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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. C onvertible bond is a hybrid attributes of both fi xed-income securities and equity - PowerPoint PPT Presentation
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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
)max( DVE
EDV
TT
TT
Firm value model (Structure model)
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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)
conversion before priceStock :
bond eConvertibl :
bondstraight of ueMarket val :
:securities three theof sum is Assume
right puttable no have investers
andCB,ofmaturityatonlyoccurwilldefaultAssume
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Brenen & Schwartz(1980)
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:$1000 is par value CB Assume
place takehas conversionafter priceStock :
conversion ofresult a as issued share ofNumber :
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is valuefirm theCB,convert After
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bondholder econvertibl Coupon to :
bondholdersenior Coupon to :
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:processes random follow& Assume
return future of gdiscountinby valueCB affects
valueconversion andy probabilitdefault through valueCB affects
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yield dividend :
rate free-risk :
where
1,,,
intensitydefault neutral-risk is
raterecovery has bond and 0 tofalls coccurs,default When
timeperiodshort each in y probabilitdefault
th motion wiBrownian geometric follows Assume
)(
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r
ea
udeu
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auePd
du
deaPu
λ
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S
Δtr-q
ttt
Reduce model (simple)
48080516708681015191
250405130
501132100750 Assume 0
., Pd., Pu., d.u
.%, Δ,%, δ, r% per yearm, λ,% per annuσ
,, S, CP, CR year, F.T
s
Reduce model
S Cox-Ross-Rubinstein (CRR model)
r Ho-Lee lognormal model
λ 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
Δtσd
Δtσu
r
r
eRR
eRR
0
0
CRR model
du
dep
ud
eu
SeSd
SeSu
Δtr
Δtσ
Δtσ
Δtσ
f
s
s
s
1
Two factor tree with correction
R,S
Ru,Su
Rd,Su
Ru,Sd
Rd,Sd
p1
p2
p3
p4
Jarrow & Turnbull Intensity Model
100
0
0
11
find , observe , Assume
rateinterest free-riskyear -One :
rateinterest risky year -One :
])1[(1 00
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eδλλe
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nodeparent for the rateinterest is ,)1/(~
treerateinterest free-risk with treeCRR adjusted Combine
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)]1()1/(
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)]1)(~1()1(~0[
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yprobabilit CRR eadjust th tonecessary isit
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Δtr
ff
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f
Reduce model (Chamber & Lu)
R,S
Ru,Su
Rd,Su
Ru,Sd
Rd,Sd
p1(1-λ i)
p2(1-λi)
p3 (1-λi)p
4 (1-λi)
δ
λi
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
)(22
222
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,)2(
)ln(,
)ln(
&Fit
)(
)()()()(
)()(
)( 0
)())(()(
optionbarrier out &down Use
imply vol form estimatecan ,
tTtVV
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tt
t
t
t
t
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tt
t
t
t
First Passage Model+KMV
VuV
Vd
Default boundary=Ke-γ(T-t)
λS(t)
S
Su
Sd
Default boundary=Ke-γ(T-t)
K1/2 long debt+ short debt (KMV), γ r
Default probability
V(t)
Assume V ~Lognormal distribution
σv
Default boundary=Ke-γ(T-t)
Default probability
The log-normal distribution has PDF
Further work
• The default boundary is given exogenously
• Use market CB to look for imply boundary