Cont_IR_Models_One

Cont_IR_Models_One - Brownian
Mo*on
and
Con*nuous


Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Brownian
Mo*on
and
Con*nuous
 Time
Interest
Rate
Models
 (Part
One)
 Outline
 •  Brownian
Mo*on
 –  Defini*on
 –  Important
Proper*es
 –  Brief
introduc*on
to
stochas*c
calculus
 •  Con*nuous
Time
Interest
Rate
Models
 Brownian
Mo*on
 Stochas*c
Process
 •  A
con$nuous $me stochas$c process
is
a
 collec*on
of
random
variables,
one
for
each
 *me
t ≥ 0.
 X = { X t | t ∈ [0, ∞)} •  Common
examples
(building
blocks
of
most
 financial
models):

 € •  Typically,
we
assume
that
the
value
of
Xt is
 known
(revealed)
at
*me
t.
 –  Poisson
process,
Brownian
Mo*on
 Brownian
Mo*on
 •  A
con*nuous
*me
stochas*c
process
W
is
a
 Brownian
mo*on
(Wiener
process)
if:

 –  W0
=
0
 –  For
any
0≤
t0
<
t1
<
…
<
tN
the
random
variables
 W t − W t , n = 1, ..., N (increments):




































are
independent
 –  For
any
t
>
s:

 n n−1 € W t − W s ~ N (0, t − s) € Simula*ng
a
Brownian
mo*on
 •  Set
W0
=
0,
define
a
step‐size
h
and
number
of
 desired
*me
steps
N.
 •  For
n=1
to
N,
set:
 W nh = W( n −1)h + h ⋅ ξ n –  Where
ξn,
n=1,…,N
is
a
sequence
of
standard
 normal
random
variables.
 € Simula*ons
of
Brownian
Mo*on
 •  Note
the
jagged
paths.
 Proper*es
of
Brownian
Mo*on
 •  A
given
realiza*on
(sample)
of
the
stochas*c
 process
Wt,
t≥0
is
called
a
sample path.
 •  The
sample
paths
of
a
Brownian
mo*on
are
 con*nuous
func*on
of
t
(with
probability
1).
 •  Any
linear
transforma*on
of
a
Brownian
 X t = µ t + σW t mo*on
of
the
form:
























is
called
 Brownian
mo*on
with dri6 μ and vola$lity (diffusion) σ.

 € Proper*es
of
Brownian
Mo*on
 •  A
Brownian
Mo*on
W
is
a
Markov process.
For
 any
0≤
t0
<
t1
<
…
<
tN:
 P W tN ≤ x W t n = w t n , n = 1, ..., N − 1 = P W t N ≤ x W t N −1 = w t N −1 •  If
W
is
Brownian
mo*on
with
zero
dri[
and
 vola*lity
σ>0:
 € ( )( ) –  σ‐1Wt
is
a
standard
Brownian
mo*on.
 –  μt+Wt
is
a
Brownian
mo*on
with
dri[
μ
and
vola*lity
 σ.
 –  (Brownian
scaling).
For
λ
>
0,
the
following
process
is
 also
Brownian
mo*on
with
zero
dri[
and
vola*lity
σ.
 1 ˜ Wt = W λ⋅ t λ Proper*es
of
Brownian
Mo*on
 •  Brownian
mo*on
will
eventually
hit
(with
 probability
one)
every
real
value,
no
ma`er
how
 large
or
how
nega*ve.
 •  No
ma`er
how
far
above
or
below
the
axis,
the
 Brownian
mo*on
process
will
be
back
to
zero
at
 some
later
*me
with
probability
one.
 •  Once
BM
hits
a
value,
it
hits
it
again
infinitely
 o[en.
 •  Brownian
mo*on
is
a
fractal;
it
doesn’t
ma`er
 what
scale
you
examine
it
on,
it
looks
the
same.
 Proper*es
of
Brownian
Mo*on
 •  Calculus
with
Brownian
mo*on
is
more
complex
 because
of
the
jagged
nature
of
the
paths
of
 Brownian
mo*on.
 •  The
Brownian
paths
are
so
erra*c
that
we
can't
 make
sense
of
dW(t)
in
the
usual
way.

 •  Close
up:

 –  Differen*able
func*ons
look
like
lines
 –  Brownian
mo*on
looks
just
as
jagged.
 •  Sample
paths
of
Brownian
mo*on
are
 con*nuous,
nowhere
differen*able
func*ons
(of
 t).
 Proper*es
of
Brownian
Mo*on
 •  Sample
paths
of
Brownian
mo*on
have
 infinite
varia*on
on
bounded
intervals
[a,b].
 •  For
a
func*on
f
on
[a,b],
the
total
varia*on
is
 defined
to
be:

 N V( f ) = sup a = t 0 < t1 < t N = b n =1 ∑ f (t n ) − f ( t n −1 ) 
where
the
supremum
is
taken
over
all
 € par**ons
of
the
interval
[a,b].
 Proper*es
of
Brownian
Mo*on
 •  For
func*ons
whose
first
deriva*ve
exists
 everywhere,
we
have:

 V( f ) = ∫ b a df ( t ) = ∫ f ' ( t ) dt a b •  The
fact
that
Brownian
mo*on
has
infinite
 varia*on
implies:

 € •  This
in
turn
implies
that
one
cannot
define
 integrals
w.r.t.
Brownian
mo*on
simply
by
 € extending
the
usual
Riemann‐S*eltjes
integral:
 ∫ b a dW ( t ) = ∞ ∫ b a f ( t,ω ) dW t (ω ) = ? Proper*es
of
Brownian
Mo*on
 •  Brownian
mo*on
has
finite
quadra$c varia$on.
 •  A
sequence
of
random
variables
Xn with
finite
 variances
is
said
to
converge
in
L2
to
a
random
 variable
X
iff:
 lim E [( X n − X ) 2 ] = 0 •  For
any
par**on
Π:
a=
t0
<
t1
<
…
<
tN
=b
:


 N n →∞ € QV (W , Π) = ∑ (W t n − W t n−1 ) n =1 2 Proper*es
of
Brownian
Mo*on
 •  It
can
be
proved
that:
 
 
in
L2
as
m
tends
to
infinity,
where
Πm
is
any
 sequence
of
par**ons
such
that:

 QV (W , Π m ) → (b − a) € m →∞ t n ∈Π m lim max t n − t n −1 = 0 € Proper*es
of
Brownian
Mo*on
 •  The
problems
with
trying
to
do
calculus
with
 Brownian
mo*on
can
be
illustrated
with
the
 simple
integral
that
is
easy
in
the
smooth
case:
 ∫ € t 0 h ( s) dh ( s) = [( h ( t )) − ( h (0)) ] t 1 2 2 2 Let's try to do this with Brownian motion. ∫ W dW 0 s s =? Simple
Integral:
Le[
Endpoint
Rule
 N −1 ∑W n= 0 N −1 n= 0 nt N (W ( n+1 ) t − W nt ) N N N −1 1 2 2 2 N N = ∑ (W ( n+1 ) t − W nt ) − ∑ W ( n+1 ) t − W ( n+1 ) t ⋅ W nt + W nt 1 2 2 n= 0 N −1 N N N ( 1 2 2 N ) = 1 2 W t2 − W 02 ) − 1 ∑ (W ( n+1 ) t − W nt ) 2 ( 2 n= 0 N N N −1 = 1 W t2 − 1 ∑ (W ( n+1 ) t − W nt ) 2 2 2 n= 0 N N Simple
Integral:
Le[
Endpoint
Rule
 N −1 N −1 2 2 E ∑W nt (W ( n+1 ) t − W nt ) = E 1 W t − 1 ∑ (W ( n+1 ) t − W nt ) 2 2 N N N N N n = 0 n= 0 t1 = − ∑ E [(W ( n+1 ) t − W nt ) 2 ] N N 2 2 n= 0 t1 t =−∑ 2 2 n= 0 N =0 N −1 N −1 Simple
Integral:
Le[
Endpoint
Rule
 N −1 ∑ (W n= 0 ( n+1 ) t N t 2 − W nt ) = ∑ X n N N n= 0 2 N N −1 Where: Xn = (W ( n+1 ) t − W nt ) N € N −1 N →∞ t ~ N (0,1) are i.i.d. N The Law of Large Numbers gives: € lim ∑ (W n= 0 ( n+1 ) t N t 2 2 − W nt ) = lim ∑ X n = tE [ X 0 ] = t N N →∞ N n= 0 2 N −1 Simple
Integral:
Le[
Endpoint
Rule
 N −1 N →∞ lim ∑W nt (W ( n+1 ) t n= 0 N N N −1 2 2 − W nt ) = lim 1 W t − 1 ∑ (W ( n+1 ) t − W nt ) 2 N N N →∞ 2 N n= 0 = 1 2 W t2 − t ) ( " ∫ W sdW s" = 0 t (W − t ) (≠ (W − W ) = 1 2 2 t 1 2 2 t 2 0 1 2 Wt 2 ) Le[
Endpoint
Rule:
Review
 •  For
smooth
func*ons:
 ∫ € h ( s) dh ( s) = [( h ( t )) − ( h (0)) ] 0 t 1 2 2 2 •  For
Brownian
Mo*on:
 " ∫ W sdW s" = 0 t (W − t ) (≠ (W − W ) = 1 2 2 t 1 2 2 t 2 0 1 2 Wt 2 ) Simple
Integral:

 Right
Endpoint
Rule
 •  With
the
right
endpoint
rule,
we
get
a
totally
 different
result.
The
expecta*ons
aren't
even
 the
same!
 N −1 N −1 ( n+1 ) t N ∑ E [W n= 0 (W ( n+1 ) t − W nt )] = ∑ E [(W ( n+1 ) t − W nt + W nt )(W ( n+1 ) t W nt )] N N n= 0 N −1 N N N N N N −1 2 N N N N N = ∑ E [(W ( n+1 ) t − W nt ) ] + ∑ E [W nt (W ( n+1 ) t − W nt )] n= 0 N −1 n= 0 =∑ t +0=t n= 0 N But E[Wt2 - t] = 0! € Simple
Integral:

 Right
Endpoint
Rule
 N −1 N →∞ N N N N −1 N →∞ N N N N N lim ∑W ( n+1 ) t (W ( n+1 ) t − W nt ) = lim ∑ (W ( n+1 ) t − W nt + W nt )W ( n+1 ) t − W nt ) n= 0 n= 0 N −1 N →∞ N N N −1 N →∞ N N N = lim ∑ (W ( n+1 ) t − W nt ) 2 + lim ∑W nt (W ( n+1 ) t − W nt ) n= 0 n= 0 = t + 1 (W t2 − t ) 2 = 1 (W t2 + t ) 2 Right
Endpoint
Rule:
Review
 •  For
smooth
func*ons:
 t 0 ∫ € h ( s) dh ( s) = 1 [( h ( t )) 2 − ( h (0)) 2 ] 2 •  For
Brownian
Mo*on:
 " ∫ t 0 W sdW s" = (W + t ) (≠ (W − W ) = 1 2 2 t 1 2 2 t 2 0 1 2 Wt 2 ) Simple
Integral
 •  Lelng
LN
denote
the
le[
endpoint
Riemann
 sum
and
RN
denote
the
right
endpoint
sum:
 N −1 N N N N −1 N N N RN − LN = ∑W ( n+1 ) t (W ( n+1 ) t − W nt ) − ∑W nt (W ( n+1 ) t − W nt ) n= 0 N −1 n= 0 = ∑ (W ( n+1 ) t − W nt ) 2 → t n= 0 N N as N→∞. € Simple
Integral:
Normal
Calculus
 •  Letting LN denote the left endpoint Riemann sum and RN denote the right endpoint sum: N −1 + + + RN − LN = ∑ h ( ( n N1)t ) ⋅ h ( ( n N1)t ) − h ( nt ) − ∑ h ( nt ) ⋅ h ( ( n N1)t ) − h ( nt ) N N N n= 0 n= 0 N −1 2 ( ) N −1 ( ) = ∑ h( n= 0 N −1 ( ( n +1)t N ) − h( )) nt N 2 t2 ≤ ∑ nt max1 ) t ( h ' ( s)) ⋅ ( N ) ≤ s≤ ( n+ N n = 0 N ≤ t2 N 0≤ s≤ t max( h ' ( s)) 2 → 0 as N→∞. Stochas*c
Calculus:
 The
Fundamental
Calcula*on
 Using the Law of Large Numbers, we computed: N −1 N →∞ lim ∑ (W ( n+1 ) t − W nt ) 2 = t n= 0 N N Δt →0 lim ∑ (ΔW ) = t = ∑ Δt 2 € (dW ) € 2 = dt Con*nuous
Time
Interest
Rate
 Models
 Con*nuous
Time
Models
 •  The
binomial
model
is
easy
to
use
and
program,
 but
it
can
be
cumbersome
to
work
with,
and
is
 not
very
realis*c
unless
we
take
a
large
number
 of
*me
steps.
 •  Con*nuous
*me
models
are
more
convenient
to
 work
with
from
a
modelling
perspec*ve,
and
can
 some*mes
result
in
analy*c
solu*ons
to
prices.
 •  Monte‐Carlo
simula*on
and/or
discre*za*on
can
 be
employed
when
analy*c
solu*ons
are
not
 available.
 Con*nuous
Time
Models
 •  We
will
build
con*nuous
*me
models
of
the
 short
rate
rt.
 •  rt(Δt)
represents
the
amount
of
interest
 earned
in
the
*me
interval
(t,t+Δt)
(with
the
 size
of
the
interval
tending
to
zero). Example
 •  (Special
case
of)
Rendleman‐Bar`er
Model:
 drt = σ ⋅ rt dW t Δrt := rt + Δt − rt ≈ σ ⋅ rt (W t + Δt − W t ) •  Since
W
is
a
Brownian
mo*on:

 € W t + Δt − W t ~ N (0, Δt ) Δrt ≈ N (0,σ 2 ⋅ Δt ) rt € Example
 •  It
can
be
shown
that
the
solu*on
of
the
 equa*on
for
rt
is:
 rt = r0 exp − •  rt
has
a
lognormal
distribu*on
(i.e.
log(rt)
is
 normally
distributed).
 € •  Cannot
fit
the
current
term
structure
exactly.
 •  Bond
prices
can
be
calculated
through:

 t P (0, t ) = E exp − ∫ 0 rsds ( σ2 2 t + σW t ) ( ) Stochas*c
Differen*al
Equa*ons
 •  In
general,
an
expression
of
the
form:

 


dXt
=
b(Xt,t)dt
+
a(Xt,t)dWt




X0=x
 is
called
a
Stochas*c
Differen*al
Equa*on.
 They
are
o[en
difficult/impossible
to
solve
 analy*cally,
but
they
are
easy
to
simulate.
 Stochas*c
Differen*al
Equa*ons
 dXt = b(Xt,t)dt + a(Xt,t)dWt X0 = x Means that, for small Δt, the change in X is (approximately) normally distributed with mean b(Xt,t)Δt and variance a2(Xt,t) Δt Xt+Δt - Xt ≈N(b(Xt,t)Δt, a2(Xt,t) Δt) This leads to a simple simulation algorithm. Pick a value for Δt (small, if you want an accurate simulation). Set X0=x and Xt+Δt = Xt + b(Xt,t)Δt + a(Xt,t) (Δt)1/2 wt Where wt is a sequence of i.i.d. N(0,1) variables. Stochas*c
Differen*al
Equa*ons:
 Simula*on
 •  Geometric
Brownian
Mo*on
(S0=1,μ=0.15,σ=0.4)
 St = exp((µ − σ2 2 ) t + σW t ) dSt = µSt dt + σSt dW t € Stochas*c
Differen*al
Equa*ons:
 Simula*on
 •  Vasicek's
Model
of
Interest
Rates:

 •  r0=0.02,
β=0.08,
α=4,
σ=0.1
 drt = α (β − rt ) dt + σdW t € Vasicek’s
Model
 •  Interest
rates
tend
to
fluctuate
through
*me
 around
a
long
average
level.
 –  This
property
is
called
mean
reversion.
 –  The
Rendleman‐Bar`er
model
does
not
have
this
 property.
 •  Vasicek’s
model
exhibits
mean
reversion
 drt = α (β − rt ) dt + σdW t € Vasicek’s
Model
 •  The
term
in
front
of
the
dt
is
referred
to
as
the
 dri[.
 •  With
Vasicek’s
model
α,β>0,
 –  When
r
<
β,
the
dri[
is
posi*ve
 –  When
r
>
β,
the
dri[
is
nega*ve
 drt = α (β − rt ) dt + σdW t € Vasicek’s
Model
 •  Vasicek’s
Model
is
explicitly
solvable:
 rt = β + e −α ⋅ t ( r0 − β ) + σ ∫e 0 t −α ( t − s ) dW s 2 rt ~ N β + e−α ⋅ t ( r0 − β ), (1 − e−2αt ) σ α 2 ( ) •  Because
interest
rates
are
normally
 €distributed,
they
can
become
nega*ve.
 Other
Interest
Rate
Models
 •  Cox‐Ingersoll‐Ross:
 drt = α (β − rt ) dt + σ rt dW t •  Heath‐Jarrow‐Morton
 €•  Hull‐White:
 drt = (Θ( t ) − art ) dt + σdW t € Pricing
European
Deriva*ves
 •  If
a
security
has
a
payoff
VT
at
*me
T,
then
its
 price
at
*me
0
(today)
is:
 T V0 = E exp − ∫ 0 rt dt ⋅ VT ( ) •  Some*mes
these
prices
can
be
computed
 € analy*cally
using
special
techniques.
In
this
 course,
we
will
focus
on
compu*ng
prices
 using
lalces
and
simula*on.
 ...
View Full Document

This note was uploaded on 01/13/2011 for the course ACTSC 445 taught by Professor Christianelemieux during the Spring '09 term at Waterloo.

Ask a homework question - tutors are online