Defaultable Bond Markets with Jumps

Abstract

We construct a model for the term structure in a market of defaultable bonds with jumps ${p^d(t,T);t\le T}$, $T\in(0,T^]$. We derive the instantaneous defaultable forward rate $f^d(t,T)$ defined by $p^d(t,T)=\bb I_{{t<\tilde\tau}}e^{-\int_t^Tf^d(t,l)dl}$ in the real world probability. We are also given default-free bonds ${p(t,T);t\le T}$, $T\in(0,T^]$ and we establish the market consisting of both the defaultable and the non-defaultable bonds. In this market we study the common equivalent martingale measure and in this arbitrage free market we derive the relationship between the forward rates $f(t,T)$ and $f^d(t,T)$ associated with the two sorts of bonds. Especially, it is proved that in a parameterized market with common equivalent martingale measure where $f(t,T)$ can be described by (\ref{eqn-forward rate-1}) the defaultable forward rate $f^d(t,T)$ can be reconstructed from the special form of the default-free forward rate $f(t,T)$ if a certain system of BSDEs has a solution.\ Finally we extend the results to a market with recovery rate and give examples where the system of BSDEs has a solution.\