Fo schung am IVW Köln, 3/2015
Ins i u ü Ve siche ungswesen
Calib a ion o Hes on's s ochas ic
ola ili y model o an empi ical
densi y using a gene ic algo i hm
U ij Dolgo
Fo schung am IVW Köln, 3
/
2015 Wählen Sie ein Elemen aus.
Dolgo
Fo schungss elle FaRis
Calib a ion o Hes on's s ochas ic ola ili y model o an
empi ical densi y using a gene ic algo i hm
Zusammen assung
In diesem A ikel schlagen wi die Ve wendung eines gene ischen Algo i hmus (GA) zu Kalib ie ung eines
S ochas ischen P ozesses an eine empi ische Dich e on Ak ien endi en o . Anhand des Hes on Models
zeigen wi wie eine solche Kalib ie ung du chge üh we den kann. Neben des Pseudocodes ü einen
ein achen abe leis ungs ähigen GA p äsen ie en wi zudem auch Kalib ie ungse gebnisse ü den DAX und
den S&P 500.
Abs ac
In his pape we p opose he use o gene ic algo i hms when i ing a s ochas ic p ocess o he empi ical
densi y o s ock e u ns. Using he Hes on Model as an example, we show how such a calib a ion can be
ca ied ou . We also p esen an easy o implemen gene ic algo i hm and p o ide calib a ion esul s o he
daily s ock e u ns o he DAX and he S&P 500.
Schlagwö e :
Ak ien endi en, Dich e unk ion, empi ische Dich e, Hes on Model, Modellkalib ie ung, S ochas ische P ozesse
Keywo ds:
Empi ical Densi y, Hes on Model, Model Calib a ion, P obabili y Densi y, S ochas ic P ocesses, S ock Re u ns
Calib a ion o Hes on’s s ochas ic ola ili y model
o an empi ical densi y using a gene ic algo i hm
U ij Dolgo
23 Decembe 2014
Abs ac
In his pape we p opose he use o gene ic algo i hms when i ing
a s ochas ic p ocess o he empi ical densi y o s ock e u ns. Using he
Hes on Model as an example we show how such a calib a ion can be ca ied
ou . We also p esen an easy o implemen gene ic algo i hm and p o ide
calib a ion esul s o he daily s ock e u ns o he DAX and he S&P
500.
1 The Hes on Model and i ’s ansi ion densi y
The Hes on Model (HM) sugges ed by Hes on (1993) is o en seen as he i s
logical ex ension o he widely known Black and Scholes (BS) app oach. I uses
a s ochas ic ola ili y ins ead o he la one sugges ed by i ’s less sophis ica ed
coun e pa .
Se e al empi ical s udies ha e shown al eady ha he cons an - ola ili y-
assump ion con adic s ma ke eali ies (see e.g. Con (2001), Guillaume e al.
(1997) ) The mos salien d awback o he B&S-model is o en conside ed o
be i ’s inabili y o eplica e he long ails which a e obse able in daily s ock-
e u ns. These howe e can be cap u ed qui e well by Hes on’s app oach. (see
Sil a and Yako enko (2003), Daniel (2003))
The model’s dynamics a e cha ac e ized by he ollowing h ee equa ions
dS =µS d +√ S dW(1)
(1)
d =−γ( −θ)d +κ√ dW(2)
(2)
dW(2)
=ρdW(1)
+p1−ρ2dZ (3)
He e Z is a Wiene p ocess independen o W(1)
.De ining by = ln(S /S0),
applying I o’s Fo mula and se ing x = −µ one a i es a
dx =−0.5 d +√ dW(1)
(4)
1
A ansi ion densi y P (x, | i) o he join ealiza ion o x and a ime
gi en an ini ial log- e u n x= 0 and a iance ia = 0 was cons uc ed
by D agulescu and Yako enko (2002). Howe e , because he a iance is no a
di ec ly obse able ma ke quan i y P (x, | i) is no sui able o he calib a ion
o he Hes on Model o empi ical da a.
Fo una ely D agulescu and Yako enko (2002) also in oduce educed den-
si ies by in eg a ing ou he a iance.
P (x| 0) = Z+∞
0
P (x, | 0)d (5)
This is he densi y o ha ing log- e u n xa ime gi en a a iance 0a ime
= 0. I we wan ed o speci y a s a ing a iance he unc ion abo e migh be
he co ec choice. Howe e deciding upon such a alue can be qui e a bi a y
o a iance can no be di ec ly obse ed. To ci cum en his unce ain y D ag-
ulescu and Yako enko (2002) use he s a iona y densi y o (2) which is gi en
by
Π∗( ) = αα
Γ(α)
α−1
θαe−α /θ, α =2γθ
κ2(6)
Inse ing (6) in o (5) and in eg a ing o e 0 hey e en ually a i e a he ol-
lowing esul :
P (x) = 1
2πZ+∞
−∞
eipx+F (px)dpx(7)
wi h
F( , px) = γθ
κ2Γ −2γθ
κ2ln cosh Ω
2+Ω2−Γ2+ 2γΓ
2γΩsinh Ω
2(8)
Γ = γ+ iρκpx(9)
Ω = pΓ2+κ2(p2
x−ipx) (10)
2 Empi ical densi y unc ion and i ing
Le a se ies o log-no mal s ock e u ns be gi en by x(∆ ) = {x1, . . . , xn}. He e
∆ deno es he ime-s ep. Thus in he case o 21 ading days pe mon h and
∆ = 1/252, x(1/252) will be a se ies o daily e u ns. To make he da a
compa ible wi h p ocess (4) we also shi e e y e u n by µ∆ =∆
nPn
i=1 xi
and hus ese
x(∆ ) = {x1−µ∆ , . . . , xn−µ∆ }(11)
Fu he mo e we se xmin = min x(∆ ), xmax = max x(∆ ) and in oduce
he bin-size ∆x. The numbe o bins is hen gi en by M= ceil xmax−xmin
∆x.
2
We hen simply de e mine he ela i e p obabili ies pi o each bin by weigh ing
he numbe o e u ns in ha bin by he o al numbe o obse a ions.
Fo a bin wi h bounda ies [a, b) we also in oduce he bin ep esen a i e
¯xi=a+1
2(b−a). Using he M ep esen a i es {¯x1. . . , ¯xM}we can now de ine
he unc ion
pemp(x) =
M
X
i=1
piδ¯xi(x) (12)
A e he ini ial empi ical unc ion has been cons uc ed one can ge id o
ou lie s by de e mining he αand 1 −αquan iles. Using hese alues o ese
xmin and xmax one hen p oceeds o de i e a new empi ical densi y (wi hou
he ou lie s).
To unde line ha he shape o he analy ical densi y depends on he choice
o (γ, θ, κ, ρ), we se P (x) = p( , x, γ, θ, κ, ρ) The dis ance unc ion be ween he
empi ical dis ibu ion and he analy ical one is hen gi en by
(γ, θ, κ, ρ) =
M
X
1=1
[pemp(¯x)−p(∆ , ¯x, γ, θ, κ, ρ))]2(13)
To conclude he i ing p ocedu e we ha e o minimize wi h espec o
(γ, θ, κ, ρ). No e ha as de ined in (13) is jus one choice o a dis ance unc ion.
Al e na i es migh be he he oo -mean-squa ed e o o he absolu e dis ance.
3 The gene ic algo i hm app oach
3.1 Mo i a ion
In his sec ion we assume ou cos unc ion o be :Rn→Rand ou goal
will be o minimize . The unc ion’s inpu is hus a ec o o he o m
~x = (x1, . . . , xn)∈Rnand an inpu -solu ion-pai is a uple (~x, (~x)) ∈Rn+1.
Fu he mo e we es ic he sea ch-space o ou algo i hm by limi ing i o he
hype cube
H(~a,~
b) = {~x ∈Rn, ai≤xi≤bi,∀∈{1,...n}} (14)
Howe e , i is se o equal (13) i ’s calcula ion mus be ca ied ou ia
nume ical in eg a ion and he in eg al migh no be well-beha ed o e e y choice
o (γ, θ, κ, ρ). This can lead o he c ash o mos con en ional op imiza ion
ou ines.
In he case o he Hes on Model is known explici ly. Howe e , o many
s ochas ic p ocesses he ansi ion densi y i sel is o en no known and has o be
es ima ed nume ically. Pede sen (1995) and B and and San a-Cla a (2001) o
example sugges he use o Mon e Ca lo coupled wi h a Maximum-Likelihood
es ima o in ode o cons uc he ansi ion densi y. In such a case app oaches
like Le enbe g-Ma qua d o s eepes descen (see Kelley (1999)) would be com-
pu a ionally expensi e, o 0will ha e o be de e mined nume ically. I would
be bes o use an op imiza ion ou ine which:
3
•does no depend on he o m o ,
•doesn’ necessi a e he compu a ion o 0,
•wo ks eliably o la ge n,
•is able o dis ega d local op ima
•and is inhe en ly pa allel which will allow o exploi mode n mul i-co e
p ocesso s.
All o he c i e ia abo e a e me by gene ic algo i hms (GA). Fo an in o-
duc ion o GA see e.g. Si anandam and Deepa (2007) o Gen (1997). Despi e
being conside ed heu is ics hey ha e been epea edly shown o pe o m well
wi h complex p oblems (see e.g. De Jong (1975) , Ma co-Blaszka and Deside i
(1999)). In he nex sec ion we a e going o in oduce a simple GA con aining
all he necessa y buildings blocks which a e inhe en o his ype o op imiza ion
p ocedu e.
3.2 The Algo i hm
The gene al s uc u e o mos gene ic algo i hms is qui e simple. One i s c e-
a es an ini ial popula ion by andomly sampling om he sea ch space. A e -
wa ds he popula ion membe s a e selec ed and exchange in o ma ion in o de
o p oduce new, i e solu ions. The popula ion is hen so ed acco ding o
he i ness le el and he wo s solu ions a e killed o . The membe s o his new
popula ion a e again selec ed o ”ma ing” and he en i e p ocedu e is epea ed.
To embed ou op imiza ion p oblem in o he GA- amewo k we mus in o-
duce some de ini ions i s . F om he e on ~x will be called a ch omosome and he
uple (~x, (~x)) ∈Rn+1 a membe o he popula ion ˜
Pwhich is simply a se o
such uples. To make he no a ion mo e compac we will e e o a popula ion
membe ia ˜m, i ’s ch omosome ia ˜m(1) and i ’s cos ia ˜m(2) . In his con ex
˜
Piwill be he i- h popula ion membe . He e we also access he ch omosome ia
˜
Pi(1) ∈Rnand he cos ia ˜
Pi(2) ∈R.
To ini ialize a GA one i s has o gene a e an ini ial popula ion by andomly
sampling Nini poin s om he space H(~a,~
b) (see Algo i hm 1)
4
inpu : The dimension o he objec i e unc ion’s inpu n, he size
o he ini ial popula ion Nini, an emp y con aine ˜
Pini and
he ec o s ~a,~
b∈Rn
ou pu : The con aine ˜
Pini illed wi h Nini elemen s
local :~x ∈Rn
o i= 1 o Nini do
o i= 1 o ndo
xi=d awUni o mRandom [ai,bi];
end
Add (~x, (~x)) o ˜
Pini;
end
Algo i hm 1: C ea ing an ini ial popula ion
The nex s ep consis s o deciding which membe s o he popula ion should
ma e. He e he bes solu ion should also ha e he bes chances o passing on i ’s
in o ma ion. To achie e ha we use a cos -weigh ed-selec ion app oach. Thus
he lowe he cos o a gi en popula ion membe he highe i ’s p obabili y o be
selec ed o ma ing. Assuming ha he cu en popula ion ˜
Pis al eady so ed
so ha ˜
P1(2) ≤˜
Pi(2) ∀˜
Pi∈˜
P he selec ion p obabili ies a e gi en by
pi=
Pi(2) −PN(2)
PN
j=1(Pj(2) −PN(2))∀i∈ {1, . . . , n}(15)
Using his p obabili ies one han cons uc s a disc e e p obabili y dis ibu ion
wi h P(X=i) = piand uses i o d aw membe s om ˜
P. The numbe o
ma ing-pai selec ions is egula ed by he a iable η∈[0,1] speci ying he ac-
ion o he popula ion ha will ma e (see Algo i hm 2) No e ha algo i hm 2
allows sel -pai ing. Thus a membe can be pai ed wi h i sel .
5
inpu : The cu en popula ion ˜
Pand i ’s size Nand η∈R
ou pu : A lis o ma ing-pai s ˜
M
local :~p ∈RN, ˜m1∈Rn+1 , ˜m2∈Rn+1
˜
P=so Popula ion (˜
P) ; // so ha ˜
P1(2) ≤˜
Pi(2) ∀˜
Pi∈˜
P
o i= 1 o Ndo
pi=
Pi(2)−PN(2)
PN
j=1 (Pj(2)−PN(2))
end
o i= 1 o b0.5Nηcdo
˜m1=d awMembe F omPopula ion (~p);
˜m2=d awMembe F omPopula ion (~p);
Add ( ˜m1,˜m2) o ˜
M;
end
Algo i hm 2: Selec ing membe s o ma ing
Now ha we ha e selec ed he ma ing pai s, we p oceed o in oduce ano he
c ucial gene ic ope a o - he c osso e . The c osso e cons i u es he s ep
whe e he exchange o in o ma ion be ween he di e en solu ions p esen in
he popula ion akes place. The key idea is o combine da a o wo membe s ( o
”ma e” hem) in o de o p oduce wo i e membe s. How his is accomplished
a ies depending on he conc e e p oblem. The app oach used he e is desc ibed
in Algo i hm 3. Using he ma ing pai s ob ained ia Algo i hm 2 one goes on
o combine he gene ic da a o he wo membe s o p oduce wo new ones. The
new membe s inhe i mos o hei pa en ’s da a bu one andomly selec ed
gene. We collec his esul s in a con aine ˜
Pch and se ˜
P=˜
P∪˜
Pch. To a oid
an exponen ial popula ion g ow h we in oduce he a iable Nmax, so ˜
Pso
ha ˜
P1(2) ≤˜
Pi(2) ∀˜
Pi∈˜
Pand emo e all ˜
Piwi h i>Nmax. This way he
popula ion size is kep cons an o educe compu a ional o e head.
Like many op imiza ion ou ines he GA-app oach migh also su e om
p ema u e con e gence. Thus i inds one local minimum and con e ges o i
neglec ing he be e op imal solu ions wi hin he sea ch-space. In he con ex
o he GA his is o en caused by he gene ic-d i . This happens when he
popula ion ˜
Pcon ains one membe ˜mbes which is signi ican ly i e han he
o he s. ˜mbes will hus be equen ly selec ed o ma ing and will end up dom-
ina ing he ”gene-pool”. A e se e al i e a ions he en i e popula ion migh
end up con e ging o ˜mbes . The usual app oach o emedy his si ua ion is
o in oduce a s eady low o new and unbiased gene ic in o ma ion o he pop-
ula ion. This can be achie ed by andomly sampling poin s om he sea ch
space and adding hem o ˜
P. Thus one uns Algo i hm 1 o p oduce a speci ied
numbe o new membe s Nmu and adds hem o he cu en popula ion. This
6
is also o en e e ed o as in oducing mu a ion.
He e i is impo an o no e ha his new membe s will be added o ˜
Pa e
i has been unca ed o Nmax. This way hey will ha e a chance o ma e in
he nex un.
inpu : A se o ma ing pai s ˜
M, hei numbe NM, ac o β∈[0,1],
he cu en popula ion ˜
Pand Nmax
ou pu : An upda ed popula ion ˜
P
local : ˜m1∈Rn+1 , ˜m2∈Rn+1 ,a∈N,~x, ~y ∈Rn,z1, z2∈R,
~z1, ~z2∈Rn,Nnew
o i= 1 o NMdo
( ˜m1,˜m2) = ˜
Mi;
a=d awIn ege ({1, . . . , n});
// Uni o m andom d aw o an in ege om {1, . . . , n}
~x = ˜m1(1) = {x1, . . . , xa,...xn};
~y = ˜m2(1) = {y1, . . . , ya,...yn};
z1=xa−β(xa−ya);
z2=ya+β(xa−ya);
~z1={x1, . . . , xa−1, z1, xa+1 . . . xn};
~z2={y1, . . . , ya−1, z1, ya+1 . . . yn};
˜m1= (~z1, (~z1)) ˜m2= (~z2, (~z2))
Add ˜m1and ˜m2 o ˜
Pch;
end
˜
P=˜
P∪˜
Pch;
˜
P=so Popula ion (˜
P) ; // so ha ˜
P1(2) ≤˜
Pi(2) ∀˜
Pi∈˜
P
Nnew=ge PopSize (˜
P);
˜
P=˜
P {˜
PNmax+1,..., ˜
PNnew };
Algo i hm 3: Ma ing/c osso e and egula ing popula ion size
Finally we ha e o de ine a s opping-c i e ia o he algo i hm. The mos
s aigh - o wa d app oach and he one we use he e, is o s op when he i es
popula ion membe ˜
P1hasn’ changed K- imes in a ow.
7
Re e ences
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ima ion o di usions wi h an applica ion o exchange a e dy-
namics in incomple e ma ke s. NBER Technical Wo king Pape
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h ps://ideas. epec.o g/p/nb /nbe e/0274.h ml.
Rama Con . Empi ical p ope ies o asse e u ns: s ylized ac s and s a is ical
issues. Quan i a i e Finance, 1:223–236, 2001.
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2003. URL h p://a xi .o g/abs/cs/0305055.
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Ad ian A. D agulescu and Vic o M. Yako enko. P obabili y dis ibu ion o e-
u ns in he hes on model wi h s ochas ic ola ili y. a Xi :cond-ma /0203046,
Ma ch 2002. URL h p://a xi .o g/abs/cond-ma /0203046. Quan i a i e
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ich A. Mlle , Richa d B. Olsen, and Oli ie V. Pic e . F om he
bi d’s eye o he mic oscope: A su ey o new s ylized ac s o he
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h p://link.sp inge .com/a icle/10.1007/s007800050018.
S. L. Hes on. A closed- o m solu ion o op ions wi h s ochas ic ola ili y wi h
applica ions o bond and cu ency op ions. Re iew o Financial S udies, 6(2):
327–343, Janua y 1993. ISSN 0893-9454, 1465-7368. doi: 10.1093/ s/6.2.327.
URL h p:// s.ox o djou nals.o g/con en /6/2/327.
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Na halie Ma co-Blaszka and Jean-An oine Deside i. Nume ical solu ion o
op imiza ion es -cases by gene ic algo i hms. Feb ua y 1999. URL
h p://hal.in ia. /in ia-00073055.
Mo en Bje egaa d Pede sen. A new app oach o maximum likelihood es i-
ma ion o s ochas ic di e en ial equa ions based on disc e e obse a ions.
Scandina ian Jou nal o S a is ics, 22:55–71, 1995.
14
A. Ch is ian Sil a and Vic o M. Yako enko. Compa ison be ween he p ob-
abili y dis ibu ion o e u ns in he hes on model and empi ical da a o
s ock indexes. Physica A: S a is ical Mechanics and i s Applica ions, 324(1):
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15
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Fo schung am IVW Köln, 3/2015
Dolgo : Calib a ion o Hes on's s ochas ic ola ili y model o an empi ical densi y using a gene ic
algo i hm
Köln, Feb ua 2015
ISSN (online) 2192-8479
He ausgebe de Sch i en eihe / Se ies Edi o ship:
P o . D . Lu z Reime s-Rawcli e
P o . D . Pe e Schimikowski
P o . D . Jü gen S obel
Ins i u ü Ve siche ungswesen /
Ins i u e o Insu ance S udies
Fakul ä ü Wi scha s- und Rech swissenscha en /
Facul y o Business, Economics and Law
Fachhochschule Köln / Cologne Uni e si y o Applied Sciences
Web www.i w-koeln.de
Sch i lei ung / Con ac edi o ’s o ice:
P o . D . Jü gen S obel
Tel. +49 221 8275-3270
Fax +49 221 8275-3277
Mail jue gen.s obel@ h-koeln.de
Ins i u ü Ve siche ungswesen /
Ins i u e o Insu ance S udies
Fakul ä ü Wi scha s- und Rech swissenscha en /
Facul y o Business, Economics and Law
Fachhochschule Köln / Cologne Uni e si y o Applied Sciences
Gus a Heinemann-U e 54
50968 Köln
Kon ak Au o / Con ac au ho :
U ij Dolgo
Ins i u ü Ve siche ungswesen /
Ins i u e o Insu ance S udies
Fakul ä ü Wi scha s- und Rech swissenscha en /
Facul y o Business, Economics and Law
Fachhochschule Köln / Cologne Uni e si y o Applied Sciences
Gus a Heinemann-U e 54
50968 Köln
Tel. +49 221 8275-3271
Fax +49 221 8275-3277
Mail [email p o ec ed]e
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Dezembe 2013 in Köln, N . 5/2014
Heep-Al ine , Hoos, K ah o s : Fai Value Bewe ung on zedie en Rese en, N . 4/2014
Heep-Al ine , Hoos: Ve ein ach e Na Ca Modellie ungsansa z zu
Rück e siche ungsop imie ung, N . 3/2014
Zimme mann: F auen im Ve siche ungs e ieb. Was sagen die P i a kunden dazu?, N .
2/2014
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2013, N . 1/2014
2013
Heep-Al ine : Ve lus abso bie ung du ch la en e S eue n nach Sol ency II in de
Schaden e siche ung, N . 11/2013
Mülle -Pe e s: Kunden e hal en im Umb uch? Neue In o ma ions- und Abschlusswege in
de K z-Ve siche ung, N . 10/2013
Knobloch: Risikomanagemen in de be ieblichen Al e s e so gung. P oceedings zum 4.
FaRis & DAV-Symposium am 14. Juni 2013, N . 9/2013
S obel (H sg.): Rechnungsg undlagen und P ämien in de Pe sonen- und
Schaden e siche ung - Ak uelle Ansä ze, Möglichkei en und G enzen. P oceedings zum 3.
FaRis & DAV Symposium am 7. Dezembe 2012, N . 8/2013
Goecke: Spa p ozesse mi kollek i em Risikoausgleich -
Back es ing, N . 7/2013
Knobloch: Kons uk ion eine un e jäh lichen Ma ko -Ke e aus eine jäh lichen Ma ko -
Ke e, N . 6/2013
Heep-Al ine e al. (H sg.): Value-Based-Managemen in Non-Li e Insu ance, N . 5/2013
Heep-Al ine : Ve ein ach es Fo melwe k ü den MCEV ohne Renewals in de
Schaden e siche ung, N . 4/2013
Mülle -Pe e s: De e ne z e Au o ah e – Akzep anz und Akzep anzg enzen on eCall,
We ks a e ne zung und Meh we diens en im Au omobilbe eich, N . 3/2013
Maie , Schimikowski: P oceedings zum 6. Diskussions o um Ve siche ungs ech am 25.
Sep embe 2012 an de FH Köln, N . 2/2013
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2012, N . 1/2013
