1 / 40

# Optimal Conversion and Put Policies - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Optimal Conversion and Put Policies ' - ailis

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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.

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

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

• 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

it is optimal to continue at , thus we have

+

+

Besides,for all .

Thus, .

It is then optimal to continue at .

in U

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

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

-

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

it is optimal to continue at , thus we have

+

+

Besides, for all .

Thus, .

It is then optimal to continue at .

in U

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

• Theorem

1.

2.

3.(put delta inequality)

Back_p20

### Part 2.B-1

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)

-

-

(the case : )

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

Using put delta inequality

,

implied by

in U

thus we have

+

+

Besides, for all .

Thus, .

It is then optimal not to convert at z.

(,t)

-

-

-

-

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

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)

-

(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

-

-

(,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

Theorem 3.A For each ,

1.

2.

Theorem 3.A For each ,

1.

2.

(,t)-

in U

Proof 3.1

If . Then as well.

According to put delta inequality,

+

Thus, ,

because 0

in U

b(,t)-

The higher the firm value, the higherthe bond

valuemust be to trigger conversion.

(the easier to trigger conversion)

If . Then as well .

According to call delta inequality,

+

Thus ,

because

in U

in U

(,t)-

in U

-

In high interest rate environments, it takes lower

firm values to make bond holders convert their

bond.

### Part 3.B

Theorem 3.B For each ,

3.

(conversion case)

( and )

4.

(put case)

( but still ) – to confirm default-free

in U

Proof 3.3

If . Then as well.

According to put delta inequality,

+

Thus, ,

because 0

in U

in U

If . Then as well.

in U

in U

- 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

Theorem 3.C For each ,

5. (conversion case)

6. (put case)

in U

Proof 3.5

If ,then .

Thus

in U

in U

Proof 3.5

If ,then .

Thus

in U

- 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.