Credit Risk:Modeling, Valuationand Hedging3.2.2 ~ 3.3.4

整理 : 田宇正

Overview

Corporate Bonds PricingDefault Time: Merton → Black & CoxCoupon / ZeroConsol / Finite Maturity

Optimal Capital StructureEquity Value Maximization(Bond Value Minimization)

Corporate Zero-Coupon BondBlack & Cox Model

- (T-t)

t

1 t { = }

2 t 1 T { =T}

3 t 2 {t< <T}

1 2 3

dV = V (r- )dt + dW

v = Ke

D (t,T) = E [ LB(t,T)I ]

D (t,T) = E [ V B(t,T)I ]

D (t,T) = E [ v B(t, )I ]

D(t,T) = D (t,T) + D (t,T) + D (t,T)

t

t

Special Cases

v = 0 Merton

v = LB(t,T) Default Free

See Proposition 3.2.1

Corporate Coupon BondT

C t { >s}

t

D (t,T) = D(t,T) + E [ cB(t,s)I ds ]

= D(t,T) + A(t,T)

T

t { >s}

t

T

t

t

T

t t

t

A(t,T) = E [ cB(t,s)I ds ]

= cB(t,s)Pr { >s}ds

c= 1 - B(t,T)Pr { T} + B(t,s)dPr { >s}

r

2 212

2t t

t

2(r- - - )/ 21t tt 2

t

2a1 t t 2

Recall

Log(V /v ) + (r- - - /2)(s-t) Pr { >s} = N

s-t

-Log(V /v ) + (r- - - )(s-t)v - N

V s-t

= N k (V ,s-t) - R N k

t(V ,s-t)

2a1 t t 2 t

a+ a-t 1 t t 2 t

c= 1 - B(t,T) N k (V ,T-t) - R N k (V ,T-t)

r

- R N g (V ,T-t) - R N g (V ,T-t)

22 212

2

2t t

1 t

2t t

2 t

Where

r- - - +2 r =

Log(v /V ) + (T-t)g (V ,T-t) =

T-t

Log(v /V ) - (T-t)g (V ,T-t) =

T-t

Proposition 3.2.2

Corporate Consol BondC C 3

T TD (t) = lim D (t,T) = lim A(t,T) + D (t,T)

2a1 t t 2 t

a+ a-t 1 t t 2 t

2a a+ a

a

-t

t

t t

+

t

Recall

cA(t,T) = 1 - B(t,T) N k (V ,T-t) - R N k (V ,T-t)

r

- R N g (V ,T-t) - R N g (V

vc1 -

r V

,T-t)

c 1 - 0 1 - R 0 - R 1 - R 0

r

=

(

2 2 21

22

(r- - - ) +2 r = )

+ -3 2 t t 7 t t 8 t

a+1+

a+

t2 t

t2 t

t

2

t

2 212

2

212

2

vv

V

Recall

D (t,T) = V R N h (V ,T-t) + R N h (V ,T-t)

v V

V

=

(r- - - ) +2 (r- )( = )

r- - -( a = )

t

a+ a+

C 2 t 2 tt t

a+

t t tt t

Special Case = 0 v = v, =

c v v cD (t) = 1 - + v = 1 - q + v q

r V V r

v( Notice that q = = - B(t,s)dPr ( >s) = E [ B(t, ) ] )

V

a+ a+

t t2 t

t t

v vc= 1 -

V + v

r V

Optimal Capital Structure #1 (1/2)Black & Cox: Consol Bond, No Dividend, Constant Barrier, Fully Recover

2 2

C tt t

2rAssume = = 0, = 1 a+ = =

c v vD (V ) = 1 - + v

r V V

C

2 2VV V

V

2 2VV V t

Notice that, u (V) = D (V) satisfies the ODE

u (v) = v < c/r1 V u + rVu + c - ru = 0, with

lim u (V) = c/r2

which is reduced form the PDE

u1

V u + rVu + u + c = ru , with 2

V

t

(v,t) = v < c/r

lim u (V,t) = c/r

u (V,t) = 0

Optimal Capital Structure #1 (2/2)Black & Cox: Consol Bond, No Dividend, Constant Barrier, Fully Recover

*

t tt t

Shareholders can choose a "bakrupcy level" v to mazimize the equity value

c v v E(V ) = V - 1 - - v

r V V

t t t

Equity Value + Bond Value = Firm Value

E(V ) D(V ) V

-1 -1

t t t t t

2*

2 212

The optimal strtegy can be solved with

c v v v v - - = 0

r V V V V V

c c 2r/ c v = = =

r 1+ r 1+2r/ r+

Optimal Capital Structure #2 (1/2)Leland: Black & Cox + Non-Fully Recover, Tax Benefits, Bankruptcy Cost

t t t 2 t t t t t 2 t

Equity Value + Bond Value = Firm Value + Tax Benefits - Bankruptcy Cost

c cE(V ) D(V ) = (1-q ) + vq V T(V ) = (1-q ) B(V ) = (1- )vq

r r

2

t t t

2r/

t t t

Total Firm Value: G(V ) = E(V ) + D(V )

Expected Discount Factor: q = v /V

Corporate Tax Rate:

t t t t

To maximize

(1- )c E(V ) = V - 1-q - vq

r

*21

2

We will have

(1- )c v =

r+

+1

*0 0 21

0 2

210 2

At time t = 0

(1- )c 1 (1- )c E (V ) = V - +

r V r+

which is maximized when

V (r + ) c =

1-

Optimal Capital Structure #2 (2/2)Leland: Black & Cox + Non-Fully Recover, Tax Benefits, Bankruptcy Cost

0

00 2

0 0 0

Protected Debt

Instead, consider the barrier is the principle value of the debt, which is assume to be D(V ).

We will have

D(V )c v v c D(V ) = 1 - + v = 1 -

r V V r V

02

0

D(V ) + v

V

2 0Special Case = 1 (No Bankruptcy Cost) D(V ) = c/r

Optimal Capital Structure #3Leland & Toft: Leland + Finite Maturity

Stationary Debt Structure: at any time t, the debt outstanding is composed of coupon bonds of maturities form the interval [t..t+T] and the face value uniformly distributed over this interval.

t+T

t C

t

1D(V ) = D (t,u) du

T

C 2

2a1 0 0 2 0

Recall (Corollary 3.2.1)

c c c = 0, v L D (0,u) = + B(0,u) L - g(u) + v - h(u)

r r r

( g(u) = N k (V ,u) - R N k (V ,u) )

a+ a-0 1 0 0 2 0 ( h(u) = R N g (V ,u) + R N g (V ,u )

)

0 2

c 1 c 1 c D(V ) = + L - G(T) + v - H(T)

r T r T r

T -rT-ru

0

Ta+ a-0 1 0 1 0 0 2 0 2 0

0

Where

1 - h(T) - e g(T)G(T) = e g(u)du =

r

TH(T) = h(u)du = R g (V ,T)N g (V ,T) - R g (V ,T)N g (V ,T)

Further Developments

State Variables: default time is in terms of some SV (operating earnings, etc).

Stratigic Debt Service: allow renegotiation in case of distress.

Probabilistic Approaches: decompose defaultable claims into (1) down-and-out call, (2) down-and-out digital claim, (3) digital claim which pays $1 when default occurs.

Comparative Studies: fixed cost of bankruptcy.

C t t 2

cD (t) = (1 - q ) + q max{ v - K,0}

r