[PDF] - Lecture 3 Finance Pro ject Somesh Jha 1 Maxim um Lik eliho PDF Document

SLIDE 1 Lecture 3 Finance Pro ject Somesh Jha 1

SLIDE 2 Maxim um Lik eliho

d

estimation (MLE)

Assume

that w e ha v e N t mortgages at time t in the mortgage p

l.
The

n um b er

f

pre-pa ymen ts at time (denoted b y c t ) follo ws a P

isson

distribution. 2

SLIDE 3 MLE (Con td)

F
rmally

, probabilit y P (c t = k ) that c t is equal to k is: e (X t ; )N t ((X t ;

)N

t ) k k

Chec

k that the follo wi ng is true: 1 X k =0 k P (c t = k ) = (X t ;

)N

t 3

SLIDE 4 F

rm
f

(X t ;

)
Recall

that (X t ;

)

has the follo wi ng form:

(t)

exp ( 1

r

f 7+

2
ln

(bur nout) +

3
season)
Meaning
f

eac h co v ariate is giv en b elo w: r f 7 (Renancing

pp
rtunities)
(t)

(Age) season (seasonal) bur nout (burnout) 4

SLIDE 5 Problem

Supp
se

w e a ha v e p

l
f

mortgages that is similar to the p

l

underlying the MBS w e are trying to price.

W

e ha v e historical data ab

ut

this mortgage. 5

SLIDE 6 Problem (Con td)

W

an t to nd parameters

1

;

2

;

3

that b est t this historical data.

W

e will use a tec hnique called Maxim um Lik eliho

d

Estimation (or MLE) for this purp

se.

6

SLIDE 7 Basic idea

f

MLE

Assume

a distribution (w e assume P

isson

distribution for the n um b er

f

prepa ymen ts).

Estimate

the probabilit y f ( )

f
bserving

the historical data. 7

SLIDE 8 Basic idea

f

MLE (Con td)

The

lo g likeliho

d

function (denoted b y L( )) is log f ( ).

The

parameters

are

giv en b y the solution to the follo wi ng global-opti mization problem: max

L(

) 8

SLIDE 9 MLE (Con td)

Assume

that w e ha v e historic pre-pa ymen t data for a p

l
f

mortgages.

Also

assume that the n um b er

f

prepa ymen ts at time t

nly

dep ends

n

the n um b er

f

mortgages in the p

l

at time t and is indep enden t

f

the history . 9

SLIDE 10 MLE (Con td)

History

is giv en for times 1; 2;

;

T , and the follo wi ng things are giv en: { c t (Num b er

f

pre-pa ymen ts at time t). { N t (n um b er

f

mortgages remaining in the p

l).
T

is the lifetime

f

the mortgage p

l

under consideration. 10

SLIDE 11 MLE (Con td)

Probabilit

y P (c t ) that the n um b er

f

prepa ymen ts is c t at time t is: e (X t ; )N t ((X t ;

)N

t ) c t c t !

The

probabilit y

f
bserving

the en tire history (using indep endence here) is: f ( ) = T Y t=1 P (c t )

Log

lik eli ho

d

function L( ) is: T X t=1 (c t ln(N t )

N

t

ln(c

t !))

F
r

notational con v enience, In the expression for L( ) I ha v e suppressed X t and

.

11

SLIDE 12 MLE (con td)

The

factor ln (c t !) is a constan t so w e ignore it in the maximization problem.

W

e ha v e to maximize the follo wing function with resp ect to

(I

ha v e suppressed the X t and

factors

for notational con v enience): T X t=1 (c t ln (N t )

N

t )

Next

w e discuss a metho d for maximization. 12

SLIDE 13 Steep est Ascen t

Let

L( ) b e the log lik eli ho

d

function.

Recall

that

is

v ector

f

three parameters ( 1 ,

2

,

3

).

The

gr adient ve ctor

f

the log lik eli ho

d

function is (denoted b y @ L( ) @

)

is: 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @ L

1

@ L

2

@ L

3

3 7 7 7 7 7 7 7 7 7 7 7 7 5

In

tuitiv ely , the gradien t v ector at

(denoted

b y g ( )) is the direction in whic h 13

SLIDE 14 the log lik eliho

d

function increases most steeply (at the p

in

t

).

14

SLIDE 15 Steep est Ascen t (Con td)

Cho
se

an initial v ector

.
Let
i1

b e the

ld

estimate. The new estimate

i

is giv en b y the follo wing equations:

i
i1

= cg ( i1 ) k i

i1

k = k

k

is the step size. Notice that c is determined b y the equations giv en ab

v

e. Norm

f

the v ector

i
i1

is denoted b y k i

i1

k. 15

SLIDE 16 Problems with Steep est Ascen t

Con

v ergence v ery slo w near a lo cal maxim um.

V

ariet y

f

metho ds for n umerical

ptimization.
Judge,

George G., Willia m E. Griths, R. Carter Hill, and Tsoung-Chao Lee. 1980. The The

ry

and Pr actic e

f

Ec

nometrics.

New Y

rk:

Wiley .

Quandt

Ric hard E. 1983. \Computational Problems and Metho ds," in Zvi Grilic hes and Mic hael D. In triligator editors., Handb

k
f

Ec

nometrics,

V

l

1. Amsterdam : North-Holland. 16

SLIDE 17 In teresting exercise

Assume

that sto c k prices follo w the ge

metric

br

wnian

motion.

Using

MLE estimate the drift and v

latili

t y

f

the sto c k.

Historical

prices for man y sto c ks are a v ailable

n

n umerous w eb-sites. 17

SLIDE 18 In teresting exercise (Con td)

Using

prices

f

v arious

ptions
n

the sto c k nd the implie d volatility curve.

Ho

w far is the implie d volatility curve a w a y from the MLE estimate?

Let

me kno w if y

u

try this exercise. 18

SLIDE 19 Summary

f

MBS cash-o ws

M

P t (mortgage pa ymen t at time t)

I

t (in terest pa ymen t at time t)

P

t (principal pa ymen t at time t)

P

P t (prepa ymen t at time t)

S

t (service c harge at time t)

N

I t (net in terest rate at time t)

M

B t (mortgage balance at time t)

C

F t (cash-o w at time t)

S

M M t (Single Mon thly Mortalit y Rate at time t) 19

SLIDE 20 Summary (Con td)

M

P t is equal to M B t1 c(1 + c) nt+1 (1 + c) nt+1

1
I

t , S t , P t , and N I t follo w the equations giv en b elo w: I t = cM B t1 S t = sM B t1 P t = M P t

I

t N I t = I t

S

t

M

B t is giv en b y the follo wi ng expression: M B t1

P

t

P

P t 20

SLIDE 21 Summary (Con td)

C

F t is giv en b y the follo wi ng form ula: N I t + P t + P P t

S

M M t is

btained

from the pre-pa ymen t mo del.

Prepa

ymen t P P t at time t is giv en b y the follo wi ng equation: S M M t (M B t1

S

t ) 21

SLIDE 22 P ass-throughs

Supp
se

a pass-through

wns

x p ercen t

f

the mortgage p

l.
Cash
w
f

the pass-through at time t is giv en b y the follo wi ng equation: C F t x 100 22

SLIDE 23 CMOs

Supp
se

there are m tranc hes T 1 ;

;

T m with par-v alues P 1 ;

;

P m .

A

t time t let the remaining par-v alue

f

tranc h T i b e P t i .

Let

j b e the least n um b er suc h that T j is not retired.

The

cash-o w

f

that tranc h is: P t1 j M B t1 I t + P t + P P t 23

SLIDE 24 CMOs(Con td)

The

new par-v alue P t j

f

tranc h T j is: P t1 j

P

t

P

P t

If

P t1 j is equal to zero, r etir e the tranc h T j .

F
r

all tranc hes T i suc h that i > j the cash-o w is P t1 i M B t1 I t

P

t i is equal to P t1 i (Wh y?) 24

SLIDE 25 Stripp ed MBSs

The

P O class gets P t + P P t min us the servicing fee.

The

I O class gets I t min us the servicing fee. 25

SLIDE 26 High-Lev el Design Do cumen t

Query

Phase Describ es the steps in whic h the user in teracts with the system. User c ho

ses

what instrumen t he/she w an ts to price and the v arious parameters.

Computation

Phase High-lev el pro cedure to price these instrumen ts. Pro vide a description

f

the general tec hnique y

u

are using (induction

n

lattices, sim ulation, nite-dierence sc hemes).

Pr

esentation Phase What is the result presen ted to the user. 26

SLIDE 27 Ho w is the result presen ted to the user. 27

SLIDE 28 Query Phase

Ask

the user what kind

f

MBS they need to price.

P

ass-throughs, CMOs,

r

Stripp ed MBSs.

Ask

the parameters

f

the mortgage p

l

asso ciated with the MBS (for description

f

parameters please see Lecture 1).

In

case

f

CMOs ask the follo wing questions: { Num b er

f

tranc hes. { P ar-v alue

f

eac h tranc h. 28

SLIDE 29 Query Phase (Con td)

Ask

user ab

ut

prepa ymen t mo dels. Supp

rt

t w

kind
f

prepa ymen t mo dels.

Prepa

ymen t Option A V ector

f

PSA sp eeds (see page 41 Lecture 2).

Prepa

ymen t Option B P

isson

pro cess based mo del (see page 46 Lecture 2). Assumption: Assume that the mo del has b een calibrated. 29

SLIDE 30 P

in

ts to notice

Notice

that I ha v en't men tioned man y details (lik e ho w the in terface will lo

k

to the user).

Details

b elong in the lo w-lev el design do cumen t.

Lo

w-lev el design do cumen t will r ene eac h step in the high-lev el design do cumen t. 30

SLIDE 31 P

in

ts to notice (Con td)

Ha

v en't committed to tec hnology

r

metho dology .

I

ha v en't said whether w e are going to use JAVA, C++.

T

ec hnology c hoice made after the high-lev el design do cumen t.

Ha

v en't ev en said whether w e are going to use

bje

ct-oriente d, imp er ative,

r

functional programming.

These

decisions will b e made after the high-lev el design do cumen t. 31

SLIDE 32 Chec k for completeness

Chec

k that all the parameters y

u

need to price the instrumen ts are there.

Nothing

should b e missing. 32

SLIDE 33 Computation phase

W

e will split this phase in to t w

phases.
Determine

whic h prepa ymen t

ption

the user has giv en.

Dep

ending

n

the prepa ymen t

ption

the algorithm is v ery dieren t. 33

SLIDE 34 Wh y the splitting?

There

is a m uc h more ecien t algorithm to price MBSs in case

f

prepa ymen t

ption

A.

F
r

example, y

u

w

uld

not use Hull-White metho d (pap er 1) to price a lo

kbac

k

ption.
The

lattice for pricing a lo

kbac

k

ption

is smal l (refer bac k to data-structures notes). 34

SLIDE 35 Pricing pass-throughs (Option A)

Notice

that the cash-o w

f

the pass-through in this case is deterministic.

Let

C F t b e the cash-o w

f

the pass-through at time t.

No

randomness in C F t . 35

SLIDE 36 P ass-throughs (Option A)

The

price

f

the pass-through at time t is giv en b y the follo wi ng equations: P T t=1 E [C F t Q t1 j =0 1 1+r j ] P T t=1 C F t E [ Q t1 j =0 1 1+r j ]

Exp

ectation tak en with resp ect to the risk-neutral

r

martingale measure. T is the lifetime

f

the mortgage p

l

underlying the pass-through.

Do

y

u

recognize the follo wi ng quan tit y? E [ t1 Y j =0 1 1 + r j ] 36

SLIDE 37 P ass-through (Option A)

The

m ystery expression is the price at time

f

a zero-coup

n

b

nd

pa ying

ne

dollar at time t.

So

w e ha v e the follo wi ng form ula for v aluing the pass-through securit y in case

f
ption

A: T X t=1 C F t P (0; t)

P

(0; t) is the price

f

a zero-coup

n

b

nd

(at time 0) pa ying

ne

dollar at time t. 37

SLIDE 38 P ass-through (Option A)

Assuming

prices P (0; 1); P (0; 2);

;

P (0; T ) are

bserv

able from the mark et we are done.

Supp
se

the prices are

nly

kno wn for some times t 1 ; t 2 ;

;

t k .

Find

the missing prices using in terp

lation.
Notice

ho w fast the algorithm is. No sim ulation required. 38

SLIDE 39 CMOs (option A)

Price

eac h tranc h separately .

Find
ut

the lifetime

i
f

eac h tranc h T i (when it retires).

Use

the form ula giv en b elo w for tranc h T i

i

X t=1 C F t;i P (0; t)

C

F t;i is the cash-o w

f

tranc h T i at time t. 39

SLIDE 40 Stripp ed MBS (option A)

Price

PO and IO classes separately .

Use

the equation giv en b efore. 40

SLIDE 41 P ass-throughs (option B)

Use

Mon te-Carlo sim ulation to price the MBSs.

Assume

that w e ha v e a pro cedure called nextPath() whic h generates a random path.

Notice

that nothing is said ab

ut

the sp ecic in terest-rate mo del. That b elongs in the lo w-lev el design do cumen t.

High-lev

el design do cumen t

nly

describ es high-lev el algorithms and tec hniques. V ery little detail ab

ut

the actual 41

SLIDE 42 implemen tation. 42

SLIDE 43 P ass-throughs (option B)

Determine

ho w man y paths to generate (sa y N ).

Let
i

b e the i-th path and r t;i b e the short-rate at time t

n

path i.

Let

C F t;i b e the cash-o w

n

path

i

at time t.

Let

V i b e the v alue

f

this cash-o w at time (giv en b y the follo wi ng equation) T X t=1 C F t;i t1 Y j =0 1 1 + r t;i 43

SLIDE 44 P ass-throughs (option B)

Recall

that the i-th path is

i

.

V

alue

f

the pass-through at time (denoted it b y V P T ) is giv en b y the follo wi ng equation (a v eraging the v alues): 1 N N X i=1 V i 44

SLIDE 45 CMOs and Stripp ed MBSs

Only

the expression for cash-o ws c hange.

Ev

erything remains the same. 45

SLIDE 46 Mon te-Carlo (Con td)

Supp
se

w e generate N paths in the Mon te-Carlo sim ulation.

Let

! b e the standard deviation

f

the v alue

f

the nancial instrumen t calculated from the sim ulation runs.

The

error

f

the estimate calculated from the sim ulation runs is appro ximately ! p N . 46

SLIDE 47 V ariance reduction

There

are tec hniques to sp e e d up the con v ergence

f

the Mon te-carlo sim ulations.

One
f

suc h class

f

tec hniques is called varianc e r e duction.

W

e will consider a sp ecial case

f

v ariance reduction called c

ntr
l

variate te chnique. 47

SLIDE 48 Con trol v ariate tec hnique

Securit

y A is the securit y to b e priced.

Consider

a similar securit y B , whic h y

u

can price b y

ther

means (sa y lattice based

r

analytical tec hniques).

In

the sim ulation runs estimate the quan tit y V (A)

V

(B ).

V

(A) and V (B ) are the v alues

f

securities A and B resp ectiv ely . 48

SLIDE 49 Con trol-v ariate tec hnique (Con td)

Let

V ? b e the estimate

f

V (A)

V

(B ) calculated from the sim ulation.

Let

V true (B ) b e the v alue

f

securit y B calculated using

ther

means (lattice based

r

analytical).

The

estimate for v alue

f

securit y A is V ? + V true (B ). 49

SLIDE 50 In teresting exercise

This

is an in teresting exercise (esp ecially for studen ts doing P ap er 1).

Supp
se

w e are in terested in pricing an eur

p

e an asian

ption

with maturit y T and strik e price K .

Europ

ean asian

ption

is the primary securit y A in this case.

T

ak e y

ur

secondary securit y B as the europ ean geometric asian

ption

with exactly the same parameters.

Recall

that geometric asian

ption

dep ends up

n

the geometric a v erage

f

the sto c k price. 50

SLIDE 51 In teresting Exercise (Con td)

Price

the europ ean geometric

ption

using lattice based tec hniques. Call this price V (G) true .

Recall

that the lattice for pricing europ ean geometric

ption

w as cubic in the n um b er

f

p erio ds.

Estimate

the dierence

f

the asian

ption

and the geometric

ption.

Call this estimate V ? .

The

v alue

f

the asian

ption

is V ? + V (G) true . 51

SLIDE 52 MBS and v ariance reduction

What

is a securit y similar to an MBS? Another MBS.

Let

us sa y w e w an t to price an MBS A.

W

e will pic k a similar MBS B .

MBS

B will ha v e deterministic c ash-ows and hence can b e priced using the closed-form form ula giv en b efore. No sim ulation required. 52

SLIDE 53

Next

w e describ e ho w to pic k MBS B . 53

SLIDE 54 In searc h

f

MBS B

Supp
se

the mortgages in the mortgage p

l

are for T mon ths.

Pic

k r times t = 1

t

1 < t 2 <

<

t r = T .

Let

S M M i (for

i

< r ) b e the S M M for the p erio d [t i ; t i+1 ).

Go

al: T

pic

k S M M ;

;

S M M r 1 so that the prepa ymen t structure

f

MBS B is close to the prepa ymen t structure

f

MBS A (our

riginal

MBS). 54

SLIDE 55 Searc h con tin ues

Generate

M random paths.

F
r

path i let S M M 1;i ; S M M 2;i ;

;

S M M T ;i b e the sequence

f

S M M s for the

riginal

MBS (securit y A).

Calculate

the distance b et w een the sequence

f

SMMs and the SMMs

f

the securit y B . 55

SLIDE 56 Searc h con tin ues

The

distance is giv en b y the follo wi ng equation: d i = T X t=1 jS M M t;i

S

M M t;B j

S

M M t;B is the S M M for the securit y B w e are trying to construct.

d

i is a function

f

the v ariables S M M ;

;

S M M r 1 . 56

SLIDE 57 Searc h ends

Add

the distances

v

er all the M paths D = M X i=1 d i

Find

v ariables S M M ;

;

S M M r 1 b y solving the follo wing global

ptimization

problem: max S M M ;;S M M r 1 D (S M M ;

;

S M M r 1 ) 57

SLIDE 58 Presen tation Phase

Presen

t the price

f

the pass-through to the user.

In

case

f

CMOs presen t the price

f

eac h tranc h.

In

case

f

stripp ed MBS presen t the price

f

PO and IO classes.

Rep
rt

an y con v ergence problems, i.e., Mon te-carlo sim ulation didn't con v erge in the required n um b er

f

steps. 58

SLIDE 59 Sc hedule

High-lev

el do cumen t due date: F eb 3, 1999, W ednesda y .

No

need to divide in to sub-teams for this do cumen t.

P

a y close atten tion to the p

in

ts suggested. 59