| A fixed-income economy, which includes defaultable securities, is developed through a potential theoretic approach to modeling the spot rate of interest. Under the assumption of an arbitrage free market, the riskless and risky state-price densities are used as inputs to generate the respective spot rates in a Markovian setting. The riskless state-price density is simply the discounted conditional expectation of the derivative of the martingale measure Q with respect to the reference probability P associated with the underlying Markov process Xt. The risky state-price density is an original modification of its riskless counterpart. If the time to default is modeled as the first jump in a generalized Poisson process with intensity lambdat = lambda(X t), then the risky state-price density is defined as the discounted conditional expectation of the derivative of the forward martingale measure F with respect to P. However, the discounting is done with respect to the default intensity lambda rather than the riskless spot rate. Furthermore, it is revealed through the resulting expression for the risky bond price, that the default intensity lambda is the risk spread between the riskless and risky spot rates.; The main example used to illustrate this procedure is the well-known Ornstein-Uhlenbeck process from which a Cox-Ingersoll-Ross model of both spot rates is derived. In addition to computing bond prices with this example, a Cauchy problem for an originally designed option on the risk spread is derived through the Feynmann-Kac Theorem. A series solution is then developed using a modern potential theoretic version of the classical parametrix method for parabolic partial differential equations. |
