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
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
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
Proof
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
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
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

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
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 supremumof that .
The point can not lie in because is open.
Thus , for all and
Then, ,
not in U
Part 2.B
1.
2.
3.(put delta inequality)
Back_p20
Part 2.B1
21 For the (defaultfree) puttableconvertible 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)


Proof 21
(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)




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.B2
22 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)

Proof 22
(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)
Proof 3.2
If . Then as well .
According to call delta inequality,
+
Thus ,
because
in U
in U
(,t)
in U

The discussion above suggests
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 defaultfree
in U
Proof 3.3
If . Then as well.
According to put delta inequality,
+
Thus, ,
because 0
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
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.