A Regime Switching Model for the Term Structure of Credit Risk Spreads ()
1. Introduction
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. [1] ). Generally, they fall into a loose hierarchy known as reduced-form models. The most ubiquitous approach involving hazard rate models wherein default risk via unexpected events is modeled by a jump process. In this framework, credit-risky securities are priced as discounted expectation under the risk neutral probability mea- sure with modified discount rate (cf. [2] , [3] ). Although conceptually simple and easy to implement, these models are limited by the appropriate calibration of the hazard rate process. More generally, spread modeling represents spreads directly and eliminates the need to make assumptions on recovery (cf. [4] , [5] ). Finally, rating based models consider the creditworthiness of the issuer to be a key state variable used to calibrate the risk-neutral hazard rate (cf. [6] - [8] ). A progressive drift in credit quality toward default (an absorbing state) is now allowed as opposed to a single jump to bankruptcy, as in many hazard rate models. Rating based models are particularly useful for the pricing of securities whose payoffs depend on the rating of the issuer.
In this paper, we consider a rating based regime switching model for the term-structure of credit risk spreads in continuous time (cf. [9] , [10] ). A unique feature of our model is the inclusion of stochastic transition pro- babilities. Credit instruments are then characterized as the solution to a ultraparabolic Hamilton-Jacobi system of equations for which we develop a methods-of-lines finite difference method. Computations are presented for a rating based callable bond which validates the applicability and efficiency of the method.
2. Model of the Economy
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. [11] , [12] ). Without loss of generality, we suppose a single risky zero-coupon bond price and continuous trading over a finite time interval
. We let
denote a continuous time Markov process on the regime (or états) space
with associated transition probabilities
, for all
; it follows that
(2.1)
for
. Let
represent the
-state transition distribution.
We define the transition probabilities as follows. The
-state we associate with default, in which case
. For
, we define the
-state transition dynamics consistent with the non- negativity constraint in (2.1) such that 
(2.2a)
(2.2b)
for
, where
![]()
and
is the mean transition level satisfying
,
is the rate of reversion to the mean,
and
is a Wiener process. From (2.1), it follows that
and so
(2.2c)
(2.2d)
We relate the transition matrix
to the regime dynamics via the infinitesimal generator
,
![]()
such that
![]()
for
, and
![]()
where
is the vector of probabilities
. Without loss of generality, we associate
with the vector
,
,
,
, subject to the dynamics
(2.3a)
(2.3b)
for
, where
is a martingale with respect to the filtration generated by
and
( [13] , Chap 4.8; [14] , Part III, App. B; [15] , Chap 8). In particular, the state of the system
is known at inception such that
, for some
.
We suppose that the risky interest rate R follows a state specific Cox-Ingersall-Ross dynamic given by
(2.4a)
for
, with mean reversion level
and rate of reversion to the mean
, such that
(2.4b)
where
is a Wiener process. In default
, otherwise
and
. The risky bond price
associated with a maturity
satisfies
(2.5a)
(2.5b)
We consider the risk-less interest rate
to satisfy
![]()
![]()
where in default
for convenience, and
otherwise.
For a given contract
, we define the value function associated with the joint Markov ultradiffusion process (2.2)-(2.5) such that
(2.6)
for
, where
.
In particular, for a non-coupon paying bond
and
otherwise, where
is the de- fault recovery rate, whereas for a callable bond
and
other- wise, for some rating based exercise price
. Generalization of (2.6) and the subsequent analysis to include early exercise features follows routinely and will not be considered here.
3. Characterization
Letting
and
(3.1a)
we recover (2.6) succinctly as
(3.1b)
for
. By Itô’s rule, the value function (2.6) is characterized via (3.1) as the solution to the ultraparabolic Hamilton-Jacobi system of equations
![]()
![]()
![]()
![]()
![]()
![]()
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where
![]()
Let
denote the temporal variable and
the spatial, we define
![]()
and
, such that the above can be written
(3.2a)
for all
, subject to the terminal constraint
(3.2b)
for
, where ![]()
4. Approximation Solvability
Towards obtaining a constructive approximation of (3.2), we consider an exhaustive sequence of bounded open domains
such that
and
as well as a sequence of monotonically increasing real numbers
, as
. Let
and
, we seek
satisfying
(4.1a)
for all
, subject to the boundary condition
(4.1b)
for
, and terminal constraint
(4.1c)
where
. As (3.2) is an infinite horizon problem in
, we remark to the necessity of intro- ducing the artificial terminal condition
along the frontier
(cf. [16] ). In particular,
as
, on any compact subset of
, for any fixed
.
We next place (4.1) into standard form by setting
,
,
, in which case
. Letting
![]()
Equation (4.1) becomes
(4.2a)
for all
, subject to the boundary condition
(4.2b)
for
, and initial condition
(4.2c)
where
, 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
and mesh sizes
, such
that
and
. Spatially, we utilize the step sizes
and mesh sizes
; we denote the value of
on the grid by
![]()
where
,
,
,
, and so forth. Notationally, we let
, where
,
,
, and
. For
![]()
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
(4.3a)
for all
, subject to the boundary condition
(4.3b)
for
, and initial condition
(4.3c)
where
,
,
![]()
![]()
and
. We solve (4.3) utilizing the pseudo-code (cf. [16] , [17] ):
do ![]()
do ![]()
solve for
via (4.3).
5. Numerical Experiment
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
. We suppose a solvent risk-free rate of return of
. For simplicity, we will con- sider the following transition matrix
![]()
in which only the default probability
is stochastic.
For
, we have the economy;
(5.1a)
(5.1b)
(5.1c)
where
![]()
and
![]()
Letting
and
, the ultraparabolic Hamilton-Jacobi system of Equations (4.1) for the value function
associated with the ultradiffusion (5.1) is then
(5.2)
for all
,
(5.3a)
for all
, such that
(5.3b)
for
and
(5.3c)
(5.3d)
and
(5.4a)
for all
, such that
(5.4b)
for
and
(5.4c)
(5.4d)
Figure 1 and Figure 2 show the value function components
and
, respectively, for
. Relative to the discretization of (5.2)-(5.4), we utilized
,
,
,
. In particular, we note the effect of the rating based exercise prices on
and
and the de- creasing value of
with increasing
, as expected.