^{1}

^{*}

^{2}

^{*}

We consider a rating-based model for the term structure of credit risk spreads wherein the credit-worthiness of the issuer is represented as a finite-state continuous time Markov process. This approach entails a progressive drift in credit quality towards default. A model of the economy is presented featuring stochastic transition probabilities; credit instruments are valued via an ultra parabolic Hamilton-Jacobi system of equations discretized utilizing the method-of-lines finite difference method. Computations for a callable bond are presented demonstrating the efficiency of the method.

When pricing of credit instruments subject to default risk, market participants typically assume that default is unpredictable, using dynamics derived from rating information in order to take advantage of credit events (cf. [

In this paper, we consider a rating based regime switching model for the term-structure of credit risk spreads in continuous time (cf. [

In this section, we introduce the dynamics of the risk-less and risky term structures of interest rates as well as the bankruptcy process. To this end, we assume the existence of a unique equivalent martingale measure such that all risk-less and risky zero-coupon bond prices are martingales after normalization by the money market account (cf. [

for

We define the transition probabilities as follows. The

for

and

We relate the transition matrix

such that

for

where

for

is known at inception such that

We suppose that the risky interest rate R follows a state specific Cox-Ingersall-Ross dynamic given by

for

where

We consider the risk-less interest rate

where in default

For a given contract

for

In particular, for a non-coupon paying bond

Letting

we recover (2.6) succinctly as

for

where

Let

and

for all

for

Towards obtaining a constructive approximation of (3.2), we consider an exhaustive sequence of bounded open domains

for all

for

where

We next place (4.1) into standard form by setting

Equation (4.1) becomes

for all

for

where

We consider the discretization of (4.2) by the backward Euler method temporally and central differencing in

space. To this end, we introduce the temporal step sizes

that

where

the difference quotients are then backward first order in time:

and central second-order in space:

and so forth, and

and so forth.

Given the above, we define the method-of-lines finite difference discretization of (4.2) such that

for all

for

where

and

do

do

solve for

In this section, we present a representative computation for the valuation of a callable bond relative to three credit ratings:

and rating’s dependent pay-off contract

with expiry

in which only the default probability

For

where

and

Letting

for all

for all

for

and

for all

for

SeungmookChoi,Michael D.Marcozzi, (2015) A Regime Switching Model for the Term Structure of Credit Risk Spreads. Journal of Mathematical Finance,05,49-57. doi: 10.4236/jmf.2015.51005