Table 1 Payoff structure of convertible bonds
Payoff | Boundary condition | Time restriction | State |
|---|---|---|---|
\(\text {C}_t\) | \(\text {V}_t\) > \(\text {C}_t\) and \(\text {C}_t\) > \(\text {n}_t\text {S}_t\) | For \(t\in \Omega _{\text {conv}}\cap \Omega _{\text {call}}\) | Conditional redemption |
\(\text {n}_t\text {S}_t\) | \(\text {V}_t\) > \(\text {K}_t\) and \(\text {n}_t\text {S}_t\) > \(\text {K}_t\) | For \(t\in \Omega _{\text {conv}}\) | Forced conversion |
\(\text {n}_t\text {S}_t\) | \(\text {n}_t\text {S}_t\) > \(\text {C}_t\) and \(\text {P}_t\) < \(\text {n}_t\text {S}_t\) | For \(t\in \Omega _{\text {conv}}\) | Voluntary conversion |
0 | \(\text {V}_t\) < \(\text {C}_t\); \(\text {V}_t\) > \(\text {n}_t\text {S}_t\) and \(\text {V}_t\) < \(\text {P}_t\) | For \(t\in \Omega _{\text {conv}}\) | Continuation |
\(\text {P}_t\) | \(\text {P}_t\) > \(\text {V}_t\) and \(\text {n}_t\text {S}_t\) < \(\text {P}_t\) | For \(t\in \Omega _{\text {put}}\) | Repurchase |
\(\kappa \text {B}\) | \(\text {n}_t\text {S}_t\) < \(\kappa \text {B}\) | For \(t=T\in \Omega _{\text {put}}\) | Redemption at maturity |
\(\theta \kappa \text {B}\) | \(\text {V}_t\) < \(\theta \kappa \text {B}\) | For \(t\in [0,T]\) | Default |