t]=M t−1), which

followsfromV definition.Inparticular,M 1=Z 1.

In addition, from Eq. (9) for t =1, we also define the lower boundfromV 0, namely, V 0low=V 0 (giventhatV 0≤V 0∗).Thislower

boundV 0 isapproximatedbysimulation.

3.1. Factorizing the martingale

Importantly,theprocessM in(10) isexplicitlydefinedbyM 0=

V 0and

Mt= t−1

j=1

Ij−Vj

×1_{j=τ (j)}×Rj,t+

1{t<τ (t)}Vt+1_{t=τ (t)}It

,

(11)

which is equal to the sum of the early-exercise premium (rein-vested in a bank account) plus the right to exercise at time t

(Appendix A ). For t ₌ T , because V T=0, the risk-neutral

expec-tationof the discounted value of Eq. (11) ’s right-hand-side(rhs) impliesthe classical factorizationof an American option into an early-exercisepremiumplustheEuropeancounterpart(if

τ

=

τ

∗_).

Thisfactorization is related to the Doob-decomposition theorem, inwhichtheBermudan-optionpriceprocessistheSnellenvelope (e.g.,Carr et al., 1992 ).

FromEq. (11) ,itfollowsfor t ≤

τ

that

Mt=Vt,ift<

τ

; andMt=It,ift=

τ

,

andtherefore,

max

1≤t≤T

It−Mt

≥I_τ−M_τ=0. (12)

Themartingaleassociatedwiththeoptimalstopping-time fam-ily,

τ

∗,isdenotedby M ∗/R andisdefinedinasimilarwayasinEq. (11) , where M ₀∗=V ₀∗. The next resultcomplements the literature (Rogers, 2001 )ondualupperboundsfortheoptimal

τ

∗.

Proposition 1. (i) An upper bound based on the optimal stopping

time

τ

∗is binding, V ₀up=V ₀∗. And (ii), for any path,

τ

∗_=inf

_argmax

1≤t≤T

{

It−M ∗ t

}

,

0= max

1≤t≤T

{

It−M ∗ t

}

,

in which the “inf” is taken in the case of multiple solutions.

Proof. SeeAppendix A .

Remark. Proposition 1 showsEq. (12) ’sinequality is bindingand

themaximumis equalto0path bypath forthe optimal martin-gale M ∗(associatedwith

τ

∗).Itfollowsthatthetermmax1≤t≤T

{

I t−

M t

}

,aswellasthe(sample)gapbetweenthelowerandtheupper

bound,haslittlevarianceiftheprocess M isbasedona good ex-ercisepolicy

τ

(andif,inaddition,V inM isestimatedwithlittle simulation error).Next,we define therecursive lower andupper bounds, andthen trytominimize therecursiveupperboundand arecursivegap.

3.2. Recursive lower and upper bounds

Wedefineanewvariable GAP attime s asfollows:

GAPs:=Ms

Rs −

Zs

Rs+

max

s≤t≤T

It−Mt

Rt

, 1≤s≤T, (13)

where M s

R_s −

M_t

R_t , s ≤t ,areDoob-martingale increments.In

particu-lar,becauseM 1=Z 1,

GAP1:=max 1≤t≤T

It−Mt

Rt

,

andtheupperboundisgivenby

V₀up:=M0+E0Q

max

1≤t≤T

It−Mt

Rt

=M0+EQ0[GAP1],

whereM 0=V 0.Thenextresultallowsustounderstandarecursive

gapbetweenlowerandupperbounds(seeBelomestny et al., 2013 , forotherrecursivestatements).

Theorem 1. The process GAP defined in Eq. (13) for 1_≤s _≤T , with GAPT+1=0, satisfies that

GAPs=−

Zs

Rs +

max

Is

Rs,

Vs

Rs+

GAPs+1

, (14)

where GAPT=0. Moreover, in Eq.(14)’s rhs, only Z s depends on the

function

τ

(

s

)

.

Proof. SeeAppendix A .

Example. ConsideraBermudan optionwiththreeexercise

oppor-tunities(s ₌1and T ₌3),fromEq. (14) ,

GAP1+

Z1

R1 =

max

I1

R1,

V1

R1 +

GAP2

=max

⎧

⎨

⎩

I1

R1,

V1

R1−

Z2

R2 +

max

⎧

⎨

⎩

I2

R2,

V2

R2 +

GAP3

=0

⎫

⎬

⎭

⎫

⎬

⎭

,

which,fromEqs. (8) and(9) ,isindependentofthestoppingtime

τ

(

1

)

_,asinTheorem 1 .

For tractability, we analyze the upper bound recursively. We considerthefollowinglowerandupperbounds,whichcorrespond to a Bermudan option that can only be exercised from s to T , 1≤s ≤T −1.Thatis,

Vlow s,0 :=E

Q 0

Vs−1

Rs−1

and V_sup_,0:=E₀Q

Vs−1

Rs−1

A.IbáñezandC.Velasco/EuropeanJournalofOperationalResearchxxx(xxxx)xxx 5

Consistentwithournotation,wehave V low

0 =V 1low,0, V up 0 =V

up 1,0,and

V _s,up₀₋V low s,0 = E

Q 0[GAPs].

Proposition2. V _s,up₀ is independent of

τ

(

s

)

_, which is the next-stage exercise decision.

Proof. FromEq. (15) andTheorem 1 ,

V_s,up₀ :=E₀Q

Vs−1

Rs−1

+EQ₀[GAPs]

=E₀Q

Vs−1

Rs−1

+EQ₀

−Zs

Rs +

max

Is

Rs,

Vs

Rs+

GAPs+1

=E₀Q

max

Is

Rs,

Vs

Rs+

GAPs+1

.

V _s,up₀ does not depend on

τ

(

s

)

because the upper bound di-rectlycompares theintrinsicvalue andan estimatedcontinuation valueattime s (i.e.,max

{

I s,V s

}

).Inparticular,atwo-period

Bermu-danupperboundisalwaysbinding.1 _{Foratwo-periodBermudan,} T =2,

V₀up=V₁up_,0=M0+E0Q[GAP1]

=V0−EQ0

Z1×

1

R1

= V0

+E₀Q

⎡

⎣

max

⎧

⎨

⎩

I1

R1,

V1

R1+

GAP2

=0

⎫

⎬

⎭

⎤

⎦

=E₀Q

max

I1,E1Q

I2×

1

R1,2

× 1

R1

=V0∗,

wherethe lastequalityfollows from V ₀∗definition(i.e.,the maxi-mumbetweenexerciseandtheEuropeanoptionat t =1).

Followingthe exampleofa Bermudanoptionwiththree exer-ciseopportunities, t ∈{1,2,3},considerafamilyofexercise strate-gies. The first-order conditions associated with maximizing the Bermudan priceat t =0 imply optimal exercise at t ∈{1, 2}, but onlyforthosepathsthatarealivefortheexercisedecisionat t ₌2 (Ibáñez & Velasco, 2018 ).Hence,ifweconsiderall pathsat t =2,

wecansolvethisproblemrecursively,whichistractableandclose to theoptimalone.Bycontrast,minimizing theupperbound de-pendsonlyontheexercisedecisionat t =2,noton t =1.Thatis,

Proposition 1 impliesthat we cannot minimize theupper bound

V _s,up₀ butratherthegap, E ₀Q[GAPs],inarecursiveway(where

τ

(

t

)

is

givenfor t >s ).

3.3. An optimal recursive gap

Define

τ

∗

₍

_s

₎

_{:=arg max}

τ (s)∈T|τ (s+1)V low

s,0, (16)

where V low

s,0 is given in Eq. (15) . The stopping time

τ

∗

(

s

)

means

optimalexerciseattime s ,conditionalon

τ

(

s ₊1

₎

andsubjecttoa givensetofstoppingtimesT (inwhichnow

τ

∈

{

s,s +1,...,T

}

). Namely, if

τ

∗

(

s

)

> s,

τ

∗

₍

_s

₎

₌

_τ

₍

_{s +1}

₎

_where

_τ

₍

_{s +1}

₎

_iscomputed

inadvance.

Proposition 3. Consider a Bermudan option that can only be exer-

cised from s to T , 1_≤s _≤T ₋1; that is, s _≤

τ

(

s

)

_∈T. Assume the stopping time

τ

(

s +1

)

is given. Then

τ

∗

(

s

)

, which is defined in Eq.

(16), satisfies

τ

∗

₍

_s

₎

_{=arg min}

τ (s)∈T|τ (s+1)E Q 0[GAPs].

1_{Kaniel, Tompaidis, and Zemlianov (2008) ,Lemma1,provedthisresultfora}

two-periodBermudanoption,t∈{1,2};theupperbounddoesnotdependonthet=1 exercisedecision,implyingatwo-periodBermudanupperboundisunbiased.

Further, if

τ

(

s +1

)

=

τ

∗

₍

_{s +1}

₎

_and

_τ

∗

₍

_s

₎

_{∈T, then}

_τ

∗

₍

_s

₎

₌

_τ

∗

₍

_s

₎

and V low

s,0 =V up s,0.

In particular, for s =1 (where R 0=1,V 0=V 0lowand M 0=V 0),

τ

∗

₍

₁

₎

_{:=arg max}

τ (1)∈T|τ (2)V low

0 =arg min τ (1)∈T|τ (2)

V₀up−Vlow 0

.

Proof. SeeAppendix A .

Minimizing E ₀Q[GAPs] corresponds to optimally exercising at

time s conditional on

τ

(

s +1

)

andissolvedby thelocalLSM ap-proach,whichyieldsacontinuation-valuefunctionV _s∗.

Thelower/upperboundbiases

From E ₀Q[GAP1]=

(

V 0∗−V 0low

)

+

(

V up

0 −V 0∗

)

, we obtain the bias

associatedwiththelowerbound,

0≤V₀∗−Vlow 0

=EQ₀

1_{1=τ∗_(1)}I1+1_{1<τ∗_(1)}V₁∗

× 1

R1

−EQ 0

1_{1=τ (1)}I1+1{1<τ (1)}V1

×_R1 1

,

and with the upper-bound (from V ₀up₌M 0+E 0Q[GAP1] and Eq.

(14) for GAP 1),

0≤V₀up−V0∗

=EQ₀

max

I1,V1+GAP2×

₁ R1

−1

× 1 R1

−EQ₀

max

{

I1,V1∗

}

×

1

R1

.

Forinstance, if GAP2=0, becauseV 1≤V 1∗ and V 0∗≤V up 0 , then

V 1=V 1∗(if I 1<V 1∗)and V up

0 =V 0∗sothattheupperboundis

unbi-asedandindependentofthe(t ₌1)next-periodexercisedecision, asinProposition 1 .Here,wehaveassumedV 1 iscomputed

with-outsimulationerror.

3.4. Computing the martingale paths and the upper bound

From Eq. (10) ,one path ofthe process M is approximated as follows,M 0=V 0and,for t ∈

{

1,2,...,T

}

,

Mt =Mt−1Rt−1,t+

1_{t<τ (t)}

Vt+

ξ

t

+1_{t=τ (t)}It

−

Vt−1+

ξ

t−1

×Rt−1,t, (17)

where

ξ

t isa zero-meanapproximationerror(i.e., E[

ξ

t]=0),and

henceM t/R t isamartingale.

BasedonEq. (9) ,V iscomputedseparatelyfrom

τ

and

Vt−1=EtQ−1

Rt−1×

I_{τ (}t)

R_{τ (}t)

.

Becausethelatterexpectation isnot analytical,V isestimatedby subsimulation. For every path, and for every t _∈

{

1_,2_,...,T ₋1

}

_, the valueV t is approximated by a newsubsimulation (from t to

τ

(

t +1

)

),where

ξ

t istheerror,whichintroducesasecondbiasin

theupperbound.

Then,wedirectlysimulatethefollowinggap,

g=EQ 0

max

1≤t≤T

It−Mt

Rt

.

Thatis,we approximateV 0 andg by twoindependentsimulations

andapproximatethetwoboundsasfollows:

V0low≈V0+

ξ

0low and V up

0 ≈V0+

ξ

0low+g+

ξ

g 0,

where

ξ

low 0 and

ξ

g

0 are the respectivesimulationerrors. Although

theupper bound is basicallyasin Andersen and Broadie (2004) ,

it isbased on an optimal recursive exercise policy (i.e.,

τ

in Eq. (5) )—andthemartingale(17) associatedwiththisoptimalrecursive policy.

Inaddition,fromEq. (11) ,themartingale(i.e.,M _/R )isexplicitly givenby

Mt= t−1

j=1

Ij−

Vj+

ξ

j

×1_{j=τ (j)}×Rj,t+

1{t<τ (t)}

Vt+

ξ

t

+1_{t=τ (t)}It

. (18)

IntheAppendix,weshowhowthecomputationalcostofM could bereduced.

4. Upperboundsbasedoncontinuationvalues

Wecomputeanupperboundbasedoncontinuation-value func-tions.First,weshowwhyanupperboundthatisbasedonagood exercisestrategyislessbiasedthanaboundbasedoncontinuation values.Inthelattercase,themartingalehasanadditionalthird er-rorterm,whichimpliesalargerbias(fromJenseninequality). Sec-ond,we showfittingthecontinuationvalue inthewaitingregion issufficient.Inthisregion,thecontinuationvalueislessnonlinear andeasierto fitthan intheentiresupport.We show thesecond resultintwodifferentways.

Consider a newfamily of continuation-valuefunctionsV t, t =

1,2,...,T −1, and V T=0. Extending (the stopping-times based) Eq. (8) ,wedefine

Zt=max

{

Vt,It

}

, (19)

where V is defined as in Eq. (9) with the new Z t (i.e., V t−1=

E _tQ₋₁[max_R{Vt ,It }

t−1,t ], t =1,2,...,T ).

We define new recursive lower and upper bounds akin to

Eq. (15) , but using the new process Z . Because Theorem 1 and

Proposition 1 triviallyholdforthisnewprocessZ (whichdepends on the max function and V ), the new recursive upper bound is givenby

V_sup_,0 :=EQ₀

Vs−1

Rs−1

+E₀Q[GAPs]=EQ₀

max

Is

Rs,

Vs

Rs+

GAPs+1

,

(20)

whichdoesnotdepend onV s attime s (i.e.,V sdependsonV s+1).

Here,minimizing thegap is not well defined, becausethe lower boundbasedonV s,

Vlow s,0 :=E

Q 0

Vs−1

Rs−1

=EQ₀

max

Is,Vs

× 1

Rs

, (21)

isnotnecessarilylowerbiased.

Hence,letusimposethebestcase E ₀Q[GAPs]=0in(thesecond

equalityof)Eq. (20) , andsearch forthe bestV guaranteeing this zeroequality.Ifweassume GAPs+1=0,

E₀Q

max

Is,Vs

× 1

Rs

=EQ₀

max

Is,Vs

× 1

Rs

,

andthesimplesolutionassociatedwiththelatterequation isthat

V s=V ssubjecttoV s>I s.Thissolutionimpliesafittingofthe

func-tion V s only in the waiting region, in which V s>I s. This same

(waiting-regionconstraint)resultisderivednextbyfactorizingthe processM .

Remark. In the case of stopping times (i.e., Z t=1_{t<τ (t)}V t+

1_{t=τ (t)}I t),thelastequationisgivenby

E₀Q

1_{{s<τ (}s)}Vs+1_{s=τ (s)}Is

× 1

Rs

=E₀Q

max

Is,Vs

× 1

Rs

,

and the same V s appears on both sides of the equality, where

theonlypossibledifferenceisbecauseof

τ

(

s

)

.

4.1. Computing a martingale based on the continuation value

ExtendingEq. (17) , we define a martingalebased on the con-tinuation value (V ) in Eq. (19) ; that is, M 0=V 0, and for t ∈

{

1,2,...,T

}

,

Mt=Mt−1Rt−1,t+max

Vt,It

−

Vt−1+

ξ

t−1

×Rt−1,t, (22)

and

Vt−1=E_tQ₋₁

#

R−1 t−1,tmax

Vt,It

$

.

Now, for every path, the process V t is computed for t ∈

{

1,2,...,T −1

}

by a one-period subsimulation (from t to t +1), where

ξ

istheone-periodsubsimulationerror, E[

ξ

t]=0.

Then(seeAppendix A )

Mt= t−1

j=1

Ij−Vj

+

×Rj,t+ t−1

j=1

Vj−

Vj+

ξ

j

×R_j,t+max

Vt,It

, (23)

wherethe second sumisa martingale error-correctingterm(i.e.,

V _j−V _j). However, V _j cancels if

{

I _j−V _j

}

+_>0_, from the first and secondsums,implyingtheerrorV j−V jonlymattersinthewaiting

region, in which I j≤V j. Hence, we define V =V co, whereV co is

giveninthelocalLSMalgorithm(seeEq. (4) ).

Two remarks .First,fromthefactorizationsinEqs. (18) and(23) ,

theformerimpliesamartingale thatiscloseto theoptimal mar-tingale M ∗/R , if

τ

is closeto

τ

∗. The latter factorizationincludes asecondsumthatdependsontheerrorbetweenV co_andthe

im-plicitV inthewaitingregion,implyingamorebiasedupperbound. Second,intheexamplesbelow,weshowastandard-LSM

stop-ping time produces only slightly worse upper bounds than the

local-LSMstoppingtime,assuminginbothcasesV =V co_.This

find-ing impliesestimating the continuation value in the waiting re-gion (i.e.,V ₌V co_{) is thekey insightfor upper boundsbased on}

acontinuation-valuefunction.Thatis,insteadoflocalizingthe es-timation (of acontinuation value) inthe exercise boundaryasin the localLSMmethod, welocalize thisestimationin thewaiting region.

Inaddition,wecanbuildathirdmartingalebasedonboth

τ

(

t

)

andV t.However, thismartingale yields a lesstight upper bound

thana martingalebasedexclusivelyonV t,andhenceisrelegated

totheAppendix.

5. Complexityanalysisandnumericalexample

For N state variables, T exercise dates,and M paths,the stan-dard LSMmethodrequires simulatingO(NTM ) samplepointsand computing T regressions. For n max iterations, the local LSM

re-quiresO(NTM )samplepoints,O(TM ×n max)kernelevaluations,and

n max×T regressions.Inpractice,afew(n max=1to3)local

regres-sionsare sufficient,ifthesolutionofstage t isusedastheinitial stepat t −1.Thelocalapproachincreasesthecomputationaleffort inalinearway,whichisgivenby thenumberofiterationsabove one(n max>1).Inaddition,thelowerbound,whichisindependent

ofthespecific LSMmethod,requiresup to M lowT intrinsicvalues

(where M low isthenumberofsimulatedpaths).

Theupperboundiscostly;itrequirescomputingtheBermudan value(V )ineveryexercisestageandineverypath,whichimplies

M upT subsimulations,where M upisthenumberofsimulatedpaths.

Inthecaseofanupperboundbasedon stoppingtimes (continu-ation values), thesubsimulation is launched until a path is exer-cisedandisboundedby T stages(is aone-periodsubsimulation).

A.IbáñezandC.Velasco/EuropeanJournalofOperationalResearchxxx(xxxx)xxx 7

Table1

Lowerandupperbounds.

Lowerbound,Vlow

0 Upperbound,V

up

0

S0 binomialprice LSM localLSMiterations τ-based Vco-based

V0∗ 1st 2nd 3rd 3rd LSM 3rd LSM

N=2assets(kernel=0.5%) 100 31.074 28.799

(.006) 30(.008.869) 30(.006.988) 31(.006.016) 31(.001.083) 31(.028.278) 31(.006.347) 31(.007.331) [Vlow

0,DFM,V

up

0,DFM] N=4assets(kernel=1%)

90 [33.011,34.989] 32.706

(.008) 34(.004.612) 34(.004.656) 34(.004.667) 34(.005.749) 34(.015.934) 34(.013.976) 34(.010.962) 100 [41.541,43.587] 40.328

(.008) 43(.003.117) 43(.004.138) 43(.004.161) 43(.004.251) 43(.024.630) 43(.011.558) 43(.010.557) 110 [48.169,49.909] 47.197

(.007) 49(.004.398) 49(.004.429) 49(.004.430) 4(9.004.482) 49(.028.998) 49(.007.780) 49(.007.851)

N=8assets(kernel=5%) 90 [44.113,45.847] 43.321

(.006) 45(.004.460) 45(.004.460) 45(.004.460) 45(.003.580) 46(.015.743) 45(.007.847) 45(.008.830) 100 [50.252,51.814] 49.523

(.007) 51(.003.357) 51(.003.360) 51(.003.360) 51(.003.433) 51(..015646) 51(.003.668) 51(.005.667) 110 [53.488,54.890] 52.319

(.006) 54(.002.525) 54(.002.527) 54(.002.527) 54(.001.564) 54(.018.898) 54(.004.697) 54(.003.744)

N=16assets(kernel=5%) 90 [50.885,52.316] 49.779

(.005) 51(.003.916) 51(.002.923) 51(.002.925) 51(.002.981) 52(.015.252) 52(.005.158) 52(.006.184) 100 [53.638,54.883] 52.574

(.002) 54(.002.601) 54(.002.603) 54(.002.603) 54(.002.633) 53(.017.806) 54(.002.718) 54(.003.800) 110 [55.146,56.201] 54.968

(.005) 55(.003.994) 55(.003.995) 55(.003.995) 56(.002.025) 56(.018.200) 56(.002.070) 56(.003.125)

Hence, in addition to M upT intrinsicvalues, the upperbound

re-quiresupto M up−stT 2K st (M up−cvT K cv)evaluationsduetothe

mar-tingale,where K stand K cv arethesubsimulation paths.Becauseof

thequadraticeffort(T 2_{), M}

up−st and K st arekept smallinthe

for-mercase.

Inthecaseofstoppingtimes,theupperboundisequaltothe

lower bound plus the gap. Hence, we directly estimate the gap

(g ),whichislessvolatile(see Proposition 1 )andrequiresamuch smallernumberofpaths, M up−stM low.

5.1. Numerical example: pricing up-and-out Bermudan max-options

Wepriceup-and-out max-callBermudanoptions.The

up-and-out barrierfeature makes callpayoffssensitive tosuboptimal ex-ercise.Wedefine I t=

{

max

{

S t

}

−K

}

+, S are lognormal-distributed

prices, K is the strike price, and B >K is the barrier.2 _{We define} Y t=1_{max{S_t }<B_}, t =1,2,...,T , inEq. (1) . Hence, b t=0indicates

theup-and-outbarrier(B )hasbeenhit(b j=0, j =t,t +1,...,T ).

TheBermudanpayoff isgivenby I t×b t.

WefollowtheMCexerciseinTable 1 inDesaietal.(DFM).This tablehasnineexamples,correspondingtothreenumbersofstocks (N ₌

{

4_,8_,16

}

) and three initial stock prices (S 0=

{

90,100,110

}

).

The strikeprice K =100 iscommonacross all scenarios.The up-and-out barrier is B =170. To derive the two exercise strategies associated withthelocalandstandard LSMmethods,we exclude those pointsthat are out ofthe moneyor hit the barrier(i.e., if max{S t}≤K or ifmax{S t}≥B ). In theAppendix, we emphasize a

fewpointsregardingtheimplementationoflocalLSMforthis bar-rierproblem.

We use the same basis of N +2 variables as DFM,

namely, a constant, every component of the price vector

S ₌

(

S (1)_,S (2)_,...,S (N)

₎

_,and

_{

_max

_{

_{S t}

_}

_−K

_}

+_;thatis,

xt=

1,St,

{

max

{

St

}

−K

}

+

,

andthesamelinearfunction,namely, f t

(

x t

)

=

β

t×x t,where

β

t∈

RN+2 _{are the parameters. Forboth bounds, we report the mean}

andstandarderrorover10independenttrials.

5.1.1. Lower and upper bounds based on stopping times

In ourTable 1 ,we providethe lower boundproduced by two regressionmethods:thestandardandthelocalLSMmethod(first

2_{SeeBallotta and Bonfiglioli (2016) orZeng and Kwok (2014) forEuropean}

op-tionsinarichjumpsetting.

tothirditerations).Fromtheexercisestrategiesassociatedwiththe standardLSMandthelocalthirditeration,wealsogenerateupper bounds.

Table 1 . Prices of Bermudan up-and-out max-call options for

N =

{

4,8,16

}

uncorrelated stocks in a lognormal setting (where

r =0.05 isthe riskfreerate,

δ

=0is thedividend yield, and

σ

= 0.20 isvolatility). K =100 isthe strike price, B =170 is the bar-rier, T =3 is maturity, and 54 exercise opportunities exist. The first columnis the stock priceand the second is the best lower andupperbound, [V low

0,DFM, V up

0,DFM], reportedby DFM(Desai et al.,

2012 ). The third tosixth andseventh totenthare thelower and upperbounds, respectively.ThethirdcolumnisthestandardLSM method, and the fourth to sixth columns are the first three

it-erations of the local LSM method. The seventh and eighth are

upper-bounds based on a stopping time

τ

, whichare associated withthestandardLSM(asAndersen–Broadie)andlocalLSM third-iterationexercisestrategies,respectively.Thelasttwocolumnsare upper boundsbased on a continuation value V co_{, which is}

rees-timated in thecontinuation region, whichis associated with the LSMandthird-iterationlocalLSMexercise strategies,respectively. AsinDSM,forbothLSMmethods,weuse200,000pathsto recur-sively compute the continuation valuesand then2 million paths tocomputetheBermudanprice.Wereportthemeanandstandard error(over10independenttrials).Forthegapoftheupperbound basedon

τ

,weuse3,000externalpathsand10,000subsimulation paths.Fortheupperboundbasedon V co_{,weuse10,000external}

pathsand500subsimulation paths. Wealsoreport thetwo-asset case,inwhichthetruepriceisderived fromthebinomialmethod andlinearextrapolation(tocorrecttheerraticbinomialprices).

The local-LSM lower bounds improve upon the reciprocal

standard-LSMlower boundsby 100–280 cents (uponDFM by 85

to160cents).Inthenineexamples,thefirstiterationofthelocal methodyields the mostsignificant improvement.For fourassets, thisboundincreasesonlybyacoupleofcentsafterthethird iter-ation;foreightand16assets,thepriceconvergesinoneiteration. Inallcases,thelocalupperboundyieldsaone-digitgap.

InTable 2 ,we increase thenumber ofpaths that are used in thelocalregressiontoimproveTable 1 numbers.Weconsiderthe hardestproblem, N ₌4stocks.Improvingthelowerboundis diffi-cult.Wereducethestandarderrorandgetsmootherprices,which isintuitive inaleast-squaressetting.InFig.1 ,we showthelower bound’srobustnesstothekernel.More(less)than1%ofthepaths thatareusedintheTable 1 kernelimplylower(slightlylargerbut erratic)prices.This1%isouroptimalkernelchoice.

Table2

Increasingthenumberofpathsintheregression.

Lowerbound,Vlow

0 Upperbound,V

up

0

paths LSM localLSMiterations τ-based Vco_-based

M 1st 2nd 3rd 4th 5th 10th 3rd 3rd

5*104 _40.327

(.010) 43(.004.106) 43(.005.128) 43(.004.147) 43(.004.153) 43(.004.160) 43(.004.178) 43(.004.251) 43(..011558) 105 _40.329

(.008) 43(.005.112) 43(.004.135) 43(.005.154) 43(.005.161) 43(.005.167) 43(.004.183) 43(.005.250) 43(..011558) 2*105 _40.328

(.008) 43(.003.117) 43(.004.138) 43(.004.161) 43(.004.168) 43(.004.175) 43(.004.191) 43(.004.251) 43(..011558) 4*105 _40.325

(.006) 43(.003.119) 43(.003.143) 43(.002.164) 43(.002.172) 43(.002.179) 43(.002.193) 43(.002.250) 43(..011559) 106 _40.328

(.002) 43(.002.118) 43(.002.143) 43(.002.163) 43(.002.170) 43(.002.177) 43(.002.193) 43(.003.250) 43(..011558) 2*106 _40.322

(.003) 43(.003.120) 43(.002.144) 43(.002.165) 43(.002.172) 43(.002.179) 43(.002.194) 43(.003.250) 43(..011559)

Fig.1. ValuesoflowerandupperboundsfordifferentkernelsandnumbersofiterationsinthelocalLSMmethod.Forthekernels,theproportionpindicatestheeffective numberofpointsusedinthelocalregression(p=0.5%inthedarkblueline,p=1%intheredline,andp=5%inthelightblueline).Forexample,LocalLSM1(LLSM3) indicatesone(three)localregressions.Wealsoshowthelower-boundvalueofthestandardLSMmethodandtwoupperboundsbasedonstoppingtimesandacontinuation valueestimatedinthewaitingregion.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Table 2 . Prices of Bermudan up-and-out max-call options for

N =4 stocksand S 0=100,in thesetup ofTable 1 .The first

col-umn(M ) is the number ofpaths inthe backward regressions to estimate the continuation values.The second columnis the LSM method,andthethirdtoeighthcolumnsarethefirstfiveandthe tenthiterationofthelocal-LSMlowerbounds.Theninthandtenth aretheupperbounds.Wereportthemeanandstandarderrorover 10trials.LowerandupperboundsareasinTable 1 .

Table 2 andFig. 1 showtherobustnessofbothupperboundsto theestimationofthecontinuationvaluebylocalleastsquares(i.e., numberofsimulationpathsandkernel).Becausetheupperbound basedonstoppingtimesisalsorobusttothenumberofiterations, thisupperboundistighter(closertothetrueprice)thanthelower bound.

5.1.2. Upper bounds

Table 3 shows the upper bound deteriorates little with the

number of subsimulated paths. We can reduce the upper-bound

costwithoutlosingaccuracy:Byreducingthenumberof subsimu-latedpathsfrom10,000to500—a5%(to100—a1%)oftheoriginal subsimulations,thegaprisesbyonly3(15)cents.

Table 3 .Gapsoflower and

τ

-basedupperboundsfor

up-and-out Bermudan max-call options for N =4 stocks, in the setup

of Table 1 . We directly compute the gap for different numbers of subsimulation paths (sub-paths): 100, 500, and 10,000. The

upper bound is defined as the lower bound plus the gap (i.e.,

V ₀up=V low

0 +Gap). We report the mean and standard error over

10 trials.Table 1 represents thecase using10,000 subsimulation paths.

FromTable 1 ,an upperbound basedoncontinuationvaluesis more biased. Yetfitting a continuation value (V co_{) in the waiting}

regionyields upperboundsthat,especially iftheoptionis at-/in-the-money,are asaccurate asthosebasedonstoppingtimesand thestandard LSMbutina fractionofthe time.Theimprovement inthisdualbound ismostlyduetothewaiting-regionconstraint andnot the subsequent policy (e.g.,if alternativelybased on the standardLSMmethod);seeTable 1 ,lasttwocolumns.3

3_{Inaddition(notreportedhereforbrevity),forfourassets,wecanreducethis}

upperboundbyanother10bpsifweuseaquadratic(insteadofalinear)function

Vco_{,andbyanotherfewpointsifV}co_{isestimatedfrompathssimulatedexactlyfrom}

S0(tocomputeV∗forthe τstopping-time,itisconvenienttosimulatepathsnot

A.IbáñezandC.Velasco/EuropeanJournalofOperationalResearchxxx(xxxx)xxx 9

Table3

Gapsbetweenlowerandτ-basedupperbounds. Lowerbound,Vlow

0 Gapτ-based Upperbound,V

up

0

localLSM 100sub-paths 500sub-paths 10,000sub-paths 10,000sub-paths

S0 LSM 3rditer LSM 3rditer LSM 3rditer LSM 3rditer LSM 3rditer

90 32.706

(.008) 34(.005.647) 2(..405015) 0(..007306) 2(..015257) 0(..006133) 2(..228015) 0(..005082) 34(..015934) 34(.005.752) 100 40.328

(.008) 43(.002.159) 3(..483025) 0(..004284) 3(..026329) 0(..004120) 3(..024302) 0(..005090) 43(..024630) 43(.004.249) 110 46.197

(.007) 49(.003.429) 3(..965029) 0(..004234) 3(..027833) 0(..003082) 3(..028801) 0(..004052) 49(.028.998) 49(.003.480)

Table4a

Relativecomputingtimesforcontinuation-valueparameters.

Method Bandwidth N

4 8 16

LSM – 1 1 1

LocalLSM1stiter Fixed 2.29 2.27 2.31 Optimal 10.75 12.34 14.43 LocalLSM3rditer Fixed 4.87 4.89 4.98

Optimal 13.34 14.96 16.59 LSM+LSMVco _{– 3.58 3.66 3.41}

LocalLSM1stiter+LSMVco _{Fixed 4.87 4.93 4.73}

Optimal 13.33 15.00 16.84 LocalLSM3rditer+LSMVco _{Fixed 7.45 7.55 7.39}

Optimal 15.92 17.62 19.00

Table4b

Relativecomputingtimesforlowerandupperbounds.

Bound Method Sub-paths N

4 8 16 LowerVlow

0 LSM – 1 1 1

LocalLSM3rditer – 1.42 1.53 1.64 UpperV₀upVco_{-based LocalLSM3rditer 500 3.77 2.85 2.88}

UpperV0upτ-based LocalLSM3rditer 10,000 113.40 125.40 197.39

500 10.83 9.75 9.20 100 6.73 5.69 4.77

InTables 4a and4b ,we reportthe effortrequiredto compute theoptimal-recursiveexercisepolicyalongwith V co_andthe

asso-ciated lowerandupperbounds. Weuseafixedandanoptimized

kernel,anduseoneandthreelocalregressions.Alltimesare rela-tivetothestandardLSM.First,toestimatetheexercisepolicyand

V co_{,the cost increasesbetween1 and19 timesinthe 30entries}

ofTable 4a .Second,tocomputethelowerbound,thetotalcost in-creasesonlybetween42%and64%(becausethelowerbounduses 2millionpaths)inTable 4b .Inthecaseoftheupperboundbased onacontinuationvalue(stoppingtimes),thecostincreasesbyone orderofmagnitude(between100 and200 times).Yet,inthe lat-ter caseofstoppingtimes,reducing thenumberofsubsimulation pathsto500,theincreaseisjust10times.

Table 4a . Relative computing times for parameter estimation timesnormalizedbycolumninthesetupofTable 1 .TheLSM,local LSM,andLSM V co _{methodsuse200,000pathstorecursively}

com-putetheparametersassociatedwithcontinuationvaluestodefine ˜

τ

or V co _{(Matlab R2017a 64-bit, HP Z620 Workstation,and Intel}

[email protected]).

Table 4b .Relativecomputingtimesforlowerandupperbounds in thesetup of Table 1 . Lower V low

0 bounds (LSMand localLSM)

use200,000paths torecursivelycomputethecontinuationvalues todefine

τ

˜andanother2millionpathstocomputetheBermudan price.Upper V ₀upV co_{-basedandUpperV} up

0

τ

-baseduse30,000and

3000outerpaths,respectively.

DSM report (see their Table 2 ) the cost of the upper bound is similar in these two methods, in their pathwise optimization and in the martingale-based continuation values. Pricing Ameri-can options with stochastic parameters, Brown et al. (2010) also

report both martingale-based upper bounds are time

consum-ing compared to upper bounds based on information relaxation.

Hence, our paper provides two extensions of martingale-based

upper bounds. First, when using continuation values, we get a

tighterupperboundby computingthe continuationvalue onlyin thewaitingregion.Second,whenusingstoppingtimes,theupper

bound is both tight and efficient if we use fewer subsimulation

paths,butanear-optimalexercisestrategyasinthelocalLSM ap-proach.Withbothmethods,webringtheoverall costofthe mar-tingaleapproachinlinewithpathwiseoptimizationorinformation relaxation.

6. Concludingremarks

In this paper, we show the exercise strategy that maximizes

theBermudan price/lowerboundalsominimizesthegapbetween

thelowerandthedualupperbound.Weassumebothboundsare

specified recursively, and show the upper bound is independent

ofthe next-stagepolicy.Upperboundsbased onthis optimal re-cursiveexercise policy arevery tight,aswe show forup-and-out

Bermudan max-options, andrequire few nested simulations.

Up-perboundsaretighterbutmoretimeintensivethanlowerbounds. In addition, a better upper bound based on continuation values, whichisnot asaccurate butismore efficientthan onebased on stoppingtimes, requiresreestimating thecontinuation value only

inthewaitingregion.Fromtheseresultsemergesthefactthat al-though a tradeoff between tighter bounds andtime effort isnot straightforward inother methods, thistradeoff exists foroptimal recursivelower/upperbounds.

Securities that provide flexibility of early exercise are

ubiq-uitous in financial markets: from single-name American-equity

options and Bermudan options to enter/cancel an interest-rate

swap, to credit-risk models (Ayadi, Ben-Ameurb, & Fakhfakh, 2016 ). Specifically, for applications of lower/dual bounds in eq-uity models with stochastic volatility, see Ibáñez and Velasco (2016) and Fabozzi, Paletta, and Tunaru (2017) ; for the ap-praisalofBermudanswaptionsprices,seeAndersen and Andreasen (2001) andSvenstrup (2005) ; forterm-structureapplications, see

Joshi and Tang (2014) ; and see Kogan and Mitra (2017) and

Bender, Schweizer, and Zhuo (2017) for extensions to other

eco-nomic problems. Lower/dual bounds applications are also

com-moninoperationsresearch(Trigeorgis & Tsekrekos, 2018 ),suchas whento launch anew productor halt a failingproject, delayed-purchaseoptions(Aydin, Birbil, & Topalo ˘glu, 2017 ),inventory prob-lems(Brown et al., 2010 ),orenergyrealoptions(Nadarajaha, Mar- got, & Secomandi, 2017 ),whichincludesmorereferencestothe lit-erature.