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\begin{document}
\title{\bf Modeling the Distance-to-Default Process of a Firm}
\author{Marco Avellaneda\\
Courant Institute of Mathematical Sciences\\
New York University\\
New York, NY 10012\\
and\\
Jingyi Zhu\\
Department of Mathematics\\
University of Utah\\
Salt Lake City, UT 84112}
\maketitle
\section{Introduction}
Credit derivatives provide synthetic protection against bond and loan
defaults.
A simple example of a credit derivative
is the credit default swap, in which one counterparty makes periodic
payments to another
in exchange for the right to be paid a notional amount if a
credit event happens. Recently,
we have seen the emergence of "wholesale" credit protection in the
form of first-to-default swaps written on a basket of underlying credits.
Pricing of credit derivatives requires quantifying
the likelihood of default of the reference entity.
Default probabilities can be estimated
from the spreads of the bond issued by the reference
entity \footnote{A more difficult problem, which often occurs in
practice, is to estimate default probabilities of a firm that has
not issued any debt.}.
%The value of default-contingent cash flows under
%different default scenarios can be calculated, in principle, from these
%probabilities.
Two kinds of mathematical frameworks for pricing credit derivatives
have been proposed to this date: the
\it{structural models}, \rm introduced by Merton \cite{Merton} and
others \cite{BC,BS,TT,KRS,STD,Geske},
and the \it{reduced form models} \rm of Duffie and Singleton \cite{DS}.
Here, we will be concerned with the structural approach, and in particular
with the "default barrier" methodology recently introduced by Hull and
White \cite{HW2}.
This paper draws from Hull and White \cite{HW2} and extends it in several
directions.
First, we consider a continuous-time version of the Hull-White model in
which the
default index follows a general diffusion process. We show that, in this
general
framework, calibration of the default barrier leads to a new free boundary
problem for the associated Fokker-Planck partial differential equation.
Second, based on
these results, we propose a new interpretation of the default barrier model
in terms of a \it{risk-neutral distance-to-default} \rm
process for the firm (or risk-neutral
debt-to-value ratio, etc.). We show that
finding a default boundary which is calibrated to a set of default
probabilities
is equivalent to specifying an appropriate ``excess drift'' for the
distance-to-default
process. This excess drift can be interpreted
as a market price of risk for the value of the firm, \it{a la }
\rm Merton \cite{Merton}
which renders the process consistent with observed credit spreads.
Finally, we discuss the numerical implementation of the model, using a
finite difference scheme for the Fokker-Planck
equation, coupled with a Newton-Raphson scheme for determining the
excess drift
at each successive time step. The theory is applied to calculating the
default boundaries
and drifts for AAA and BAA1 credits under different assumptions about
default probabilities
and volatility functions.
\section{The Barrier Diffusion Model}
Following Hull and White \cite{HW2}, we define the function
$P(t)$ to be the probability that the firm has defaulted by time $t$,
as seen today \footnote{We assume henceforth this probability has been
determined from bond spreads or other procedure \cite{HW1}.}.
The default probability density is given by $P'(t)$. In particular,
$P'(t) \Delta t$ represents the probability of default between times
$t$ and $t+\Delta t$, as seen at time zero. Just as in \cite{HW2}, we
consider a ``default index'' associated with the firm, represented
by an Ito process
$\{X(t),\ X(0)=X_0\}$
\be
dX(t) = a(X(t),t) dt + \sigma(X(t),t) dW(t),
\ee
where $W(t)$ is the standard Wiener process.
We pronounce that the firm defaults at time $t$ if
\be
X(t) = b(t), \ \ \mbox{and} \ \
X(s) > b(s), \ \ s0, \ \ \ x>b(t);
\label{eq:heat}
\ee
with initial and boundary conditions
\be
f(x,t)_{|_{t=0}} = \delta(x-X_0), \label{eq:init}
\ee
\be
f(x,t)_{|_{x=b(t)}} = 0. \label{eq:bc}
\ee
We note that the integral
\be
\int_{b(t)}^\infty f(x,t) dx
\ee
represents the survival probability up to time $t$, $1-P(t)$.
Therefore, the default probability $P(t)$
is related to the survival probability density $f(x,t)$ and the barrier
$b(t)$
by the equation
\be
P(t) = 1 - \int_{b(t)}^\infty f(x,t) dx. \label{eq:Pt}
\ee
We see from this equation that
the barrier function $b(t)$ must be chosen appropriately so
that the pair $\{f(x,t), b(t)\}$ is consistent with
the given default probabilities $\{P(t), t>0\}$.
To see this in a more explicit way, we differentiate the
default probability with respect to time in (\ref{eq:Pt}),
and use the equations satisfied by $f$, whence
\begin{eqnarray}
P'(t) & = & - \int_{b(t)}^\infty \frac{\partial f}{\partial t} dx
+ f(b(t),t) b'(t) \nonumber \\
& = & -\frac{1}{2} \int_{b(t)}^\infty \left( \sigma^2 f \right)_{xx} dx
+
\int_{b(t)}^\infty \left(a f\right)_x dx \nonumber \\
& = & \frac{1}{2} {\frac{\partial}{\partial x} (\sigma^2 f)
}_{|_{x=b(t)}}.
\label{eq:flux}
\end{eqnarray}
Thus, aside from (\ref{eq:bc}), the survival
density function must satisfy an additional boundary condition at the
barrier
$x=b(t)$. This gives rise to a \it{free boundary problem} \rm for the forward
Fokker-Planck equation, since the boundary $b(t)$ is unknown and
must be determined consistently with the two boundary conditions
(\ref{eq:bc}) and (\ref{eq:flux}).
\section{Similarity Transformation}
%As we explained above, this model will be calibrated to the
%market data (represented by $P(t)$) by choosing an appropriate barrier
%function $b(t)$ that matches the market-implied default
%probabilities. It remains to determine
%what kind of roles played by the volatility and drift in the process
%followed
%by $X(t)$.
%In the following, in order to illuminate the nature of the problem,
%we assume that both $\sigma$ and
%$a$ are only time dependent. The model itself is in no way restricted by
%this simplification.
We observe that the model is invariant
under a scaling, or similarity, transformation.
In fact,
let $\sigma_0$ be a positive number, and consider the transformation
\be
\tilde{x} = \frac{x}{\sigma_0}, \ \ \tilde{b}(t) = \frac{b(t)}{\sigma_0},
\ \ \tilde{\sigma}(\tilde{x},t) = \frac{\sigma(x,t)}{\sigma_0},
\ \ \tilde{a}(\tilde{x},t) = \frac{a(x,t)}{\sigma_0}.
\ee
It follows immediately that the new function $\tilde{f}(\tilde{x},t)$
given by
\be
\tilde{f}(\tilde{x},t) = \sigma_0 f(x,t)
\ee
also satisfies Eqs.(\ref{eq:heat}-\ref{eq:bc}) and Eq.(\ref{eq:flux}).
In particular, if $\sigma(x,t)$ is a constant, we can ``scale out''
the volatility parameter -- solutions with arbitrary
constant volatility $\sigma$ can be derived from the solution with
$\sigma=1$. The latter case is, in essence, the Hull and White model \cite{HW2},
in which the default index is taken to be a standard Brownian motion.
If $\sigma$ is not a constant, the proposed diffusion model
allows us to incorporate
volatility functions that depend on the index value as well as
time.
This observation can be useful if one expects
the volatility of the credit
default index to increase as the firm approaches default, for example.
%Another example would be a scenario in which the default index is inactive
%for a period time.
%In this case the default probability density can be very small for
%those times, but $f_x$ needs not to approach zero since the time dependent
%volatility will have
%to reflect the inactivity by taking a rather small value to offset the
%fact that $P'(t)$ can approach zero.
\section{Distance-to-Default}
Let us make the following
\medskip
\noindent
\bf{Definition}: \it
Let $P(t)$ denote the market-implied default probability of the firm.
A risk-neutral distance-to-default process (RNDD) is a diffusion process
satisfying
\be
dY(t) = \tilde{a}(Y,t) dt + \tilde{\sigma}(Y,t) dW(t),
\ee
such that (i) $Y(0) > 0$, and (ii) $\mbox{Prob} \left[ \inf_{s < t}
Y(s) \leq 0 \right] = P(t)$.
\rm
This definition is consistent with the notion that a firm defaults when the
value of its assets falls below the value of the debt.\footnote{ The RNDD
$Y(t)$ can
be
interpreted as the difference between the value of the assets and
the debt, or the log of the debt-to-equity ratio, etc.,
seen in a ``risk-neutral'' world.}
\rm
We notice that if we set
\be
Y(t) = X(t) - b(t),
\ee
where $X(t)$ is the default index process discussed in the previous section,
then $Y(t)$ is a RNDD process with $\tilde{a}(Y,t)=a(X,t)-b'(t)$ and
$\tilde{\sigma}(Y,t)=\sigma(X,t)$.
Furthermore, we have
\be
dY(t) = dX(t) - b'(t) dt.
\ee
Therefore, the problem of finding the barrier in the continuous-time
analog of the Hull-White model is
equivalent to the problem of finding the \it{excess drift} \rm
in the distance-to-default process that makes the latter ``risk-neutral''
(calibrated to data on default probabilities).
The RNDD survival density $u(y,t)$ is related to the default
index survival density $f(x,t)$ by
\be
u(y,t) = f(y+b(t),t).
\ee
It follows that the Fokker-Planck equation for the survival probability of the RNDD
process is given by
\be
u_t = b' u_y - (a u)_y + \half ( \sigma^2 u)_{yy},
\ \ \ y>0, \ t>0,
\label{eq:model_eq}
\ee
\be
u_{|_{t=0}} = \delta(y-Y_0), \label{eq:model_ini}
\ee
\be
u_{|_{y=0}} = 0, \ \ \ t>0, \label{eq:model_bd}
\ee
\be
\half \left[ \frac{\partial}{\partial y}
\left( \sigma^2 u \right) \right]_{|_{y=0}} = P'(t),
\ \ \ t>0.
\label{eq:model_fb_cond}
\ee
Here, $b'$ must be chosen, adaptively, in such a way that the
second boundary condition (\ref{eq:model_fb_cond}) is satisfied at all
times.
We conclude that the
free boundary problem for the default index is transformed into a
control problem for the RNDD.
%So in summary, \it{the Hull-White default barrier approach
%is equivalent to finding a time-dependent risk-neutral drift for the
%DDRN process}. \rm
In this reformulation, $b'(t)/\sigma$ can be viewed as a ``market price of risk''
associated with the firm's perceived creditworthiness, consistently with
Merton \cite{Merton}.
\section{Initial Layer and Matching of Solutions}
%To focus on the essence of the model, we will work on the simplified case
%where the volatility is a constant and adopt the setting of the equations
%(\ref{eq:model_eq}-\ref{eq:model_fb_cond})
%in the stretched coordinate system.
We first consider the special case where the coefficients $\sigma$ and
$a$ are constants and $b(t)$ is an affine function. Without any loss of
generality
we set $a=0$. \footnote{ The case of Brownian motion with drift is analogous,
with the only difference being that the slope of the line defining
the barrier must be modified.}
This special case will be used later to construct the
general solution.
%As we see in this model, the input to the problem will be a
%default probability $P(t)$ for $t>0$ and the output is the function $b(t)$.
%It turns out that only the default
%probability density $P'(t)$ is required since
%\be
%P(0) = 1 - \int_0^\infty u(x,0) dx = 1 - \int_0^\infty
%\delta (x-X_0+b(0)) dx = 0,
%\ee
%as long as $X_0 > b(0)$, which means that currently the firm is
%not defaulting
%\footnote{In a more general case, where not enough information has
%been gathered about the firm, the initial condition for $u$ may be a
%smooth function and
%$\int_0^\infty u(x,0) dx < 1$ (which means that the firm is not for
%sure to have survived up till now), we may have a nonzero $P(0)$ to start
%with.}.
Assume accordingly that the default barrier is given by the equation
\be
\bar{b}(t) = - \alpha - \beta t,\ \ \ \alpha>0, \ \ \ 0< t < t_0.
\ee
In this case, it can be shown that the corresponding default probability
is given by
\be
\bar{P}(t)
= 1 - \int_{\bar{b}(t)}^\infty f(x,t) dx
= N \left( \frac{-\alpha - \beta t - X_0}{\sigma \sqrt{t}} \right)
+ e^\frac{-2(\alpha+X_0)) \beta}{\sigma^2} N
\left( \frac{-\alpha + \beta t - X_0} {\sigma \sqrt{t}} \right),
\ee
where $N(x)$ is the standard cumulative normal distribution,
and the density is
\be
\bar{P}'(t) = \frac{\alpha + X_0}{t \sqrt{2 \pi t} \sigma}
e^{-\frac{(\alpha+\beta t + X_0)^2} {2 \sigma^2 t}}.
\ee
This follows from standard properties of Brownian motions.
Under the same assumptions, the survival probability density is
given by
\be
f_{\alpha,\beta} (x,t) = \frac{1}{\sigma \sqrt{2\pi t}}
e^{-\frac{(x-X_0)^2} {2 \sigma^2 t}}
\left[1 - e^{-\frac{2(\alpha+X_0)}{\sigma^2 t} (x+\alpha+\beta t)} \right]
\label{eq:layer}
\ee
for $-\alpha - \beta t \leq x < \infty$.
The RNDD density is given by
\be
u_{\alpha,\beta} (y,t) = f_{\alpha,\beta} (y-\alpha-\beta t, t) =
\frac{1}{\sigma \sqrt{2\pi t}} e^{-\frac{\left(y-\beta t - Y_0
\right)^2} {2 \sigma^2 t}}
\left[1 - e^{-\frac{2 Y_0}{\sigma^2 t} y} \right]
\label{eq:ini_lay}
\ee
for $y \geq 0$, where $Y_0 = X_0 + \alpha > 0$ is the initial
distance-to-default.
In Figure \ref{fig:def_prob},
the default probability $\bar{P}(t)$ and the default
probability density $\bar{P}'(t)$ as functions of
$t$ are shown for several sets of positive $\alpha$ and $\beta$ values.
In all these cases, we take $X(t)$ to be a standard Brownian motion with drift zero, $\sigma=1$ and $X_0=0$. In this case, $\alpha > 0$ is the initial
distance-to-default. We observe that
$\bar{P}'$ is unimodal and drops to zero exponentially after reaching its
maximum.
The reason is that the barrier, which is linear in time,
outgrows the square root scaling in time of the Brownian motion,
therefore as time increases and passes over a certain level, the default
probability density will decrease towards zero.
\begin{figure}[!hb]
\caption{Default Probability Density for Some $\alpha$ and $\beta$ Values}
\label{fig:def_prob}
\begin{center}
\input{fig_def_prob}
\end{center}
\end{figure}
We now consider the solution of the model for arbitrary data $P(t)$.
The idea is to use the straight line model for a finite but small
time $t_0$, and then to match this solution to a numerically computed
$b(t)$. The need for an ``initial layer'' arises from the fact that
the $\delta$-function initial data vanishes to all orders for $t=0$ and
must be regularized consistently with the boundary conditions that we
want to impose for small values of $t$.
For a given default probability data $P(t)$, let us choose the
parameters $\alpha$ and
$\beta$ in such a way that
\be
\bar{P}(t_0) = P(t_0), \label{cond:def1}
\ee
\be
\bar{P}'(t_0) = P'(t_0), \label{cond:def2}
\ee
where $P(t_0)$ and $P'(t_0)$ are estimated from the market data.
A simple Newton-Raphson solver can lead to a solution of $\alpha$ and
$\beta$
for small values of $P(t_0)$ and $P'(t_0)$.
In Figure \ref{fig:surv}, we consider an example where $t_0=0.5$ is chosen, and
$P(0.5)=0.01$, $P'(0.5)=0.02$,
which lead to $\alpha=1.044$ and $\beta=1.949$. The survival
distribution at $t=0.5$ is plotted in the graph.
Once the initial survival distribution at $t=t_0$ has been determined,
we use it as an initial condition for the PDE problem
(\ref{eq:model_eq}-\ref{eq:model_fb_cond}).
Since this distribution is derived from the default probability conditions
(\ref{cond:def1}) and (\ref{cond:def2}), the compatibility condition
(\ref{eq:model_fb_cond}) is automatically satisfied at $t=t_0$.
A second-order finite difference algorithm is described below to solve the
PDE for time beyond
$t_0$.
\begin{figure}[!hb]
\caption{Survival Distribution at $t=0.5$} \label{fig:surv}
\begin{center}
\input{fig_surv}
\end{center}
\end{figure}
\section{Numerical Algorithm for General Default Probabilities}
%Our solutions consist of two parts: an analytic solution (\ref{eq:ini_lay})
%for time $t \leq t_0$ and a finite difference solution for time beyond
%$t_0$.
%As we explained earlier in
%section 3, this problem is a control problem where a
%drift coefficient $b'(t)$ is to be determined in the PDE problem with
%a fixed boundary. Our numerical algorithm is based on solving the PDE
%over this fixed domain.
The numerics are based on the ``RNDD formulation'', i.e. on solving
a control problem for the unknown drift coefficient $b'(t)$. For
simplicity, we write down the scheme for the case $\sigma=1$, and $a=0$.
The extension of the algorithm to variable coefficients is obvious.
We start from $t=t_0$ with the initial condition from the initial
layer solution, that is,
\be
u(y,t_0) = u_{\alpha,\beta} (y,t_0), \ \ \ y \geq 0.
\ee
A second-order finite difference algorithm for
Eq.(\ref{eq:model_eq}-\ref{eq:model_bd}) can be constructed as follows.
Define $y_j = (j-\half)h, \ t^n = n \Delta t$, and let $u_j^n$
represent the numerical approximation to $u(y_j,t^n)$.
We consider a Crank-Nicholson scheme
\be
\frac{u_j^{n+1} - u_j^n}{\Delta t} = \lambda^{n+\frac{1}{2}}
\frac{u_{j+\half}^{n+\half} -u_{j-\half}^{n+\half}}{h} + \frac{u_{j+1}^n
- 2 u_j^n + u_{j-1}^n}{4h^2} +
\frac{u_{j+1}^{n+1} - 2 u_j^{n+1} + u_{j-1}^{n+1}}{4h^2}
\ee
with boundary condition $u_0^n = 0, \ n \geq 0$,
where $\lambda^{n+\half}$ is an undetermined drift which depends on the
time-step ($n$) and which will be determined inductively.
Here
$u_{j+\half}^{n+\half}$ is computed from a predictor step, which involves
Taylor extrapolations in space and time with an upwind scheme approximation
for the spatial derivative.\footnote{We note that this
numerical scheme is second-order accurate in both space and time,
and all of our calculations are unconditionally stable with respect to the
choices of $h$,
$\Delta t$ and $\lambda^{n+\half}$.}
For each value of $\lambda^{n+\half}$, the resulting tridiagonal
system is solved using a standard linear algebra package.
The value of $\lambda^{n+\half}$ which matches the
extra boundary condition (\ref{eq:model_fb_cond}), $\lambda^{n+\half}_*$,
is found by using the Newton-Raphson iteration method.
Finally, we equate the computed $\lambda^{n+\half}_*$ with the drift, i.e.
\be
b'(t^{n+\half}) = \lambda^{n+\half}_*,
\ee
and extend the barrier in one time step
\be
b(t^{n+1}) = b(t^n) + \lambda^{n+\frac{1}{2}}_* \Delta t.
\ee
The numerical stability of the algorithm, {\it i.e.} the continuous dependence
of the function $b'(t)$ on the probability density $P'(t)$, is
an important consideration, considering the fact that credit default
data is discrete and that, consequently,
the probability density needs to be constructed
by interpolation. We mention here without further proof that the
free-boundary problem and the algorithm
admit a unique, stable solution on any interval where the probability density
$P'(t)$ is positive. Further comments on stability and the issue of
interpolation of probabilities are made in the study of concrete examples in
the next section.
\section{Examples}
For the numerical example, we introduce a finite
domain ($0 \leq x \leq 20$) which is large enough to cover the dynamics of
the solutions studied in this section.
Also, we use an initial layer with
$t_0=0.5$. Unless explicitly noted, a constant volatility $\sigma=1$
is assumed. In the finite difference calculations, we use 400 points
in the $x$ direction and choose $\Delta t = 0.05$.
\begin{table}[!hb]
\caption{Default Probability for Banks}
\begin{center}
\begin{tabular}{|r|r|r|r|r|} \hline
& \multicolumn{3}{c|}{\bf{AAA}} & \bf{BAA1} \\ \cline{2-5}
\bf{year} & \multicolumn{4}{c|}{expected recovery rate} \\ \cline{2-5}
& 30\% & 50\% & 70\% & 50\% \\ \hline
1 & 0.0052 & 0.0073 & 0.0122 & 0.0222 \\
2 & 0.0097 & 0.0136 & 0.0227 & 0.0285 \\
3 & 0.0119 & 0.0166 & 0.0277 & 0.0315 \\
4 & 0.0135 & 0.0190 & 0.0316 & 0.0339 \\
5 & 0.0150 & 0.0210 & 0.0351 & 0.0360 \\
6 & 0.0164 & 0.0229 & 0.0382 & 0.0380 \\
7 & 0.0176 & 0.0246 & 0.0410 & 0.0396 \\
8 & 0.0188 & 0.0264 & 0.0439 & 0.0415 \\
9 & 0.0203 & 0.0284 & 0.0473 & 0.0437 \\
10& 0.0220 & 0.0307 & 0.0512 & 0.0466 \\ \hline
\end{tabular}
\end{center}
\end{table}
In the first example
we consider default probabilities for the bank industry with Standard and
Poor's AAA and BAA1 ratings
\footnote{data and sources are available from the authors upon request.}.
The default probabilities for several recovery rates are shown in Table 1,
where a bank's default probabilities in each of the next 10 years are
listed. For instance, 0.0073 means that a AAA-rated bank would have a
0.73\% probability to default within the next year, and likewise 0.0136
means that it would have a 1.36\% probability to default within the
second year.
These default probabilities were estimated based on an
expected recovery rate of 30\%, 50\% and 70\%. As discussed in \cite{HW1},
different expected recovery rates cause very different default
probability estimates. As a consequence, our default barriers will exhibit
a strong dependence on the expected recovery rate assumed. In our
calculations, the data from the table is expanded to generate a piecewise
constant function of time $P'(t)$. In Figure \ref{fig:AAA_BAA1}, default
barriers for AAA and BAA1 banks from our model
are plotted for
$0 \leq t \leq 10$, based on data sets in Table 1 with a $50\%$ expected
recovery rate. As we
notice, the shapes of the curves for these two ratings are quite similar,
since they both belong to the same industry and therefore bear
similar characteristics as when the firm is more likely to default in the
future. The barrier
for the lower rating (BAA1) is always above the barrier with the higher rating
(AAA), indicating that it is much more likely for the firm with a
lower rating to default.
\begin{figure}[!hb]
\begin{center}
\caption{Default Boundaries for AAA and BAA1 companies} \label{fig:AAA_BAA1}
\input{fig_AAA_BAA1}
\end{center}
\end{figure}
One of the advantages of the generalized model presented here
is the ability to
incorporate general volatility structures. Here we consider
an example where the
volatility is increased to a higher level once the Brownian path
gets near to the default boundary, and compare the result to the
result with
a constant volatility ($\sigma=1$). In particular, we choose
\be
\sigma (x) = \left \{
\begin{array}{ll}
1 & 0 \leq x \leq 2, \\
1 - \frac{1}{4} (x-2) & 2 < x \leq 4, \\
\frac{1}{2} & x > 4. \\
\end{array} \right.
\ee
\begin{figure}[!hb]
\caption{Barriers from Different Volatility Structures}
\label{fig:vol}
\begin{center}
\input{fig_vol}
\end{center}
\end{figure}
In Figure \ref{fig:vol}, default
barriers from our model are plotted for
$0 \leq t \leq 10$, based on data set with expected recovery rate 0.5 in
Table 1. Two barrier curves represent the cases with the constant
volatility and the variable volatility, respectively.
We find that the barrier with variable volatility
lies above the barrier with constant volatility. This is
because the average volatility in the variable case is lower than the
contant volatility level chosen for the problem. To achieve the same exit
probability, the barrier has to move up to accommodate a lower volatility.
Next, we compare this PDE model with the original
Hull-White model in this application. We implemented the Hull-White model
according to \cite{HW2} with the same discretization and numerical
parameters. The results are shown in Figure \ref{fig:comp},
where default probabilities assuming an expected recovery rate of 50\%
are used. In regard to the shape and location of the barriers,
the main difference between the
models is that in Hull-White paths are allowed to exit from the
barrier only at a discrete times, whereas the paths can exit any time
in the PDE model. This explains the fact that
the barrier from the PDE model lies slightly below the Hull-White barrier.
%Another difference is that
%in the PDE model the initial layer gives a particular default probability
%structure that is not fitted to a constant density as required by the
%data.
\begin{figure}[!hb]
\caption{Comparison with Hull-White Model} \label{fig:comp}
\begin{center}
\input{fig_comp}
\end{center}
\end{figure}
Barriers for default probabilities with other expected
recovery rates can also be computed. In Figure \ref{fig:rec}, barriers
corresponding to default probabilities for AAA banks listed in Table 1 for
recovery rates of 30\%, 50\% and 70\% are plotted. As expected,
since default probabilities for a lower recovery rate are smaller
than the corresponding default probabilities for a higher recovery
rate, the barrier for this low recovery rate will lie below a barrier with
a higher recovery rate.
\begin{figure}[!hb]
\caption{Default Boundaries for Different Expected Recovery Rates}
\label{fig:rec}
\begin{center}
\input{fig_rec}
\end{center}
\end{figure}
In general, default probability data is discrete, as shown in Table 1.
Since the PDE method requires interpolation of the probabilities, it
is important to verify that different interpolation methods for the
default probability density function do not produce significant changes
in the barriers generated by the model. In all of the above calculations,
we used a piecewise constant default probability density $P'(t)$ computed
in a straightforward way from cumulative default probability data.
To study the sensitivity to different interpolation schemes, we considered
a piecewise linear
interpolation scheme for $P'(t)$, requiring
that $P(t)$ generated be consistent
with the data
at the original data points. In Figure \ref{fig:stab}, we plot the barriers resulted
from these two default probability densities. The numerical results indicate
that the scheme is stable with respect to small perturbations of the
probability density function representing the data.
\begin{figure}[!hb]
\caption{Different Interpolation Methods for Default Probabilities}
\label{fig:stab}
\begin{center}
\input{fig_stab}
\end{center}
\end{figure}
In Figure \ref{fig:drift}, we display the default probability density (input)
and the drift function $b'(t)$ (output) for the case of AAA-rated
banks. This graph shows qualitatively the way in which the drift
``responds'' to the default probability density data:
an increase in the default probability will certainly lead to an
increase in the drift, which moves the barrier up, making the paths
more likely to exit the barrier.
\begin{figure}[!hb]
\caption{Default Probability and Corresponding Drift}
\label{fig:drift}
\begin{center}
\input{fig_drift}
\end{center}
\end{figure}
Once the default barrier for a firm has been computed for a period $(0 \leq t
\leq T)$, it is possible to calculate {\it forward default probabilities}
at certain future time $T_0 > 0$ as well using this model. In fact, we
just
need to solve the PDE starting from $T_0$ with the default barrier fixed, and
start the survival density from $T_0$
\be
u (x,T_0) = \delta(x-X_0),
\ee
so the initial distance-to-default at $T_0$ is the same as today.
As we discussed the equivalence between the drift $a(t)$ and the
default boundary $b(t)$ in sections 3 and 4,
here they will have different roles to play.
If there is enough information available, it is possible to
fit the drift to a forward default probability structure. In Table 2,
we present the 5 year forward default probabilities from the results in
Figure \ref{fig:AAA_BAA1}.
Intuitively speaking, these are the default probabilities
for the next 6 to 10 years, given that the firm has survived the
first 5 years and the
default probability for the next instant is the same as today.
It is observed that the forward default probabilities
are much larger than the spot probabilities, due to the fact that
the shape of our barrier function is concave upward.
\begin{table}[!hb]
\caption{Forward Default Probabilities}
\begin{center}
\begin{tabular}{|r|r|r|} \hline
year & AAA & BAA1 \\ \hline
1 & 0.244002 & 0.301495 \\
2 & 0.165700 & 0.182113 \\
3 & 0.096906 & 0.103299 \\
4 & 0.068161 & 0.071312 \\
5 & 0.053285 & 0.055123 \\ \hline
\end{tabular}
\end{center}
\end{table}
Finally, as a verification of the numerical scheme,
it is mathematically interesting to study the case where
the default probability reaches a level where a default is certain to happen
by certain time $T$, as predicted by the input default probability.
This should be reflected in the fact that the default boundary
will be exited before this particular time $T$ by virtually all Brownian
paths $X(t)$, which can only happen when $b'(t)$ blows up at this time
and the curve $b(t)$ becomes ``vertical'' as $t$ approaches $T$.
To verify this,
we choose a uniform default probability density $P'(t)=0.1$. In this
case, the cumulative default probability $P(10)=1$, which means that
the firm will necessarily default before $T=10$ with probability one.
In Figure \ref{fig:blowup}, we see that the barrier function $b(t)$ indeed
becomes vertical as $t$ approaches 10.
Also shown in the same figure is the result of another
experiment where the default probability density is increased to 0.2, where
the blow-up time is pushed to approximately
$T=5$, as predicted from the fact that $P(t)$ reaches 1 as $t$ approaches
5.
\begin{figure}[!hb]
\caption{Blow Up of Default Boundary} \label{fig:blowup}
\begin{center}
\input{fig_blowup}
\end{center}
\end{figure}
\section{Conclusions}
Generalizing the Hull-White model \cite{HW2} to continuous-time default
index models, we propose a general framework for modeling default indices
as diffusions and default events as first-passages across barriers that
generalizes the Hull-White discrete model based on a discrete random
walk. We show that the
calibration of such continuous-time default index models to default
probability data
leads to a free-boundary problem for the corresponding
Fokker-Planck equation.
We also established an isomorphism between the
default index formulation of Hull and White and the concept of a
risk-neutral distance-to-default (RNDD)
index. This isomorphism allows us to
reinterpret the derivative of the Hull-White default boundary as an
``market price of risk'' that has to be added to the distance-to-default process of
the firm to make it consistent with observed default probabilities extracted
from bond spreads or
credit ratings. We proposed a simple numerical algorithm for
finding the
unknown drift, based on a discretization of a control problem.
Several examples and tests were presented,
indicating that the algorithm produces reasonable
results and is stable with respect to
small perturbations of the input probability densities.
Finally, we point out that it is also possible to construct ``non-parametric''
models that implement the concepts of risk-neutral default index and RNDD.
These models would be based on fitting the first-passage times of
random paths across a barrier to given default probabilities.
For example, a Monte Carlo simulation of different scenarios for the
distance-to-default of a firm can be generated
using econometric data on the volatility of the firm.
In a second step, the probabilities of the different scenarios
can be appropriately re-calibrated so as to reflect contemporaneous data on
cumulative default probabilities, as in the Weighted Monte Carlo method
\cite{ABFGKN}.
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\end{document}