an efficient and accurate lattice for pricing derivatives under a jump-diffusion process
TRANSCRIPT
An Efficient and Accurate Lattice for Pricing Derivatives
under a Jump-Diffusion Process
1. Introduction
• In a lattice, the prices of the derivatives converge when the number of time steps increase.
• Nonlinearity error from nonlinearity option value causes the pricing results to converge slowly or oscillate significantly.
• Underlying assets : Stock• Derivatives : Option
• CRR Lattice Lognormal diffusion process• Amin’s Lattice• HS Lattice Jump diffusion process• Paper’s Lattice
}
CRR Lattice
uP
dP
Lognormal Diffusion Process Su with Sd with =1-d < uud = 1
uP dP
Problem !?
• Distribution has heavier tails & higher peak ----------------------------------------------------------------
Jump diffusion process !!! ↙ ↘ Diffusion component Jump component ↓ ↓ lognormal diffusion process lognormal jump (Poisson)
Amin’s Lattice
less accurate !?Volatility(lognormal jump > diffusion component)
Hilliard and Schwartz’s (HS) Lattice
Diffusion nodesJump nodesRate of )( 3nO
Paper’s Lattice
Trinomial structurelower the nodeRate of )( 5.2nO
2. Modeling and Definitions
The risk-neutralized version of the underlying asset’s jump diffusion process
)()()5.0(0
2
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1
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)1()(
25.0
2
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ek
Nk
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i
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i i
magnitudejumprandomk
ensityjump
processPoissontn
componentdiffusionofvolatility
ratefreeriskr
MotionBrowniandardstz
i :
int:
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:
:
tan:)(
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0
)(
)1ln(
)(
)()5.0(
)(
0
2
gcompoundinPoissonundernormalcomponentjump
kY
processItocomponentdiffusion
tztkrX
tn
iit
t
• Financial knowledge (Pricing options)
)(:
)(:sin
)0),(max()(:
TPoptionbarrierdouble
TPoptionBarriergle
XSTPoptionVanilla T
otherwise
LSifXS TT
,0
),0),(max(
{
{
otherwise
LSHSifXS TTT
,0
&),0),(max(
)]([' TPEevalussOption rT
3. Preliminaries
• (a) CRR Latticejump diffusion process (λ=0)
tVar
tr
S
S
withXbyrepesentedbecanS
S
t
tt
tti
2
2
0
)5.0(
)ln(
0)ln(
ud
ueP
du
deP
ed
eu
ud
PP
VardPuP
dPuP
tr
d
tr
u
t
t
du
du
du
1
1
)(ln)(ln
lnln22
tt
tr
tenoughsmallWith
implies
)5.0( 2
)3(
)2(.
deg2
structureTrinomial
LatticeCRRge
iesprobabilitbranching
factorstivemultiplicaprice
freedomofrees
latticenomial
• (b) HS Lattice
jhtcVV
Vnodefollowingisteptimeat
nodesmofpricesS
Xcomponentdiffusiontkr
S
S
titi
ti
t
t
tt
)1(
0
2
1
)12(2log
)()5.0(
)ln(
22
,...,2,1,0,1
h
mjc
1
2,...,2,1),,...,2,1,0(
])1ln([)()(
0
'
mj
mjj
j
itn
wwij
mj
mj
i
q
mimjq
kEqjh
Continuous-time distribution of the jump component
→ obtain 2m+1 probabilities from solving 2m+1 equations
jdtito
ti
jutito
ti
qPjhtVV
qPjhtVV
)ln()ln(,0),(:
]))1(,(
))1(,([),(
)(),(
)(:),(
00
0
S
HVor
S
LVifiVFoptionBarrier
qPtijhtVF
qPtijhtVFeiVF
TPnVF
eSSisteptimeatvalueoptioniVF
tititi
mj
mj jdti
jutitrti
tn
Vti
ti
• Pricing option
• Complexity Analysis
0)ln(0
00 S
SV t
},...,2,1{,,...,2,1,0,1 ikmjc kk
ht
hjjjtcccV
eS
eS iit
0
)...()...(0
21210
mimiiiisteptimeAt ,,
n
inOmiicountnodetotal
iOmiiisteptimeatcountnodethe
0
3
2
)()12)(12(
)()12)(12(
Problem !? 1. time complexity 2. oscillation
4. Lattice Construction
• Trinomial Structure
→ Cramer’s rule
1
0
111
222 Var
P
P
P
d
m
u
• fitting the derivatives specification
CRReven
WO.
• Jump nodes
jhjump
jhjump
SueSd
SueSu
• Complexity analysis
t
mh
X
d
2
)&( ZY
5. Numerical Results
optionVanilla optionBarrier