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Pricing No-Negative-Equity-Guarantee for Equity Release Products under a Jump ARMA-GARCH Model. Presenter: Sharon Yang Co-authors: Chuang-Chang Chang Jr-Wei Huang National Central University, Taiwan. Outline. Introduction. Investigation of House Price Return Dynamics With Jumps. - PowerPoint PPT Presentation
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112/04/211
Pricing No-Negative-Equity-Guarantee for Equity Release Products under a Jump ARMA-GARCH Model
Presenter: Sharon Yang Co-authors: Chuang-Chang Chang Jr-Wei Huang
National Central University, Taiwan
112/04/212
Outline Introduction.
Investigation of House Price Return Dynamics With Jumps.
Valuation Framework for No-Negative-Equity-Guarantee.
Numerical Analysis.
Conclusion.
112/04/213
Introduction
112/04/214
A kind of home equity conversion that allows the elder persons to borrow money with their home as the collateral .
The loans accrue interest are only repaid once the people is died or leave the house.
Such products are needed for “equity rich and cash poor” persons. For example: a rolled-up mortgage
4
Loan Period
Die(x+s)Age x
What are Equity Releasing Products?
0
Loan Value: --- at ti
Property Value:
me
--->
tvtt
t
K K
H H
Ke
112/04/215
The Risk from Lender Prospective The loan value may exceed the value of the property.
How to deal with such risk? Using Insurance. Ex: HECM program in the united states. Securitization Writing a no-negative-equity-guarantee(NNEG)
Payoffs:
an European put option on the mortgaged property
vKe H
[( ),0]vMax Ke H
112/04/216
Purpose of this study
Can Black & Sholes option pricing formula apply to value NNEG?
No! We built up a general framework which
considers the dynamics of the house price return with jumps.
112/04/217
Purpose of this study-Con’t Li et al . (2010) conclude that the Nationwide House Price
Index has the following statistical properties: there is a strong positive autocorrelation effect
among the log-returns the volatility of the log-returns varies with time; a leverage effect is present in the log-return series
ARMA-EGARCH Model
Chen et al.(2010) use the ARMA-GARCH model to price reverse mortgage for the HECM program in the U.S..
112/04/218
Purpose of this study-Con’t
We consider a jump model that incorporate both autocorrelation effect and volatility cluster.
a Jump ARMA-GARCH Model
112/04/219
An Investigation of House Price Return Dynamics with Jumps
112/04/2110
Jumps in House Price Returns?
According to the quarterly data from 1952 to 2008, it can show that the quarterly housing price changed more than three standard deviations.
112/04/2111
Jumps in HousePrice or Equity Returns ? Chen et al. (2009) study U.S. mortgage insurance premium using
Merton jump diffusion process for house price returns.
Merton (1976) build a jump diffusion model with a continuous-time basis.
1
( 1)T
tt t
t
N
T jj
dHdt dW dJ
H
J V
112/04/2112
Jumps in House Price or Equity Returns ? Kou (2002) also considers the leptokurtic feature and
proposes a double exponential jump-diffusion model. The return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution.
1 21 { 0} 2 { 0} 1 2( ) 1 1 , 1, 0,y y
y yf y p e q e
112/04/2113
Jumps in House Price or Equity Returns ? Chan and Maheu (2002) and Duan et al. (2006, 2007) both
examine the jump effect with equity returns under a GARCH model Dynamic jumps in return v.s. Constant jumps in both
returns and volatility.
112/04/2114
Jumps in House Price or Equity Returns ? Chan and Maheu (2002)
Dynamic jumps in return
,
1
11 1
2
1 1
1
0 1 1
,
exp( )( | ) , 0,1, 2...
!
Nt
t k
k
s m
t i t j t j ti j
q p
t i t i j t ji j
jt t
t t
t t t
VY c Y
h w h
P N j jj
112/04/2115
Jumps in House Price or Equity Returns ? Duan et al. (2006, 2007)
Constant jumps in both returns and volatility.
(0) ( )
1
2
10 1 1 2 1 2 2
(0) ( ) 2
1 ( )
where
~ (0,1), ~ ( , )
~ ( )
t
t t t t
Nj
t t tj
tt t t
jt t
t
r h J
J z z
Jh h h c
z N z N
N Poisson
112/04/2116
Jumps in House Price or Equity Returns ? We extend Chan and Maheu (2002) to
consider the dynamic jump effect with house price returns under an ARMA-GARCH model and develop a framework for pricing the NNEG.
112/04/2117
ARMA-GARCH Model
follows an ARMA process. follows a GARCH process.
1
1 1
2
1 1
1 1
1
tYt
t
s m
t i t i j t j ti j
q p
t i t i j t ji j
q p
i ji j
He
H
Y c Y
h w h
tY
th
112/04/2118
Dynamic Jump ARMA-GARCH Model
( )
,1
2
11 1
2
1 1
Return jump size : ( , )
Number of jumps between t-1 and t: ( ) ( )
s m
t i t j t j t ti j
q p
t i t i j t ji j
N t
t t jj
t t t
t
J V
V N
N t Pois
Y c Y J
h w
son
h
The case for a dynamic jump: 0 1 1t t t
112/04/2119
A Comparison of Model Fitting Model Selection, 1953Q4~2008Q4
Model Log-Likelihood AIC BIC
Geometric Brownian Motion
499.1072 -4.5192 -4.4883
ARMA-GARCH 567.8156 -5.4861 -5.2542
ARMA-EGARCH 586.4799 -5.4871 -5.2303
Merton Jump 516.2469 -4.6477 -4.5706
Double Exponential Jump
Diffusion
506.3450 -4.5481 -4.4555
Constant Jump ARMA-GARCH
592.8361 -5.6193 -5.3149
Dynamic Jump ARMA-GARCH
607.5076 -5.6512 -5.3014
112/04/2120
A Comparison of Model Fitting Model Selection, 1958Q4~2008Q4
Model Log-Likelihood AIC BIC
Geometric Brownian Motion
448.9902 -4.4699 -4.4369
ARMA-GARCH 498.4404 -5.3309 -5.0791
ARMA-EGARCH 505.4725 -5.2110 -4.9395
Merton Jump 465.1300 -4.6013 -4.5188
Double Exponential Jump
Diffusion
452.3087 -4.4907 -4.4061
Constant Jump ARMA-GARCH
519.8619 -5.3594 -5.0330
Dynamic Jump ARMA-GARCH
522.0326 -5.3817 -5.0016
112/04/2121
A Comparison of Model Fitting Model Selection, 1968Q4~2008Q4
Model Log-Likelihood AIC BIC
Geometric Brownian Motion
343.6102 -4.2701 -4.2317
ARMA-GARCH 405.9601 -5.2722 -4.9021
ARMA-EGARCH 397.5060 -5.0637 -4.7417
Merton Jump 345.2055 -4.2526 -4.1565
Double Exponential Jump
Diffusion
345.4001 -4.2425 -4.1272
Constant Jump ARMA-GARCH
415.0801 -5.3314 -4.9211
Dynamic Jump ARMA-GARCH
416.0165 -5.3679 -4.9124
112/04/2122
The Valuation Framework for No-Negative-Equity-Guarantee
112/04/2123
Pricing No Negative Equity Guarantee Let us define the following notation: K : the amount of loan advanced at time zero; : the value of the mortgaged property at time t; r : the constant risk-free interest rate; v: the roll-up interest rate; g : the rental yield; : the average delay in time from the point of home exit until
the actual sale of the property.
tH
112/04/2124
Pricing No Negative Equity Guarantee Assuming the person dies in the middle of the
year Considering the delaying time Payoff
Valuation
1/ 2 1/ 2[( ),0]s kMax K H
( 1/ 2 )1/ 2 1/ 2[ [( ),0]Q r s
s kE e Max K H
112/04/2125
Pricing No Negative Equity Guarantee
1( 1/ 2 )
1/ 2 1/ 2 00
0
1
0
(0) [ [( ),0] | ]
1
(0, , , , , , )2
w xQ r s
NNEG s x x s s kt
w x
s x x st
V s H K v r
V p q E e Max
p q g
K H
0
1where ( , , , , , ) is calculated using simulations.
2V k H K v r g
The value of P under measure Q can be obtained using conditional Esscher transform.
112/04/2126
Pricing No Negative Equity Guarantee Under the risk-neutral measure Q, the return processes of and to
characterize the jump ARMA(s,m)-GARCH(p,q) model become
Special Case: Constant Jump
,1
2
1 1
2
( )
Q NtQ Qt
t t kk
q pQ Q Qt i t i t i j t j
i j
hY r g V
h w h h
2
1 1
2
( ) ,
QQ t
t
q pQ Q Qt i t i t i j t j
i j
hY r g
h w h h
112/04/2127
Pricing No Negative Equity Guarantee Black and Sholes
Merton Jump
0
( - )( 1/2 ) (- ( 1/2 ))2 0 1
(0, 1/ 2 , , , , , )
= (- ) - (- ),
BSM
v r s g s
V s H K v r g
Ke N d H e N d
*( 1/2 )
0
*( 1/2 )( - )( 1/2 ) (- ( 1/2 ))
2 0 10
(0, 1/ 2 , , , , , )
exp ( )(- ) - (- ),
!
s
MJ
MJ
s jv r s MJ g s MJ
j
V s H K v r g
Ke N d H e N dj
112/04/2128
Making Numerical Analysis
112/04/2129
Numerical Analysis
112/04/2130
Numerical Analysis
112/04/2131
Numerical Analysis
112/04/2132
Conclusion This article contributes to the literature in the
following ways. Dynamic Jump ARMA-GARCH model can better
capture the dynamics of house price return. The estimation of the proposed jump ARMA-
GARCH model is carried out and presents a better fitting result compared with various house price return models proposed in the literature.
112/04/2133
Conclusion This article contributes to the literature in the
following ways. The risk neutral pricing framework for the jump
ARMA-GARCH model is derived using the conditional Esscher transform technique.
Numerical result shows that incorporating the jump effect in house price returns is important for pricing NNEG.
112/04/2134
The End.Thanks!The End.Thanks!