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.

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Introduction

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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

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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

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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.

1

( 1)T

tt t

t

N

T jj

dHdt dW dJ

H

J V

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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.

1 21 { 0} 2 { 0} 1 2( ) 1 1 , 1, 0,y y

y yf y p e q e

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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

,

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

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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

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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

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

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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

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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

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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

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

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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

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

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The Valuation Framework for No-Negative-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; : 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

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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

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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.

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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

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

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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

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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 ARMA-

GARCH 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!The End.Thanks!