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
ud
,
( 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
ud
,
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
ud
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
ud
-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
Rt
e
r f Δt
/( 1   )  d
ud
ue
r f Δt
 (1   )
ud
adjusted
/( 1  λi )  d
ud
(1   )  Sd
S
u-e
ud
, R is interest
/( 1   )
ud
(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)
K1/2 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