Directory UMM :Data Elmu:jurnal:J-a:Journal Of Banking And Finance:Vol24.Issue1-2.2000:

Journal of Banking & Finance 24 (2000) 329±352
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Determination of the adequate capital for
default protection under the one-factor
Gaussian term structure model
Daisuke Nakazato

*

The Industrial Bank of Japan Ltd, IBJ-DL Financial Technology, 5-1, Ootemachi 1-chome, Chiyodaku, Tokyo 100-0004, Japan

Abstract
In practice, credit risk is measured by one of the two di€erent methodologies. One
measures the prices and sensitivities of the credit linked instruments. Another measures
the required collateral or capital needed to cover a potential default loss. This paper
introduces a pricing methodology, which also determines the required capital.
Conventionally the value-at-risk method is used to determine collateral requirements.
Artzner et al. demonstrated that the resulting adequate capital measures may fail to
capture the credit diversi®cation e€ect, which is critical in credit risk management. (cf.
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. 1997. De®nition of coherent measures

of risk. Paper presented at the Symposium on Risk Management, European Finance
Association, Vienna, August 27±30). In order to overcome this problem, this paper
introduces a new approach, which combines closed form solutions and the Hull±White
trinomial tree. This combined approach is computationally faster than a naively implemented Monte-Carlo-based VAR methodology. The pricing model is based on the
rating-based Gaussian term structure model developed by Nakazato.
This approach is applicable to a wide variety of credit derivatives and their portfolios
in a coherent fashion. In this paper, however, attention is concentrated on determining
the collateral requirement for the counter party risk when there is the risk of credit
rating change on an issue and also risk of default on that issue or on any protection
already written on the issue. Ó 2000 Elsevier Science B.V. All rights reserved.

*

Tel.: +813-5200-7611; fax: +813-3201-0698.
E-mail address: d.nakazato@alum.mit.edu (D. Nakazato).

0378-4266/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 6 2 - X

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

JEL classi®cation: G33; G21; E43
Keywords: Capital adequacy; Credit; Term-structure; Risk-management

1. Introduction
Collateral is required to cover losses resulting from a potential default event.
If collateral is not present, then any loss must be withdrawn from capital.
Regulatory authorities require adequate capital to be available for holdings of
credit linked contingent claims, and recommend the use of certain formulas for
the calculation of this capital requirement. These methods are fairly simplistic,
hence when volumes of credit transactions increase, the accumulated required
capital becomes unreasonably large. Therefore a rational computational
methodology for capital adequacy is essential for survival in credit related
businesses.
The conventional method for determining adequate capital is the Value-atRisk Quantile method. From the plot on the probability distribution for the
present value of loss, the collateral is set at the required con®dence (quantile)
level. This method is know in practice to be time consuming, especially when
the distribution is generated by a Monte Carlo technique without any sophisticated acceleration scheme. An alternative method is to use the contingent

claim method. This method considers a contingent claim which covers the
excess loss that is not covered by the collateral at the time of default. The
collateral amount is then set so that the price of the contingent claim is suciently small compared to the insurance premium against the underlying default. This method may also be time consuming if the contingent claim price
does not have an analytic solution.
Artzner et al. (1997a,b) de®ne the term coherence for risk measurement.
Essentially, a credit risk methodology will give a coherent risk measurement if
the capital required to protect a portfolio of two positions is no greater than
the sum of the capitals required for each position. Hence coherent risk measures capture the credit diversi®cation e€ect. Artzner et al. criticize in particular
the quantile method for failing this test. They also point out that replacing
percentile criteria with standard deviation does not solve this problem. Other
contingent claim methods are not yet known to be coherent. The new methodology proposed in this paper is a contingent claim method which gives analytic solutions and is shown to be coherent.
One of the bene®ts of solving the capital adequacy as a valuation problem is
that, rather than inventing a separate scheme, it uses tools which must already
be in place to conduct any credit related business, namely the pricing and
sensitivities of credit related instruments. Also, the valuation model is ratings
based, an advantage of which compared to other approaches is the availability

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331

of ratings information. Furthermore such an approach ®ts comfortably with
the way credit risk is controlled in most ®nancial institutions.
In general, solving the pricing-based capital adequacy problem requires a
¯exible model. In particular, determination of the adequate capital for default
protection requires a dual-party model, where both the issuer of the bond and
the protector face separate credit rating changes as well as default risks.
In this paper, the model employed is a one-factor Gaussian term structure
model with credit rating classes, as developed in Nakazato (1997a). Lando
(1998) independently developed a similar model, however this model was not
addressed as a generalized Heath, Jarrow and Morton term structure model see
e.g., Heath et al. (1992). Hence its one-factor implementation is not completely
designed to ®t to the observed individual credit spread term structures for each
credit rating class while simultaneously maintaining the arbitrage-free conditions. The Nakazato model extends to the multiple party case with ane
Gaussian multi-factors while keeping the integrity of the single party HJM
equilibrium as a special case of the multiple party extension.
Because of the complex nature of this problem, it is necessary to use tensor
notation. This notation, although straightforward to use and common in
physics, is rarely met in ®nance, hence an explanation is given in Appendix A.
In this paper, it is shown that, in the one-factor case especially, algorithmic

implementation is straightforward. The method is essentially this: a Hull±
White (1990) tree is prepared, however, at each node of the tree, instead of a
single value, a two-dimensional array representing a matrix of corresponding
credit ratings is determined before discounting backwards. When accumulating
stepwise discounted values, occasional matrix multiplication is necessary. To
speed up the calculation, coding of the closed form expression may be used,
which is typical in the Hull±White approach. This numerical evaluation takes a
fraction of a second on a standard PC.
The essence of the Nakazato (1997a) is as follows: analytical tractability
comes from the use of spectrum decomposition and the choice of the term
structure model being Gaussian. The expression of the credit rating transition
in a matrix form reduces the term structure model to a matrix form. When
attention is restricted to a subset of matrix term structure models that have
common eigenvectors, then the matrix form of the equilibrium condition becomes an equilibrium condition of each individual eigenvalue (spectrum). Thus
for an individual spectrum, it is possible to develop an arbitrage-free Gaussian
term structure model. An advantage of the Gaussian model is that when
pricing discount bonds with di€erent maturities and credit ratings, their processes are driven by common Gaussian factors independent of their maturity
and credit rating classes. This approach is extended to the multiple party case,
by exploiting the simple structure of eigenvalues and spectra of the matrix
discount bond prices. Further simpli®cation is given by reducing the number of

Gaussian factors to one. It is then possible to use the Hull±White trinomial tree

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

methodology to evaluate the contingent claim analytically and algorithmically.
In the case of multi-factor evaluation, an ecient high-dimensional lattice
generation technique must be used. For details refer to Nakazato (1998).
The approach in this paper is general, however, the focus is on determining
the adequate capital for default protection when both a bond and also a credit
default option on that bond have been purchased. In this case there is not only
the risk of credit rating changes or default of the bond, but also the risk that
the writer of the option (known as the protector) may default on their obligation. Hence it is necessary to determine the price of a contingent claim which
protects against excess loss due to credit protection default.

2. Coherent risk measures
Artzner et al. de®ne a risk measure K(X), for a future net loss X from a
portfolio, to be coherent when the following four conditions are satis®ed:
‰Sub-additivityŠ K…X ‡ Y † 6 K…X † ‡ K…Y †:

‰HomogeneityŠ K…cX † ˆ cK…X †:
‰MonotonicityŠ K…X † 6 K…Y †;

if X 6 Y :

‰Risk-free conditionŠ K…X ÿ Rc† ˆ K…X † ÿ c;
fR : risk-free money market discount bondg:
Among these conditions, Sub-additivity is the most important. This constraint
is the mathematical restatement of the diversi®cation e€ect, which is at the
heart of credit risk management. The failure of the VAR method to satisfy
these criteria was illustrated in the article by Artzner et al. (1997b). For example, suppose a company has a policy to protect against any risk that has
more than 5% chance of occurring, and that the same criterion is applied
during approval of any individual position with such risks. A problem could
occur if the company individually approved two short positions in out-of-themoney options. A short position in an out-of-money call option, with a strike
in the upper 4% tail of the distribution, would be approved with no required
protection or collateral. Similarly, an out-of-money put option with a strike in
the lower 4% tail of the distribution would also be approved with no required
protection. However, together the two positions represent an 8% risk,
breaching the policy as no collateral protection has been allocated.
Not all pricing-based methods have been proven to be coherent. The method

introduced in this paper determines adequate capital by pricing a contingent
claim, which covers the excess loss that is not covered by the capital at the time
of default. The capital amount is then set so that the price of the contingent

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

333

claim is suciently small, when compared to an insurance premium that would
cover the total loss against the underlying default. The insurance premium
represents the expected loss amount. When expressed mathematically, the
adequate capital K(X) for the loss distribution X is de®ned as follows, for some
given suciently small e > 0:
K…X † ˆ inf K :

‡

E…X ÿ K† 6 eEX :

This function K(X) satis®es the most essential part of Artzner et al.'s de®nition

of the coherent risk measurement, which is sub-additivity, i.e. K…X ‡ Y † 6
K…X † ‡ K…Y †. The proof is as follows: Sub-additivity is clearly satis®ed if
‡

E…X ‡ Y ÿ K…X † ÿ K…Y †† 6 eE…X ‡ Y †;
since
K…X ‡ Y † ˆ inf K : E…X ‡ Y ÿ K†‡ 6 eE…X ‡ Y †:
By convexity
‡

‡

‡

E…X ‡ Y ÿ K…X † ÿ K…Y †† 6 E… X ÿ K…X †† E…Y ÿ K…Y †† :
Then by the de®nition
E… X ‡ Y ÿ K…X † ÿ K…Y ††‡ 6 eEX ‡ eEY ˆ eE… X ‡ Y †:
In this proof, it was assumed that EX > 0. Otherwise, replacing X with the
positive part X ‡ gives the general result, provided that the negative part (pro®t)
of this particular loss is not used for netting purposes.

The Homogeneity condition is trivially satis®ed by this measure. But there
are some pathological cases when the Monotonicity condition is violated. See
Appendix C for an example. In these rare occasions, this violation may cause
minor discrepancies in setting collateral levels too high for individual low-risk
positions, but when the credit business grows and diversi®cation is in e€ect,
then these discrepancies can be ignored.
The Risk-free condition states that risk-free assets can go in and out of the
portfolio freely, causing a change in risk measure proportional to the change in
risk-free asset. This measure does not satisfy the Risk-free condition as stated
above, however it does satisfy the criteria when equality is replaced with inequality. Unlike the violation of sub-additivity of VAR, this relaxation does
not have serious economic consequences. Hence the following modi®ed Riskfree condition is satis®ed:
‰Risk-free conditionŠ K…X ÿ Rc† 6 K…X † ÿ c;
f R: risk free money market discount bond: 0 6 cg:
This condition is, however, debatable because of its treatment of the riskfree asset, typically capital. The question is whether reinvestment of capital is

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

allowed. In practice, if reinvestment is allowed, then liquidity can become an

issue. In the event of default, the collateral must be easily liquidated, but this
has not always proved to be possible in practice. For this reason most prudent
risk managers prefer to inhibit reinvestment of collateral. However capital
managers prefer to reinvest, thus avoiding the depreciation of dormant capital
over time, and also reducing their capital adequacy requirement. Hence the
capital adequacy problem should be formulated di€erently, depending on the
company policy on capital reinvestment treatment.
In essence, the pricing-based capital adequacy determination should be
considered as a coherent risk measure for credit derivative risk management,
despite minor violations of the technical conditions, because it does capture the
credit diversi®cation e€ect.

3. The pricing model
In order to determine the adequate capital for default protection, a ¯exible
pricing model is required. In practice, a rating-based model is preferable as,
conventionally, credit risk is monitored by rating-based categorizations. The
term structure of yield spreads of each credit class must be re¯ected in the
model as well as the credit-risk-free interest rate term structure. Also the dualparty risk must be considered simultaneously. The two parties are the issuer of
the bond and the provider of protection in the event of the issuerÕs default.
These are the minimum requirements for a model to approach this particular
problem. The model should be further required to ®t the observed volatility
term structure of the yield curve and spreads. Each term structure of every
rating class must evolve consistently with the speci®cations in a no arbitrage
manner as in Heath et al. (1992). To the best of the authorÕs knowledge, the
Nakazato (1997a) is the only model at present that meets all these requirements. In this paper, for ease of algorithmic implementation, the requirement
for the volatility term structure is dropped and the term structures are modeled
by a simple one-factor extended Vasicek type model.
Now, the results from the Gaussian term structure model with credit rating
classes developed by Nakazato (1997a) are summarized. For this particular
problem, the dual-party one-factor ane Gaussian model is assumed. It is also
assumed that both parties come from the same credit rating category; this
means that when they are in the same credit class, they become statistically
indistinguishable. This assumption is imposed because of computational simplicity, for a more general treatment of individual credit transition, refer to
Nakazato (1997a). The credit rating classes are 1; . . . ; n and 0, the last one
being the default state. For the dual-party case, the state space is then a
product space of classes of each party. Now the basic building block of pricing
instruments, the discount bond price, is de®ned. In tensor notation, the dis-

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

335

count bond price from an initial state (a, b) at t to a terminal state (a, b) at
maturity T is denoted by Da; ab ;b (t,T). The bond pays the principal value of 1
only when the terminal state is reached at maturity. The discount bond price
function is assumed to be of the form

 Z T
a;b
n1 ;n2
a;b
exp ÿ
F…n1 ;n2 † …t; s† ds U n1 ;n2 :
Da;b …t; T † ˆ Ua;b
t

P n Pn
Here, EinsteinÕs convention is used, i.e. the summation signs n1 ˆ0 n2 ˆ0 are
omitted. An explanation for the tensor notation is given in Appendix A. From
this formula it can be seen that, for a ®xed maturity, each tensor discount bond
can be expressed as a linear combination
of the common n2 hypothetical assets
RT
(spectrum discount bonds) exp…ÿ t F…n1 ;n2 † …t; s† ds†. The basic tensors U are
constant. These are computed from the eigenvectors of the single-party transition probability matrix. This is explained in more detail below. The bene®ts
of tensor notation become apparent when evaluating complex credit contingent
claims, which are contingent on the credit states of multiple parties at various
cash ¯ow timings, in which case keeping track of indices can easily get out of
hand. Each spectrum then satis®es the equilibrium condition:

 Z T

F…n1 ;n2 † …t; s† ds
Et d exp ÿ
t
 Z T

F…n1 ;n2 † …t; s† ds dt;
ˆ R…n1 ;n2 † …t† exp ÿ
t

R…n1 ;n2 † …t† ˆ F…n1 ;n2 † …t; t†:

Or


Et exp



ÿ

Z

T

R…n1 ;n2 † …t; s† ds
t



ˆ exp



ÿ

Z

T
t


F…n1 ;n2 † …t; s† ds :

Each spectrum discount bond behaves as if it has its own closed economy. In
other words, for ®xed indices (n1 ; n2 ), the spectrum discount bonds have a
unique yield curve and volatility term structure.
Before explaining the notation used, it is necessary to review the single party
case, which gives a basis for the construction of the dual-party case. Suppose
that in practice, from historical observation, a state transition rate matrix is
estimated as a constant Q. The problem of calibrating parameters from the
market and historical data is explained in detail in Nakazato (1997a). Then Q
should be of the form
"
#
0 ~
0
;
Q ˆ ~0
k
k q ÿ~
q~
10 ˆ ~
00 :

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

Each element of ~
k is the default rate multiplied by one minus the recovery rate.
x when its diagonal elements form the vector ~
x.
A diagonal matrix is denoted by ~
All vectors are row vectors, unless stated otherwise; the transpose ~
x is explicitly
used to indicate a column vector. The matrix U of row eigenvectors of the
10 ˆ ~
10 . This ask may now be computed, normalized to U~
minor matrix q ÿ ~
sumes that all eigenvalues are distinct and non-zero, however if this is not the
case then perturb Q slightly. The basic tensors are now constructed as follows:
2 0
3
U 0


Un 0
2
3
~
6
7
0
6 ..
7 4 1
..
5;
6 .
7ˆ
.
4
5
ÿ~
10 U
U0 n

2

U0 0

6
6 ..
6 .
4

Un 0

Un n






..

U0 n

.
Un n

3

2
3
~
7
7 41 0 5
:
7ˆ
5
~
10 U ÿ1

In tensor notation, the order of subscripts and superscripts plays an important
role. Elements in the inverse matrix are expressed by reversing the order of
subscripts and superscripts. In the dual-party case, the basic tensors are obtained by multiplication of the two basic tensors for the single-party case:
U a1 ;a2 b1 ;b2 ˆ U a1 b1 U a2 b2 ;
Ua1 ;a2 b1 ;b2 ˆ Ua1 b1 Ua2 b2 :
These are the linear weights for the hypothetical assets when constructing the
tensor discount bond, and are simply the product of two elements from the
eigenvector matrix of the transition rate matrix of the single-party credit rating
transition.
Next, the model is ®tted to the initial yield and spread term structure. For
each credit class 1 through n as well as the ``credit'' risk-free case, smooth
discount yield curves are determined from the market data. Prices at time t of
~ n …t; T †,
~ 1 …t; T †; . . . ; D
pure discount bonds maturing at time T are denoted D
where the subscripts denote credit rating class. In the risk-free case, the price is
denoted D…t; T †.
The instantaneous forward spectra are de®ned as
F…0† …t; T † ˆ f …t; T † ˆ ÿ
F…m† …t; T † ˆ ÿ

o
log
oT

o
log D…t; T † …m ˆ 0†;
oT
!
n
X
a ~
U m Da …t; T †
…m ˆ 1; . . . ; n†;
aˆ1

F…n1 ;n2 † …t; T † ˆ F…n1 † …t; T † ‡ F…n2 † …t; T † ÿ f …t; T †:

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

337

The dual-party instantaneous forward spectra are simply linear combinations
of the single-party instantaneous forward rates.
In the Hull and White model, i.e. the extended Vasicek model, where a onefactor ane Gaussian model is assumed, it is natural to use the risk-free spot
rate rt as the unique common factor. However, for the one-factor Nakazato
model, when coecients of the spot rate are expressed in terms of the current
term structure of the yield and spreads, the calculation of these coecients is
non-trivial. The use of a one-factor ane Gaussian model implies that
Z T
F…n1 ;n2 † …t; s† ds ˆ a…n1 ;n2 † …t; T † ‡ b…n1 ;n2 † …t; T †rt :
t

The single-party case is described in the same way except with one less index. It
is feasible but dicult to ®nd the coecient functions a, b for both single- and
dual-party cases which satisfy all the constraints discussed above. In this paper,
an alternative formulation of the Vasicek model is given which yields simpler
expressions for the coecient functions. The formulation uses the martingale
part oY …0; t† of the spot rate as the common factor, rather than the spot rate rt
itself.
The extended Vasicek model is now brie¯y reviewed. Under this model, the
process for the risk-free spot rate is given by
drt ˆ …h…t† ‡ Q…t†rt †dt ‡ n…0† dwt ;
ÿ


dwt  N 0; r2 …t†dt ;

where the mean reversion coecient is denoted Q…t†. The following terms are
now de®ned:
Z T

Q…t† dt ;
P…t; T † ˆ exp
t

D…t; T † ˆ

Z

T

P…t; s† ds;
t

also
rYY …t; T † ˆ

Z

T

P…s; T †2 r2 …s† ds;

t

rYy …t; T † ˆ
ryy …t; T † ˆ

Z

Z

T

D…s; T †P…s; T †r2 …s† ds;
t
T

2

D…s; T † r2 …s† ds:
t

The Gaussian factor oY …t; T † and the supplementary factor oy…t; T † are de®ned
as

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

oY …t; T † ˆ

Z

T

P…t; T † dwt ;
t

Z

T

oy…t; T † ˆ
D…t; T † dwt ;
t
 


rYY …t; T †
oY …t; T †
N ~
00 ;
rYy …t; T †
oy…t; T †

rYy …t; T †
ryy …t; T †


:

From the equilibrium condition:

 Z T

 Z
f …t; s† ds
ˆ rt exp ÿ
Et d exp ÿ
t

rt ˆ f …t; t†:

T
t


f …t; s† ds dt;

Or


Et exp



ÿ

Z

T

rs ds
t



ˆ exp



ÿ

Z

T



f …t; s† ds :
t

The processes for the accumulative instantaneous forward rates and the accumulative spot rate are now obtained by integration:
Z T
Z T
1
f …s; s† ds ˆ
f …t; s† ds ‡ n2…0† D2 …s; T †rYY …t; s†
2
s
s
‡ n2…0† D…s; T †rYy …t; s† ‡ n…0† D…s; T †oY …t; s†;

Z

T

rt dt ˆ
t

Z

T
t

1
f …t; s† ds ‡ n2…0† ryy …t; T † ‡ n…0† oy…t; T †:
2

These results are now extended to the spot spectra.
Since the assumption is that the each discount function has a common
Gaussian single factor oY …0; t†, which is independent of the maturity T and the
state (credit rating class), the processes of the spot spectra become
ÿ


dR…n† …t† ˆ h…n† …t† ‡ Q…t†R…n† …t† dt ‡ n…n† dwt ;
ÿ


ÿ


dR…n1 ;n2 † …t† ˆ h…n1 ;n2 † …t† ‡ Q…t†R…n1 ;n2 † …t† dt ‡ n…n1 † ‡ n…n2 † ÿ n…0† dwt ;
whereas

R…n† …t† ˆ F…n† …t; t†;
R…n1 ;n2 † …t† ˆ R…n1 † …t† ‡ R…n2 † …t† ÿ rt :
All spot spectra move instantaneously parallel with di€erent ampli®cations n.
All the processes have a common mean reverting coecient. The spectra satisfy
the equilibrium conditions:

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352





Z

T





Et d exp ÿ
F…n† …t; s† ds
ˆ R…n† …t† exp ÿ
t

 Z T

F…n1 ;n2 † …t; s† ds
Et d exp ÿ
t
 Z T

F…n1 ;n2 † …t; s† ds dt:
ˆ R…n1 ;n2 † …t† exp ÿ

Z

T



F…n† …t; s† ds dt;
t

t

Ampli®cation constants n of the spectrum volatility are determined by the
relationship


df …t; T † ÿ Et ‰df …t; T †Š dF…1† …t; T † ÿ Et dF…1† …t; T †
ˆ
ˆ



8t; T :
n…0†
n…1†


dF…n† …t; T † ÿ Et dF…n† …t; T †
:
ˆ
n…n†
Similarly, the processes for the accumulative instantaneous forward spectrum
and the accumulative spot spectrum are given by
Z T
Z T
1
F…n† …s; s† ds ˆ
F…n† …t; s†ds ‡ n2…n† D2 …s; T †rYY …t; s†
2
s
s
‡ n2…n† D…s; T †rYy …t; s† ‡ n…n† D…s; T †oY …t; s†;

Z

Z

T
t
T
s

T

1
F…n† …t; s† ds ‡ n2…n† ryy …t; T † ‡ n…n† oy…t; T †;
2
t
Z T
F…n1 ;n2 † …s; s†ds ˆ
F…n1 ;n2 † …t; s†ds
R…n† …t† dt ˆ

Z

s


2
1ÿ
n ‡ n…n2 † ÿ n…0† D2 …s; T †rYY …t; s†
2 …n1 †
ÿ

2
‡ n…n1 † ‡ n…n2 † ÿ n…0† D…s; T †rYy …t; s†
ÿ


‡ n…n1 † ‡ n…n2 † ÿ n…0† D…s; T †oY …t; s†;

‡

Z

T

R…n1 ;n2 † …t†dt ˆ
t

T


2
1ÿ
F…n1 ;n2 † …t; s†ds ‡ n…n1 † ‡ n…n2 † ÿ n…0† ryy …t; T †
2
t


ÿ
‡ n…n1 † ‡ n…n2 † ÿ n…0† oy…t; T †:

Z

Note that the accumulative forward spectra for the discount bonds have the
common Gaussian factor oY …0; t†. The tensor discount functions are now
completely determined. The money market discount process in tensor form is
obtained by taking the exponential, then multiplying the basic tensors.
 a;b
 Z T

 Z T

R…t† dt 
ˆ Ua;b n1 ;n2 exp ÿ
R…n1 ;n2 † …t† dt U a;b n1 ;n2 :
exp ÿ
t

a;b

t

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D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

This is a tensor extension of the money market discount bond in a sense
that
"
 Z T
 a;b #

a;b
Da;b …t; T † ˆ Et exp ÿ
R…t† dt 
:
t

a;b

Now all the necessary pricing tools are de®ned.
3.1. Use of the Hull and White tree

An innovation in option pricing came when Cox, Ross and Rubinstein introduced the binomial tree as a discretization of the Black and Scholes continuous model. This innovation was extended to interest rate term structure
modeling by Hull and White, (1990), who discretized the extended Vasicek
model. The major contribution of the Hull±White approach is the combining
of trinomial trees and analytical solutions. This allows the ecient evaluation
of complex interest rate derivatives. In this subsection, the Hull±White approach is extended to include credit rating changes and default.
In this one-factor setting, it is possible to evaluate the whole structure by the
Hull±White tree. Given a Hull±White tree, that is a trinomial lattice of the riskfree spot rate r…n ot†, then for each lattice point, there is a simple conversion
formula to the state variable:
oY …0; n ot† ˆ

1
…r…n ot† ÿ f …0; n ot†† ÿ n…0† rYy …0; n ot†:
n…0†

For interest rate derivative evaluation, it is sucient to store the price of the
contingent claim at each node of the tree. The price is then discounted back by
the risk-free spot rate r…n ot†, multiplied by the appropriate branching equilibrium probability. With credit classes, the prices become a tensor. Thus the
tensor, i.e. the two-dimensional array for the dual-party case, must be stored at
each node. The branching probability stays the same, however, the prices are
now discounted by the tensors.
For tensor discounting of each time step, if the time step ot is chosen to be
small enough, then the following approximation holds:
 b1 ;b2
 Z …n‡1†ot

1
U b1 ;b2 n1 ;n2
R…t† dt 
 Ua1 ;a2 n1 ;n2
exp ÿ
…n


ot†ot
1
‡
R
…n
;n
†
n
ot
1 2
a1 ;a2
!2
2
X
n…nk † ÿ n…0† rYy …0; n
ot†
R…n1 ;n2 † …n
ot† ˆ F…n1 ;n2 † …0; n
ot† ‡
kˆ1

‡

2
X
kˆ1

!

n…nk † ÿ n…0† oY …0; n
ot†:

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

341

Hence it is not necessary to compute the supplementary factor oy…0; n
ot†. In
general, Nakazato (1997a) demonstrated that this model is extendable to the
multi-factor Gaussian model so that individual movements of the risky yields
corresponding to the di€erent credit classes can be re¯ected. The multi-factor
model can be evaluated eciently using a high-dimensional lattice. For more
details, refer to Nakazato (1998) for algorithmic implementation of the multifactor ane Gaussian models.

4. Evaluation of the excess loss protection by the Hull±White tree
In the case of the protectorÕs default it is necessary to purchase protection
from the ``credit'' risk-free party. Thus the loss due to the protectorÕs default is
equivalent to the insurance fee for secondary protection. Suppose that default
takes place at time t and the credit rating of the issuer is n, then the loss is given
by

 a
Z T
 Z s


R…s† ds  ga …s†ds ;
exp ÿ
Ln …t† ˆ Et
t

exp



ÿ

Z

s

t

t



R…s† ds 

n

a

m

ˆ Un exp

n



ÿ

Z

s



R…m† …s† ds U m a ;

t

ÿ


ga …t† ˆ Ua m R…m† …t† ÿ r…t† U m b ;

where the hazard rate ga …t† can be considered as the instantaneous insurance
fee to cover the principal of the bond. Here the credit rating class of the issuer
of the bond prior to default is a, and the maturity of the underlying bond is T.
Note that the tensor indices run through non-defaulting states only, because it
is only necessary to consider the case when neither party defaults. Hence the
loss can be interpreted as a ``tensor discounted'' total of the instantaneous
insurance fee.
The loss can be expressed in a closed form:
 Z s
Z T
1
exp ÿ
F…m† …0; s† ds ÿ n2…m† D2 …t; s†rYY …0; t†
Ln …t† ˆ Un m
2
t
t
n
ÿ


2
F…m† …0; s† ÿ f …0; s†
ÿ n…m† D…t; s†rYy …0; t† ÿ n…m† D…t; s†oY …0; t†


ÿ


‡ n…m† ÿ n…0† P…t; s†oY …0; t† ‡ n2…m† ÿ n2…0† P…t; s†rYy …0; t†
o


ÿ
‡ n…0† n…m† ÿ n…0† rYy …t; s† ds:

The proof is given in the appendix. The excess loss over the collateral K is
de®ned as:

342

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

· when reinvestment is prohibited:
‡

…Ln …t† ÿ K† ˆ max … Ln …t† ÿ K;

0 †:

This restriction is a conservative one and it comes from the limitation of the
Hull±White tree, because the tree spans in oY …0; t† direction only.
· when reinvestment is allowed:

Z t
‡

Z t


rt dt
ˆ max Ln …t† ÿ exp
rt dt K; 0 :
Ln …t† ÿ K exp
0

0

In the latter case, when building the Hull±White treeRit is necessary
to keep

t
track of the accumulated money market value exp 0 rt dt in each node.
Therefore, for more general applications, one additional dimension oy…0; t†
should be added when the evaluation lattice is generated.
In Appendix B, the closed form solution is given when the collateral can be
invested in the money market. The tree approach is recommended because of
the complexities in the closed form solution. Therefore in the rest of the paper,
it is assumed that no reinvestment is allowed.
Now suppose the issuer is in credit class n1 , and the protector is in n2 , then
the instantaneous insurance fee to cover the excess loss is given by
gn2 …t†…Ln1 …t† ÿ K†‡ ;
where the hazard rate is given as
gn …t† ˆ Un m gm …t†;


ÿ


gm …t† ˆ F…m† …0; t† ÿ f …0; t† ‡ n2…m† ÿ n2…0† rYy …0; t† ‡ n…m† ÿ n…0† oY …0; t†:

From the state variable oY …0; t† at each node of the Hull±White tree, the insurance fee is then analytically determined. The prices of the contingent claims
which cover the excess loss are:
#
"Z
 Z t
 n1 ;n2
T

‡
exp ÿ
R…t† dt 
gn2 …t†…Ln1 …t† ÿ K† dt :
Va1 ;a2 …0; K† ˆ E0
0

0

a1 ;a2

This can be interpreted as a ``tensor discounted'' total of the instantaneous
insurance fee to cover the excess loss. Thus at each node of the tree, this formula can be recursively evaluated as
‡

Va1 ;a2 …t; K† ˆ ga2 …t†…La1 …t† ÿ K† ot
"
 Z
t‡ot

‡ Et exp

ÿ

t



R…t† dt 

b1 ;b2

#

Vb1 ;b2 …t ‡ ot; K† :
a1 ;a2

The algorithm to evaluate this expression ®ts naturally with the tree construction, as the ®rst term is associated with a single root node, and the second

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

343

term is associated with subsequent branch nodes. These prices are at time t
when the issuer is in the credit class a1 , and the protector is in a2 . Further
reduction of the computation time is possible by eliminating tensor multiplication at each node; discounting the tree backward in the spectra, then computing the price at the root of the tree from the accumulated spectrum:
‡

V m1 ;m2 …t; K† ˆ gm2 …t†U n m1 …Ln …t† ÿ K† ot ‡

Et ‰V m1 ;m2 …t ‡ ot; K†Š
;
1 ‡ R…m1 ;m2 † …t†ot

Va1 ;a2 …0; K† ˆ Ua1 ;a2 m1 ;m2 V m1 ;m2 …0; K†:
Using this price, the collateral K is set by the criteria Va1 ;a2 …0; K† ˆ eVa1 ;a2 …0; 0†
for suciently small e. The value K can be found numerically using the secant
method or the bisection method.
4.1. Numerical example
The pricing model must be ¯exible enough to allow the pricing-based capital
adequacy determination to be applied to a wide variety of instruments. This
subsection illustrates the pricing model ¯exibility and shows that the adequate
capital required can be numerically solved, given the credit rating classes,
current yield and spreads. The adequate capital required with and without
reinvestment is also compared.
The plots in Fig. 1 were calculated using a Hull±White model with a constant price volatility of 1% and mean reversion coecient of 2%. For conciseness of the illustration, the transition probabilities and the credit risk-free
yield curve data are omitted. The numbers used are hypothetical. See Fig. 1 for
the spread curves used.

Fig. 1. Spread curve.

344

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

For di€erent credit rating classes of the protector and the issuer with a risk
tolerance level of 0.1% and maturity 5 years, the adequate collateral was
computed as given in Fig. 2.
With these 3-D plots, it is dicult to show the re-investment e€ect of the
collateral. For di€erent risk tolerance levels with the 5 year ``A'' bond protected by the ``Aa'' protector, Fig. 3 illustrates the re-investment e€ect.
And for varying maturities with a 0.1% tolerance level, the same comparison
is made (see Fig. 4).
5. Conclusion
The purpose of this paper is to provide a practical algorithmic solution to
the problem of determining the adequate level of capital for complex credit

Fig. 2. Collateral vs. credit ratings.

Fig. 3. Collateral vs. risk tolerance.

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

345

Fig. 4. Collateral vs. maturity of the bond.

derivatives. The example given in this paper concerns the most common
problem in credit derivatives, namely default protection. However, this contingent claim is suciently complex to demonstrate the novelty of the approach, since the price depends on the credit ratings and default risk of both
the protector and the issuer of the protected bond.
As far as the algorithmic construction is concerned, the Hull and White tree
is one of the simplest term structure models available. The additional practical
implementation is also straightforward, because of the credit evaluation model.
The model naturally combines transition data between di€erent credit rating
classes with current yield curves observed in the market for di€erent credit
classes.
A rational computational methodology alternative to the VAR quantile
method is introduced in this paper. The method is based on coherent analytical evaluation of the protection required against the excess default loss over
and above the coverage provided by the collateral. The advantage of a coherent approach is that the risk measurement captures the diversi®cation effect. This is the essence of credit business and credit risk management. In
addition, the computational speed is increased over the conventional Monte
Carlo method, using the recombining Hull±White tree and the bene®t of the
tensor-spectrum decomposition. It is, in fact, possible to solve the problem in
closed form without using the tree. The computation time then reduces further.
Unlike the adequate capital criteria recommended by the regulators, the
methodology developed in this paper does not unreasonably increase the
capital requirement when it is applied to a large portfolio of the credit linked
contingent claims.

346

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

Acknowledgements
The author is grateful for Richard Bateson, Michael Dempster, Toshifumi
Ikemori, Patricia Jackson, Farshid Jamshidian, Masaaki Kijima, Hiroshi
Shirakawa, Akihiko Takahashi and Domingo Tavella for comments, and most
importantly Alan Ambrose of the Quantz Ltd. for patiently coding the complex formulae. The author owes to Mark Davis of the Tokyo Mitsubishi International for showing the counter example. The author especially would like
to thank Stephen Hancock and Alex McGuire of IBJ International for careful
reading and clari®cation of the paper. Of course any remaining errors are those
of the author alone.

Appendix A. Introduction to matrix term structure model
In this appendix, the general theory behind the rating-based model developed by Nakazato (1997a) is introduced. The initial breakthrough in this area
came from Due and Singleton (1997) when they formulated the price of a
defaultable discount bond as


 Z T
D…t; T † ˆ Et exp ÿ
frt ‡ kt g dt ;
t

where t is the evaluation time, T the maturity, rt the risk-free spot rate and kt is
the mean loss rate (e€ective hazard rate), which is de®ned as the product of the
hazard rate and one minus recovery rate. This formulation is powerful because
not only does this give a direct economic interpretation of the yield spread as
the mean loss rate, but also the mathematical elegance of the result allows
extensions and applications to more complicated problems. The hazard rate is
the continuous state Markov chain analogy to the transition rate matrix of a
discrete state Markov chain. Thus it is a natural extension to substitute the
e€ective hazard rate with the e€ective transition rate matrix. This allows the
inclusion of credit rating classes. In this technique the risky discount bond
becomes a matrix discount bond. The economic justi®cation for this matrix
extension is rather lengthy, therefore the interested reader should refer to
Nakazato (1997a). The meaning of the matrix discount bond is clear since each
element is de®ned as a tensor in the pricing model section of this paper.
Caution is required when substituting matrices in such a formula since
commutativity does not hold:
exp…A ‡ B† 6ˆ exp …A† exp …B† 6ˆ exp …B† exp …A†:
To express the matrix discount bound price correctly it is necessary to use
product integration.

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

D…t; T † ˆ Et

"

T
a

347

#

exp f ÿ frt ÿ Q…t†g dtg :

t

`t
As usual the product integral Y …o; t† ˆ o expfAt dtg is the unique solution of
the ordinary di€erential equation d=dtY …o; t† ˆ Y …o; t†At , Y …o; o† ˆ I.
Extension to the multiple party case is straightforward. Using the decomposition result of Markov chains, the transition rate matrix is decomposed into
the sum of the transition rate matrices of each party. For example, in the dualparty case:
"
#
T
n n
o o
a
exp ÿ rt ÿ Q1 …t†
I ÿ I
Q2 …t† dt ;
D…t; T † ˆ Et
t

where the symbol
is the direct product in the tensor algebra.
In tensor notation, the discount bond price from an initial state (a, b) at t to
a terminal state (a; b) at maturity T is denoted by Da;b a;b …t; T †. The bond pays
the principal value of 1 only when the terminal state is reached at maturity. The
discount bond price function is assumed to be of the form
 Z T

a;b
n1 ;n2
exp ÿ
F…n1 ;n2 † …t; s† ds U a;b n1 ;n2 :
Da;b …t; T † ˆ Ua;b
t

P n Pn
Here, EinsteinÕs convention is used, i.e. the summation signs n1 ˆ0 n2 ˆ0 are
omitted.
In general, a certain handling rule for subscripts and superscripts needs to be
agreed. The state space of D…t; T † is indexed by a vector coordinate (n1 ; . . . ; nm ),
this means that party i is in credit class ni for i ˆ 1; . . . ; m. If ni ˆ 0 then party i
is in the default state. For notational simplicity, an element f i;ni …t; T † is written
f …ni † …t; T †, and the 0th element f …0† …t; T † is written f (t,T). Under this convention, tensor notations are now reviewed.
In tensor notation, an element of the column vector is written as a covariant
vector xi indexed by subscript; examples are price vectors, or terminal cash ¯ow
vectors. An element of the row vector is written as a contravariant vector xi
indexed by superscript; examples are probability vectors, or Arrow±Debreu
price vectors. In order to distinguish originally non-vector elements, all nontensor subscripts are parenthesized; e.g. f …ni † …t; T †. It is also assumed that any
sub-subscript is a non-tensor index. Using EinsteinÕs convention, namely
omitting the summation symbol; the right-hand side of an expression is summed over all sub-(super-)scripts not appearing in the left-hand side for appropriate ranges.
The Heath, Jarrow and Morton model, Heath et al. (1992) can also be extended to matrix form. The instantaneous forward transition rate matrix is
de®ned as

348

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

F…t; T † ˆ ÿD…t; T †

ÿ1

o
D…t; T †:
oT

Alternatively,
D…t; T † ˆ

T
a
t

n
o
exp ÿ F…t; s† ds :

De®ne
R…t† ˆ F…t; t† ˆ Irt ÿ Q…t†:
Then the matrix form of the HJM model is
Z T

ÿ1
ÿ1
D…t; T † D…t; s†dWt …s†D…t; s† D…t; T †dWt …T †ds
dF…t; T † ˆ Et
t

‡ dWt …T †:

Unfortunately, this result is too general. Although these expressions are
compact, in order to obtain analytical solutions for the contingent claim in
question, further assumptions are necessary, so it follows that a simpler version
of the matrix term structure model is obtained as in the main body of this
paper. This model, called the ane Gaussian model, can be viewed as an extension of the Vasicek±Jamshidian model.

Appendix B. The closed form solution
Using the de®nition of the hazard rate, the expression for the protection fee
can be simpli®ed to

 Z s


Z T
ÿ


m
exp ÿ
R…m† …s† ds R…m† …s† ÿ r…s† ds
Ln …t† ˆ Et Un
t
t


Z s
Z T 


ÿ
d
Et exp k R…m† …s† ÿ r…s† ÿ
R…m† …s† ds ds:
ˆ Un m lim
k!0 dk t
t

Since



R…m† …s† ÿ rs ˆ F…m† …t; s† ÿ f …t; s† ‡ n2…m† ÿ n2…0† rYy …t; s†
ÿ


‡ n…m† ÿ n…0† oY …t; s†;
Z s
Z s
1
R…m† …t† dt ˆ
F…m† …t; s† ds ‡ n2…m† ryy …t; s† ‡ n…m† oy…t; s†;
2
t
t
substitution gives

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

349

Z T
n ÿ




d
Ln …t† ˆ Un lim
exp k F…m† …t; s† ÿ f …t; s† ‡ k n2…m† ÿ n2…0† rYy …t; s†
k!0 dk t

Z s

 ÿ


1
ÿ
F…m† …t; s† ds ÿ n2…m† ryy …t; s† Et exp k n…m† ÿ n…0† oY …t; s†
2
t

ÿ n…m† oy…t; s† ds
Z T
n ÿ




d
m
exp k F…m† …t; s† ÿ f …t; s† ‡ k n2…m† ÿ n2…0† rYy …t; s†
ˆ Un lim
k!0 dk t


Z s

2
1
1 2ÿ
ÿ
F…m† …t; s† ds ÿ n2…m† ryy …t; s† exp
k n…m† ÿ n…0† ryy …t; s†
2
2
t

ÿ


1
ÿ kn…m† n…m† ÿ n…0† rYy …t; s† ‡ n2…m† ryy …t; s† ds
2
 Z s

Z T
ÿ


m
ˆ Un
exp ÿ
F…m† …t; s† ds
F…m† …t; s† ÿ f …t; s†
m

t

t


ÿ


‡ n…0† n…m† ÿ n…0† rYy …t; s† ds:

This can be expressed in terms of the Gaussian factor oY …0; t† using the
relations


F…m† …t; s† ÿ f …t; s† ˆ F…m† …0; s† ÿ f …0; s† ‡ n2…m† ÿ n2…0† P…t; s†rYy …0; t†
ÿ


‡ n…m† ÿ n…0† P…t; s†oY …0; t†;
Z s
Z s
1
F…m† …t; s† ds ˆ
F…m† …0; s† ds ‡ n2…m† D2 …t; s†rYY …0; t†
2
t
t
‡ n2…m† D…t; s†rYy …0; t† ‡ n…m† D…t; s†oY …0; t†:
Hence
T



s

1
F…m† …0; s†ds ÿ n2…m† D2 …t; s†rYY …0; t†
2
t
t

2
ÿ n…m† D…t; s†rYy …0; t† ÿ n…m† D…t; s†oY …0; t†

Ln …t† ˆUn

m

Z

exp

ÿ

Z

nÿ


ÿ


F…m† …0; s† ÿ f …0; s† ‡ n…m† ÿ n…0† P…t; s†oY …0; t†
o




ÿ
‡ n2…m† ÿ n2…0† P…t; s†rYy …0; t† ‡ n…0† n…m† ÿ n…0† rYy …t; s† ds:


Suppose the initial collateral K is invested in the money market, then the excess
loss is

350

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352


Z t
‡
Ln …t† ÿ K exp
rs ds
0


Z t
 
Z t

ˆ Ln …t† ÿ K exp
rs ds
Prt Ln …t† P K exp
rs ds :
0

0

The loss occurs in the event of default by the protector, thus the price of the
contingent claim which covers the excess loss is
Z T
 Z t

b1 ;b2
exp ÿ
U a1 ;a2
R…b1 ;b2 † …s† ds U n1 ;n2 b1 ;b2 gn2 …t†
Va1 ;a2 …0; K† ˆ E0
0

0


Z t
‡ 
rs ds
dt :
 Ln1 …t† ÿ K exp
0

The insurance premium for the excess loss is
Z T
 Z t


b1 ;b2

n1 ;n2
Ua1 ;a2
Va1 ;a2 …0† ˆ E0
exp ÿ
R…b1 ;b2 † …s† ds U
b1 ;b2 gn2 …t†Ln1 …t† dt :
0

0

Using these prices, the collateral K is set by the coherent criteria
Va1 ;a2 …0; K† ˆ eVa1 ;a2 …0† for suciently small e. The value K is then numerically
determined by the Newton±Raphson method:
K ˆ K1 ;
Ki‡1 ˆ Ki ‡

eVa1 ;a2 …0† ÿ Va1 ;a2 …0; Ki †
:
…Va1 ;a2 =oK†…0; Ki †

The closed form solution of Va1 ;a2 …0† is given by
 Z t
Z T Z T
ÿ


exp ÿ
F…b2 † …0; s† ÿ f …0; s† ds
Va1 ;a2 …0† ˆ Ua1 ;a2 b1 ;b2
0

ÿ

Z

s

0

t

0

n 
F…b1 † …0; s† ds
F…b1 † …0; s† ÿ f …0; s†

ÿ


ÿ


‡ n…0† n…b1 † ÿ n…0† rYy …0; s† ÿ n…b2 † n…b1 † ÿ n…0† P…t; s†rYy …0; t†



ÿ n2…b1 † ÿ n2…0† P…t; s†D…t; s†rYY …0; t†



ÿ
ÿ
ÿ
 F…b2 † …0; t† ÿ f …0; t† ‡ 2n…0† ÿ n…b1 † n…b2 † ÿ n…0† rYy …0; t†


ÿ


ÿ n…b1 † n…b2 † ÿ n…0† D…t; s†rYY …0; t†
o
ÿ

ÿ


‡ n…b1 † ÿ n…0† n…b2 † ÿ n…0† P…t; s†rYY …0; t†

ÿ



 exp n…b1 † n…b2 † ÿ n…0† D…t; s†rYy …0; t† ds dt:

D. Nakazato / Journal of Banking & Finance 24 (2000) 329±352

351

A proof of this formula is omitted. For interested readers, the proof for the
more complicated cases Va1 ;a2 …0; K† and oVa1 ;a2 =oK…0; K† is given in the original
working paper Nakazato (1997b).

Appendix C. Counter example of monotonicity in the coherent measure
This ingenious counter example is due to Mark Davis.
Let ([0,1], du) be the unit interval probability space, and Y(u) ˆ u.
1
Then EY ˆ :
2
1
And E…Y ÿ k†‡ ˆ …1 ‡ k 2 † ÿ k:
2
‡

For example; when k ˆ kY ˆ 0:8; then E…Y ÿ k† ˆ 0:02:
‡

Hence E…Y ÿ kY † ˆ 0:02 ˆ 0:04EY :
Thus kY ˆ K…Y † when e ˆ 0:04:
Now let
X …u† ˆ

1

u;
u

2

u 6 k0
u > k0

…so X 6 Y †:

Then
‡

E…X ÿ kY † ˆ E…X ÿ k0 †
ˆ 0:04EY

‡

f* k0 ˆ kY gg

> 0:04EX :
For this example X 6 Y but K…X † > K…Y †.

References
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1997a. De®nition of coherent measures of risk.
Paper presented at the Symposium on Risk Management, European Finance Association,
Vienna, August 27±30.
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1997b. Thinking coherently. Risk 10 (11), 68±71.
Due, D., Singleton, K. 1997. Modeling term structures of defaultable bonds. Working paper,
Graduate School of Business, Stanford University, Stanford CA.
Heath, D., Jarrow, R., Morton, A., 1992. Bond pricing and the term structure of interest rates: A
new methodology for contingent claims valuation. Econometrica 60, 77±106.

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Hull, J., White, A., 1990. Numerical procedures for implementing term structure models I: Single
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Lando, D., 1998. On Cox processes and credit risky securities. Working paper, Department of
Operations Research, University of Copenhagen. Forthcoming in the Review of Derivatives.
Nakazato, D., 1997a. Gaussian term structure model with credit rating classes. Working paper,
Financial Engineering Department, The Industrial Bank of Japan, Tokyo.
Nakazato, D., 1997b. Determination of the adequate capital for default protection under the onefactor Gaussian term structure model. Working paper, Financial Engineering Department, The
Industrial Bank of Japan, Tokyo.
Nakazato, D., 1998. Lattice generation and hedging methodology for ane Gaussian model.
Working paper, Financial Engineering Department, The Industrial Bank of Japan, Tokyo.