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

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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)

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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

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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

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imply vol form estimatecan ,

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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