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A Numerical Method to Price Defaultable Bonds Based on the Madan and Unal Credit Risk Model

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A numerical method to price defaultable bondsbased on the Madan and Unal credit risk model
Luca Vincenzo Ballestra
Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +39-071-2207251, FAX N. +39-071-2207150, E-mail: l.v.ballestra@univpm.it
Graziella Pacelli
Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +39-071-2207050, FAX N. +39-071-2207150, E-mail: g.pacelli@univpm.it
Abstract
We propose a numerical method to price corporate bonds based on themodel of default risk developed by Madan and Unal in [23]. In particularusing a perturbation approach we derive two semi-explicit formulae that allowsto approximate the survival probability of the ﬁrm issuing the bond veryeﬃciently. More precisely we consider both the ﬁrst-order and the second-order power series expansions of the survival probability in powers of themodel parameter
c
. The zero-order coeﬃcient of the series is evaluated usingan exact analytical formula. The ﬁrst-order and the second-order coeﬃcientsof the series are computed using an approximation algorithm based on theLaplace transform.Extensive simulation is carried out on several test cases where the pa-rameters of the model of Madan and Unal are chosen as in [15], [23], andbonds with diﬀerent maturities are considered. The numerical experimentsperformed reveal that the numerical method proposed in this paper is accurateand computationally eﬃcient.
JEL Classiﬁcation Codes:
G13, C63.
Key Words:
Credit Risk, Defaultable Bonds, Asymptotic Expansion.
1 Introduction
One of the ﬁrst and most popular models to price defaultable bonds has been devel-oped by Madan and Unal in [23]. This model is considered particularly interesting1
since it incorporates the most attractive features of both the reduced-form modelsand the structural models of credit risk. In fact in [23] the probability of defaultis related to the value of the equity of the ﬁrm issuing the bond. This is typicalof the structural models, where the default event is linked to ﬁrm-speciﬁc variables(see for instance [3], [5], [6], [14], [17], [20], [21], [22], [28]). Moreover in [23] defaultis modeled as a random event that can occur unexpectedly at every time. Such anapproach allows to obtain high-short term spreads, and is peculiar to the reduced-form models (see [9], [10], [12], [18], [19]). According to these reasons the model[23] is considered a middle-way approach between the reduced-form models and thestructural models of default risk (for a detailed description of diﬀerent models of default risk see for example [11]).Let
V
(
t,T
) denote the price at time
t
of a defaultable zero-coupon bond withface value 1 maturing at time
T
,
T
≥
t
. Using the model of Madan and Unal [23]
V
(
t,T
) is evaluated as follows:
V
(
t,T
) =
Q
(
t,T
)(Ψ(
t,T
) + (1
−
Ψ(
t,T
))
R
)
,
(1)where
Q
(
t,T
) is the price at time
t
of a Treasury (riskless) zero-coupon bond withface value 1 maturing at time
T
, Ψ(
t,T
) is the probability that the ﬁrm issuing thebond survives up to time
T
given no default at time
t
, and
R
is the expected payoﬀ in case of default:
R
=
10
yq
(
y
)
dy,
(2)where
q
(
y
) is the probability density function of the bond recovery value. In [23]
q
(
y
) is modeled as the probability density function of a Beta distribution.Let us consider the “relativized” equity value of the ﬁrm:
s
(
t
) =
e
(
t
)
B
(
t
)
,
(3)where
e
(
t
) is the ﬁrm’s equity value and
B
(
t
) is the money market account (see[16]). In [23]
s
(
t
) is modeled as the stochastic process:
ds
(
t
) =
σs
(
t
)
dW
(
t
)
,
(4)where
σ
is a volatility parameter and
W
(
t
) is a standard Wiener process under therisk-neutral measure (see [16]). Note that the “relativized” equity value is modeledas a stochastic process with constant volatility, which is a very common practice inmathematical ﬁnance.2
Let
φ
(
t
) denote the so-called instantaneous intensity of default, that is
φ
(
t
)
dt
isthe probability that default occurs in the time interval [
t,t
+
dt
]. Madan and Unalmodel
φ
(
t
) as a function of
s
(
t
), in particular they assume:
φ
(
t
) =
c
(
log
(
s
(
t
))
−
log
(
δ
))
2
,
(5)where
c
and
δ
are positive constants. This choice is widely justiﬁed in [23], here weonly observe that according to (5) the probability of default is measured by the dis-tance of
s
(
t
) from the critical value
δ
. In particular when
s
(
t
) reaches the thresholdlevel
δ
the intensity
φ
(
t
) becomes inﬁnite, and default occurs with certainty.As shown in [23] Ψ(
t,T
) is a function of the time to maturity
T
−
t
and
s
(
t
).Then we set:Ψ(
t,T
) =
P
(
x
(
t
)
,τ
)
,
(6)where
x
(
t
) =
log
(
s
(
t
))
−
log
(
δ
)
,
(7)and
τ
=
T
−
t
. It is shown in [23] that the function
P
(
x,τ
) must satisfy the followingpartial diﬀerential equation:
∂P
(
x,τ
)
∂τ
−
σ
2
2
∂
2
P
(
x,τ
)
∂x
2
+
σ
2
2
∂P
(
x,τ
)
∂x
+
cP
(
x,τ
)
x
2
= 0
,
(8)with boundary conditions:
P
(0
,τ
) = 0
,
lim
x
→
+
∞
P
(
x,τ
) = 1
,
(9)and initial conditions:
P
(
x,
0) = 1
.
(10)Madan and Unal [23] have solved the partial diﬀerential problem (8)-(10) usingcertain changes of variables that reduce equation (8) to a ﬁrst-order ordinary dif-ferential equation. This ordinary diﬀerential equation can be easily approximatednumerically using a ﬁnite diﬀerence scheme. However, as pointed out by [15], theapproach of Madan and Unal lacks of mathematical rigor, since in one of the changesof variables used to transform the partial diﬀerential equation (8) a diﬀerential termhas been neglected. In particular in [15] it is shown that the solutions of (8)-(10)3
computed using a ﬁnite diﬀerence scheme are signiﬁcantly diﬀerent from those ob-tained using the approach of Madan and Unal.In this work we propose a perturbation approach to derive two approximateformulae that allows to compute the solution of the partial diﬀerential problem (8)-(10) very eﬃciently. Let
n
denote a positive integer, we consider the asymptoticexpansion of the solution of (8)-(10) in powers of the model parameter
c
with basepoint in
c
= 0:
P
(
x,τ
) =
n
j
=0
c
j
P
j
(
x,τ
) +
O
(
c
n
+1
)
, c
→
0
.
(11)We approximate
P
(
x,τ
) using both the power series expansion (11) truncatedat ﬁrst-order and the power series expansion (11) truncated at second-order. Thezero-order coeﬃcient of the series is evaluated using an exact analytical formula.The ﬁrst-order and the second-order coeﬃcients are computed using a fast andaccurate numerical algorithm based on the Laplace transform. More precisely theLaplace transform of
P
1
(
x,τ
) done with respect to the variable
τ
is evaluated usingan exact analytical formula that contains the so-called exponential integral function.Moreover the Laplace transform of
P
2
(
x,τ
) done with respect to the variable
τ
isaccurately approximated using only elementary functions and a limited number of exponential integral functions. Finally the Laplace transforms obtained are invertednumerically using an ad-hoc technique based on contour integration.Extensive numerical simulation is carried out on several test cases where theparameters of the model are chosen as in [15], [23], and bonds with diﬀerent maturi-ties are considered. The numerical experiments performed reveal that the numericalmethod is very accurate. In fact, when the ﬁrst-order power series expansion isused, the relative error obtained is always smaller than 1
.
0
×
10
−
2
, is often of order10
−
3
, 10
−
4
, and is sometimes of order 10
−
5
, 10
−
6
. Instead, when the second-orderpower series expansion is used, the relative error obtained is always smaller than9
.
8
×
10
−
4
, is often of order 10
−
5
, and is even of order 10
−
6
, 10
−
7
.The numerical method presented in this paper is also computationally fast. Infact the computer time necessary for the simulation is equal to 0
.
028
s
when thepower series expansion (11) is truncated at ﬁrst-order, and varies from 0
.
24
s
to 0
.
45
s
when the power series expansion (11) is truncated at second-order. Note that theseexecution times are experienced when the simulation is carried out on a computerwith a Pentium 4 Processor 1700 MHZ 256 MB Ram and the software programs arewritten using Matlab.We remark that the numerical method presented in this paper is well suited forparallel computing, since the algorithm used to invert the Laplace transforms of
P
1
(
x,τ
) and
P
2
(
x,τ
) is fully parallelizable.We point out that our numerical method proposed can also be used to forecast theparameters of the model of Madan and Unal. For instance, following an approach4
similar to the one used in [23] (we recall that the method used by [23] to solveproblem (8)-(10) is not mathematically correct), the semi-explicit formulae derivedin this paper can be applied to determined the model parameters
σ
,
c
,
δ
by maximumlikelihood ﬁtting to observed data.In addition our formulae can be used to obtain
implied
model parameters. Apossible approach to determine the parameters
δ
and
c
implied by model (1)-(5) isbrieﬂy described in the concluding section of this manuscript.The paper is organized as follows: in the next section we describe the numericalmethod used to compute the solution of problem (8)-(10) (some mathematical detailsabout the numerical method are given in the Appendix). In Section 3 we presentand discuss the results obtained using the numerical algorithm developed in Section2. Finally some conclusions are drawn in Section 4.
2 The numerical method
In this section we present the numerical method used to compute the solution of problem (8)-(10). For the sake of clarity this section is divided in two subsections.In Subsection 2.1 we write down the diﬀerential problems that must be solved inorder to obtain the coeﬃcients of the power series expansion (11) and determinesuitable expressions for the Laplace transforms of
P
0
(
x,τ
),
P
1
(
x,τ
),
P
2
(
x,τ
) donewith respect to the variable
τ
. In Subsection 2.2 we show how to numerically invertthe Laplace transforms obtained.
2.1 The power series expansion approach and the Laplacetransforms
Substituting the power series expansion (11) in equations (8)-(10), and equating tozero the terms of the same order, we obtain for the zero-order terms:
∂P
0
(
x,τ
)
∂τ
−
σ
2
2
∂
2
P
0
(
x,τ
)
∂x
2
+
σ
2
2
∂P
0
(
x,τ
)
∂x
= 0
,
(12)
P
0
(0
,τ
) = 0
,
lim
x
→
+
∞
P
0
(
x,τ
) = 1
,
(13)
P
0
(
x,
0) = 1
,
(14)and for the higher order-terms:
∂P
j
(
x,τ
)
∂τ
−
σ
2
2
∂
2
P
j
(
x,τ
)
∂x
2
+
σ
2
2
∂P
j
(
x,τ
)
∂x
=
−
P
j
−
1
(
x,τ
)
x
2
, j
= 1
,
2
,...,n,
(15)5

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