Intensity Model
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Transcript Intensity Model
Convertible bond pricing
model
資管所 蘇柏屹
指導老師 戴天時
Agenda
•
•
•
•
Introduction
Credit risk model
Convertible bond pricing model
Our convertible bond pricing model
Introduction
• Convertible bond is a hybrid attributes of
both fixed-income securities and equity
• In specific period, convertible bond can be
converted into equity with predetermined
convert ratio
• Convertible bonds have call features,
which provide the issuer a way to force
conversion or redemption of the bonds
Credit risk model
• Firm value model (Merton,1974)
– Credit risk is considered equity as call option on firm's
assets
• First passage time model (Black & Cox ,1976)
– Solve the problem of premature bankruptcy
• Intensity Model (Jarrow & Turnbull ,1995)
– Use an arbitrage-free bankruptcy process that
triggers default
Firm value model (Structure model)
• Assume
– Firm has only one class of bond that has no
coupon payment and the risk-free interest
rate is constant
– Bankruptcy is triggered at the maturity and
the cost for bankruptcy is zero
VT D E T
E T max( V T D )
Firm value model (Structure model)
B/S formula
calculate
E 0 V 0 N ( d 1 ) De
E0
rT
N (d 2 )
(1)
ln V 0 D ( r V / 2 )T
2
where d 1
V
,
T
Under risk neutral probabilit
y, default
d 2 d1 V
probabilit
T
y is N ( d 2 )
To calcute d 1 & d 2 , we require V 0 & V
From Ito ' s lemma :
0E0
E
V
V V0
or 0 E 0 N ( d 1 ) V V 0
(2)
First passage time model
Assume
B Ke
bankruptcy
(T t )
occurs if the firm' s value crosses a specied boundary
, with K and as an exogenous
If we are at time t 0 and default
constant
has not been trigg
ered yet and V t B
is first time to reach boundary
inf s t V s B
Using reflection
principle,
B
ln(
P ( T t ) N (
we can infer the
Vt e
2
V
B
1
exp 2 ( r
) ln(
)
2
2
V
t
V
(T t )
V
N (
)(
V
default
2
)( T t )
2
)
(T t )
B
ln(
Vt e
probabilit
(T t )
V
)(
V
2
2
(T t )
)( T t )
)
y from time t to T :
Intensity Model
p ( t , T ) : Time t dollar value of default free zero coupon bond
paying 1 dollar at time T t
B ( t ) : Time t dollar value of money market account
initialize
d
with 1 dollar at time 0
v ( t , T ) : Time t dollar value of default
zero coupon bond
paying 1 dollar at time T t
Assume
no arbitrage,
exist pseudoprob ability , t such that
p ( t , T ) / B ( t ), v ( t , T ) / B ( t ) are martingale
Market completene
ss is equivalent
these pseudoprob abilities
s
to uniqueness
of
Intensity Model
Default
bond Default
v ( 0 ,1) e
r (0)
- free bond payoff
[ 0 (1 0 )]
p ( 0 ,1)[ 0 (1 0 )] find 0
rate in default
Intensity Model
Default
bond Default
v (0,2 ) e
e
r (1 ) d
r (0)
{e
r (1 ) u
- free bond payoff
0 e
r (1 ) d
rate in default
(1 ) 0 e
(1 )( 1 0 )[ 1 (1 1 )]}
p ( 0 , 2 ){ 0 (1 0 )[ 1 (1 1 )]}
find 1
r (1 ) u
(1 0 )[ 1 (1 1 )]
Paper survey
• Structure model: Assume stochastic processes
for S&r, and use Ito’s lemma to derive PDE, then
exploit boundary condition to solve PDE
– Brenen & Schwartz(1977)
– Brenen & Schwartz(1980)
• Reduce model: Use tree model to simulate S&r,
and calculate each node price then rollback
– Hung & Wang(2002)
– Chambers & Lu(2007)
Brenen & Schwartz(1980)
Assume
default
will occur only at maturity
investers
have no puttable
Assume
V is sum of the three
V N BB NCC N OS
S
BC
securities
BC
B : Market val ue of straight
C : Convertibl
right
bond
e bond
: Stock price before conversion
:
of CB, and
Brenen & Schwartz(1980)
After convert CB, the firm value
V N B B ( N O N )S
is
AC
N : Number of share issued as a result of conversion
S
AC
: Stock price after conversion
Assume
CB par value
q ( Conversion
has take place
is $1000 :
Ratio ) 1000 /Conversio n price
N N C q
Fraction
of total shares owned by holder of each CB after conversion
Z q/ ( N O ΔN )
:
Call & conversion strategy
Conversion
Optimal
when
Strategy
to convert
C qS
AC
if CB value
( conversion
C Z [V-N B B ]
Call Strategy
Minimize
when
the value
of CB
C Call price
C Call price
max(min(ho
ld, call), convert)
falls
below
value ) Z [V-N B B ]
its conversion
value
Random process
V affects CB value through
default
r affects CB value by discountin
Assume V & r follow
probabilit
y and conversion
g of future return
random processes :
dr α ( μ r r ) dt rσ r dZ r
dV [V μ V Q (V , t )] dt V σ V dZ V
Q (V , t ) : Total cash payment to
security
Q (V , t ) I B I CB D (V , t )
I B : Coupon to
senior bondholder
I CB : Coupon to
D (V , t ) : Dividend
convertibl
e bondholder
to stockholde
r
holders
value
CB’s PDE
By Ito' s lemma, value of CB satisfies
1
2
C VV V V C Vr Vr V r
2
2
1
2
the PDE :
C rr r r
2
2
C r [ ( r r ) r r ] C V [ rV Q (V , T )] rC cF C t 0
: Market price of interest
rate risk
F : CB par value
c : Coupon rate
: Instantane ous correlatio n between dZ V and dZ r
Boundary condition
The Conversion
Condition
C (V,r,t ) Z (V-N B B (V,r,t ))
The Call Condition
C (V,r,t ) CP
The Bankruptcy
Condition
C (V,r,T ) kF
if V N B B
The Maturity
O
kN C F
Condition
Z (V N B B (V , r , T ))
F
C (V,r,t )
0
1
N
(
V
N
B
)
C
B
0
if Z (V N B B (V , r , T )) F
if Z (V N B B (V , r , T )) F 1 N C (V N B B )
0
if F 1 N C (V N B B ) 0
0
if V N B B
0
Reduce model (simple)
Assume
default
When
S follows
probabilit
default
Pu
where
ud
,
( r-q ) Δt
r : risk - free rate
q : dividend
y λΔt in each short
default
t
a e
Brownian
coccurs, S falls
λ is risk - neutral
a de
geometric
yield
Pd
motion wi
period
time
to 0 and bond
th
t
has recovery rate δ
intensity
ue
t
a
ud
,
u e
(
2
) t
,
d
1
u
Reduce model (simple)
Assume
T 0 .75 year, F
100 , CR 2 , CP 113 , S 0 50 ,
σ s 30 % per annu m, λ, 1% per year , r 5%, δ 40 %, Δ, 0 .25
u 1.1519 , d 0 .8681 , Pu 0 .5167 , Pd 0 .4808
Reduce model
S
r
λ
Cox-Ross-Rubinstein
(CRR model)
Ho-Lee lognormal model
Jarrow & Turnbull
Intensity Model
S & r without correction Hung & Wang two factor
model
S & r with correction
Das & Sundaram two
factor model
Ho-Lee(1986) lognormal model
Ru R0 e
σr
Rd R0e
σr
Δt
Δt
CRR model
Su Se
σs
Sd Se
σ s
u e
d
σs
Δt
Δt
Δt
1
u
p
e
rf
Δt
d
ud
Two factor tree with correction
p1
R,S
p2
p3
p4
Ru,Su
Rd,Su
Ru,Sd
Rd,Sd
Jarrow & Turnbull Intensity Model
1 e
R0
*
[( 1 λ1 ) δλ 1 e
*
R 0 : One - year risky
R0
]
interest
rate
R 0 : One - year risk - free interest
Assume
δ , observe
*
R 0 R 0 , find λ1
rate
Adjust CRR probability
In order to
develop
it is necessary
~
p
e
r f Δt
a risk
neutral
non - arbitrage
to adjust th e CRR
probabilit
tree,
y
/( 1 λ ) d
ud
-r Δt
e f [ 0 λ Su ~
p (1 λ ) Sd (1 ~
p )( 1 )]
e
-r f Δt
e
[ Su
-r f Δt
[S
Combine
~
p
e
Rt
e
r f Δt
/( 1 ) d
ud
ue
r f Δt
(1 )
ud
adjusted
/( 1 λi ) d
ud
(1 ) Sd
S
u-e
ud
, R is interest
/( 1 )
ud
(1 ) de
CRR tree with
r f Δt
(1 )]
r f Δt
] S
risk - free interest
rate for the
parent
rate tree
node
Reduce model (Chamber & Lu)
Ru,Su
Rd,Su
R,S
λi
δ
Ru,Sd
Rd,Sd
Our pricing model
• Improve default probability which is
unrelated to stock price
• Improve default only occur in maturity date
• Structure model + down & out barrier
option + FPM + KMV
Structure model + down & out
barrier option
E t N s S t , σ E t can estimate
form imply vol
Use down & out barrier option
r (T t )
(V ( t ) -De
E (t )
0
)
if V ( t ) B
if V ( t ) B
E t V t N ( x ) e qT De rT N ( x V T )
t
qT B t
2
rT B t
2 2
V
e
(
)
N
(
y
)
De
(
)
N ( y Vt
t
Vt
Vt
σ E t E t N ( x ) V t V t
ln(
x
Vt
V
Bt
t
)
ln(
T
(r q V
2
t
Vt
y
T,
V
2 ) V t , B t ke
2
Bt
Vt
t
)
T
(T t )
Vt
T
T)
Fit V t & V t
First Passage Model+KMV
V
S
Su Vu
Sd Vd
λS(t)
Default boundary=Ke-γ(T-t)
Default boundary=Ke-γ(T-t)
K1/2 long debt+ short debt (KMV), γ r
Default probability
Assume V ~Lognormal distribution
σv
The log-normal distribution
has PDF
Default boundary=Ke-γ(T-t)
Default
probability
V(t)
Further work
• The default boundary is given
exogenously
• Use market CB to look for imply boundary