1

Recent Advances in Mathematical Finance

Chicago, December 8, 2007

Yevgeny Goncharov

Department of Mathematics

Florida State University

goncharov @ math.fsu.edu

Prepayment Prepayment and and Mortgage Rate Mortgage Rate ModelingModeling

partially supported by NSF grant DMS-0703849

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Mortgage Securities

Pooled cash flow: interest + principal payments

Rule for distribution of cash flow

Investors

A B C

prepayment

default

curtailment

? ? ????? ? ????

3

s

{ } [ I ( ) ]θ θ

t tT - r dθ - r dθ

t t T ttM c e ds P eE

G

A Mortgage• m – mortgage rate • P(t) – outstanding principal • t – t -intensity of prep.

(“prepayment rate”)

Borrower Lender

P0 at time “0”

$$

cash flow up to time min( , )T

( )s

θt

T - r dθ

t s ttM c P t e ds= E

F ( ) ( )

s

θt

T - r dθ

t s ttM P t P s m r e ds= E

F

• c – payment rate ($/time)

• – prepayment time

• t – “relavant” information( ) ( )c mP t P t

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0 , comparison of mortgage rates:

t

t

m m

Prepayment (Intensity) Specification

0 , comparison of mortgage rates:

( ), "market price" of profitability

t

t

t

m m

L P t

( )tX

Empirical

market factors

:t

Prepayment:

( )t

Model-based

( )tX

refinancingincentive

measure of borrower’sreaction

or

or

Note: the incentive translates “market” to “resident” money-language

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Intensity Modeling : ( )t t

Prepayment or

Refinancing incentive t

(“High frequency”)Estimates “usefulness” of therefinancing from the borrower’spoint of view. Based on the “market” information(interest, unemployment rates, home prices).

Intensity function (.) (“Low frequency”)Estimates the borrower’s “response”(in probabilistic terms) to certain market situations.

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Why to Model?

Empirical

Modeling

1tX time4

tX3tX2

tX

1 1( )tX 4 4( )tX3 3( )tX2 2( )tX

1( )t 4( )t 3( )t 2( )t

sensitive to time

not sensitive to time

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Mortgage Modeling

, , etc...t tr X

Numericalimplementation

Calibration

t

Mortgage Model

t

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Mortgage Model Classification

• Data, calibration• Computational Method• Interest Rate Model, • Prepayment intensity function• Additional predictors (house prices, media effect, etc)

• Refinancing incentive

Mortgage Model = Ref. Inc.

Statistics

Not mortgage-specific.“Standard” problems

1. Mort-Rate-Based

2. Option-Based

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Implied Implied Mortgage Rate Process

Let be mortgage rate at time :

(0) ( ) 0s

θt

t

t T - r dθnew tt s tt

m t

M P P s m r e ds= E

F

( ; )

( ; )

s

θt

s

θt

t T - r dθtt st

t

t T - r dθtt t

r P s m e ds

m

P s m e ds

E

E

( { } )t T tt s t s t Tm L m

The rate mt implied by the prepayment process t:

: ( { } )T tt s t s t TL m

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Mortgage-Rate-Based Approaches0 0

0 0( )Examples: , , or , i.e., ( , )

( )t t

t tt t

m c mm m m m

m c m

1. The process the 10-year Treasury yield+const.tm

[ ( , )]

[ ( , )]

[ ( ; ) ]2. ( { } )

[ ( ; ) ]

in general !

s tθ

t

s tθ

t

t T - r m m dθtst T t t

t s t s t Tt T - r m m dθt

t

t t

r P s m e dsm L m

P s m e ds

m m

E

E

3. Endogenous mortgage rate :

( { ( , )} )t T t t st t s t Tm L m m m

0{ }ttm Pliska/Goncharov

MOATS

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A Simple Example

Consider a simple market which is completely described

by a Markovian time-homogenous process . Then

tr

( ) ( , ( ) ( ) ) m r L r m r m

0

0

- ( ( ), ( ))

0

- ( ( ), ( ))

0

[ ( ; ( )) ]i.e., ( )

[ ( ; ( )) ]

s

s

T r m r m r d

s 0

T r m r m r d

0

r P s m r e ds r rm r

P s m r e ds r r

E

E

0 00 0( { ( , )} ) ( { ( , )} )t T t t s T s

t t s t T s Tm L m m m L m m m

Thus ( ), where ttm m r

12

13

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

MRB Mortgage Rate0, ( ) / ( ) 1 0.3

, otherwise

t

t

c m c m

0

2 0

0.05, 2

0.6

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Option-Based Mortgage Rate

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

, (1 0.3) ( )

, otherwiset

t

L P t

0

0.6

2 to 0

0.05, 2

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Citigroup’s MOATS(generalized)

moats

moatsmoats moats

moats

1. For from 1 to : ( { ( , )} )

2. For from 1 to 0 : ( { ( , )} )

T tt t t st t s T

t T t t st t s t T

t T T T m L m m m

t T T m L m m m

0 Tmoats

T T T

0 Tmoats

T

t

t

Citigroup: • T=360 (30 yr), Tmoats=720 (60yr) comlpexity: (361*360/2+360*360)*N*I=194,580*N*I• Interest only? One factor only? • Historical dependence dropped, “calibrated” later…

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0.02 0.040.06

0.080.1

020

4060

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

time to Tmoats (quarterly)

interest rate

MO

AT

S m

ortg

age

rate

s

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MOATS convergence

0.04

0.05

0.06

0.07

0.08

0 30 60 90 120 150 180 210 240

time to Tmoats (quarterly)

MO

AT

S m

ortg

age

rate

s

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MOATS convergence

0.054

0.059

0.064

0.069

0.074

0.079

0.084

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

time: 45/ term:15

time: 30/ term:30

time:22.5/term:30

time: 15/term:30

time: 0 / term:30

interest rate

MO

AT

S m

ortg

age

rate

s

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MOATS convergence (interest only)

0.054

0.059

0.064

0.069

0.074

0.079

0.084

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

time: 45/term:15

time: 30/term:30

time:22.5/term:30

time: 15/term:30

time: 0/ term:30

MO

AT

S m

ortg

age

rate

s

interest rate

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Endogenous Mort Rate Iteration

• The result of L()-estimation is used at x=r only, other values discarded?• Curse of dimensionality with growth of r-dimension?

• mi+1() requires estimation of L() for “every” r?

0

0

( ( ), ( ))

00

( ( ), ( ))

00

( , ( ))

( )

( , ( ))

s

s

T - r m r m r dθ

s

T - r m r m r dθ

x r

r P s m r e ds r x

m r

P s m r e ds r x

E

E

1( ) , ( ) ( )i i ix r

m r L x m r m

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Mortgage Rate “Iterations”

r0 r1 r2

3( )im r

r3

3( )m r

1( )m r0( )m r

4( )im r

2( )m r

4( )m r

1( ) ( , ( ) ( ) )i i im r L r m r m

r4

Fix r then solve for m(r):

the same…

Refinancing region controlled

( , )L r m

r0 r1 r2

3m

r3

4m

1m0m

2m

40r

4ir

4r

0km m k m

Fix m then solve for r(m):

1( , ( ) )i im L r m m

the same!

r0 r1 r2

3m

r3

4m

1m0m

2m

40r

4ir

4r

0km m k m

Fix m then solve for r(m):

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Computation with Level Sets

1x

2x

1m

2m

1 2( , )m x x

rr0 r1 r2

3m

r3

4m

1m0m

( )m r

2m

40r 4

ir

0km m k m

3m4m

, ( )m x m L x m m

4r

• No need for iterations if

• The conditional expectation in L()

is used on a hypersurface (level set),

i.e., “waste of one dimension” only• Number of L()-estimations is

independent of the

dimension/number

of the underlying factors

"transaction costs"m£V

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Conclusion Endogenous mortgage rate is defined

far from or implied by 10yr Treasury yield accented nonlinear behavior

MOATS transparent definition efficient implementation convergence to MRB is shown

A general ‘level set’ method is proposed flexibility of implementation: [RQ]MC or PDE reduces/eliminates the burden of iterations complexity of the same order as the underlying problem efficient and simple for the computation of implied

mortgage rate given any prepayment model