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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

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Optimal Conversion and Put Policies

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  1. 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.

  2. 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,

  3. Part 1

  4. 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

  5. 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

  6. it is optimal to continue at , thus we have + + Besides,for all . Thus, . It is then optimal to continue at . in U

  7. 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

  8. Part 2.A

  9. 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 -

  10. 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

  11. it is optimal to continue at , thus we have + + Besides, for all . Thus, . It is then optimal to continue at . in U

  12. Step 2 Let be the supremumof that . The point can not lie in because is open. Thus , for all and Then, , not in U

  13. Part 2.B

  14. Bond Valuation • Theorem 1. 2. 3.(put delta inequality) Back_p20

  15. Part 2.B-1

  16. 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) - -

  17. Proof 2-1 (the case : ) Suppose it is optimal “NOT” to convert (continue) at . Using put delta inequality , implied by

  18. in U thus we have + + Besides, for all . Thus, . It is then optimal not to convert at z. (,t) - - - -

  19. 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)

  20. Part 2.B-2

  21. 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) -

  22. 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

  23. - - - (,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. -

  24. Part 3.A

  25. Theorem 3.A For each , 1. 2.

  26. Theorem 3.A For each , 1. 2.

  27. (,t)- in U Proof 3.1 If . Then as well. According to put delta inequality, + Thus, , because 0 in U b(,t)-

  28. The higher the firm value, the higherthe bond valuemust be to trigger conversion. (the easier to trigger conversion)

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

  30. The discussion above suggests In high interest rate environments, it takes lower firm values to make bond holders convert their bond.

  31. Part 3.B

  32. Theorem 3.B For each , 3. (conversion case) ( and ) 4. (put case) ( but still ) – to confirm default-free

  33. in U Proof 3.3 If . Then as well. According to put delta inequality, + Thus, , because 0 in U in U

  34. Proof 3.4 If . Then as well. in U in U

  35. - 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.

  36. Part 3.C

  37. Theorem 3.C For each , 5. (conversion case) 6. (put case)

  38. in U Proof 3.5 If ,then . Thus in U

  39. in U Proof 3.5 If ,then . Thus in U

  40. - 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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