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Transcript of 1 Recent Advances in Mathematical Finance Chicago, December 8, 2007 Yevgeny Goncharov Department of...
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
2
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
4
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
5
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.
6
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
7
Mortgage Modeling
, , etc...t tr X
Numericalimplementation
Calibration
t
Mortgage Model
t
8
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
10
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…
16
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
17
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
20
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):
22
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
23
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