Lecture 4 Finance Pro ject Somesh Jha 1 Lo w-Lev el Design - - PDF document

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Lecture 4 Finance Pro ject Somesh Jha 1 Lo w-Lev el Design - - PDF document

Lecture 4 Finance Pro ject Somesh Jha 1 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 on. Be v ery detailed. Programming should b e a v ery small


slide-1
SLIDE 1 Lecture 4 Finance Pro ject Somesh Jha 1
slide-2
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
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
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
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
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
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
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
SLIDE 9 The parameter SMM determines the prepa ymen t for this time p erio d. 9
slide-10
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
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
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
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
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
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
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
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
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
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
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
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
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
SLIDE 23 metho d nextPath() in the
  • b
ject
  • f
t yp e InterestRate. 23
slide-24
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
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
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
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
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
SLIDE 29 required MBS
  • b
ject.
  • Step
6: Return the result to the user. 29
slide-30
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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