Chapter 4
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1 Chapter 4 Financial Engineering --- Nesting GARCH and GARCH Option Pricing Theory
description
Chapter 4. Financial Engineering --- Nesting GARCH and GARCH Option Pricing Theory. A. GARCH Family History. (A) ARCH(R.Engle,1982). ARCH(1):. (B) GARCH(T.Bollerslev,1986). GARCH(1,1). (C) GARCHM (Engle and Bollerslev,1986). GARCHM(1,1). (D) EGARCH (D. Nelson, 1991). EGARCH(1,1). - PowerPoint PPT Presentation
Transcript of Chapter 4
1
Chapter 4
Financial Engineering --- Nesting GARCH and GARCH Option Pricing
Theory
2
A. GARCH Family History
3
(A) ARCH(R.Engle,1982)
ARCH(1):
tsCoefficien
ttperiodoftermsError
tperiodofVariancelConditionah
tperiodofturnR
h
hNR
tt
t
t
tt
tttt
:,,
1,:,
:
Re:
),0(~
10
1
2110
4
(B) GARCH(T.Bollerslev,1986)
GARCH(1,1)
tsCoefficien
tperiodofVariancelConditionah
hh
hNR
t
ttt
tttt
:,,,
1:
),0(~
210
1
212110
5
(C) GARCHM (Engle and Bollerslev,1986)
GARCHM(1,1)
tsCoefficien
hh
hNhR
ttt
ttttt
:,,,,
),0(~
21010
212110
10
6
(D) EGARCH (D. Nelson, 1991)EGARCH(1,1)
tsCoefficien
ofvalueabsoluteExpectedE
ofvalueabsolute
tperiodofVariancelconditionah
tperiodofVariancelconditionah
Ehh
hNR
tt
tt
t
t
tttt
tttt
:,,,
:
:
1log:)ln(
log:)ln(
][ln)ln(
),0(~
2100
1
2110
0
7
(E) NAGARCH (Engle and Ng, 1993)
NAGARCH(1,1)
tsCoefficien
tconsb
bhhh
hNR
tttt
tttt
:,,,
tan:
)(
),0(~
2100
212110
0
8
B. GARCH Option Pricing
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(A) Assumptions1. The spot price S(t) follow NAGARCH-M process
• p=q=1 and β1+α1r12＜ 1
•
• • Continuous-time limit
tZththrtStS loglog
2
11
q
ii
p
ii ithritZithwth
ht
thrthrtStSthwth
21
11
loglog
thrtSthCovt 112log,
dzdtd
10
• Risk-neutral form
where,
2.The value of a call option with one period to expiration obeys the Black-Sholes-Rubinstein formula.
tZththrtStS *
2
1loglog
q
ii
p
ii itZithwth
21
2*
1*
1
2thrtZithri
thtZtZ
2
1*
2
11
*1 rr
11
(B) Model
• Proposition I:
The risk-neutral process takes the same NAGARCH form with λ replaced by –1/2
and r1 replaced by r1*= r1+λ+1/2
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• Proposition II: The generating function takes the log-linear form
where
for the single log(p=q=1) version and these coefficients can be calculated recursively from the terminal conditions: A(T;T,ψ)=0
B1(T;T,ψ)=0
p
ii ithTtBTtAtSf
1
2,;,;exp
1
1
2,;
q
iii ithritZTtC
,;21ln2
1,;,;,; 111 TtwTtBrTtATtA
,;21
2/1,;
2
1,;
11
21
112
111 Tt
rTtBrrTtB
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• Proposition III:
If the characteristic function of the log spot price is f(iψ), then
where Re[ ] denotes the real part of a complex
number.
0,KTSMaxEt
00Re
1
2
1
1
1Re
1
2
11
di
ifKKd
fi
ifKf
i
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European call option:
Et*[ ]: denotes the expectation under risk-neutral
distribution.
0,* KTSMaxEeC ttTr
di
ifKetS
itTr
0
* 1Re
2
1
0
*
Re1
2
1
di
ifKKe
itTr
