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
2015/4/9
1
Outline

Introduction.

Investigation of House Price Return Dynamics With
Jumps.

Valuation Framework for No-Negative-Equity-Guarantee.

Numerical Analysis.

Conclusion.
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Introduction
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What are Equity Releasing Products?




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
Loan Value: K ---  Kt  Kevt at time t
Property Value: H 0 ---> H t
Age x
Die(x+s)
Loan Period
4
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The Risk from Lender Prospective

The loan value may exceed the value of the property.
v
Ke  H

How to deal with such risk?



Using Insurance. Ex: HECM program in the united states.
Securitization
Writing a no-negative-equity-guarantee(NNEG)

Payoffs:
v
Max[(Ke  H ),0]
 an European put option on the mortgaged property
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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.

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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..
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Purpose of this study-Con’t

We consider a jump model that incorporate
both autocorrelation effect and volatility
cluster.
 a Jump ARMA-GARCH Model
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An Investigation of House Price Return
Dynamics with Jumps
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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.
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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.
dH t
 dt   dWt  dJ t
Ht
NT
J T   (V j  1)
j 1
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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.
f ( y)  p 1e1 y1{ y0} q 2e2 y1{ y0} ,
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1  1,2  0,
12
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.
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Jumps in House Price or Equity Returns
?

Chan and Maheu (2002)

Dynamic jumps in return
s
m
Nt
i 1
j 1
k 1
Yt  c   iYt 1    j  t  j  t   Vt , k
q
ht  w    
i 1
2
i t i
p
   j ht  j ,
j 1
exp(t )t j
P( N t  j |  t 1 ) 
, j  0,1, 2...
j!
t  0  t 1   t 1
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Jumps in House Price or Equity Returns
?

Duan et al. (2006, 2007)
 Constant jumps in both returns and volatility.
rt   t  ht J t
Jt  z
(0)
t
Nt
  zt( j )
j 1


J t 1  
ht   0  1ht 1   2 ht 1 
c
2
2
 1   (   )



where
2
zt(0) ~ N (0,1), zt( j ) ~ N (  ,  2 )
N t ~ Poisson( )
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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.
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ARMA-GARCH Model
Ht
 eYt
H t 1
Yt  c 
 Y
i 1
ht  w 
i 1
i
t i
  j
j 1
q
p
i 1
j 1
t j
 t
2


 i t i    j ht  j
p
q

m
s
i
  j  1
j 1
Y t follows an ARMA process.
ht follows a GARCH process.
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Dynamic Jump ARMA-GARCH Model
s
m
Yt  c   iYt 1    j 
i 1
j 1
q
ht  w    i
i 1
J t 
t j
 t  J t
p
2
t i
   j ht  j
j 1
N (t )
V
j 1
t, j
Return jump size : Vt
N (t , t2 )
Number of jumps between t-1 and t: N (t )
The case for a dynamic jump:
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Poisson( t )
t  0  t 1   t 1
18
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
506.3450
-4.5481
-4.4555
Constant Jump ARMAGARCH
592.8361
-5.6193
-5.3149
Dynamic Jump ARMAGARCH
607.5076
-5.6512
-5.3014
Diffusion
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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
452.3087
-4.4907
-4.4061
Constant Jump ARMAGARCH
519.8619
-5.3594
-5.0330
Dynamic Jump ARMAGARCH
522.0326
-5.3817
-5.0016
Diffusion
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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
345.4001
-4.2425
-4.1272
Constant Jump ARMAGARCH
415.0801
-5.3314
-4.9211
Dynamic Jump ARMAGARCH
416.0165
-5.3679
-4.9124
Diffusion
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The Valuation Framework for NoNegative-Equity-Guarantee
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Pricing No Negative Equity Guarantee
Let us define the following notation:
 K : the amount of loan advanced at time zero;
 H t : 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.
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Pricing No Negative Equity Guarantee



Assuming the person dies in the middle of the
year
Considering the delaying time
Payoff
Max[( Ks 1/ 2  H k 1/ 2 ),0]

Valuation
EQ [er ( s 1/ 2 ) Max[( Ks 1/ 2  Hk 1/ 2 ), 0]
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Pricing No Negative Equity Guarantee
w x 1

VNNEG (0) 

t 0
w x 1

t 0
Q
 r ( s 1/ 2  )
p
q
E
[
e
Max[( K s 1/ 2  H k 1/ 2 ), 0] | 0 ]
s x xs
1
  , H 0 , K , v, r , g )
s px qx  sV (0, s 
2
1
where V (k    , H 0 , K , v, r , g ) is calculated using simulations.
2
The value of P under measure Q can be obtained using
conditional Esscher transform.
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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
Q
Nt
h
YtQ  r  g  t  VtQ,k
2 k 1
q
p
i 1
j 1
htQ  w   i (t i   htQi ) 2    j htQ j

Special Case: Constant Jump
htQ
Yt  r  g 
2
Q
q
p
i 1
j 1
htQ  w   i (t i   htQi )2    j htQ j ,
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Pricing No Negative Equity Guarantee

Black and Sholes
V (0, s  1/ 2   , H 0 , K , v, r , g ) BSM
=Ke( v-r )( s 1/2 ) N (-d 2 ) - H 0e (- g ( s 1/2 )) N (-d1 ),

Merton Jump
V (0, s  1/ 2   , H 0 , K , v, r , g ) MJ


exp
  *( s 1/2 )
j 0
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( *( s 1/2 ) ) j ( v-r MJ )( s 1/2 )
Ke
N (-d 2MJ ) - H 0e (- g ( s 1/2 )) N (-d1MJ ),
j!
27
Making Numerical Analysis
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Numerical Analysis
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Numerical Analysis
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Numerical Analysis
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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 ARMAGARCH model is carried out and presents a better
fitting result compared with various house price return
models proposed in the literature.
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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.
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The End.
Thanks!
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