1 Introduction 1.1 The Danish mortgage market The Danish mortgage loan system is among the most complex of its kind intheworld

Multi Stage Stochastic Programming

Kourosh MarjaniRasmussen

∗

and Jens Clausen

†

August 24,2004

Abstract

We consider the dynamics of the Danish mortgage loan system and

proposeseveralmodelstoreectthechoicesofamortgagoraswellas

his attitude towardsrisk. The models are formulated as multi stage

stochasticintegerprograms,whicharediculttosolveformorethan

10 stages. Scenario reduction and LP relaxation are used to obtain

near optimal solutionsfor large problem instances. Our results show

that the standard Danish mortgagor should hold a more diversied

portfolioofmortgageloans,andthatheshouldrebalancetheportfolio

morefrequentlythancurrentpractice.

1 Introduction

1.1 The Danish mortgage market

The Danish mortgage loan system is among the most complex of its kind

intheworld. Purchaseof mostpropertiesinDenmarkis nancedbyissuing

xedrate callable mortgagebondsbasedon an annuity principle.It is also

possibletoraiseloans,whicharenancedthroughissuingnoncallable short

term bullet bonds. Such loans may be renanced at themarket rate on an

ongoingbasis. Theproportion of loansnanced byshortterm bulletbonds

hasbeen increasing since 1996. Furthermore it is allowed to mix loans in a

mortgageloan portfolio, but this choice hasnot yetbecomepopular.

Callable mortgage bonds have a xed coupon throughout the full term of

theloan. Thematurities are10, 15,20 or 30 years.Theborrower hasacall

∗

Informatics and Mathematical Modelleing, Technical University of Denmark, Bldg.

305,DK-2800Lyngby,Denmark.kmr@imm.dtu.dk

†

Informatics and Mathematical Modelleing, Technical University of Denmark, Bldg.

305,DK-2800Lyngby,Denmark.jc@imm.dtu.dk

thelifetimeof theloan. When theinterestrates are lowthecall option can

be usedto obtaina new loanwith lessinterest payment inexchange for an

increaseintheamountofoutstandingdebt.Theborrowerhasalsoadelivery

option.When the interest rates are highthis option can be used to reduce

theamount ofoutstanding debt,inexchangefor paying higherinterest rate

payments. There are both xed and variable transaction costs associated

withexercisinganyof theseoptions.

Noncallable shortterm bullet bonds are used to nance the adjustable

rate loans. The bonds' maturities range from one to eleven years and the

adjustablerate loans are oered as 10,15, 20 or 30year loans. Since 1996

themostpopularadjustablerateloanhasbeentheloannancedbytheone

yearbond.From 2001,however, therehasbeena newtrend, where demand

for loans nanced by bullet bonds with 3 and 5year maturities has risen

substantially.

1.2 The mortgagor's problem

It is known on the investor side of the nancial markets that investment

portfolios shouldconsistofavarietyofinstrumentsinorderto decreaserisk

whilemaintainingprotability. Theportfoliois alsorebalanced regularlyto

take best advantage ofthemoves inthemarket.

Theportfolio diversicationprinciple and rebalancingis, however, notcom-

moninthe borrowerside ofthe mortgage market. Mostmortgagors nance

their loans in one type of bond only.Besides they do not always rebalance

theirloan whengoodopportunitiesfor this have arisen.

There aretwomajor reasonsfor the mortgagors reluctance to better taking

advantage of their options (that they have fully paid for) through the life

timeofthe mortgage loan.

1. The complexity of the mortgage market makes it impossible for the

average mortgagorto analyzeallthealternativesandchoosethebest.

2. Themortgagecompaniesdonotprovideenoughquantitativeadviceto

theindividualmortgagor. Theyonlyprovide generalguidelines,which

are normally not enough to illuminate all dierent options and their

consequences.

Thecomplexityof the mortgageloan systemmakesit a nontrivial taskto

decide on an initial choice of mortgageloan portfolioand on nding a con-

tinuingplanto readjust theportfoliooptimally.There existsasof today no

functionaloptimization modeltoprovidedecisionsupportfortheindividual

mortgagoron hischoice ofloan.

a mortgage loan market such as the Danish one, aswell as the basic ideas

behind themathematical modeling concept ofstochasticprogramming.

TheDanishmortgagor'sproblemhasbeenintroducedbyNielsenandPoulsen

(N&P,[12]).Theyhavedevelopedatwofactortermstructuremodelforgen-

eratingscenariosandontopofithavebuiltamultistagestochasticprogram

to nd optimal loan strategies. The article, however, does not describe the

details necessary to have a functional optimization model, and it does not

dierentiate between dierent types of risks in the mortgage market. The

main contribution of this article is to make Nielsen & Poulsen's model op-

erational byreformulating parts of their model and adding new featuresto

it.

We reformulate the Nielsen & Poulsen model in section 2. In section 3 we

modeldierent options available to the Danish mortgagor, and insection 4

wemodelmortgagor'sriskattitudes.Hereweconsiderbothmarket riskand

wealth risk.

Inthe basic modelweincorporate xedtransaction costsusing binaryvari-

ables. We usea noncombining binomial tree to generate scenarios ina 11

stageproblem. Thisresultsin51175binaryvariables,makingsome versions

of the problem extremely challenging to solve. Dupa

c ˇ

^ová, GröweKuska, Heitsch and Römisch (Scenred, [7 , 8]) have modeled the scenario reduc-

tionproblem asa setcovering problem andsolvedit using several heuristic

algorithms. We review these algorithms in section 5 and use them in our

implementation to reducethesizeof the problemandherebyreducetheso-

lutiontimes. Anotherapproachtogettingshortersolutiontimesisproposed

insection6,where we solveanLPapproximated versionoftheproblem.In

section7wediscuss andcommenton ournumericalresults andweconclude

thearticle withsuggestions forfurther research insection8.We useGAMS

(General Algebraic Modeling Language) tomodeltheproblemand CPLEX

9.0astheunderlying MPandMIPsolver.Forscenario reductionwe usethe

GAMS/SCENRED module(scenredmanual,[9 ]).

TheobtainedresultsshowthattheaverageDanishmortgagor wouldbenet

fromhisoptionofmixingloansinhisloanportfolio.Likewiseheshouldread-

justtheportfoliomoreoftenthanisthecasetoday.Thedevelopedmodeland

softwarecanalsobeusedtodevelopnewloanportfolioproducts.Suchprod-

uctswillconsidertheindividualcustomerinputssuchasbudgetconstraints,

riskprole,expectedlifetimeof theloan, etc.

Even thoughwe considertheDanish mortgageloanmarked, theproblemis

universalandthepractitionersinanymortgageloansystemshouldbeableto

usethemodels developed inthisarticle,possiblywithminor modications.

In this section we develop a riskneutral optimization model which nds a

mortgageloanportfoliowiththeminimumexpectedtotalpayment.Consider

thescenariotree ingure(1). We assumethat we have sucha treeat hand

withinformation on price andcoupon ratefor all mortgage bondsavailable

at each node together with the realization probability of a given node at a

given time.

n=2

n=4

n=5 t = 1

t = 0 t = 2

n=3 n=1

n=6

n=7

t = 3 n=8

n=9

n=10

n=11

n=12

n=13

n=14

n=15 4:C32−04/98.3

1:C29−05/105.4

1:C28−05/93.7 2:C31−06/98.4 1:C28−05/93.7 3:C31−06/98.4

1:C28−05/96.9 3:C31−06/100.7

1:C28−05/93.7 3:C31−06/98.4

1:C28−05/96.9 3:C31−06/100.7

1:C28−05/96.9

1:C28−05/108.4 2:C31−06/92.5 1:C28−05/84.4

4:C31−04/101.4 4:C31−04/94.2 1:C31−05/96.8

1:C30−05/92.35

1:C30−05/101.8

2:C32−06/95.3 1:C29−05/88.8

3:C32−06/98.7 1:C29−05/95.4

3:C32−06/98.7 1:C29−05/95.4

Figure 1: Abinomial scenario tree,representing ourexpectation offuturebond prices

andcouponrates.Allbondsarecallablexedratebonds.

In the basic model we only consider xedrate loans, i.e. loans where the

interest rate does not change during the lifetime of the loan. For the sake

ofdemonstration we consider anexample with4 stages,

t ∈ { 0, 1, 2, 3 }

^,and

15 decisionnodes,

n ∈ { 1, · · · , 15 }

^{,with the}probability

p _n

^{for being at the}

node

n

^{.Inorderto make our model}independent of theunderlying scenario structure we capture the dependency between any time step and its nodes

inthe set

tn(t, n)

^,where

n ∈ { 2 ^t , · · · , 2 ^t+1 − 1 }

^forall

t

^{.Similarlywe dene}

aset

tree(n ⁰ , n)

^{to dene a}parent-child relationship.

1

We want the basic model to be able to nd an optimal portfolio of bonds

1

Most modelinglanguages facilitate aneasy way to declare suchsets. In GAMSfor

exampleonecandenetwosets

t

^and

n

^{andthendeclare}

tn(t, n)

^and

tree(n, n)

^as:

settn(t,n)timenodemapping/t0.(n1),t1.(n2*n3),t2.(n4*n7)/,

tree(n,n)parentchildmapping/n1.(n2*n3),n2.(n4*n5),n3.(n6*n7)/;

the4 bondsshown ingure (1). Each bond isrepresented as (Index:Type

Coupon/Price), so(3:C3206/98.7) isa xedratecallablebond withmatu-

rityafter 32 periods,a coupon rate of6% and a price of 98.7. Theletter

C

indicates that thebond is callable,so theprepayment price is never higher

than par.

To generate bondsinformation we can useterm structure and bond pricing

theories (see [10, 11, 4] for an introduction to these topics and further ref-

erences to articles.). It is also possible to use expert knowledge to predict

possible bondpricesinthefuture. A combination oftheoreticalpricing and

expertinformation can also be usedtogenerate suchscenariotrees. Nielsen

and Poulsen (N&P,[12])have proposedone way togenerate ascenario tree

for the Danishmortgagorproblem.

Given that we have a scenario tree with

N

^{stages at hand we can now}

formulate theoptimization problem.We rstconsidertheparameters ofthe

optimization model:

p _n

^:Theprobabilityof being atnode

n, ∀ n ∈ { 1, · · · , 2 ^N − 1 }

^.

τ _n

^{: Discount factorat node}

n

^.

IA

^{:Theinitial amount of loanneededbythemortgagor.}

r _in

^{:Coupon rate forbond}

i

^atnode

n

^.

k _in

^{:Price ofbond}

i

^{at node}

n

^.

Callk _in

^{:Priceofacallablebond}

i

^atnode

n

^.Wehave

Callk _in = min { 1, k _in }

for callablebondsand

Callk _in = k _in

^fornoncallable bonds.

γ

^{:Tax reductionratefrom interestrate payment.}

β

^{:Tax reductionratefrom} administrationfees.

b

^: Administration fee rates given as a percentage of outstanding debt for bond

i

^{at node}

n

^.

η

^:Transaction fee ratesfor sale and purchaseof bonds.

m

^{:Fixed costsassociated withrenancing.}

Next we dene the variables usedinour model:

B _tn

^{:Total netpayment at node}

n

^,time

t

^.

X _itn

^:Outstanding debtofbond

i

^{at node}

n

^,time

t

^.

S _itn

^{:Unitssold ofbond}

i

^{at scenario}

n

^,time

t

^.

P _itn

^{:Unitspurchasedof bond}

i

^{at node}

n

^,time

t

^.

A _itn

^{: Principalpayment ofbond}

i

^atnode

n

^,time

t

^.

L _itn :

1

^{ifthereare anyxed costsassociated withbond}

i,

^node

n

^,time

t.

0

^otherwise.

min X

tn(t,n)

p _n · τ _n · B _tn

⁽¹⁾

X

i

k _i1 · S _i01 ≥ IA

⁽²⁾

X _i01 = S _i01 ∀ i

⁽³⁾

X

tree(n ⁰ ,n)

X _i,t−1,n 0 − A _i,t−1,n 0 − P _itn + S _itn − X _itn

= 0 ∀ i, tn(t \ 0, n \ 1)

⁽⁴⁾

X

i

(k _in · S _itn ) = X

i

(Callk _in · P _itn ) ∀ tn(t \ 0, n \ 1)

⁽⁵⁾

A _itn = X _itn h r _in

1 − (1 + r _in ) ^{−N +t} − r _in i

∀ i, tn(t, n)

⁽⁶⁾

B _tn = X

i

A _itn + r _in · (1 − γ)X _itn +

b · (1 − β)X _itn + η · (S _itn + P _itn ) + m · L _itn

∀ tn(t, n)

⁽⁷⁾

BigM · L _itn − S _itn ≥ 0 ∀ i, tn(t, n)

⁽⁸⁾

X _itn , S _itn , P _itn ≥ 0 , L _itn ∈ { 0, 1 } ∀ i, tn(t, n)

⁽⁹⁾

Theobjective isto minimize the weighted averagepayment throughout the

mortgageperiod.Thepayment ateachnodeisdenedinequation(7)asthe

sum of principal payments, tax reduced interest payments, taxed reduced

administration fees, transaction fees for sale and purchase of bonds and -

nallyxedcostsforestablishingnewmortgageloans.Theprincipalpayment

isdened inequation(6) asanannuitypayment.

The dynamics of the model are formulated in constraints (2) to (5). Con-

straint (2)makessurethatwe sellenough bondsto raiseaninitial amount,

IA

^{, neededbythe mortgagor. In equation (3)we initialize the}outstanding debt. Equation (4) is the balance equation, where the outstanding debt at

anychildnodeforanybondequalstheoutstandingdebtattheparent node

minus principal payment and possible prepayment (purchased bonds), plus

possiblesoldbondstoestablishanewloan.Noticethatthemodelisindepen-

dent oftheunderlyingscenariotree.Useoftheset

tree(n ⁰ , n)

^inthebalance

equation(4)guaranteesthatanytreestructure,denedintheset

tree(n ⁰ , n)

canbeusedtoformthebalanceequations. Thisformulationofthemodelis

very usefulwhen we want to reduce the number of scenarios intheoriginal

scenariotree.Equation(5)isacashowequationwhichguaranteesthatthe

money usedto prepaycomes fromthesale of newbonds.

Finallyconstraint(8)addsthexedcoststothenodepayment,ifweperform

anyreadjustingofthe mortgageportfolio.The

BigM

^{constant might be set}

toavalueslightly greaterthan theinitial amountraised.Ifatoolargevalue

isused, numerical problems mayarise.

Themodelof section2hasthree implicitassumptionswhich limit itsappli-

cability:

1. Weassumethataloanportfolioisheldbythemortgagoruntiltheend

ofhorizon.

2. We assumethatall bondsarexedrateand callable,i.e. theycan be

prepaidat anytime at a priceno higher than 100.

3. Themortgagor isassumedto be riskneutral.

We will relaxthe rst two assumptions in this section and the third inthe

following section.

Therstassumptioncanbeeasilyrelaxedbyintroducingaconstant

T

^asthe

number of decision stages until the prepayment of the loan portfolio, such

that

T ≤ N

^,where

N

^{is the horizon.The decision nodesrepresent only the}

rst

T

^{stages, while the horizon remains as}

N

^{.It means that the principal}

and interest payments are calculated with the horizon

N

^{, whereas we only}

considerthe rst

T

^{stagesintheproblem.}

Thesechangesmean thatthe outstandingdebtat stage

T

^{is now apositive}

amount. We dene thisprepayment amount (

P P _{T n}

^{) as:}

P P _{T n} = X

i

(X _{iT n} − A _{iT n} ) ∀ n ∈ { 2 ^{T −1} , · · · , 2 ^T − 1 }

⁽¹⁰⁾

We add this equation to the model and we update the object function as

follows:

min X

tn(t,n)

p _n · τ _n · B _tn +

tn(t=T,n=2 X ^T −1) tn(t=T,n=2 ^(T−1) )

p _n · τ _n · P P _tn .

⁽¹¹⁾

Withthis objectfunction we are now minimizing theweightedpayment at

allnodesplus the weightedaverage amount of prepayment attime

T

^.

The problem with the second assumption is more subtle. Consider thesce-

nariotreeat gure(2),where twoadjustablerateloans have been addedto

oursetofloansattime0.Loan5(F1)isanadjustablerateloanwithannual

renancingandloan6(F2)isanadjustablerateloanwithrenancingevery

secondyear

For adjustablerate loans (Fmloans) the underlying

m

^{year bond is com-}

pletelyrenancedevery

m

^{yearsbysellinganother}

m

^{yearbond.Butunlike}

normalrenancingthis kindofrenancing doesnot incuranyextraxedor

n=2

n=4

n=5 t = 1

t = 0 t = 2

n=3 n=1

n=6

n=7

t = 3 n=8

n=9

n=10

n=11

n=12

n=13 n=14

n=15 1:C30−05/92.35

5:F1−06/101.2 6:F1−04/95.8

1:C31−05/96.8 5:F1−04/99.1 6:F2−04/98.2

1:C30−05/101.8 5:F1−02/99.1 6:F1−04/104.7

1:C29−05/88.8 2:C32−06/95.3 5:F1−08/102

6:F2−08/100.6

1:C29−05/95.4 5:F1−04/99.2

1:C29−05/95.4 3:C32−06/98.7

3:C32−06/98.7 5:F1−04/99.2

6:F2−05/100.1

6:F2−05/100.1

5:F1−01/99.2 6:F2−03/101.2 4:C32−04/98.3 1:C29−05/105.4

1:C28−05/93.7 2:C31−06/98.4

5:F1−07/99.6 6:F1−08/103.2 1:C28−05/93.7

3:C31−06/98.4 5:F1−07/99.6

1:C28−05/96.9 3:C31−06/100.7

5:F1−03/99.9 6:F1−05/105.4 1:C28−05/93.7

3:C31−06/98.4

5:F1−07/99.6

1:C28−05/96.9 3:C31−06/100.7

5:F1−03/99.9 6:F1−05/105.4 1:C28−05/96.9 5:F1−03/99.9

6:F1−03/98.6

1:C28−05/108.4 5:F1−01/102.4 6:F1−03/108.4 5:F1−10/101.5 6:F1−08/97.8 2:C31−06/92.5 1:C28−05/84.4

4:C31−04/101.4 6:F1−05/102.9

6:F1−05/102.9

4:C31−04/94.2

Figure2:Abinomialscenario treewherebothxedrateandadjustablerate loansare

considered.

variable transaction costs. We modelthatbyusing thesame loan index for

anadjustablerateloanthroughout themortgageperiod.Forexampleindex

5isusedfor theloanwithannualrenancing,eventhough theactualbonds

behind the loanchange every year. Since we usethesame index, themodel

doesnot register anyactual sale or purchase ofbondswhen renancing oc-

curs.Weshould,however,readjusttheoutstandingdebtgiventhatthebond

price isnormally dierent from par.To take this into account we introduce

theset

i ⁰

^of noncallable adjustablerate loans. For these loans we use the following balance equationinstead ofequation (4).

X

tree(n ⁰ ,n)

X _i 0 ,t−1,n ⁰ − A _i 0 ,t−1,n ⁰ − P _i 0 tn + S _i 0 tn − k _i 0 n · X _i 0 tn

= 0 ∀ i ⁰ , tn(t \ 0, n \ 1).

(12)

Notethat variables

P _i 0 tn

^and

S _i 0 tn

^{remain0 aslongwekeep an adjustable}

rateloan

i ⁰

^{inourmortgageportfolio.The}outstandingdebtinthechildnode ishoweverrebalanced bymultiplying the bondprice.

Whenweconsiderthe adjustablerateloanswe shouldrememberthatthese

loansarenoncallable,soforprepayment purposeswehave:

Callk _i 0 n = k _i 0 n

^.

Anotherissuetobedealtwithisthatifabondisnotavailableforestablishing

a loan at a given node, we have to set the corresponding value of

k _in

^{to 0}

to make sure thatthe bond isnot sold at thatnodeinan optimal solution.

prepayment.

4 Modeling risk

Sofar wehaveonly consideredariskneutral mortgagorwhoisinterestedin

theminimumweightedaverageoftotalcosts.Mostmortgagorshoweverhave

anaversiontowardsrisk.There aretwokindsofriskwhichmostmortgagors

areawareof:

1. Marketrisk:Inthemortgage marketthis istheriskofextrainterest

ratepaymentfor amortgagorwhoholdsanadjustablerateloanwhen

interestrateincreases,ortheriskofextraprepayment foramortgagor

withanykindofmortgageloanwhentheinterestratedecreasessothe

bond price increases.

2. Wealth risk: In the mortgage market this is a potential risk which

can be realized if themortgagor needs to prepay themortgagebefore

a planned date or ifhe needs to use thefree value of thepropertyto

take another loan. It can be measured asa deviation froman average

outstandingdebt atanygiven time duringthelife timeoftheloan.

We will in the following model both kinds of risk. To that end we use the

ideas behind minmax optimization and utility theory with use of budget

constraints.

4.1 The minmax criterion

Anextremely riskaversemortgagorwantsto payleastintheworstpossible

scenario.Inotherwordsifwedene the maximumpayment as

MP

^thenwe

havethefollowing minmax criterion:

min MP,

⁽¹³⁾

MP ≥ X

tn(t,n∈NP f )

B _tn ∀ f ∈ { 1, · · · , 2 ^{T −1} } ,

⁽¹⁴⁾

where

NP _f

^{is a set of scenarios with}

f ∈ { 1, · · · , 2 ^{T −1} }

^{. Each scenario is}

describedastheunique pathofnodesfromthe rootofthetreeto oneofthe

NP ₁ = { 1, 2, 4, 8 } NP ₂ = { 1, 2, 4, 9 }

· · ·

NP ₈ = { 1, 3, 7, 15 }

4.2 Utilityfunction

Insteadofminimizingcostswecandeneautilityfunction,whichrepresents

asaving andmaximizeit.NielsenandPoulsen (N&P[12 ])suggestaconcave

utilityfunction withthesame formasinsketch (3).

Utility

Saving

Figure 3: A concave utility function. An increase of an already big saving is not as

interestingasanincreaseofasmallersaving.

The decreasing interest for bigger savings is based on the idea that bigger

savingsaretypically riskierthansmall savings.Nielsenand Poulsen suggest

alogarithmic objectfunction, whichcan beformulatedasfollows:

max X

tn(t,n)

p _n · log(τ _n · (B _tn ^max − B _tn )),

⁽¹⁵⁾

where

B _tn ^max

^{is themaximumamount a mortgagor iswillingto pay.Nielsen}

and Poulsen x

B _tn ^max

^{to a big value sothat the actual payment will never}

riseabove thislevel.

Adding this nonlinear object function to our stochastic binary problem

makes the problem extremely challenging to solve. There are no eective

general purposesolvers for solving large mixedinteger nonlinear programs

(seeBussieckandPruessner, [6]).Therearethreewaysofcircumventingthe

problem: Either we use a linearutilityfunction inconjunction with budget

lem (nlp) or both (lp). We demonstrate the rst approach in the following

andcomment onthe second andthird approach insection6.

Instead of maximizing the logarithm of the saving at each node we can

simplymaximize thesaving:

B _tn ^max − B _tn

^.If

B _tn ^max

^{issolargethatthesaving}

is always positive, then we are in eect minimizing the weighted average

costs similar to the risk neutral case presented in section 2. However ifwe

allowthesaving to benegative at timesand add a penaltyto theobjective

function whenever we get a negative saving,we can introduce riskaversion

into the model. For this reason we need to have a goodestimate for

B _tn ^max

^.

The risk neutral model can be solved to give us these estimates. Then we

can usethefollowing objective functionand budgetconstraints.

max X

tn

p _n · τ _n (B _tn ^max − B _tn ) − P R _tn · BO _tn

(16)

B _tn ^max + BO _tn − B _tn ≥ 0 ∀ tn(t, n)

⁽¹⁷⁾

BO _tn ≤ BO ^max _tn ∀ tn(t, n).

⁽¹⁸⁾

We allow crossing the budget limit in constraint (17) by introducing the

slack variable

BO _tn

^{. This value will then be penalized by a given penalty}

rate(

P R _tn

^{)intheobjectivefunction(16).Thepenaltyratecanfor example}

be ahighone timeinterestratefortakingabankloan.Thebudgetoverow

(

BO _tn

^{)isthencontrolledinconstraint(18)wheretheoverowisnotallowed}

to be greaterthan a maximum amount

BO ^max _tn

^.

4.3 Wealth risk aversion

So far we have only considered the marketrisk or theinterest rate risk. In

thefollowing we willmodel theotherimportant riskfactor inthemortgage

market, namelythewealth risk.

Wealthriskistheriskthattheactualoutstandingdebtbecomesbiggerthan

theexpectedoutstandingdebtatagiventimeduringthelifetimeoftheloan.

Forexamplesellinga30yearbondatapriceof80,wehaveabigwealthrisk

given thata smallfall inthe interestrate can causea considerable increase

inthebondprice,which meansaconsiderableincreaseintheamountofthe

outstandingdebt.

We consider the deviation from the average outstanding debt, which we

dene as

DX _tn

^:

DX _tn = X _t − X

i

X _itn , ∀ tn(t, n),

where

X _t

^{is the average}outstanding debtfor agiven time

t

^:

X _t = X

i,tn(t,n)

p _n · X _itn , ∀ t.

Apositivevalueof

DX _tn

^{means thatwe have asavinganda negative value}

meansalossascomparedtotheaverageoutstandingdebt

X _t

^{.Weintroducea}

surplusvariable

XS _tn

^{torepresent theamountofsavingandaslackvariable}

XL _tn

^{to represent theamount of loss:}

X _t − X

i

X _itn

− XS _tn + XL _tn = 0 ∀ tn(t, n),

⁽¹⁹⁾

To make themodel both market riskand wealth risk averse we update the

objective functionwith weightedvalues of

XS _tn

^and

XL _tn

^asfollows:

max X

tn(t,n)

p _n · τ _n

(B _tn ^max − B _tn ) − P R _tn · BO _tn + P W _n · XS _tn − NW _n · XL _tn

,

(20)

where

P W _n

^{isshortforpositiveweightandcanbeusedtoencouragesavings}

and

NW _n

^{,shortfor negative weight,isthere topenalizealossascompared}

to the average outstandingdebt. If we set

P W _n = NW _n

^{,itmeans that the}

model isindierent towards wealth risk. Onthe other hand

P W _n < NW _n

^,

meansthatthe modeliswealthriskaverse,sinceitpenalizesapotentialloss

harderthan itencourages apotential saving.

5 Scenario reduction

Sincethenumberofscenariosgrowsexponentiallyasafunctionoftimesteps

the stochasticbinarymodelisno longer tractable whenwe have more than

10timesteps.Foran11stage modelwehavethescenariotreeingure(4).

Asof todaythere are nogeneral purpose solvers which can solve stochastic

integerproblemsofthissize inareasonableamount oftime.Noticehowever

that a great number of nodes in the last 3-4 time steps have such a close

distance that a reduction of nodes for these time steps might not eect

the rststage results. We arein otherwordsinterested innding a wayto

optimallyreducethenumberofscenarios.Ifwegetthesamerststageresult

for a reduced and a nonreduced problem, it suces to solve the reduced

problem, and then at each step resolve the problem until horizon. In that

casethe nalresultofsolvinganyofthetwoproblemswillbethesame.The

reasonforthisisthatweinitiallyonlyimplementtherststagesolution.As

the time passesby and we get more information we have to solve the new

problemandimplement the newrst stagesolution eachtime.

0 1 2 3 4 5 6 7 8 9 10 100

200 300 400 500 600 700 800 900 1000

Time

Scenarios

Figure4:Abinomialscenariotreewith11stages.

NicoleGröweKuska, Holger Heitsch,Jitka Dupa

ˇ c

^{ová and Werner Römisch}

(see [7 , 8 ]) have dened the scenario reduction problem (SRP) asa special

setcovering problemand have solvedit usingheuristic algorithms.

TheauthorsbehindtheSCENREDarticleshaveincooperationwithGAMS

SoftwareGmbHand GAMSDevelopmentCorporation,developedanum-

ber ofC++ routines, SCENRED, for optimal scenario reduction ina given

scenariotree.Likewisetheyhavedevelopedalink,GAMS/SCENRED,which

connects the GAMS program to the SCENRED module (see [9]). The sce-

nario tree in gure 5 is obtained after using the fast backward algorithm

of the GAMS SCENRED module for a 50% relative reduction, where the

relative reductionis measured asan average of nodereductions for all time

step. If we for example remove half of the nodes at the last time step, we

geta50%reductionfor thattimesteponly.Thenwemeasurethereduction

percentages for all other time steps in the same way. The average of these

percentagescorrespondstothe relativereduction(see[7 ,8]).Inour example

thenumberof scenarios isreducedfrom 1024 to 12.

We useGAMS/SCENRED and SCENRED modules for scenarioreduction,

and compare the results with those found by solving the LPrelaxed non

reducedproblem.

6 LP relaxation

Wheneverwe renancethe mortageportfolioweneed to paya variableand

a xed transaction cost. The variable cost is

100 · η

^{percent of the sum of}

0 1 2 3 4 5 6 7 8 9 10 100

200 300 400 500 600 700 800 900 1000

Time

Scenarios

Figure5: Abinomialscenario treewith11stages aftera50% scenarioreductionusing

thefastbackwardalgorithmoftheSCENREDmoduleinGAMS.

thesold and purchased amount of bondsand thexed cost issimply DKK

m

^{(see constraint 7 and 8). The binary variables in the problem (1 to 9)}

aredue to incorporation of xed costs

m

^{.The numeric value of these xed}

costsisaboutDKK2500whereas

η = 0.15%

^{.Whilethevalueofthevariable}

transaction costs decreases as the time passes by, the xed costs remain

the same. Therefore, if we simply drop the xed costs or just add a small

percentageto thevariabletransaction costs,wenormallygetadierent rst

stage solution than what we get if we solve the problem with the actual

xed transaction costs. We therefore suggest an iterative updating scheme

forthevariabletransactioncosts,sothatwecanapproximate thexedcosts

without using binary variables. We do that by iteratively solving the LP

problem

k

^{times asfollows.}

We dene a ratio

ψ ^k _itn

^{and initialize it to}

ψ _itn ⁰ = 0

^{. Theratio}

ψ _itn ^k

^{can then}

be used inthe denition of a node payment (7) in the

k + 1

^{st iteration as}

follows:

B _tn = X

i

A _itn + r _in · (1 − γ )X _itn +

b · (1 − β)X _itn + η · (S _itn + P _itn ) + ψ _itn ^k+1 · S _itn

∀ tn(t, n)

⁽²¹⁾

SolvingtheLPproblemat eachiteration

k

^weget

S _itn ^∗k

^{astheoptimal value}

ofthe soldbondsatthe

k

^{th iteration.Before each iteration}

k > 0

^,theratio

ψ _itn ^k

^{isthen updatedaccordingto thefollowing rule:}

ψ _itn ^k = ( _m

S ^∗k _itn ∀ i, tn(t, n)

^if

S _itn ^∗k > 0,

ψ ^k−1 _itn

^{otherwise. (22)}

This brings us to our approximation scheme for an LP relaxation of the

problem:

1. Dropthe xed costsand solve the LP relaxedproblem.

2. Find theratios

ψ _itn

^{according to(22).}

3. Incorporate the ratios in the model so that DKK

m

^{is added to the}

objectivefunctionforeachpurchasedbond,giventhesamesolutionas

theone inthe last iterationis obtained.Solve theproblemagain.

4. Stop if the solution in iteration

k + 1

^{has not changed more than}

α

percent as compared to the solution in iteration

k

^{. Otherwise go to}

step5.

5. Update

ψ _itn

^{accordingto (22).}

6. Repeatfromstep 3.

Ourexperimentalresultsshowthatfor

α ' 2%

^{wendnearoptimalsolutions}

whichhavesimilarcharacteristicstothe solutionsfromtheoriginalproblem

withthexedcosts.

7 Numerical results

We consider an 11 stage problem, starting with 3 callable bonds and 1 1

year bullet bond at the rst stage. We then introduce 7 new bonds every

3 years. An initial portfolio of loans has to be chosen at year 0 and it may

be rebalanced oncea year the next 10 years. We assume thatthe loan is a

30year loanandthat itisprepaid fullyat year11.

The 24 callable bonds used in our test problem are seen in table 1. The

tableonlypresentsthe average couponrateand priceforthese bondsatthe

date of issue. Besides these 24 callable bonds we use a 1year noncallable

bullet bond, bond 25. The eective interest rate on this bond is about 2%

to start with.Using a BDTtree (see [5, 3 ]) withthe inputgiven in table 2

theeectiveratecanincreaseto21%ordecreasetoslightlyunder1%atthe

10th year. The BDT tree has also been used for estimating the prices and

rates ofthe 24 callable bondsduring thelife timeofthemortgage loan.

A practical problem arises when writing the GAMS tables containing the

stochastic data. The optimization problem is a path dependent problem,

1 6% 103.06 3/10-02 3/10-35

2 5% 98.5 3/10-02 3/10-35

3 4% 89.4 3/10-02 3/10-35

4 9% 107.33 3/10-05 3/10-38

5 8% 103.16 3/10-05 3/10-38

6 7% 103.09 3/10-05 3/10-38

7 6% 100.51 3/10-05 3/10-38

8 5% 94.01 3/10-05 3/10-38

9 4% 84.55 3/10-05 3/10-38

10 3% 74.46 3/10-05 3/10-38

11 9% 105.4 3/10-08 3/10-41

12 8% 101.98 3/10-08 3/10-41

13 7% 100.3 3/10-08 3/10-41

14 6% 96.19 3/10-08 3/10-41

15 5% 89.5 3/10-08 3/10-41

16 4% 80.74 3/10-08 3/10-41

17 3% 71.32 3/10-08 3/10-41

18 9% 104.41 3/10-11 3/10-44

19 8% 100.9 3/10-11 3/10-44

20 7% 98.51 3/10-11 3/10-44

21 6% 94.07 3/10-11 3/10-44

22 5% 87.49 3/10-11 3/10-44

23 4% 79.25 3/10-11 3/10-44

24 3% 70.26 3/10-11 3/10-44

Table1:Thecallablebondsusedasinputtotheproblem.

whereastheBDTtreeispathindependent.GAMSisnotwellsuitedforsuch

programmingtasksasmapping thedata from acombining binomial tree(a

lattice) to a noncombining binomial tree.A general purpose programming

language is better suited for this task. We have used VBA to generate the

input data to the GAMS model, and we have run the GAMS models on a

sunsolaris machine.

Thepurposeof our testscan be summarizedasthefollowing:

1. Validatingthe4versionsofour model,i.e.weinvestigate ifthemodels

represent the actual dynamics oftheDanishmortgagemarket.

2. Observing theeectsof usingthe GAMS/SCENRED module.

3. Tryingour LP approximationon theproblem.

For each of these objectives we consider all four versions of themodel and

compare theresults.

(Year) (%) (%) (Year) (%) (%)

1 2.23% 16 4.87% 17.25%

2 2.35% 32.20% 17 4.93% 17.00%

3 2.73% 32.10% 18 4.99% 16.85%

4 3.08% 29.50% 19 5.05% 16.75%

5 3.41% 27.00% 20 5.11% 16.70%

6 3.68% 25.00% 21 5.16% 16.65%

7 3.92% 23.00% 22 5.21% 16.60%

8 4.12% 22.00% 23 5.25% 16.56%

9 4.30% 20.90% 24 5.29% 16.52%

10 4.44% 20.10% 25 5.34% 16.48%

11 4.56% 19.40% 26 5.37% 16.45%

12 4.62% 18.80% 27 5.40% 16.42%

13 4.68% 18.30% 28 5.43% 16.39%

14 4.74% 17.90% 29 5.46% 16.36%

15 4.80% 17.55% 30 5.49% 16.34%

Table2: TheinputtermstructuretotheBDTmodel.

7.1 The original stochastic MIP problem

Figure (6) shows the solutions found for the rst three stages of the prob-

lem for all four instances of our model, namely the risk neutral model, the

minimaxmodel, themodelwithinterestrate riskaversionwithbudgetcon-

straints and nally the model with interest rate and wealth risk aversion

withbudgetconstraints.Notice, however, thatno feasiblesolution couldbe

foundfor themodelwithinterestrate andwealth riskaversionwithbudget

constraints withina timelimit of10 hours.

A full prescription of the solution with all 11 stages will not contribute to

a better understanding of the dynamics of the solution, which is why we

presentthesolutiontotherstthreestagesoftheproblemonly.Itisthough

enoughto giveus anindicationofthebehaviourofeachsolution. Intherisk

neutral casewestart bytakinga1yearadjustablerateloan.Iftheinterest

rate increases after a year, the adjustablerate loan is prepaid by taking a

xedrateloan.Evenifitmeansanincreaseintheamountoftheoutstanding

debt, it proves to be a protable strategy since if the rates increase again

in the next stage we can reduce the amount of outstanding debt greatly

by renancing theloan to another xedrate loan with a higher price. The

minmax strategychoosesa xedrateloanwitha priceclose to par tostart

with. Thisloanisnot renanced until the 9th stage of theproblem.

The risk neutral and the minimax model represent the two extreme mort-

gagors as far as the risk attitude is concerned. The third model reects a

0 1 2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000 s3=1128 p25=975

s1=916 p3=1107

Time (Year)

Scenarios

Risk neutral model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s2=1018

Time (Year)

Scenarios

Mimimax model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s2=1018

s3=1040 p2=1003

p2=987 s1=880

s25=952 p3=968

Time (Year)

Scenarios

Interest risk averse model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

NO SOLUTION WITHIN 10 HOURS

Time (Year)

Scenarios

Interest and wealth risk averse model

Figure6: Presentationofthesolutions forthe rst3stages oftheproblem.Variable

s

isforsale and

p

^{forpurchaseandtheunitsare givenin1000DKK,so}

s3 = 1128

^means

thatthemortgagorshouldsellapproximately1.128.000DKKatthegivennode.Theshort

ratesfromtheBDTtreeareindicatedusingtheletter

r

^.

mortgagor witha riskattitude between therst two mortgagors. The solu-

tion to this model guarantees that the mortgagor will not pay more than

whathisbudgetallows atanygivennode.Table (3)indicatesthedierence

inthe characteristics ofthe solutions forthe threedierent models.

The risk neutral model gives the lowest average total cost. The standard

deviationfromthisaveragecostis,however,ratherhigh.Theminimaxmodel

hasamuch smallerstandard deviation.Thishigher levelofsecurityagainst

variationhasthoughanaveragecostofabout72000 DKK.Thethird model

hasreducedtheriskconsiderablywithouthavingincreasedthetotalaverage

costwithmore than about 7000 DKK.

For the sake of budget constraints in model 3 and 4 we use the following

constants:

SCALAR BMAX /570842/; // An average level of payment for a scenario

1-Riskneutral 1.281.857 92.289 1.502.042 1.004.583 276s

2-Minimax 1.353.713 19.729 1.374.183 1.117.084 10h

3-Int.rateriskaverse 1.288.405 66.019 1.431.857 1.005.412 10h

4-Int./Wealthriskaverse Nosolutionfoundwithin 10h.

Table 3:Comparisonofthefourstrategiesfortheoriginalproblem.

SCALAR IBMAX /711015/; // An average level of prepayment at year 11

SCALAR BOMAX /50000/; // Maximum extra payment for a scenario

SCALAR IBOMAX /100000/; // Maximum extra prepayment at year 11

Theseconstants arechosenafter considering the average paymentsand the

standard deviationsfrom theseinthe risk neutral model.

Themajor problemwiththesesolutions isthecomputingtimetakento nd

nearoptimalsolutionsbyCPLEX.Exceptfortherstmodel,wecannotnd

solutions within 1% of a lower bound after 10 hours of CPU time. For the

fourthmodelnofeasible solutionisfound atall.Inthefollowing wewill see

theresults foundfor the reduced problem.

7.2 The reduced stochastic MIP problem

Afterreducing thenumberofscenarios from1024 to12wegetthesolutions

ingure(7).

Exceptforthe riskneutral modelwedonotobtainthesamerststagesolu-

tions aswesawfor theoriginalproblem. It seemsthatthereducedproblem

hasamore optimistic viewofthefutureascompared to theoriginalmodel.

Thiscanalso beseenintable (4)where the totalcostsareconsiderablyless

than the onesfor theoriginal problem.

Modeltype Totalcosts Std.dev. max min time

1-Riskneutral 1.169.173 49.765 1.274.079 1.064.525 12s

2-Minimax 1.187.938 0.00 1.187.938 1.187.938 52.2s

3-Int.rateriskaverse 1.171.926 24.270 1.229.897 1.136.655 300s

4-Int./Wealthriskaverse 1.172.479 25.610 1.229.742 1.128.412 300s

Table4:Comparisonofthefourstrategiesforthereducedproblem.

By testing the scenario reduction algorithms for dierent levels of reduc-

tion onour problem we notice that thereduced problemhasalmost always

an overweight ofscenarios with lowerinterest rates.Thisexplains the more

optimistic view of the future given by scenario reduction. Even much less

aggressive scenarioreductions donotguarantee usthesame initialsolutions

as found for the original problem. What we need is therefore a method to

optimally reduce the number of scenarios while the tree remains balanced.

0 1 2 r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000

s3=1011 p25=975

s1=1009 p25=953

s3=1049 p25=953 s25=900 p3=992

Time (Year)

Scenarios

Risk neutral model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s3=144 s25=868

s2=264 p25=245

s3=877 p25=846

s6=340 p2=259,p3=138

s3=924 p25=259 s25=907 p3=999

s25=395 p3=402

Time (Year)

Scenarios

Mimimax model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000

s3=1011 p25=975

s1=380 p25=358

s3=1049 p25=953 s25=900 p3=992

Time (Year)

Scenarios

Interest risk averse model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000

s3=1011 p25=975

s1=378 p25=356

s3=1049 p25=953 s25=900 p3=992

Time (Year)

Scenarios

Interest and wealth risk averse model

Figure 7: Presentation of the solutions for the rst 3 stages of the reduced problem.

Unitsaregivenin1000DKK.

For the purposes of this article however, we continue withthe reducedtree

found by GAMS/SCENRED. The reason for this is our focus on the op-

timization part of the problem in this article. We suggest that the market

practitioners experiment with dierent reduced scenario trees ranging from

verypessimisticto optimisticonesto ndrobust solutionsfor eachriskcat-

egory ofmortgagors.

We can seein table(4) thatthe behaviourof thesolutions for thedierent

models is similar to that of theoriginal problem. Notice also thatwe get a

feasiblesolutionhereforthefourthmodelwithinterestrateandwealth risk

aversion.

For the sake of budget constraints in model 3 and 4 we use the following

constants:

SCALAR BMAX /565915/; // An average level of payment for a scenario

SCALAR IBMAX /601983/; // An average level of prepayment at year 11

SCALAR BOMAX /50000/; // Maximum extra payment for a scenario

7.3 The reduced and LPapproximated problem

When we use our LPapproximation algorithm on this problem we getthe

solution aspresentedintable (5)and gure(8).

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000

s3=1011 p25=975

s1=1009 p25=953

s3=1049 p25=953 s25=900 p3=992

Time (Year)

Scenarios

Risk neutral model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s2=174 s25=829

s3=203 p25=175

s3=102 p25=808 p2=171

s25=159 p2=169,p3=22

s3=681 p25=618 s25=905 p3=997

Time (Year)

Scenarios

Mimimax model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000

s3=1011 p25=975

s2=441 p25=372

s3=1049 p25=953 s25=900 p3=992

Time (Year)

Scenarios

Interest risk averse model

0 1 2

r=2.23%

r=3.23%

r=1.70%

r=5.99%

r=3.15%

r=3.15%

r=1.66%

s25=1000

s3=1011 p25=975

s2=552 p25=465

s3=1049 p25=953 s25=900 p3=992

Time (Year)

Scenarios

Interest and wealth risk averse model

Figure 8: Presentation of the solutions for the rst3 stages of the LP approximated

reducedproblem.Unitsaregivenin1000DKK.

The algorithm uses 1018 runs for the dierent problems to nd solutions

whichareoverall lessthan 2%dierentfrom thesolutions foundinthelast

iteration.

It is important to point out that simply dropping the xed costsresults in

solutionswhichdeviateconsiderablyfromtheproblemswiththexedcosts,

whereasapproximatingthexedcostsusingouralgorithmgivesverysimilar

results asfoundbytheMIPmodel.

1-Riskneutral 1.169.147 49.775 1.274.078 1.064.524 25s

2-Minimax 1.179.654 11.150 1.185.795 1.154.602 22s

3-Int.rateriskaverse 1.172.364 26.436 1.239.168 1.130.196 28s

4-Int./Wealthriskaverse 1.174.038 29.128 1.249.520 1.131.185 44s

Table5:ComparisonofthefourstrategiesforthereducedproblemwithLPapproxima-

tion.

7.4 Comments on results

The results presented in this section are in agreement with the nancial

arguments used in the Danish mortgage market. Even though the original

problemis hardto solvewe have shown thatuseful results canbe found by

solvingthereducedproblems.Thereducedscenariotreesrepresentedamore

optimisticpredictionofthefuture,buttheresultsfoundarestillquiteuseful.

Inpracticethemortgage portfoliomanager shouldtry severalscenariotrees

with dierent risk representations as an input to the model. This way the

optimizationmodelcanbeusedasan analyticaltoolfor performing what

if analyseson a highabstraction level.

8 Conclusions

Wehave developed afunctionaloptimization modelthatcanbeusedasthe

basis for a quantitative analysis of the mortgagors decision options. This

modelin conjunction with dierent term structures or marketexpertopin-

ions on the development of bond prices can assist market analysts in the

following ways:

Decisionsupport:Insteadofcalculatingtheconsequencesofthesingleloan

portfoliosforsingleinterestratescenarios,theoptimizationmodelallowsfor

performingwhatifanalysisonahigherlevelofabstraction.Theanalystcan

provide the systemwith dierent sets of information such asthe presumed

lifetimeof theloan, budgetconstraints and riskattitudes.The systemthen

nds the optimalloan portfoliofor each setof inputinformation.

Product development: Traditionally, loan products are based on single

bondsor bondswithembedded options. In some mortgagemarkets such as

theDanishone itisallowed to mixbondsinamortgageportfolioandthere

are even some standard products which are based on mixing bonds. The

product

P ₃₃

^{isforexamplealoanportfoliowhere 33%oftheloanisnanced}

in3yearnoncallablebondsandtherestinxedratecallablebonds.These

mixedproductsarecurrently notpopularsince therationale behindexactly

this kind of mixis not well argued. The optimization model givesthe pos-

sibility to tailor mixed products that, given a set of requirements, can be

The greatest challenge insolving the presented models ison decreasing the

computing times. We have experimented with scenario reduction (scenred,

[7,8,9])andwe have suggestedanLP approximation method toreducethe

solution times while maintaining solution quality. It is, however, an open

problem to develop tailored exact algorithms such as decomposition algo-

rithms (see [1, 2]) to solve the mortgagors problem. Another approach for

gettingreal time solutions is to investigate dierent heuristic algorithmsor

makeuse ofparallel programming(see [13 , 14])to solve theproblem.

Integrationofthetwodisciplinesofmathematicalnanceandstochasticpro-

grammingcombinedwithuseofthestateof theartsoftwarehasagreatpo-

tential,whichhasnotyetbeenrealizedinallnancialmarketsingeneraland

inmortgage companies inparticular. There isa need for more detailed and

operationalmodels andhighperformingeasy to useaccompanying software

to promoteuseofthe mathematical modelswithspecialfocusonstochastic

programming.

References

[1] BirgeJ.R.1985.Decompositionandpartitioningmethodsformultistage

stochasticlinear programs. OperationsResearch,33(5): 989-1007.

[2] Birge J.R.and Louveaux F. 1997.Introduction to Stochastic Program-

ming. ISBN0-387-98217-5, Springer-Verlag NewYork,Inc.

[3] Bjerksund P. and Stensland G. 1996. Implementation of the Black

DermanToyInterestRateModel.TheJournalofFixedIncome,Volume

6 Number2,September1996.

[4] BjörkT.1998.ArbitrageTheoryinContinuousTime.ISBN0-19-877518-

0,OxfordUniversity Press.

[5] BlackF.,DermanE. andToyW. 1990.A OneFactor Modelof Interest

RatesanditsApplicationtoTreasuryBondOptions.FinancialAnanlysts

Journal/JanuaryFebruary 1990.

[6] BussieckM.,pruessnerA.2003.MixedIntegerNonlinearProgramming.

Overviewfor GAMSDevelopment Corporation,February19,2003.

[7] Dupa

ˇ c

^{ová J., GröweKuska N. and Römisch W. 2003. Scenario reduc-}

tion instochasticprogramming: An approach usingprobability metrics.

Mathematical Programming, Ser.A 95 (2003), 493-511.

stochasticprogramming. Computational Optimization and Applications

24 (2003),187-206.

[9] GAMS/SCENRED Documentation. Available from

<www.gams.com/docs/document.htm>.

[10] Hull J.C. 2003.Options, Futures and Other Derivatives.Fifth edition,

Prentice hall international.

[11] Luenberger D. G.1998. Investment Science.OxfordUni. Press.

[12] NielsenS.S.andPoulsenR.2004.A TwoFactor, StochasticProgram-

mingModelofDanishMortgageBackedSecurities.JournalofEconomic

Dynamicsand Control, Volume 28, Issue7,1267-1289.

[13] NielsenS.S.andZeniosS.A.1996.SolvingMultistage StochasticNet-

work Programs on Massively Parallel Computers. Mathematical Pro-

gramming 73,227-250.

[14] Ruszczynski A. 1988. Parallel decomposition of multistage stochas-

tic programming problems.Working paperWP-88-094,IIASA, October.

1988.