Optimal conversion and put policies
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Optimal Conversion and Put Policies . The first theorem establishes the existence of a boundary of critical host bond prices . The second theorem describes the boundary in terms of critical firm value. The third theorem characterizes the shape and relation of

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Optimal Conversion and Put Policies

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Optimal conversion and put policies

Optimal Conversion and Put Policies

  • The first theorem establishes the existence of a boundary of critical host bond prices .

  • The second theorem describes the boundary in terms of

  • critical firm value.

  • The third theorem characterizes the shape and relation of

  • the boundaries for the different types of bonds.


Optimal conversion and put policies

Remark

The continuation region for conversion, put, and puttable-

convertible option is the open set

Note that for all , .

If the subscript Y is CB, ;

ifthe subscript Y is P, ;

if the subscript Y is PCB,


Part 1

Part 1


Optimal conversion and put policies

Theorem (given the firm value)

Let and If there is any bond price

such that it is optimal to exercise the embedded

optionat time , then there exists a critical bond

price such that it is optimal to

exercisethe option if and only if .

Intrinsic Value

(,t)-

b(,t)-

in the money


Optimal conversion and put policies

Proof

  • Let and are two states of and

    Step 1

    Supposeit is optimal to continue at and.

    We show that it is then optimal to continue at .

    According to the call delta inequality


Optimal conversion and put policies

it is optimal to continue at , thus we have

+

+

Besides,for all .

Thus, .

It is then optimal to continue at .

in U


Optimal conversion and put policies

Step 2

Let be the infimumof that .

The point can not lie in because is open.

Thus , for all and

Then, .

This theorem implies that the increase of interest rate can not only trigger bond put but also trigger conversion.

not in U


Part 2 a

Part 2.A


Optimal conversion and put policies

Theorem (given the host bond price)

Let and

1. For the pure convertible bond, there exists a critical

firm value such that it is optimal to

default if and only if

(,t)-

Intrinsic Value

in the money

-


Optimal conversion and put policies

Proof

Let and are two states of and .

Step1

Supposeit is optimal to continue at and.

We show that it is then optimal to continue at .

Using put delta inequality

Above result is implied by

Review


Optimal conversion and put policies

it is optimal to continue at , thus we have

+

+

Besides, for all .

Thus, .

It is then optimal to continue at .

in U


Optimal conversion and put policies

Step 2

Let be the supremumof that .

The point can not lie in because is open.

Thus , for all and

Then, ,

not in U


Part 2 b

Part 2.B


Bond valuation

Bond Valuation

  • Theorem

    1.

    2.

    3.(put delta inequality)

Back_p20


Part 2 b 1

Part 2.B-1


Optimal conversion and put policies

2-1 For the (default-free) puttable-convertible bond,

there exists a critical firm value ,satisfying

(implied by z)

, and such that it is optimal to convert

if and only if .

Intrinsic Value

(,t)-

(,t)

-

-


Optimal conversion and put policies

Proof 2-1

(the case : )

Suppose it is optimal “NOT” to convert (continue) at .

Using put delta inequality

,

implied by


Optimal conversion and put policies

in U

thus we have

+

+

Besides, for all .

Thus, .

It is then optimal not to convert at z.

(,t)

-

-

-

-


Optimal conversion and put policies

Therefore, there exists a critical value such that

it is optimal to convert ,

Note

(1) . Otherwise

(2) (implies ).

Otherwise, there exists a firm value that makes less

than at which is optimal to convert, which is impossible.

(put rather than convert)


Part 2 b 2

Part 2.B-2


Optimal conversion and put policies

2-2 If there exists any firm value , at which it is

optimal to put at time t, then there exists a critical firm value

and such that it is optimal

to put if and only if

the case of optimal to convert

the case of optimal to put

Intrinsic Value

(,t)-

-

(,t)

-

(,t)

-


Optimal conversion and put policies

Proof 2-2

(the case : )

Suppose it is optimal “NOT” to put at .

We want to show it is also optimal “NOT” to put at .

( i.e. )

It follows

By Thmof PCB, part 2

Review

in U


Optimal conversion and put policies

-

-

-

(,t)-

Note that it must be optimal to put at .

Thus, based on the discussion above, there exists a critical value , such that it is optimal to put ,

as , it is optimal to put.

-


Part 3 a

Part 3.A


Optimal conversion and put policies

Theorem 3.A For each ,

1.

2.


Optimal conversion and put policies

Theorem 3.A For each ,

1.

2.


Optimal conversion and put policies

(,t)-

in U

Proof 3.1

If . Then as well.

According to put delta inequality,

+

Thus, ,

because 0

in U

b(,t)-


Optimal conversion and put policies

The higher the firm value, the higherthe bond

valuemust be to trigger conversion.

(the easier to trigger conversion)


Optimal conversion and put policies

Proof 3.2

If . Then as well .

According to call delta inequality,

+

Thus ,

because

in U

in U

(,t)-

in U

-


Optimal conversion and put policies

The discussion above suggests

In high interest rate environments, it takes lower

firm values to make bond holders convert their

bond.


Part 3 b

Part 3.B


Optimal conversion and put policies

Theorem 3.B For each ,

3.

(conversion case)

( and )

4.

(put case)

( but still ) – to confirm default-free


Optimal conversion and put policies

in U

Proof 3.3

If . Then as well.

According to put delta inequality,

+

Thus, ,

because 0

in U

in U


Optimal conversion and put policies

Proof 3.4

If . Then as well.

in U

in U


Optimal conversion and put policies

- exercise means conversion.

- the higher the firm value, the higher the bond

valuemust be to trigger conversion.

- exercise means put.

- at lower firm values, it takes higherbond value to

trigger a bond put.


Part 3 c

Part 3.C


Optimal conversion and put policies

Theorem 3.C For each ,

5. (conversion case)

6. (put case)


Optimal conversion and put policies

in U

Proof 3.5

If ,then .

Thus

in U


Optimal conversion and put policies

in U

Proof 3.5

If ,then .

Thus

in U


Optimal conversion and put policies

- when both options are present, the value, the

value of preserving one option can make it

optimal for issuer to continue servicing the

debt in states in which it would otherwise

exercise the other option.


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