Hull Penzugy 9
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Transcript of Hull Penzugy 9
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Wiener folyamat s az It lemma
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 1
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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 2
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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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 3
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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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 5
ltalnostott Wiener folyamatokDrift: tlagos vltozsa x-nek egysgnyi id alatt a
Variancia: egysgnyi id alatt b2
tbtax +=
dzbdtadx +=
atxx += 0
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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 6
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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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 7
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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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 8
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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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 9
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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Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 10
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.
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 11
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Taylor sorfejts:
Egy G(x, t) fggvny Taylor sora
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 12
K+
+
+
+
+
=
2
2
22
22
2
t
t
Gtx
tx
G
xx
Gt
t
Gx
x
GG
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Levgs rendje: t
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 13
!!~
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:
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 14
tbx
GttGx
xGG
t
tbtax
dztxbdttxadx
++=
+=
22
2
2
:levgsnl-
+=
iddiszkrt
),(),(
ekkor
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Az2t tag
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 15
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
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 16
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
Options, Futures, and Other Derivatives, 8th Edition,Copyright John C. Hull 2012 17
dzSS
G
dtSS
G
t
G
SS
G
dG
tSG
zdSdtSSd
:fggvnye)s(samegvltozfggvny
folyamatItorrszvnyA
22
2
2
+
++=
+=
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