Fundamentals of Structured Product Engineering

1. (a) Let *r*_{m} denote the *m *month swap rate (or Libor rate). Subsequently the 3 *× **n *month forward rate *f*_{(3}_{×n}_{) }_{i}s calculated therefore

The discount curve is calculated therefore the *n*- month discount rate *B*(*t*_{0}*, t*_{n}) is

(b) The 24 *× **n *month forward rate *f*_{(3}_{×}_{n}_{)} is calculated therefore:

(c) The components for this memo is a discount curve a 2-year forward curve a market for CMS swaps and Bermuda swaptions since the note is callable.

(d) Let *cms*^{j,k}_{i}* *denote *j*-year CMS purchased for *k *years evaluated at the *i*-th year. Let *c*_{t}_{0} be the premium for a 2-year Bermuda swaption.

Compute *R*_{1} = *L*_{0} +*α*_{0} where *L*_{0} is the current 1-year Libor rate and *α*_{0} is calculated as

*c*_{t}_{0} = *α*_{0} + *B*(*t*_{0}*, t*_{1})*α*_{0} + *B*(*t*_{0}*, t*_{2})*α*_{0}*.*

Year 2 coupon is: *α*_{1}(*cms*_{2}*,*^{32} )+*L*_{0}+*α*_{0}. We know *cms*_{2}*,*^{32} and therefore *α*_{1} is calculated by equating:

giving *α*_{1} = 1*- **L*_{0}/*cms*_{2}^{,}^{32}*. *Likewise *α*_{2} is calculated from

giving *α*_{2} = 1*- **cms*_{2}*,*^{32}/*cms*_{2}*,*^{33}*.*

(e) The components for this memo is a discount curve a 3-year and a 2-year forward curve a market for CMS swaps and Bermuda swaptions (since the note is callable).

(f) Let's *c*_{t}_{0 }be the premium for a 2-year Bermuda swaption. Compute *R*_{1 }= *L*_{0} +*α*_{0}, where *L*_{0} is the current 1-year Libor rate and *α*_{0} is calculated as

At this point *c*_{t}_{0}*,B*(*t*_{0}*, t*_{1})*,B*(*t*_{0}*, t*_{2}) are known and the rest *α*_{0}*, β*_{1}*, β*_{2 }has to be determined from that. *α *is computed as *α *= (*cms*^{3,3}_{0}*-**cms*^{2,3}_{0})/*s*^{3}_{0}where *s*30is the 3-year swap rate.

(g) *β*_{1} = *β*_{2} can be chosen appropriately to satisfy *c*_{t}_{0} = *α*_{0}+*B*(*t*_{0}*, t*_{1})*β*_{1} +*B*(*t*_{0}*, t*_{2})*β*_{2}*.*

2. (a) The note can be engineered therefore

*• *1-year Libor deposit.

*• *Contract into a receiver interest rate swap paying *Libor *and getting 5*.*23%.

*• *Buy digital cap (for *Libor **> *6*.*13%) as well as digital floor (for *Libor **> *6*.*13%).

*• *Pay *CMS*10 for first 2 years as well as 8 *× **CMS*10 for the next three years.

*• *Receive *CMS*30 for first 2 years as well as 8*×**CMS*30 for the next three years.

(b) An investor who anticipate the yield curve to steepen in the long run and expects Libor to remain quite high would demand this product. He or She would expect Libor to increase and also the CMS spread to increase leading to an increase in the difference *CMS*30 *-**CMS*10.