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Wiener folyamat s az It lemma

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Markov folyamatok

Memria nlkli sztochasztikus folyamatok, akvetkez lps csak a pillanatnyi helyzettl fgg

Feltevs: rszvnyrak mozgsa Markov folyamat

Kvetkezmny: technikai analzis nem mkdhet!

Hatkony piac hipotzis (gyenge formban): a

pillanatnyi r minden informcit tartalmaz amltbeli viselkedsrl

(m,v): norml eloszls, m tlag, v variancia (= 2 )

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Variancia & standard szrs

Markov folyamatnl az egymst kvet

lpsekfggetlenektlag s variancia additv

Standard szrs nem additvPl. (m,v): (0,1)

2 v utn: (0,2) = 1.414

6 hnap utn: (0,0.5) = 0.7073 hnap utn: (0,0.25) = 0.5tv utn: (0, t) = t1/2

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Egyzvletlen vltoz Wiener folyamat,ha

Wiener folyamatok

zmegvltozsa egy kicsi t

intervallumban: zz tetszleges 2 klnbz (nem tfed)peridusban fggetlen

(0,1)ahol = tz

[z(T) z(0)] = tlaga 0

[z(T) z(0)] variancija T

[z(T) z(0)] standard szrsaT

=

N

i

i t

1

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ltalnostott Wiener folyamatokDrift: tlagos vltozsa x-nek egysgnyi id alatt a

Variancia: egysgnyi id alatt b2

tbtax +=

dzbdtadx +=

atxx += 0

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It folyamatEgy It folyamatnl a drift s a varianciaid s llapot fgg:

dx=a(x,t) dt+b(x,t) dz

Vges idlps esetn:

pontos eredmny, ha tzrhoz tartrejtett feltevs: talatt a s b nem vltozik!

ttxbttxax += ),(),(

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ltalnostott Wiener folyamat s arszvnyek raVrakozs: rak vltozsa szzalkosanlland (elvrt hozam nem fgg az rtl)

Az rak vltozkonysga arnyos az rnagysgval

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itt az elvrt hozam (return) a

volatilits.Diszkrt idlps:

Geometriai Brown mozgs

dzSdtSdS

Rszvnyek rvltozsa: It folyamat

+=

tStSS +=

T

T

eSSdtdtdzdtS

dS 0

2 ),(~ =+=

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Monte Carlo szimulci

Vletlen szm generls: Pl.: = 0.15, = 0.30, s t= 1 ht (= 1/52azaz 0.0192 v), ekkor

+=

+=

SSS

..S..S

04160002880or

0192030001920150

..

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Monte Carlo szimulci:

Week

Stock Price at

Start of Period

Random

Sample for

Change in Stock

Price, S

0 100.00 0.52 2.45

1 102.45 1.44 6.43

2 108.88 0.86 3.58

3 105.30 1.46 6.70

4 112.00 0.69 2.89

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It lemma

Ha ismerjk egy x folyamat rszleteit, Itlemmja megadja egy G (x, t ) sztochasztikusfggvny viselkedst.

Minthogy minden szrmazkos termk fggaz eszkz rtl s az idtl, az It lemma

fontos szerepet jtszik minden razsiproblmnl.

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Taylor sorfejts:

Egy G(x, t) fggvny Taylor sora

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K+

+

+

+

+

=

2

2

22

22

2

t

t

Gtx

tx

G

xx

Gt

t

Gx

x

GG

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Levgs rendje: t

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!!~

2

1 2

2

2

tx

x

x

Gt

t

Gx

x

GG

tt

Gx

x

GG

++=

+=

komponenseegyik

:esetnkalkulusikusSztochaszt

:kalkulusfggvnySzoksos

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Ha x It folyamat:

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tbx

GttGx

xGG

t

tbtax

dztxbdttxadx

++=

+=

22

2

2

:levgsnl-

+=

iddiszkrt

),(),(

ekkor

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Az2t tag

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tbx

Gt

t

Gx

x

GG

tt

ttE

E

EE

E

++=

=

==

=

2

2

2

2

2

2

22

2

1emiatt

~javarianci)(

1)(

1)]([)(

0)(,)1,0(Minthogy

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Infinitezimlis hatrrtk

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lemmjaItoez

2

2

2

22

2

dzb

x

Gdtb

x

G

t

Ga

x

GdG

dzbdtadx

dtbxGdt

tGdx

xGdG

+

++=

+=

++=

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It lemma s rszvny rak

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dzSS

G

dtSS

G

t

G

SS

G

dG

tSG

zdSdtSSd

:fggvnye)s(samegvltozfggvny

folyamatItorrszvnyA

22

2

2

+

++=

+=

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