Cont_IR_Models_Two

Cont_IR_Models_Two - Con$nuous
Time
Interest
Rate
...

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Unformatted text preview: Con$nuous
Time
Interest
Rate
 Models:
Part
Two
 Pricing
European
Deriva$ves
 •  If
a
security
has
a
payoff
VT
at
$me
T,
then
its
 price
at
$me
0
(today)
is:
 T V0 = E exp − ∫ 0 r( t ) dt ⋅ VT ( ) •  Some$mes
these
prices
can
be
computed
 € analy$cally
using
special
techniques.
In
this
 course,
we
will
focus
on
compu$ng
prices
 using
laJces
and
simula$on.
 Monte‐Carlo
Simula$on
 •  ct
:
cashflow
at
$me
t
of
the
security
to
be
 priced.
 •  Time
steps:
t
=
Δ,
2Δ,
3Δ,…
 •  Choose
a
short‐rate
model
(e.g.
one
defined
 by
a
stochas$c
differen$al
equa$on).
 •  Simulate
the
values:

 {r(t ), 0 ≤ t ≤ NΔ} •  Discount
factor
for
present
valuing
cf
at
$me
t:
 v = exp(− ∫ r( s) ds) € t t 0 Monte‐Carlo
Simula$on
 •  The
value
of
the
security
at
$me
zero
is:
 N C0 = E ∑ c nΔ v nΔ n =1 •  Note:
This
assumes
the
expecta$on
is
taken
 with
respect
to
the
risk‐neutral
probabili$es.
 € •  The
above
expecta$on
can
be
computed
 analy$cally
only
in
very
special
circumstances.

 Monte‐Carlo
Simula$on
 •  Simulate
S
paths
for
r(Δ),
r(2Δ),…,r(NΔ).
 Denote
the
ith
simulated
path
by
ri(Δ),
ri(2Δ) …,ri(NΔ),
i=1,…,S.
 •  Es$mate
C0 by:

 1S N ˆ C0 = ∑ ∑ c i ( nΔ )v i ( nΔ ) S i=1 n =1 •  Where
ci(nΔ)
is
the
cashflow
at
$me
nΔ based
 € on
the
ith
simulated
interest
rate
path
and:
 v i ( nΔ ) = exp(−( ri (0) + ri (Δ ) + + ri ( nΔ ))Δ ) Monte‐Carlo
Simula$on
 •  Rendleman‐Bar+er Model (Wt
is
a
standard
 Brownian
mo$on):
 r( t ) = r(0) exp − ( σ2 2 t + σ ⋅ Wt ) •  Discre$za$on:
 € r( nΔ ) = r(0) ⋅ exp− σ ⋅ nΔ + σ Z , Z ~ N (0, Δ ) i.i.d. 2 ∑ j j n 2 j =1 r( nΔ ) = r(( n − 1)Δ ) ⋅ exp − ( σ2 2 ⋅ Δ + σ ⋅ Zn ) € Monte‐Carlo
Simula$on
 •  Vasicek Model (Wt
is
a
standard
Brownian
 mo$on):
 dr( t ) = α (β − r( t )) dt + σ ⋅ dW t •  Discre$za$on
(r0
given):
 € € r( nΔ ) = r(( n − 1)Δ ) + α (β − r(( n − 1)Δ ))Δ + σ ⋅ Z n , Z n ~ N (0, Δ ) i.i.d. Stochas$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. Mortgage
Backed
Securi$es
 •  c
:
Monthly
mortgage
payment
from
 underlying
collateral
pool
of
mortgages.
 •  N
:
Length
(in
months)
of
the
mortgages.
 •  r0
:
Fixed
monthly
mortgage
rate.
 1 α k = × balance on mortgage at (end) of month k (before payment) c € Mortgage
Backed
Securi$es
 •  wk :
frac$on
of
remaining
mortgages
that
will
 prepay
(en$re
balance)
at
the
end
of
month
k.
 •  fk :
frac$on
of
remaining
mortgages
at
the
 beginning
of
month
k.
 f1 = 1, f 2 = 1 − w1, f 3 = f 2 (1 − w 2 ) f k = (1 − w1 ) (1 − w k −1 ) € Mortgage
Backed
Securi$es
 •  Cashflow
at
the
end
of
month
k:
 c k = c ⋅ f k ( w k ⋅ α k + (1 − w k )) •  Simple
prepayment
model:

 w k = b1 + b2 arctan(rk b3 + b4 ) € •  One
of
b2
or
b3
less
than
zero,
so
that
wk decreases
with
rk.
 € Mortgage
Backed
Securi$es
 w k = b1 + b2 arctan( rk b3 + b4 ) b1 = 0.01, b2 = −0.005, b3 = 10, b4 = 0.5 0.009
 Prepayment Rate € w 0.008
 0.007
 0.006
 0.005
 0.004
 0.003
 0.002
 0.001
 0
 0
 0.007
 0.014
 0.021
 0.028
 0.035
 0.042
 0.049
 0.056
 0.063
 0.07
 0.077
 0.084
 0.091
 0.098
 0.105
 0.112
 0.119
 0.126
 0.133
 0.14
 0.147
 0.154
 0.161
 0.168
 0.175
 0.182
 0.189
 0.196
 r Monte
Carlo
for
MBS
 •  Pseudo‐Code:
 For
i=1
to
S
 
Set
ri(0)=r(0)
 
Set
pv(i)=0
 
For
j=1
to
N
 
 
Generate
ri(jΔ)
from
ri((j‐1)Δ)
 
 
Compute
fi(jΔ),
αi(jΔ),
wi(jΔ),
vi(jΔ)
 
 
Compute
ci(jΔ)
 
 
Set
pv(j)
=
pv(j)
+
ci(jΔ)vi(jΔ)
 
End
 End
 1 Return

 S ∑ pv ( j) S j =1 € Example:
Cancellable
Swaps
 •  •  •  •  •  Consider
a
swap
with
payment
dates
t1,…,tN.
 Pay
fixed,
receive
floa$ng.
 Payments
in
arrears.
 Denote
rj=r(tj),
and
let
L
be
the
swap no7onal.
 At
$me
tj receive:
 L( rj −1 − k ) € Cancellable
Swaps
 •  Have
a
cancellable
op$on
of
Bermudan type:

 –  European
op$ons
can
only
be
exercised
at
the
maturity
 date.
 –  American
op$ons
can
be
exercised
at
any
$me
up
to
and
 including
the
maturity
date.
 –  Bermudan
op$ons
can
be
exercised
on
a
sequence
of
 predefined
dates.
 •  At
each
payment
date
tj
we
have
the
op$on
to
cancel
 the
swap
(i.e.
to
stop
all
future
payments).
 •  Intui$vely,
the
swap
should
be
cancelled
when
interest
 rates
are
declining.
 •  How
can
we
determine
the
op$mal
cancella$on
$me
 using
Monte‐Carlo
simula$on?
 Cancellable
Swaps
 •  As
with
bond
op$ons,
we
should
compare
the
 exercise value (intrinsic value)
with
the
 con7nua7on value (value of wai7ng).
 •  Let
Vj
be
the
value
of
the
swap
at
$me
tj.
 •  Value
at
maturity
of
the
swap:
 VN = L ⋅ ( rN −1 − k ) •  The
con$nua$on
value
at
$me
tj is
denoted
by

 Cj and
is:



 − r ΔV € Cj = E e ( j j +1 rj ) Cancellable
Swaps
 •  Recall
that
when
using
the
binomial
laJce
to
 price
a
bond
op$on,
we
had:

 C j ,n 1 1 = [q j,nV j +1,n +1 + (1 − q j,n )V j +1,n ] = E 1 + r V j +1 rj = rj,n 1 + r j ,n j ,n € •  The
challenge
in
using
Monte‐Carlo
is
that
rather
 than
having
two
possible
future
states,
we
have
 infinitely
many
possible
future
values
of
rj+1 given
 rj.

 –  Rather
than
a
sum,
the
condi$onal
expecta$on
is
an
 integral
 Cancellable
Swaps
 •  The
integral
defining
the
condi$onal
 expecta$on:
 Cj = E e ( − r j ΔV j +1 rj ) 




is
generally
not
analy$cally
tractable.
 Numerical
methods
are
needed
to
 € approximate
it.
 Cancellable
Swaps
 •  How
can
we
use
Monte‐Carlo
simula$on
to
price
 the
cancellable
swap?
 •  Can
we
es$mate
Cj
by
using
a
set
of
S
paths?
 ri,1, ri,2 , ..., ri,N ri,n = ri ( t n ) •  Here
ri(tn)
denotes
the
value
of
the
short
rate
at
 $me
tn
on
the
ith
simulated
path,
i=1,…,S.
 € Cancellable
Swaps
 •  As
with
the
binomial
tree,
we
can
do
a
“backward
 recursion”,
but
we
can’t
simply
use
the
value
of
 the
swap
Vi,j+1
on
the
current
path
i
to
compute
 Ci,j as
there
is
only
one
such
future
value
on
this
 simulated
path.
 •  This
would
be
like
trying
to
compute
the
 condi$onal
expecta$on
with
a
sample
size
of
one.
 •  We
are
trying
to
avoid
doing
a
“nested
 simula$on”.
 Cancellable
Swaps
 •  Least
Squares
Monte‐Carlo
method.
 •  A
sort
of
backward
induc$on.
 •  At
$me
tj we
have:

 –  S
current
values
for
the
short
rate
(one
for
each
 simulated
path):
 r1, j , r2, j , ..., rS, j –  S
values
for
the
instrument
at
the
next
$me


 V1, j +1,V2, j +1, ...,VS , j +1 € –  Recall
that
at
the
maturity
date:

 VN = f ( rN −1 ) € Cancellable
Swaps
 •  At
$me
tj we
have
S
observa$ons
of
the
pairs
 (rj,Vj+1):
 {(r1, j ,V1, j +1),(r2, j ,V2, j +1),...,(rS, j ,VS, j +1)} •  Use
these
to
approximate,
using
regression,
 how
the
response Vj+1 depends
on
the
factor
 €j.
 r •  For
example,
we
could
es$mate
the
regression
 equa$on:
 e − rj Δ V j +1 = β 0, j + β1, j rj + β 2, j rj2 + ε Cancellable
Swap
 •  Es$mate
the
con$nua$on
value
Cj by:
 ˆ ˆ ˆ ˆ C j = β 0, j + β1, j rj + β 2, j rj2 •  Does
this
work?
 •  We
considered
a
quadra$c
func$on
for
the
 € approxima$on.
 •  It
can
be
shown
that
by
allowing
more
and
 more
terms
in
the
model
for
Cj
we
can
get
 very
close
to
the
true
con$nua$on
value:
 Cj = E e [ − r j ΔV j +1 rj = lim ∑ β p rjp k →∞ p =1 k Cancellable
Swaps
 •  Summary
of
the
Least
Squares
Monte‐Carlo
 Algorithm.
 •  Step One:
Generate
S
paths
for
the
short
rate
 according
to
the
chosen
model.
 •  Step Two: For
each
path,
compute
 Vi,N = L ⋅ ( ri,N −1 − k ) –  Set
 t i* = t N –  As
we
move
backward
in
$me,
we
will
update
the
 t es$mated
op$mal
exercise
$me


 i* € € Cancellable
Swaps
 •  Step Three:
Going
backward
from
j
=
N‐1
to
1
 ˆˆˆ –  Obtain
the
regression
coefficients





















by
 β 0, j , β1, j , β 2, j es$ma$ng
the
following
equa$on
using
least
 −r Δ e V j +1 = β 0, j + β1, j rj + β 2, j rj2 + ε squares:
 –  Es$mate
the
con$nua$on
value
on
the
ith
path
 € using
the
equa$on:
 j € ˆ ˆ ˆ ˆ Ci, j = β 0, j + β1, j ri, j + β 2, j ri,2j ˆ <0 t i* = t j Ci, j –  If













,

cancel
the
swap
and
set
 •  Otherwise,
set:




 € € Vi, j = L ⋅ ( ri, j −1 − k ) + Vi, j +1e € − rj Δ Cancellable
Swaps
 •  Step Four: Return
the
following
es$mator
for
 the
value
of
the
callable
swap
at
$me
0:
 ˆ = 1 ∑ ∑ L ⋅ exp(−( r + + r )Δ ) ⋅ ( r − k ) V0 i, 0 i, j i, j S i=1 j =1 * S t i −1 € ...
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