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.

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, ;

if the 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 option at time , then there exists a critical bond price such that it is optimal to exercise the option if and only if .

Intrinsic Value

(𝜆 (𝑉 𝑡 , 𝑡 )−𝑃 𝑡 )+¿¿

𝑷 𝒕

b(,t) -(,t) -

in the money

Proof • Let and are two states of and Step 1

Suppose it 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

Step 2 Let be the infimum of 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

𝑽 𝒕

(,t) - -

Intrinsic Value

(𝑧𝑉 𝑡−𝑃 𝑡❑)+¿¿

in the money

Proof Let and are two states of and .

Step1 Suppose it 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

Step 2 Let be the supremum of 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)

Bond Valuation

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 .

(𝑧𝑉 𝑡∨𝑘𝑡

𝑃− 𝑃 𝑡❑ )+¿ ¿

Intrinsic Value 𝑽 𝒕

--

(,t) -(,t)

Proof 2-1 (the case : ) Suppose it is optimal “NOT” to convert (continue) at .

Using put delta inequality , implied by

𝑽 𝒕

thus we have

+ +

Besides, for all . Thus, . It is then optimal not to convert at z.

in U

-

---

(,t)

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

𝑽 𝒕

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

-

(,t) -

- -

(,t)

(,t)

Intrinsic Value

(𝑧𝑉 𝑡∨𝑘𝑡𝑃− 𝑃 𝑡

❑ )+¿ ¿

the case of optimal to convertthe case of optimal to put

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

in U

By Thm of PCB, part 2 Review

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.

𝑽 𝒕

-

(,t) - ---

Part 3.A

Theorem 3.A For each , 1. 2.

Theorem 3.A For each , 1. 2.

Proof 3.1

If . Then as well.

According to put delta inequality, + Thus, , because 0

𝑷 𝒕

b(,t) -(,t) -

in U

in U

𝑉 𝑡(1)<𝑉 𝑡

(2)⟹𝑏𝐶𝐵 (𝑉 𝑡(1 ) , 𝑡)≤𝑏𝐶𝐵 (𝑉 𝑡

(2 ) ,𝑡 )The higher the firm value, the higher the bondvalue must be to trigger conversion.(the easier to trigger conversion)

Proof 3.2 If . Then as well . According to call delta inequality, +

Thus , because

in U

𝑽 𝒕

(,t) -

-

in U

in U

The discussion above suggests

𝑃 𝑡(1 )>𝑃 𝑡

(2)⟹𝑣𝐶𝐵 (𝑃 𝑡(1 ) ,𝑡 )≥𝑣𝐶𝐵(𝑃 𝑡

(2 ) ,𝑡)In high interest rate environments, it takes lowerfirm values to make bond holders convert theirbond.

Part 3.B

Theorem 3.B For each , 3. (conversion case)

( and ) 4. (put case) ( but still ) – to confirm default-free

Proof 3.3

If . Then as well.

According to put delta inequality, + Thus, , because 0

in U

in U

in U

Proof 3.4

If . Then as well.

in U

in U

•

- exercise means conversion.- the higher the firm value, the higher the bond value must be to trigger conversion.

•

- exercise means put.- at lower firm values, it takes higher bond value to trigger a bond put.

Part 3.C

Theorem 3.C For each , 5. (conversion case)

6. (put case)

Proof 3.5

If , then .

Thus

in U

in U

Proof 3.5

If , then .

Thus

in U

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.