[PDF] - Lecture 4 Finance Pro ject Somesh Jha 1 Lo w-Lev el Design PDF Document

SLIDE 1 Lecture 4 Finance Pro ject Somesh Jha 1

SLIDE 2 Lo w-Lev el Design do cumen t

W

e will call the Lo w-Lev el Design do cumen t LLD from no w

n.
Be

v ery detailed. Programming should b e a v ery small step after this do cumen t is

v

er.

State

the language upfron t. W e will b e using JAVA and hence Ob ject-Orien ted Programming.

Men

tion eac h

b

ject with its purp

se

and description

f

the constructors and the metho ds whic h can b e called from

utside

(these will b e the public metho ds in JA V A). 2

SLIDE 3 LLD (Con td)

Order

y

ur
b

jects b y uses relation. F

r

example, if O 1 is used b y

b

ject O 2 describ e O 1 rst. In case

f

recursion, pic k an

rder

arbitrarily .

Indicate

if a class extends some

ther

class

r

if a class is going to b e abstract. Also, giv e a short rationale wh y did y

u

c ho

se

to mak e a certain class abstract.

Revisit

the LLD while/after doing th implemen tation. Y

u

will b e stra ying from the design a little bit. W e will use LLD and additional information to understand and test y

ur

co de. 3

SLIDE 4 Logistics

Due

Date: Feb 12,1999.

Remem

b er more detail the b etter. This will mak e y

ur

job while implemen tation v ery easy .

Also

try to distribute w

rk

so that dieren t p eople w

rk
n

descriptions

f

dieren t

b

jects.

Men

tion clearly who is resp

nsible

for whic h

b

ject. The p erson describing the design

f

an

b

ject will also implemen t it. In the implemen tation the author

f

eac h

b

ject should b e men tioned v ery clearly in the header les. 4

SLIDE 5 Logistics (Con td)

There

should b e

ne

system in tegrator who will tak e descriptions/implemen tation

f

the

b

jects and see whether they all t together. The main loop

f

the program will b e put together b y the system in tegrator.

Men

tion the roles

f

the team mem b ers in the do cumen t.

After

a team mem b er implemen ts an

b

ject, another team mem b er should r eview the co de. Y

u

will b e surprised ho w man y silly errors y

u

will catc h during review. Men tion the review er in the header le with the

b

ject (along with the author

f-course).

5

SLIDE 6 Implemen tation

Due

date: March 1, 1999.

Mak

e sure y

u

ha v e clear instructions describing ho w to use the system.

State

limiting assumptions y

u

made while implemen ting.

Please

giv e a phone n um b er

f

a p erson w e can call in case w e ha v e dicult y running y

ur

system. This p erson should preferably b e the system in tergrator b ecause he/she has the

v

erall idea ab

ut

y

ur

system. 6

SLIDE 7 MortgagePool

b

ject

Obje

ctName: MortgagePool

b

ject.

Extends:

Object.

Implements:

None.

Uses:

None.

Constructor

A single constructor whic h tak es v arious parameters

f

the mortgage p

l.

Please see Lecture 2. Assumption: I am going to assume a homogeneous mortgage p

l.

7

SLIDE 8 MortgagePool

b

ject (Con td)

Metho

d: double[] cashFlows(double SMMS[]) T ak es as parameter arra y

f

SMMs for v arious times and returns arra y

f

cash-o ws for eac h time up=-to the lifetime

f

the mortgage. Assume that arra y

f

SMMs has same size as the lifetime

f

the mortgage p

l.
Metho

d: double next-cash-flow(double SMM) This allo ws the

b

ject to b e used in an iter ative mo de. Whenev er, this metho d is called the cash-o w in the curren t time p erio d is returned and the curren t time p erio d in the mortgage p

l

is incremen ted. 8

SLIDE 9 The parameter SMM determines the prepa ymen t for this time p erio d. 9

SLIDE 10 BondObject

b

ject

Name:

BondObject.

Extends:

Object.

Implements:

None.

Uses:

None.

Constructor

Name

f

the le/database (with the b

nd

data) is passed to the constructor. W e will assume that the b

nd

data is in a le with all the required quan tities. 10

SLIDE 11 BondObject

b

ject (Con td)

Metho

ds: double yield(t,T) Yield at time t

f

a zero-coup

n

b

nd

maturing at time T . double price(t,T) Price at time t

f

a zero-coup

n

b

nd

maturing at time T . double volatility(t,T) V

latilit

y at time t

f

a zero-coup

n

b

nd

maturing at time T .

Note:

This

b

ject will b e used in pricing MBS with deterministic cash-o ws. Please see Lecture 3 for a closed form expression. 11

SLIDE 12 NormalRandom

b

ject

Name:

NormalRandom.

Extends:

None.

Uses:

java.util.Random.

Constructor:

T ak e the mean and v ariance as parameters and record it in ternally . W e will generate t w

n

um b ers with Normal distribution with mean m and v ariance v. 12

SLIDE 13 NormalRandom

b

ject (Con td)

Metho

d: double[] nextRandom() Generate two random n um b ers with standard normal distribution. Metho d w e will follo w is due to Bo x-Muller-Marsagli a. Please see next slide for a description

f

the metho d. Before returning the random n um b ers apply the appropriate transform to matc h the mean and the v ariance giv en in the constructor. Using the follo wi ng transformation (x + m) p v 13

SLIDE 14 Bo x-Muller-Marsaglia metho d .

This

algorithm is also called the p

lar

metho d.

Step

1 Generate t w

random

v ariables U 1 and U 2 (use the metho d nextDouble) in the class java.util.Random. T ransform these v ariables according to the equations giv en b elo w: V 1 = 2U 1

1

V 2 = 2U 2

1

14

SLIDE 15 Bo x-Muller-Marsagalia (Con td)

Step

2 Compute S according to the equation giv en b elo w: V 2 1 + V 2 2

Step

3 If S

1,

go to step 1.

Step

4 Return X 1 and X 2 giv en b y the equations: X 1 = V 1 v u u t 2 ln S S X 2 = V 2 v u u t 2 ln S S

Note:

Steps 1 through 3 are executed 1.27 times

n

the a v erage with standard deviation

f

0.587. So w e are not returning to Step 1 to

man

y times. 15

SLIDE 16 InterestRate

Name:

InterestRate.

Extends:

None.

Implements:

None.

Uses:

NormalRandom.

Constructor

W e will use the Co x-Ingersoll-Ro ss mo del. Constructor will tak e all the parameters as argumen ts. Please see the SDE giv en b elo w: dr = (

r

)dt +

p

r dW The parameters , ,

,

and the initial short rate r are passed to the constructor. Assumption: The mo del has already b een calibrated. 16

SLIDE 17 InterestRate

b

ject (Con td)

Metho

d void instantiate(double N,double T) P arameter T is the time horizon. N is the n um b er

f

discrete time steps w e will divide the time in terv al [0,T] in to. Assumption: W e assume that T is in mon ths and N has gran ularit y

f

at-least a mon th.

Metho

d: double[] nextPath() Generates a random path where the t-th elemen t in the arra y is the short rate at the t-th time. Let h b e the step size giv en b y 17

SLIDE 18 the follo wi ng expression: T N Generate random path according to the follo wi ng recurrence equation: r (i + 1) = (

r

(i))h +

s

r (i) N (0; h) where r (i) is the short rate at the discrete time step i and N (0; h) is a random n um b er with normal distribution (mean and v ariance h). Use metho d in

b

ject NormalRandom is used to generate this n um b er. 18

SLIDE 19 PrePayment

b

ject

Name:

PrePayment.

Extends:

None.

Implements:

None.

Uses:

None.

Constructor

P ass a ag indication whic h

ption
f

prepa ymen t function is going to b e used (see Lecture 3 for explanation

f

the

ptions).

F

r
ption

A pass a v ector

f

PSAs and for

ption

B pass the v arious parameters

1

;

2

;

3

. 19

SLIDE 20 PrePayment (Con td)

Metho

d: double[] smmVector(int T) Use

nly

with

ption

A. Returns the v ector

f

SMMs upto time horizon T .

Metho

d: double smmRandom(double pi,double rf7, double burnout,double season) Giv es the random SMM giv en the required parameters. Please see Lecture 3 for the explanation. Need a random n um b er with P

isson

Distribution (Haven't described it here). 20

SLIDE 21 PassThrough

Name:

PassThrough.

Extends:

None.

Implements:

None.

Uses:

MortgagePool, InterestRate, BondObject, and PrePayment.

Constructor

P ass an

b

ject

f

t yp e MortgagePool, InterestRate, Prepayment, BondObject, and time horizon to the constructor. MortgagePool is the underlying mortgage p

l

for the pass-through securit y . 21

SLIDE 22 PassThrough (Con td)

Metho

d double priceDeterministic() If the Prepayment generates deterministic SMMS, use the closed form expression giv en in Lecture 3.

Metho

d double price() Determine if the prepa ymen t mo del is deterministic

r

random. If prepa ymen t mo del is deterministic call metho d priceDeterministic(). If prepa ymen t mo del is random (option B) use mon te-carlo sim ulation. Generate a co v ariate based

n

the general tec hnique describ ed in Lecture 3. Price using mon te-carlo sim ulation. Use 22

SLIDE 23 metho d nextPath() in the

b

ject

f

t yp e InterestRate. 23

SLIDE 24 CMOobject

Name:

CMOobject.

Extends:

None.

Uses:

MortgagePool, InterestRate, BondObject, PrePayment.

Implements:

None.

Constructor

P ass an

b

ject

f

t yp e MortgagePool, InterestRate, BondObject, Prepayment and the time horizon to the constructor. MortgagePool is the underlying mortgage p

l

for the pass-through securit y . Also pass the n um b er

f

tranc hes and par-v alue

f

eac h tranc h to the constructor.

Metho

d 24

SLIDE 25 double[] priceDeterministic() If the PrepaymentObject generates deterministic SMMS, use the closed form expression giv en in Lecture 3. Returns price

f

eac h tranc h.

Metho

d double[] price() Determine if the prepa ymen t mo del is deterministic

r

random. If prepa ymen t mo del is deterministic call metho d priceDeterministic. If prepa ymen t mo del is random (option B) use mon te-carlo sim ulation. Generate a co v ariate based

n

the general tec hnique describ ed in Lecture 3. Price using mon te-carlo sim ulation. Rep

rt

price

f

eac h tranc h. 25

SLIDE 26 StrippedMBS

Name:

StrippedMBS.

Extends:

None.

Uses:

InterestRate, MortgagePool, PrePayment, and BondObject.

Implements:

None.

Constructor

P ass an

b

ject

f

t yp e MortgagePool, InterestRate, Prepayment, BondObject, and time horizon to the constructor. MortgagePool is the underlying mortgage p

l

for the stripp ed MBS.

Metho

d double[] priceDeterministic() 26

SLIDE 27 If the PrepaymentObject generates deterministic SMMS, use the closed form expression giv en in Lecture 3. Returns the price

f

PO and IO class.

Metho

d double[] price() Determine if the prepa ymen t mo del is deterministic

r

random. If prepa ymen t mo del is deterministic call metho d priceDeterministic. If prepa ymen t mo del is random (option B) use mon te-carlo sim ulation. Generate a co v ariate based

n

the general tec hnique describ ed in Lecture 3. Price using mon te-carlo sim ulation. Rep

rt

price

f

PO and IO class. 27

SLIDE 28 Describ e the

w
Step

1: Ask the user for parameters

f

the underlying mortgage p

l.

Create a MortgagePool

b

ject.

Step

2: Ask the user the prepa ymen t

ption

he/she w an ts to use. Instan tiate an

b

ject

f

t yp e PrePayment

b

ject.

Step

3: Create

b

jects

f

t yp e PrePayment and BondObject.

Step

4: Ask the user what MBS he/she w an ts to price. Instan tiate a PassThrough, CMOobject,

r

StrippedMBS

b

ject based

n

this.

Step

5: Call the metho d price() in the 28

SLIDE 29 required MBS

b

ject.

Step

6: Return the result to the user. 29

SLIDE 30 Discretizing an SDE

SDE

stands for sto c hastic dieren tial equation.

Supp
se

y t follo ws the SDE giv en b elo w: dy t = (y ; t)dt +

(y

; t)dW t

Drift

term is (y ; t) and the v

latili

t y term is

(y

; t). 30

SLIDE 31 Goal

Our

goal is to build a lattic e structure corresp

nding

to the pro cess y t .

Supp
se

w e are

nly

in terested in time-in terv al [0; T ].

Step

1: Discretize File: slides-4.tex the in terv al in to N time steps. Eac h step-size is

f

the size h = T N . The discrete time steps are [0; h; 2h;

;

N h]. 31

SLIDE 32 Goal (Con td)

Step

2: Supp

se

w e are at a no de in the lattice where the pro cess has v alue y . Successors

f

the lattice and the probabilit y

n

edges is giv en b y the follo wi ng equations: Y + h = y + p h (y ; t) Y

h

= y

p

h (y ; t) q h = 1 2 + p h (y ; t) 2 (y ; t)

Y

+ h and Y

h

is the v alue

f

the pro cess in the up and down no des resp ectiv ely .

Probabilit

y

f

an up-move is q h . 32

SLIDE 33 Problem (no recom bination)

T
tal

displacemen t for up-move follo w ed b y down-move is: p h[ (y ; t) +

(Y

+ h ; t + h)]

T
tal

displacemen t for down-move follo w ed b y up-move is: p h[ (y ; t) +

(Y

+ h ; t + h)]

In

general the t w

quan

tities are not equal.

No

r e c

mbination.

W e w an t a recom bining lattice.

When

do es the lattice recom bine? 33

SLIDE 34 When do es it recom bine?

When

the v

latilit

y

f

y (giv en b y

(y

; t)) is constan t.

Basic

ide a: T ransform Y to a new pro cess X with constan t v

latili

t y .

Dene

X (y ; t) as follo ws: Z y 1

(Z

; t) d Z

Assuming

that X (y ; t) is t wice dieren tiable in y and

nce

in t, w e can use Ito's lemma 34

SLIDE 35 to sho w that X (y ; t) satises the follo wing SDE: dX (y t ; t) =

X

(y t ; t)dt + dW t F

llo

wing equalit y should b e easy to see: @ X @ y = 1

(y

; t) The v

latilit

y term in the SDE for X is giv en b y the follo wi ng form ula: @ X @ y

(y

; t) 35

SLIDE 36 Basic Idea

Also

assume that w e can invert X , i.e. there exists a function Y (x; t) suc h that: Y (X (y ; t); t) = y

Most
f

the time w e will

nly

consider cases where w e can nd analytic expressions for X and Y . This is the case in the example w e will consider. 36

SLIDE 37 General Algorithm

Warning:

I am glossing

v

er lot

f

tec hnical details. Will giv e reference at the end.

Build

a lattice for the tr ansforme d X pro cess. The lattice for X is recom bining.

Using

the in v erse Y function to deriv e a lattice for y pro cess. It is not as simple as this. There are few tec hnicalities. 37

SLIDE 38 Recom bining lattice for CIR

The

SDE for the short-rate in the Co x-Ingersoll-Ross mo del (kno wn as CIR) from here

n

is giv en b y the follo wi ng SDE: dr = (

r

)dt +

p

r dW

Initial

v alue

f

the short rate is r .

Goal:

T

build

a recom bining lattice for the CIR mo del.

T

ransform the r pro cess X (r ) = Z r 1

p

Z d Z = 2 p r

38

SLIDE 39 CIR (Con td)

If
0,
0,

and r > 0, then is lo w er b

undary

for r (This can b e pro v ed formally). In terest rate can nev er go negativ e. This is an attractiv e feature

f

the CIR mo del.

In

v erse transform for the X pro cess is: R (x) = 8 > > > > > < > > > > > :

2

x 2 4 if x >

therwise
W

e nev er w an t R (x) (whic h is the short rate) to go negativ e. Therefore w e ha v e the

therwise

clause.

Is

there me an r eversion in the mo del? 39

SLIDE 40 Algorithm

Build

the lattice for the X pro cess. Deriv e SDE for X using Ito's lemma. Use the simple construction.

A

t eac h no de in the X

lattice

w e ha v e the v alue

f

X . W e w an t to transform this lattice in to the lattice for short-rate.

Supp
se

w e are at a no de with the v alue

f

X pro cess x. The r v alue corresp

nding

to this is giv en b y

2

x 2 4 . 40

SLIDE 41 Algorithm (Con td)

No

w w e ha v e to decide the successors

f

no de with short rate R (x) and also w an t to decide the probabilit y

f

up-mo v e. These quan tities are giv en b y the follo wi ng equation: x + h = x + J + h (x) p h x

h

= x + J

h

(x) p h R

h

= R (x

J
h

p h)

The

probabilit y q h

f

making an up move is if R + h (x) = and if R + h (x) > the expression for q h is giv en b elo w: h(

R

(x)) + R (x)

R
h

(x) R + h (x)

R
h

(x)

W

e need to c hose the jump sizes J

h

(x) 41

SLIDE 42 suc h that that the follo wi ng constrain ts are satised: (L e gal pr

b

ability):

q

h (x)

1

L

c

al drift c

nver

genc e: As n um b er

f

p erio ds N tends to innit y , the lo cal drift should con v erge to the drift in the SDE. 42

SLIDE 43 Constrain ts

L

e gal pr

b

ability F

llo

wing equations ha v e to hold: q h

1

h(

R

(x)) + R (x)

R

+ h q h

R
h
h(
R

(x)) + R (x)

L
c

al drift Lo cal drift

f

the short-rate has to matc h with the drift in the SDE. F

llo

wing equation is trivially true: q h R + h + (1

q

h )R

h
R

(x) = h(

R

(x)) 43

SLIDE 44 Jump sizes

The

argumen t to deriv e jump-sizes is quite tec hnical. See the reference.

J

+ h (x) is giv en b y: the smallest,

dd,

p

sitiv

e in teger j suc h that 4h

2

+ x 2 (1

h)

< (x + j p h) 2

J
h

(x) is giv en b y: the smallest,

dd,

p

sitiv

e in teger j suc h that 4h

2

+ x 2 (1

h)
(x
j

p h) 2

r

x

j

p h

0.

44

SLIDE 45 Bac k to MBS

Assume

that w e ha v e built the lattice mo del for the CIR mo del. W an t to price MBSs

n

it.

In
rder

to price MBSs need SMMs

n

all the no des

f

the lattice. Once w e ha v e the SMMs w e can use the Hull-White metho d (P ap er 1) to price MBSs.

Hull-White

use a simple prepa ymen t mo del where the SMM

nly

dep ends

n

the in terest rate at the no de. The mo del is simple and unrealistic.

W

e will use sim ulation to estimate SMMs at eac h no de in the lattice. 45

SLIDE 46 MBSs

n

a lattice

Generate

M random paths through the lattice.

On

eac h no de in the path calculate the SMMs. This will b e estimated from the prepa ymen t mo del.

Let

us sa y a no de N is touc hed k times during the sim ulation run. Let S M M 1 ;

;

S M M k b e the SMMs at the no de N for these k paths. S M M at no de N is giv en b y the follo wi ng equation: 1 k k X j =1 S M M i 46

SLIDE 47 MBSs

n

a lattice (Con td)

There

are some no des that will not b e touc hed b y the sim ulation runs. What do w e do? P erform in terp

lation/extrap
lation

to nd the S M M s at the no de.

Supp
se

there is a no de N that is not touc hed b y the sim ulation runs. Find t w

ne

ar est no des N U and N L suc h that the follo wi ng inequalit y holds: N U (r )

N

(r )

N

L (r ) Short rate at no de N is denoted b y N (r ).

SMM

for no de N is giv en b y the follo wi ng equation: S M M (N L )+ (N (r ))(S M M (N U )

S

M M (N L )) 47

SLIDE 48 Where (N (r )) is giv en b y the follo wing equation: (N (r )) = N (r )N L (r ) N U (r )N L (r )

What

if w e can't nd N L

r

N U , use extrap

lation.
Rest
f

the metho d same as P ap er 1. 48

SLIDE 49 Hull-White metho d

A

t eac h no de in the in terest-rate lattice store M B max and M B min .

Interpr

etation M B max (M B min ) is the maxim um (minim um) mortgage balance that can b e realized at that no de. Notice that there are man y paths leading upto a no de. On eac h

f

these paths the mortgage balance

n

a no de can b e dieren t.

F
rwar

d Induction A t the ro

t

no de M B max and M B min are b

th

the same (equal to the Mortgage balance). 49

SLIDE 50 F

rw

ard Induction (Con td)

Supp
se

w e are going to compute M B max and M B min for a no de N . Let N 1 and N 2 b e the predecessors

f

this no de.

Let

maxim um and minim um mortgage balances at no des N 1 and N 2 b e giv en b y the follo wi ng quan tities: M B max ;1 , M B min ;1 , M B max ;2 , M B min ;2 ,

M

B max is giv en b y the maxim um

f

the follo wi ng quan tities: S M M (N )(M B max ;1

S

max ;1 ) S M M (N )(M B max ;2

S

max ;2 )

Similar

explanation applies to M B min . 50

SLIDE 51 Hull-White metho d (Con td)

A

t eac h no de N w e ha v e M B max and M B min (the maxim um and minim um p

ssible

mortgage balance at that no de).

If

M B min < M B min , split in to m equally spaced v alues.

W

e will write a no de as (r ; M B ; S M M ) where r is the short rate, M B the mortgage balance at that no de, and S M M determines the prepa ymen t at that no de.

Let

the successors

f

the no de (r ; M B ; S M M ) b e (r + ; M B + ) and (r

;

M B

).
The

v alue

f

the MBS at no de (r ; M B ; S M M ) is giv en b y the bac kw ard 51

SLIDE 52 equation: V (r ; M B ; S M M ) = 1 1+r (C F + q V (r + ; M B + ) + (1

q

)V (r

;

M B

)
Pr
blem

No des (r + ; M B + ) and (r

;

M B

)

migh t not exist.

Solution

Estimate M B + and M B

using

in terp

lation.

F

r

example, nd t w

no

des (r L ; M B L ) and (r U ; M B U ) suc h that r L and r U are nearest to r and the follo wi ng equation is true: r L

r
r

U Estimate M B + b y in terp

lation.

52

SLIDE 53 Additional Information

F
rm

ula giv en in Lecture 3 w as correct.

D.B.

Nelson and K. Ramasw am y , Simple Binomial Pro cesses as Diusion Appro ximations in Financial Mo dels, The R eview

f

Financial Studies, V

l

3, No 3, 1990. 53