1. An Analysis of Credit RiskWith Risky
Collateral:
A Methodology for Haircut Determination
Didier Cossin¤
& Tomas Hricko¤¤
March 2001
Comments Welcome
¤
HEC, University of Lausanne, CH-1015 Lausanne, Switzerland
Tel: 41 21 692 34 69
Fax: 41 21 692 33 05
Email: Didier.Cossin@hec.unil.ch
¤¤
Email: Tomas.Hricko@hec.unil.ch
Acknowledgments:
We thank Aydin Akgun, Sanjiv Das, Jerome Detemple, Mark Broadie,
Hugues Pirotte and Suresh Sundaresan for their help.
1

2. An Analysis of Credit RiskWith Risky
Collateral:
A Methodology for Haircut Determination
March 2001
Acknowledgments:
We thank Aydin Akgun, Sanjiv Das, Jerome Detemple, Mark Broadie,
Hugues Pirotte and Suresh Sundaresan for their help.
2

3. Abstract
Although many credit risk pricing models exist in the academic literature, very
little attention has been paid to the impact of risky collateral on credit risk. It is
nonetheless well known that practitioners often mitigate credit risk with collateral,
using so-called haircuts for collateral level determination. The presence of collateral
has a complex e¤ect that can not be analyzed simply with existing models. We ana-
lyze the value of credit risk when there is collateral in a range of di¤erent situations,
including dual-default in a simple setting, stochastic collateral, stochastic bond col-
lateral with stochastic interest rates, continuous and discrete marking-to-market and
margin calls. The models con…rm many practical intuitions, such as the impact on
the haircut level required of the risks of the collateral asset and of the underlying
asset to the forward as well as the impact of their correlation. Moreover the model
supports the intuition that the frequency of marking-to-market and collateral are
substitutes. The models also stress the possibly unexpected magnitude of these fac-
tors. More importantly, they give actual solutions to determining the value of the
credit risk depending on the haircut chosen and the frequency of marking-to-markets,
results not presented before in the literature. The models are also a good basis to
understand the portfolio e¤ect of collateral management. Finally they illustrate how
di¤erences in prices may arise from pure di¤erences of credit risk management, as
illustrated here in the case of futures and forwards.
3

4. I. Introduction
Credit risk has been a major topic of academic research during the last few years.
Credit risk has also become recognized by regulators and practitioners alike as one
of the major elements of …nancial markets risks. Financial crises keep reminding
banks and regulators of the importance of credit risk. For example, in a recent ISDA
(International Swap Dealer Association) paper, the self regulating body puts under
scrutiny (and under question) the Basle Committee on Banking Supervision’s rules
on credit risk capital, which still date back from 1988 and are outdated in many
dimensions. The paper calls for major reform of the credit risk guidelines, notably as
far as collateralization is concerned.
Indeed, since 1988, both the size and the complexity of the credit risk issue have
grown tremendously. The exponential growth of the over-the-counter (OTC) deriva-
tives markets (which all present some credit risk in a much more complex form than
standard loans), the apparition of credit risk derivatives, the widespread use of col-
lateral by …nancial intermediaries and the use of risky securities such as corporate
bonds or stocks as collateral, all have dramatically changed the face of the credit risk
exposure of …nancial intermediaries such as banks.
For example, OTC derivatives have represented the major share of the derivatives
market, both in growth and in absolute, during the last few years. While exchange
traded derivatives do not present credit risk, OTC derivatives do. With an estimate
of the OTC market at $65 trillion (Source : Risk Magazine, July 1998), it becomes
obvious that the issue of credit risk management, either through pricing or through
other management forms such as collateralization becomes essential.
Collateralization has become the favorite way for practitioners and regulators alike
to handle credit risk. As stressed in Cossin and Pirotte (1997 and 1998) for example,
collateralization a¤ects swap and other derivative instrument credit risk and thus puts
into question the academic models of credit risk pricing that tend not to incorporate
collateralization.
It is thus interesting to notice that while many theoretical models of credit risk
pricing have arisen lately, much less work has been done in order to achieve a good
theory of pricing credit risk with risky collateral. Determination of haircuts on col-
laterals asked by banks, notably in a portfolio setting, has been left to rules of thumb
rather than to advanced analysis. The goal of this research is to participate in …lling
that theoretical gap and to analyze the issue of pricing credit risk with risky collateral.
Many recent results on credit risk have attracted the attention of the …nance aca-
demic community and been published in top academic journals during the last few
years, such as, amongst others, Longsta¤ and Schwartz (1995), Jarrow and Turn-
bull (1995), Anderson and Sundaresan (1996), Leland and Toft (1996), Du¢e and
Huang (1996), Du¢e and Singleton (1997), Mella-Barral and Perraudin (1997), Jar-
row, Lando and Turnbull (1997)). Many of these papers approach the credit risk
issue from the pricing standpoint. It has been a classical way to approach the prob-
4

5. lem since the seminal Merton paper of 1974 that derives a simple, option-based model
of a zero-coupon bond pricing with credit risk. This elegant model prices credit risk
endogenously under some assumptions of …rm value dynamics.
Recently, two trends in the credit risk literature have emerged: The …rst one
extends the Merton framework in many dimensions, such as stochastic interest rates
(Shimko and alii (1993)), di¤erent bankruptcy rules (Longsta¤ and Schwartz (1995)),
gaming behaviors (Anderson and Sundaresan (1996), Mella-Barral and Perraudin
(1997)). Some of the work in progress of these di¤erent authors still go in the direction
of extending the framework further.
The second trend, the so-called reduced-form approach, although it still focuses
on arbitrage free models, gives up on endogeneizing the bankruptcy process in itself
and considers it as an exogenous process. From a theoretical point of view, this is
not a welcome concession. On the other hand, it allows for an easier treatment of
practical cases (with the weakness of ignoring the …nancial economics behind the
determination of the bankruptcy process). Many papers have recently appeared that
follow this underlying assumption (Du¢e and Huang (1996), Du¢e and Singleton
(1997), Jarrow and Turnbull (1995), Jarrow, Lando and Turnbull (1997)).
Banks and regulators have tended to use still a di¤erent approach, that is based
on actuarial calculations rather than the models developed in academia, revealing a
widely di¤ering way of analyzing the problem (See for example Iben and Litterman
(1991), Altman and Kao (1992), Lucas and Lonski (1992), Iben and Brotherton-
Ratcli¤e (1994), and Sorensen & Bollier (1994). See also for a critical approach
Du¤ee (1995a and 1995b)).
The academic literature has addressed much less often and more indirectly the
question of determining the impact of risky collateral on the price of a credit risky
instrument. Notice that the issue of pricing an instrument that is collateralized with
another risky instrument is not trivial and becomes complex when marking-to-market
or margin calls are considered. Margrabe(1978) has mentioned the analogy between
an exchange option and a margin account and provides the pricing for a very simple
framework with no marking-to-market. Stulz and Johnson (1985) have priced secured
debt using contingent claim analysis and study the use of collateralization in a cor-
porate …nance framework, analysing the impact of collateralization on the value of
the …rm. The rest of the economic literature has addressed the rationale behind the
use of collateral in debt contracts and is an extension of the questions arising in the
theory of debt (see Benjamin (1978), Plaut (1985), Bester (1994)) but has not been
concerned with pricing the credit risk with collateral or with evaluating the impact
of haircut levels on the credit risk value. Our goal of pricing credit risky instruments
with risky collateral is thus fundamentally di¤erent and has not been fully addressed
yet.
While the …nance academic world has focused on the issue of pricing credit risk,
practitioners and regulators have used collateralization very commonly rather than
pricing to manage credit risk exposures. Collateralization is an elegant way to trans-
5

6. form a credit risk issue into a market risk one. For example, it is well known in
practice that pricing of swap contracts for example bears little or no dimension of
credit risk (a problem examined in Cossin and Pirotte (1997)), notably because col-
lateralization is today used extensively. Many securities can in practice be used for
collateralization of risky contracts. The amount of the collateral required typically
di¤ers depending on the risk of the security being used in the collateral. This is why
practitioners traditionally use so-called haircuts, that determine how much collateral
is required depending on the type of security used as collateral, a phenomenon not
well incorporated in current academic research (nor in current regulation). We aim
here at providing with this research a framework to analyze haircut determination
and the impact of risky collateral on credit risk and look precisely at the case of risky
forwards, an analysis that could be generalized to swaps and other instruments.
The way this paper proceed in order to establish more …rmly a theory of credit
risk pricing with risky collateral is by analyzing di¤erent stylized situations. Firstly,
we study the simple situation of a non stochastic collateral and present our set-up in
this case. We then present an analysis of collateralizing an instrument with stochastic
equity when there is no marking-to-market, followed by an analysis of collateralizing
with bonds when interest rates are stochastic. Finally, we approach the problem of
pricing a credit risky instrument with collateral when there are marking-to-market
and margin calls.
6

7. II. Model with dual default and non-stochastic
collateral
A. Assumptions
In this very simple set-up, and before studying the more pertinent situation of
risky collateral, we analyze the value of the collateralized credit risk (CCR) for two
credit-risky agents that engage in a contract subject to credit risk (for example,
one takes the long side of a forward while the other takes the corresponding short
side). In order to manage their respective credit risk exposure both agents require
the counterparty to deposit some collateral either with their counterparty or with a
neutral institution, for example a clearing house. There is no intermediate marking-
to-market between initiation of the contract and expiration. Hence the only time
when default can occur is at maturity. The collateral deposited by both agents is
cash (and thus is not risky itself).We assume that the decision to default depends
only on the value of the original contract and the value of the collateral. We thus
consider default to be endogenous. Extension to exogenous default could be the
work of future research but the assumption of endogenous default may be stronger
that it appears to be …rst. Indeed, we assume here that an agent defaults if the loss
of the collateral and the underlying contract is worth less than the future payments
s/he had committed to make. One might object, that a company or an individual
might default for a wide variety of reasons. Some of them might be totally unrelated
to the speci…c contract considered. For example a company might have to …le for
bankruptcy for some liquidity reasons. In the case of bankruptcy though, creditors
will take over control and behave optimally, by maximizing the contract value, as
described in our model. Therefore they will take the same decisions as the company
would have taken, namely they will not default on the speci…c contract if default is
costlier than honoring the contract. We also assume no external costs to default (for
simpli…cation), which means that both agents will exercise their options if they are
in the money at expiration.
We consider a forward contract to illustrate the decomposition. At time 0, the
contract is initiated. The agents …x the price at which the stochastic asset will be
traded in the future. This price remains …xed until the expiration of the contract.
We will refer to it as the forward price. Agent 1 has the obligation to buy the asset at
the forward price at expiration (hereafter called time T), while agent 2 must sell the
asset at the same time. The forward price is set at time 0 so as to make the value of
the forward contract equal to zero. We assume that the storage cost for this asset can
be neglected and that it pays no dividend (and that there is no convenience yield).
By using the classical cash-and-carry arbitrage argument we know that the forward
price that yields a forward contract value of zero at time 0 is given by:
H = S0 ¢ ert
(1)
7

8. The collateral posted by each agent is denominated M1 resp. M2.
The following variables are used in the ensuing section.
S0 Value of the underlying of the forward contract at time 0
ST Value of the underlying of the forward contract at time T
H Forward price
M1 Cash amount that agent 1 has to give as collateral
M2 Cash amount that agent 2 has to give as collateral
We assume that the price of the underlying of the forward contract follows a
geometric Brownian motion process. The interest rate is assumed to be constant for
the time being.
B. The basic model
Under the above assumptions, the value of the credit risky forward with collateral
to the agent having the long position in the forward is:
Risky forward = riskless forward + Put(St; H ¡ M1) ¡ Call(St; H + M2) (2)
The term risky (respectively riskless) in the above formula means with (without)
default risk.
Stock price S 6 H ¡ M1 H ¡ M1 6 S 6 H + M2 H + M2 6 ST
riskless forward ST ¡ H ST ¡ H ST ¡ H
risky forward ¡M1 ST ¡ H M2
short Call 0 0 ¡(ST ¡ (H + M2))
long Put (H ¡ M1) ¡ ST 0 0
Table I: Payo¤s at maturity. This table provides payo¤s for various values of
the underlying of the forward contract at maturity.
The value of the forward contract with two-sided default risk can be smaller, larger
or equal to the situation without the possibility of default. Agent 1 with the long
position gains because he has the possibility to default if the price of the underlying
of the forward contract drops. On the other hand he looses because the counterparty
might not honor its obligation if the contract evolves in agent 1’s favor. He looses due
to the presence of the implicit call option. The resulting value of the long position in
the credit risky forward contract can be bigger or smaller than the risk free forward
contract.
As mentioned before, the amount of collateral demanded from both agents must
not be the same. In order to obtain a risky forward contract with a value of zero at
time 0, the value of both options must be the same. As a consequence of the limited
liability assumption the price of the underlying of the forward contract can not be
8

9. negative. This implies that agent 1’s loss is bounded. On the opposite side, the loss
of agent two with the short position is unbounded. Hence the possibility to default
is of greater value to him. In order to obtain the same value for both options agent
2 must deposit a higher amount.
The following example illustrates the decomposition:
S0 = 100
H = 100 ¢ exp (r ¤ (T ¡ t)) = 102:532
M1 = 20:5063 (20% of H)
M2 = 20:5063
r = 0:1
¾ = 0:3
T-t = 0:25
If the forward price is chosen in a way as to make the value of the forward contract
at time zero equal to zero, the value of the risky forward contract at time 0 is just
the sum of the two options.
Risky forward = Put(St; H ¡ M1) ¡ Call(St; H + M2) (3)
Risky forward = 0:403599 ¡ 0:891276 = ¡0:487677
In order to make the value of the credit risky forward contract equal to zero at
initiation, i.e. to give both options the same value, the party with the short position
in the forward contract would need to deposit a higher margin. In the above example
the necessary value of M2 for a 0 contract value at initiation is equal to 27% of the
forward price (versus 20% for the long position).
9

10. III. Model with stochastic collateral
A. Assumptions
We now generalize our model to the more realistic situation of one credit risky
agent giving a risky asset as a collateral to a third party, considered risk free, corre-
sponding to a bank or another …nancial intermediary. The credit risk free assumption
for the bank could for example be justi…ed by the fact that the forward contract con-
stitutes only a small fraction of the bank’s obligations. Reputational damage of
defaulting on one contract when it has many others prevents the bank from default-
ing. In this setting the credit risky agent can be seen as the client. For example, the
buyer of a forward on an exchange rate can post a collateral of a certain amount of
a portfolio of stocks with the bank it is doing the forward with. The only cost of de-
fault for the client is considered to be here, for simpli…cation, the loss of the collateral.
Hence the client will always choose to default, when the expected gain (resp. loss)
from the forward contract is bigger than the collateral he had to put up until that
moment. We assume that the price of the underlying asset to the forward contract
and the price of the asset given as collateral follow two separate Geometric Brownian
motion processes. The two processes are correlated. There is no marking-to-market
and hence no default before maturity.
B. The model
The client’s position corresponds to the following decomposition
Risky forward = riskless forward + Put (S; H ¡ M) (4)
where the put option is an option to exchange the loss of the collateral for the
forward contract. M stands for the value of the asset given as collateral, St stands
for the value of the underlying of the forward contract and H is the forward price.
The payo¤ of the implicit option at expiration is given by
Option = (¡M ¡ (S ¡ H))+
= (H ¡ S ¡ M)+
(5)
The price of the underlying and the price of the collateral follow Geometric Brow-
nian motion. The processes are given by
ds = S r dt + ¾s S
p
¿ du
dm = M r dt + ¾M M
p
¿ dv
½ = Correlation between dv and du
The collateralized credit risk can be compared to an exchange option. The most
simple situation involving the possibility to exchange one stochastic asset for another
10

11. stochastic asset was analyzed by Margrabe (1978). He mentioned the situation of a
margin account as a possible application. The situation described above is di¤erent.
In the case of Margrabe the client simply exchanges one asset for another, here he
exchanges the di¤erence of the forward and the spot price for the collateral. Hence
the implicit option is a spread option. The value of the collateralized credit risk
option is:
CCR option =
Z +up(v)
¡1
Z ¡d¡Á(v)
¡1
"
H ¡ Se(¹s¡1
2
¾2
s)¿+¾s
p
¿u
¡Me(¹M ¡ 1
2
¾2
M )¿+¾M
p
¿v
#
(6)
¢f (v) ¢ f (u j v) du dv
The value of collateralized credit risk at time zero is given by the following formula
CCR = He¡r¿
A3 ¡ SA1 ¡ MA2 (7)
A1 =
Z +up(v+½¾S
p
¿)
¡1
f (v) N
Ã
¡d ¡ Á (v + ½¾S
p
¿) ¡ ½v ¡ ¾S
p
¿
p
1 ¡ ½2
!
dv (8)
A2 =
Z +up(v+¾M
p
¿)
¡1
f (v) N
Ã
¡d ¡ Á (v + ¾M
p
¿) ¡ ½v ¡ ½¾M
p
¿
p
1 ¡ ½2
!
dv
A3 =
Z +up(v)
¡1
f (v) N
Ã
¡d ¡ Á (v) ¡ ½v
p
1 ¡ ½2
!
dv
Assume H is the risk free forward price. We will compare di¤erent collaterals
as well as di¤erent levels of collateral demanded and analyze the in‡uence of the
various parameters on the option value and interpret the di¤erent parameters in the
context of collaterals and the setting of optimal haircuts. The benchmark is the use
of non-stochastic collateral.
Figures 1 and 2 illustrate the in‡uence of changes in the various parameters.
11

12. 50
60
70
80
90
100 50
60
70
80
90
100
0
0.01
0.02
0.03
0.04
50
60
70
80
90
100
0
0.01
0.02
0.03
0.04
S M
Increasing values
of M
Increasing values
of S
CCR
Figure 1: Collateralized credit risk for various levels of S and M. This
…gure gives the value of collateralized credit risk (called CCR) for di¤erent levels of
the values of S (the underlying asset of the forward) and M (the collateral) at time
0. The values for the …xed parameters are: r = 0:1, ¾1 = 0:3, ¾2 = 0:3, ½ = 0:2 and
T-t = 0:5.(called CCR)
0.2
0.25
0.3
0.35
0.4
0.2
0.3
0.4
0.5
0
0.1
0.2
0.2
0.25
0.3
0.35
0.4
0
0.1
0.2
S
M
CCR
= -0.2
= 0
= 0.2
.
Figure 2: Collateralized credit risk and the correlation of collateral and
underlying. This graph shows the collateralized credit risk (called CCR) for di¤erent
levels of ¾S (the volatility of the underlying asset of the forward), ¾M (the volatility
of the collateral) and ½ (the correlation of the underlying asset and collateral). The
values for the …xed parameters are: S = 100;H= 100 ¢ exp (r ¤ (T ¡ t)) = 102:532,
M= 20:5063 (20% of H), r = 0:1 and T-t = 0:5.
The value of the collateralized credit risk is comparable to the value of a spread
12

13. exchange option. An intuitive result is that the value of the credit risk increases
if the amount of collateral that is requested decreases. The e¤ect of the value of
the underlying security to the forward may seem less obvious. One would expect
that the credit risk would be decreasing if the value of the underlying increases.
This would be true if the strike price (the amount of collateral asked for) of the
option would be …xed. In the case of the forward contract the bank will increase
the strike price with a rising level of the underlying, by keeping in our example a
constant proportional value of the forward as collateral (i.e., proportional haircuts).
The value of the collateralized credit risk increases if the maximum possible loss from
the underlying contract increases. This e¤ect dominates if the bank uses the cash
and carry arbitrage price in setting the forward price (resp. the strike price of the
option). Figure 2 shows the credit risk for di¤erent levels of ¾S, ¾M and ½: The riskier
the underlying contract, the higher is the value of the possibility to default. This also
implies that regulations on margin requirements have to be di¤erent for various risky
assets underlying the original contract. The riskier the asset taken as collateral the
higher the credit risk. This justi…es that assets that exhibit little or no market risk
(which is captured by ¾M ) tend to require lower haircuts. However it is important
to stress the fact that the in‡uence of the volatility of the underlying seems to be at
least as important as the in‡uence of the volatility of the collateral assets. This fact
is highlighted with the following …gure.
M
S
0.2 0.22 0.24 0.26 0.28 0.3
0.2
0.22
0.24
0.26
0.28
0.3
Figure 3: Collateralized credit risk for various values of volatilities of
the underlying and the collateral. This graph shows the collateralized credit
risk (called CCR) for di¤erent levels of ¾S (the volatility of the underlying asset
of the forward),and ¾M (the volatility of the collateral) and (the correlation of the
underlying asset and collateral). The values for the …xed parameters are: S = 100;H =
100 ¢ exp (r ¤ (T ¡ t)) = 105;127, M= 50, ½ = 0:4, r = 0:1 and T-t = 0:5.
13

14. The lines and degrees of shading represent constant levels of credit risk. We see
that there is a nearly linear relationship between the two volatilities. If one would
be signi…cantly more important than the other the slope of the lines would be sig-
ni…cantly di¤erent from -1. If the regulation on collateral is based solely on the
characteristics of the collateral, one is holding implicitely constant the volatility of
the underlying. It is often the case in practice that the volatility of the collateral asset
seems to be considered of a greater importance than the volatility of the underlying
contract for haircut determination. One can see further that the value of the possi-
bility to default is higher for more positive values of the correlation coe¢cient and
can be highly sensitive to the correlation. This underlines the necessity to consider
correlation e¤ects in order to determine collateral requirements.
If the asset given as collateral is negatively correlated with the asset underlying
the forward contract the collateralized credit risk can be lower than with a riskfree
collateral of the same size. Collateralized credit risk is lower (in our example) in the
case of stochastic collateral than in the case of non stochastic collateral with negative
correlations of about -0.2 and a volatility of 15% of the collateral asset. This means
that negative haircuts may indeed be optimal in this situation.
0.12 0.14 0.16 0.18 0.2 0.22 0.24
0.07
0.08
0.09
0.11
0.12
0.13
0.14
= -0.1
= -0.15
= -0.2
CCR with
non-stochastic
collateral
M
CCR Area of negative
haircuts
Figure 4: Comparison of riskless and risky collateral. This graph shows
the collateralized credit risk (called CCR) for di¤erent collaterals (various values of
correlation with the underlying of the forward contract) for di¤erent values of ¾M .
The collateralized credit risk is smaller if one uses the risky collateral as compared
to riskless collateral in the gray shaded area. For points in this region one obtains
negative haircuts if riskless collateral is used as a benchmark. The values for the …xed
parameters are: S = 100;H = 100 ¢ exp (r ¤ (T ¡ t)) = 102:532, ¾s = 0.4 ,M= 50,
r = 0:05 and T-t = 0:5.
14

15. The in‡uence of ¾S and ½ suggests that it is important to take into account equally
the source of risk in the underlying asset as well as its relationship to the risk of the
collateral asset.
The present framework also allows us to address one more practical issue related
to the design of collateral determination. In most of the circumstances the client will
not have only one contract with the bank. On the other hand the collateral will be a
portfolio of assets rather than a single asset. How should the bank structure collateral
use when asset portfolios are concerned? Should one asset serve as collateral for some
use or should all the assets of the collateral portfolio serve jointly as collateral for all
the liabilities? In the context of our framework the answer would depend on the value
of the implicit options. We have seen that the collateralized credit risk is in‡uenced
by the volatility of both the underlying contracts and the collateral portfolio. Due to
diversi…cation e¤ects, the variance of both sides will be reduced when the portfolio
view is adapted. The e¤ect on the correlation can not be unambiguously given a
sign. Therefore the optional structure has to be designed case by case. The present
framework is well suited to perform this analysis.
15

16. IV. Bonds as collateral
It has been shown in the previous section that a lower volatility of the asset
used as collateral may lead to a lower haircut required by the bank. Because of this
phenomenon, the asset class which is used most often as collateral are government
bonds. The preceding set up is not well adapted to pricing collateralized credit risk
when bonds are used as collateral as we considered non stochastic interest rates and
a geometric Brownian motion for collateral value.
We address these issues in this section. We still assume for the time being that
there are no intermediate marking-to-markets. The asset given as collateral is a
bond of maturity U, where U > T. We want to present in this section the valuation
equation for the collateralized credit risk in order to show that the structure of the
solution remains the same. The valuation methodology is based on results by Geman
et alii (1995) for the change of numeraire and Harrison and Pliska (1981,1983) for the
martingale pricing.
A. Assumptions
We assume that the price process of the underlying of the forward contract follows
a GBM process. The interest rate follows a Vasicek process
dr = a ¢ (b ¡ r) dt + ¾rdWr (9)
The dynamics of a bond with maturity u at time t are given
dB (U; t)
B (U; t)
= (r + ¸r ¢ ¾u) dt + ¾u ¢ dWr (10)
or written equivalently under a di¤erent probability measure
dB (U;t)
B (U; t)
= r ¢ dt + ¾u ¢ dWr (11)
where
¾u =
¾r (1 ¡ exp (¡a ¢ (U ¡ t)))
a
(12)
The price of the underlying asset of the forward contract follows a GBM process
with two sources of randomness
dSt
St
= r ¢ dt + ½ ¢ ¾s ¢ dWr +
p
1 ¡ ½2 ¢ ¾s ¢ dWs (13)
We also assume the existence of a riskless bank account or accumulation factor,
which is given by
¯ (t) = exp
µZ t
0
r (s) ds
¶
(14)
16

17. B. Valuation of the collateralized credit risk option
The gain from defaulting corresponds to
CCR (t) = ¯ (t) IEt
"µ
1
¯ (T)
(F ¡ ST ¡ M ¢ B (T; U))
¶+
#
(15)
where B(T,U) is the value of a zero-coupon bond with a face value of 1. M
represents the number of zero-coupon bonds. We split the above expression into
three parts and evaluate them separately.
Part1 = ¯ (t) IE
·µ
1
¯ (T)
F ¢ 1IA
¶¸
(16)
Part2 = ¯ (t) IEt
·µ
1
¯ (T)
ST ¢ 1IA
¶¸
(17)
Part3 = ¯ (t) IEt
·µ
1
¯ (T)
M ¢ B (T; U) ¢ 1IA
¶¸
(18)
By using the forward measure approach (change of numeraire) and some ordinary
changes of measure we derive the above value as
CCR (t) = ¯ (t) IEt
"µ
1
¯ (T)
(F ¡ ST ¡ M ¢ B (T; U))
¶+
#
(19)
= F ¢ B (t; T) ¢ IN1 + B (t; T) Fs (t; T) ¢ IN2 + M ¢ B (t; T) Fb (t; T; U) ¢ IN3
where the INi’s are de…ned in the Appendix III.
Note that the mean level of the interest rate does not enter in the valuation
equation. This result can be intuitively understood by realizing that the collateralized
credit risk is derived in a classical contingent claims setting.
There is no fundamental di¤erence in the structure of the above result from the
one in the previous section. The value of the option would certainly be lower for a
comparable set of parameters. However the values obtained can not be compared
directly as the model in this section works with stochastic interest rates whereas
the previous results assumed constant interest rates. This result allows us to extend
all the conclusions of the preceding sections to situations when bonds are used as
collateral. The only major di¤erence from the previous setup lies in the in‡uence of
the correlation between the interest rate and the underlying of the forward contract
17

18. on the collateralized credit risk. Figure 5 shows the collateralized credit risk for
di¤erent values of ½ for a forward contract that has a slightly positive value.
-0.2 0.2 0.4
0.0006715
0.0006725
0.000673
0.0006735
0.000674
CC R
Figure 5. Collateralized credit risk for di¤erent levels of ½. This graph
shows the collateralized credit risk under stochastic interest rates for di¤erent values
of ½. The values for the …xed parameters are:S0 = 100; F = 98; B0:5 = 0:9728;
M = 50; B0;6 = 0:9520; ¾s = 0:3; r0 = 0:05; ¾r = 0:01; a = 0:04; U = 0:82; T = 0:5.
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.00055
0.0006
0.00065
0.0007
0.00075
0.0008
0.00085 U = 1
U = 0.9
U = 0.8
U = 0.7
U = 0.6
CCR
18

19. Figure 6. Collateralized credit risk for di¤erent levels of ½. This graph shows
the collateralized credit risk under stochastic interest rates for di¤erent values of ½.
The values for the …xed parameters are: S0 = 100; F = 98; B0:5 = 0:9728; M = 50;
B0;i = e:¡0:06¢i
for i = f0:6; 0:7;0:8; 0:9; 1g; ¾s = 0:3; r0 = 0:05; ¾r = 0:01; a = 0:04;
T = 0:5.
The value of the option depends in two ways on the interest rate. The interest
rate in‡uences the value of the collateral and the stochastic discount factor. The
bond values are negatively correlated with the interest rate. Hence an increase of the
interest rate lowers the value of the bond and increases the value of the discount factor.
A lower value of the bond, which serves as collateral, increases the payo¤ of the option
to default at maturity. However higher interest rates lead to a higher discounting,
hence this larger payo¤ has a smaller present value. These two e¤ects work in opposite
directions. The bond e¤ect is stronger for longer maturity bonds relative to the
maturity of the underlying contract. If the collateral bond matures shortly after
the expiration of the underlying contract the e¤ect related to the discount factor
dominates. Hence the value of CCR can be a decreasing function of the correlation. If
the collateral bond has a much longer time to maturity than the underlying contract,
the bond e¤ect dominates. In this case the CCR can be an increasing function
of the correlation. For some values in between, the two e¤ects o¤set each other
and we observe the relationship depicted in …gure..This leads to the result that the
collateralized credit risk can be an increasing or decreasing function of the correlation
of the interest rate and the underlying of the forward contract, depending on the
maturity of the collateral bond.
19

20. V. On forward and futures contracts
Up to this point, all the analysis has been done by assuming that there was no
marking-to-market. In practice credit risk is often managed by requiring collateral
and by introducing marking-to-market and margin calls. The rest of the paper is
devoted to the study of the in‡uence of these dynamic strategies. In the literature
(see for example V. France et al. 1996 ) the e¤ect of marking-to-market has been
most often seen as merely replacing the multi-period decisions by a series of one
period decisions. Within this series all the decisions have been considered as being
identical. It will become clear that this view is not correct. It is true that one of
the bene…ts of marking-to-market and margin calls is to replace the option to default
by the sum of options of shorter maturity. Therefore the bene…t of introducing a
marking-to-market procedure stems from the fact that the period during which the
credit risk exposure builds up is shorter. This is however not the only bene…t. Agents
are not myopic. Therefore they consider all the e¤ects of their actions. When taking
the decision whether to default or not the agent will take into account the fact that
s/he will loose the contract and all the possibilities to default in the future. This
will cause the probability of default to be lower than in the myopic case. Technically
speaking the range of integration will be smaller than in the myopic case.
We will use our framework to show that price di¤erences between forward and
futures prices can arise from di¤erences in the management of credit risk. It has
been shown by Cox,Ingersoll and Ross(CIR) (1981) that without the possibility of
default under non-stochastic interest-rates the forward price is the same as the futures
price. We will show that this is only true in a setting without credit risk. Futures
contracts have been invented especially to deal with credit risk. An analysis which
is conducted in a framework without the possibility to default is lacking the most
important feature of the contract. This section is organized as follows. We will …rst
repeat the classical argument of CIR. Afterwards we will show that the payo¤ of
the two strategies contracts is not the same, even in a setting without stochastic
interest-rates.
We model the situation between a bank and a client. The bank is assumed to
be free of default. Therefore it does not have to provide any collateral. The client
has to provide a cash amount M. If the client increases her position she will have
to provide additional collateral. Hence a strategy with one change in the size of the
futures position will involve one marking-to-.market.
The following variables will be used:
Si Price of the underlying of the forward contract at time i
r continuously compounded interest rate at time i
Fi Value of the futures contract maturing at T at time i
Gi Value of the forward contract maturing at T at time i
A. The CIR argument without credit risk
20

21. The CIR argument is based on the following strategy:
1.At time 0, take a long position of Exp
¡T
2
¢
(r) in the futures contract maturing
at time 2.
2. At time T/2 increase the futures position to Exp((T) r)
The resulting cash ‡ows and positions are shown in Table 2.
Time 0 T/2 T
Futures price F0 FT=2 FT
Futures position e(T=2)r
e(T)r
0
Gain/Loss 0
¡
FT=2 ¡ F0
¢
e(T=2)r
¡
FT ¡ FT=2
¢
e(T )r
FV Gain/Loss 0
¡
FT=2 ¡ F0
¢
e(T)r
¡
FT ¡ FT=2
¢
e(T )r
Table II: Resulting positions and cash ‡ows from the futures strategy
At time T the payout of this strategy will be given by
¡
FT=2 ¡ F0
¢
e(T)r
+
¡
FT ¡ FT=2
¢
e(T )r
= (FT ¡ F0) e(T)r
(20)
The value of the futures at time T with delivery at time T is obviously ST , the spot
price of the underlying at time T. If you combine this strategy with an investment of
F0 in the riskfree bond at time 0 you obtain:
(ST ¡ F0) e(T)r
+ F0e(T)r
= ST e(T )r
(21)
Consider now the alternative strategy. Take a position of e(T)r
in a forward con-
tract with maturity T and buy a bond with a face value of G0 with the same maturity.
The payo¤ at time T is
(ST ¡ G0) e(T)r
+ G0e(T)r
= ST e(T )r
(22)
The payo¤ from both strategies is the same. Therefore the capital required to
implement them must be the same. Hence we conclude that
F0 = G0 (23)
B. Taking into account credit risk
In this step we introduce default risk. We use the strategy proposed by CIR and
show the structure of the implicit options. We assume, that the amount that needs to
be given as collateral is proportional to the size of the contract. Doubling the number
of forward contracts thus leads to a doubling of the required collateral. In the case of
the futures contract the client can default at the intermediate marking-to-market at
21

22. time T/2 and at maturity, while default is only possible at maturity for the forward
contract.
Assume that the client has to provide a cash amount of M as collateral. The
present value of the credit risk implicit in the forward strategy is obtained as
e¡r(T )
IE
h¡
¡M ¡
¡
ST e(T)r
¡ G0e(T )r
¢¢+
i
(24)
= Put
¡
ST e(T )r
; G0e(T)r
¡ M
¢
Hence this value can be seen as a put option on Se(T)r
with maturity T and a
strike price equal to G0e(T)r
¡ M:
Now we turn to the futures strategy. We will again assume, that the client has to
give an amount of cash equal to M as collateral. At time T/2 he faces the following
decision: He can exchange the payment that he is supposed to make (which is given
by
¡
FT=2 ¡ F0
¢
e((T )=2)r
) for the loss of the collateral, the second futures contract and
the possibility to default later. At the time when the decision is taken, the second
futures position has a value of zero. The client will choose to default, if the money
she saves is more than the loss of the collateral and the implicit default option.
¡
FT=2 ¡ F0
¢
e((T)=2)r
6 ¡M ¡ (forthcoming cash ‡ows) (25)
The term forthcoming cash ‡ows refers to the value of the possibility of defaulting
later. It is important to note that the strategy requires us to increase the number of
futures contracts and therefore also the amount of collateral. The gain from defaulting
at time T is
¡
¡e(T)r=2
M ¡
¡
ST ¡ FT=2
¢
e(T )r
¢+
(26)
This value is equal to the following option at time T/2
Put
¡
ST e(T )r
; FT=2e(T)r
¡ e(T=2)r
M
¢
(27)
The range of integration for ST=2 is obtained by …nding the value of ST=2 that
makes inequality 25 a strict equality. The gain from defaulting at time T/2 does
not include the value of the default at time T. The distinction of the equation that
determines the relevant range and the payo¤ of the option is quite important. This
is the e¤ect of the non-myopic behavior of the agent. When taking the decision to
default he takes into account that he will lose the future cash‡ows. This implies that
he will not default in some range even if the value of the option at time T/2 is in the
money.
22

23. The value of the option to default at time T/2 is a function of ST=2: This price
however is not known at time 0. The expected value at time 0 is
Put1 = e¡r((T )=2)
Z Scrit
0
¡
¡M ¡
¡
FT=2 ¡ F0
¢
e((T)=2)r
¢+
f
¡
ST=2
¢
dST=2 (28)
= e¡r((T )=2)
Z Scrit
0
¡
¡M ¡
¡
ST=2er((T)=2)
¡ F0
¢
e((T)=2)r
¢
f
¡
ST=2
¢
dST=2
At time T the client has the possibility to default given that she didn’t default at
time T/2. The value of this option is given by
Put2 = e¡r((T)=2)
Z 1
0
Put
¡
ST e(T)r
; FT=2e(T )r
¡ e((T )=2)r
M
¢
(29)
¢1INo def at T=2f
¡
ST=2
¢
dST=2
= e¡r((T)=2)
Z 1
Scrit
Put
¡
ST e(T )r
;FT=2e(T )r
¡ e((T )=2)r
M
¢
f
¡
ST=2
¢
dST=2
In order to compare the result to the one obtained without default risk we look
at the future value of the strategy.
Futures Position =
¡
FT=2 ¡ F0
¢
e(T )r
+ PutT=2e(T )r
+ (30)
¡
FT ¡ FT=2
¢
e(T )r
1INo def at T=2
¡Put2
¡
FT ; FT=2 ¡ e((T)=2)r
M
¢
e(T)r
+ F0e(T)r
We can do the following simpli…cations: FT is again equal to ST : We know that the
futures contract from T/2 to T has no more intermediate marking-to-market periods:
it is identical to a forward contract. The default free futures price at time T/2 is
therefore
FT=2 = ST=2e((T )=2)r
(31)
The future value of the forward strategy is
Forward Position = (ST ¡ G0) e(T )r
+ Put
¡
ST e(T )r
; G0e(T)r
¡ e(T )r=2
M
¢
e(T )r
+ G0e(T )r
(32)
Without credit risk the value of the forward price is:
G0 = S0e(T )r
(33)
It is obvious that the value of the two positions will not be the same anymore if
the forward price is equal to the futures price. In order to determine explicitly the
value of the two strategies at time 0, we will take the expectation of the above ‡ows
under the riskneutral probability and calculate its present value.
23

24. The discounted expected values of the various positions in the futures strategy
are given by:
IE
£
ST=2e(T )=2r
¡ F0
¤
= S0e(T )r
¡ F0
(34)
IE[Put1] = Put
¡
ST=2e(T)r
; F0e((T )=2)r
¡ M
¢
(35)
IE
£¡
ST ¡ ST=2
¢
1INo def
¤
= 0
(36)
e¡r((T )=2)
IE
£
Put
¡
FT e(T)r
; FT=2e(T )r
¡ e(T)r=2
M
¢¤
= (37)
e¡r((T)=2)
Z 1
uc
Put
¡
ST e(T)r
; FT=2e(T )r
¡ e(T)r=2
M
¢
f (u) du (38)
IE[F0] = F0 (39)
The value of the futures position at time 0 is equal to
V alue futures position = IE
£
ST=2e(T)=2r
¡ F0
¤
+ IE[Put1] (40)
+e¡r((T)=2)
IE
£
Put
¡
FT e(T )r
; FT=2e(T)r
¡ e(T )r=2
M
¢¤
+ Fo
The value of the forward position at time zero is given by
V alue forward position = Put
¡
ST e(T )r
; G0e(T)r
¡ e(T )r=2
M
¢
+ G0 (41)
The value of the two positions will not be the same anymore if forward and futures
prices are equal.
The following table shows the values of the forward and futures position, given
that the futures price would be set equal to the forward price at time 0.
V alue futures position = 106:318
Collateralized credit risk = 1:191
V alue forward position = 106:717
Collateralized credit risk = 1:589
Table III. Values of the forward and futures positions. The parameters
used in the above computation are: S0 = 100, ¾ = 0:3 , F0 = G0 = 105:127,
M = 21:0254, T = 0:5:
The di¤erence in the value of the two strategies comes uniquely from the di¤erence
in managing the credit risk exposures. The credit risk implicit in the forward strategy
is higher compared to the credit risk in the futures strategy, despite the amount of
24

25. collateral set at the beginning being the same. We would assume that the amount of
collateral required will be lower in a situation with marking-to-market. In order to
obtain the same value of the option to default, the required margin for the futures
contract needs to be nearly 9% lower then for the forward contract. This shows
clearly how collateral can be replaced by dynamic strategies like the simple marking-
to-market procedure used in the above example.
We have demonstrated in a simple framework that futures and forward prices need
not to be the same in the presence of credit risk. We have also modelled the decision
of the agent by taking explicitly into account that s/he will consider the loss of the
second option when s/he decides whether to default or not. From this consideration
it follows that it is incorrect to model the situation with marking-to-market as a
series of independent put options. The e¤ect of neglecting the more restricted ranges
of the underlying variable is to overestimate the value of the default option under
marking-to-market. The value of the marking-to-market procedure comes from the
fact that it splits the longer maturity option into options of shorter maturities and
that it introduces the dependence among the successive decisions to default.
25

26. VI. Dynamic collateral management
In the following sections we analyze a model with non-stochastic collateral but
dynamic collateral management. We have analyzed in the previous section a situ-
ation with discrete marking-to-market using the contingent claim framework. The
resulting collateralized credit risk values have a structure which is comparable to
non-standard compound options. In order to generalize the results one could use the
same kind of setup and increase the number of marking-to-market times, but pricing
becomes cumbersome quickly as option numbers increase exponentially. In this sec-
tion we want to take a di¤erent viewpoint starting from a situation with continuous
marking-to-market. This will have a deep impact on the structure of the result. We
show how the time dimension in the continuous framework becomes less pertinent
than the underlying value dimension, an intuition that leads us to a simpler pricing
methodology for non-continuous marking-to-market.
A. Continuous marking-to-market case
In this part we want to outline the structure and valuation of collateralized credit
risk when the bank can monitor the value of the contract continuously. The bank is
assumed to be able to issue a margin call whenever it considers it to be necessary. We
will show that there is no credit risk in this setting, an intuitive result. The setting
provides a starting point for the next section where we will show that collateralized
credit risk arises from the fact that the bank will not monitor the value of the contract
continuously and where we value collateralized credit with frequent margin calls,
something not doable with the previous methodology.
A.I. Assumptions and variables
We make all the standard assumptions on the asset processes. The bank can
monitor the value of the forward contract continuously. It will issue a margin call
when the value of the contract has decreased by the amount given as collateral. The
bank will require a cash amount M as collateral. In order to clarify the setup we will
further assume that the client can only default at maturity. This means that s/he
will always make the required margin payments until time T. The di¤erence to the
model without marking-to-market is that the collateralized credit risk is comparable
to a put option with a strike price that is strongly path-dependent. This implicit
option corresponds to a number of barrier options. The variables used in this section
are:
St Price of the underlying of the forward
F Forward price
r Interest rate
Lit ith barrier
M Collateral (cash amount)
26

27. A.II. The model
The value of the credit risky contract can be modelled as a riskless forward contract
plus a number of barrier options. The package of options consists of one down-and-
out put option, a number of couples of long and short positions in down-and -in put
options and one down-and-in put option. The couples of long and short positions of
options are composed of options with the same strike price but di¤erent barriers. The
…rst barrier option is a down-and-out put option with a strike price equal to F-M.
The bank monitors the value of the forward contract continuously and issues a margin
call if the loss is not covered anymore by the collateral. Therefore the out-barrier of
the …rst option will be set at the level of St, which makes the value of the forward
contract equal to the negative value of the collateral. Hence the out-barrier of this
option is given by
L1t = F ¢ e¡r(T ¡t)
¡ M (42)
We assume further that the bank requires an amount of M as new collateral.
Hence the client restores the net level of collateral to M. The overall collateral is
now given by 2M. The next option is a long position in a down-and-in put option
with a strike price equal to F- 2¢M and the same in-barrier as the out-barrier of the
down-and-out put option. The next option is a short position in a down-and-in put
option with a strike price of F- 2¢M and a barrier given by
L2t = F ¢ e¡r(T¡t)
¡ 2 ¢ M (43)
The following down-and-in option has a strike price equal to F-3¢M and a barrier
equal to the out-barrier of the preceding out option. The next position is a short
position in a down-and-in put option with a strike equal to F-3¢M and a barrier
given by
L3t = F ¢ e¡r(T¡t)
¡ 3 ¢ M (44)
The two barriers of the nth couple of a long and a short position on options are :
higher barrier / long position: Lnt = F ¢ e¡r(T¡t)
¡ n ¢ M
lower barrier / short position: L(n+1)t = F ¢ e¡r(T¡t)
¡ (n + 1) ¢ M
As the price of the underlying of the forward contract is bounded to be greater
or equal to zero the barrier can never be lower than zero. The couple of options that
would have a lower barrier (resp. the in barrier of the short position would be lower
than zero) is just a down-and-in put option. There is no o¤setting short position in
a put option with the same strike price.
Hence the number of barrier options needed to replicate a risky forward contract in
the above framework depends on the relative size of the margin requirement. Relative
27

28. as compared to the price of the underlying of the forward contract and the strike price.
An analysis of the payo¤s yields the conclusion that there is no credit risk present
in this situation. The collateral will always be at least equal to the loss at time T.
Hence the possibility of default has no value in this setting.
A.III. Default at any margin call
In this section we will relax the assumption that the agent can default only at
maturity. The bank will issue a margin call, if the value of the underlying of the
forward contract is smaller than the negative value of the margin payment M. Hence
the bank will ask for additional collateral if the potential loss arising from credit risk
is bigger than the collateral to cover it. The critical level of St at which a maintenance
call will be triggered is given by
Scritical ¡ F ¢ e:r(T ¡t)
= ¡M (45)
Scritical = L1 = F ¢ e:r(T ¡t)
¡ M
this level can again be modeled as a barrier (which shall be called L1). If the
barrier is hit, the client can choose if he wants to provide additional collateral and
thus keep the contract alive or default. The gain resp. loss for the client depending
on his default decision is
gain=loss = ¡M ¡
¡
S¿ ¡ F ¢ e¡r(T¡¿)
¢
¡ forthcoming cash ‡ows (46)
The term forthcoming cash ‡ows correspond to the gains from defaulting later
if s/he chooses not to default at this point. In the case of no default the client
has to provide new collateral M. By doing this he keeps the contract alive with the
possibility to default later. He then has a second option which is conditional on the
fact that he didn’t default when the …rst barrier has been breached. The level of the
second barrier is the same as in the previous section. Without dwelling to much on
the structure of the decomposition we obtain the following result.
Credit risk is totally eliminated under continuous marking-to-market even with
default at any margin call. The following argument will prove the above statement.
At the moment when the barrier is hit, we know the level of S¿ . Replacing this value
in the default condition yields
gain=loss = ¡M ¡
¡¡
F ¢ e:r(T¡t)
¡ M
¢
¡ F ¢ e¡r(T ¡¿)
¢
(47)
¡future cash flows
gain=loss = ¡future cash flows < 0 (48)
This implies that the agent will never …nd it advantageous to default or will at
least be indi¤erent between defaulting or not if the value of the future cash ‡ows will
be zero.
28

29. This result holds for all the options forming the barrier option structure. None of
the option has ever a greater value than zero if the bank can issue a margin call at
any time when a barrier is hit.
In reality the client is allowed for some time to deliver the additional collateral.
The risk arising from the uncertain evolution of the value of the underlying contract
during the time between the margin call and the time of delivery of the additional
collateral is managed by adopting a double trigger strategy. When the …rst barrier
is reached the margin call is issued. The client has a prespeci…ed amount of time
to react. However if in the meantime the second (lower) barrier is reached the same
procedure is applied as in the case of default, meaning the clients position is closed
out.
The above calculations let us calculate in closed form the cumulative value of the
margin calls, in other words, the total value of the collateral required.
B. Discrete marking-to-market
We will relax now the assumption of continuous marking-to-market in order to
investigate the more realistic setting of discrete marking-to-market.
The bank is assumed to monitor at regular points in time (e.g. daily). Therefore it
can happen that the price of the underlying breaks the barrier between two marking-
to-market times. The bank uses the same rule as before when setting the critical level
at which to issue a margin call. The same kind of problem arises in the valuation of
ordinary barrier options. Broadie et al. (1997) have developed an approximation for
the case of ordinary barrier options. The basic idea is that the value of a discretely
monitored barrier option corresponds to a continuously monitored barrier option with
a lower (for a down option) barrier. Therefore the uncertainty about the price at the
next marking-to-market instant is translated into a lower barrier. The methodology
for pricing barrier options is well explained in Rich (1994). The valuation of curved
boundaries is due to Kunitomo et al. (1992). The actual valuation obtained then is
original.
B.II. Valuation using a correction of the barrier for lower
marking-to-market frequency
In this section we calculate the value of the option to default under the above
described rules for collateral management. The amount of collateral taken at the
initiation of the contract is assumed to be at least 50% of the initial maximum possible
loss. If the next barrier is hit, the bank will require again the same amout. Therefore
it is fully hedged against a loss arrising from this contract if the client provides the
collateral. This implies that there is only one barrier option as the barrier option
with the lower strike price would always have a value of zero.
The structure of the problem remains the same as in the previous section. The
29

30. payo¤ from the …rst option is again given by
¡M ¡
¡
S¿ ¡ F ¢ e¡r(T¡¿)
¢
(49)
The maximum possible loss is given by -F. If the amount required as collateral is
at least 50% of F. This implies that the option the agent receives if he decides to
continue has no value.
The e¤ect of discrete marking-to-market can be approximated by using a corrected
barrier. The corrected barrier is obtained by multiplying the original barrier with the
factor
Exp
³
¡¯¾
p
¢t
´
(50)
as shown in Broadie et al. (1997). The parameter ¯ is a constant and ¾ is the
volatility of the underlying of the forward contract. The corrected density function
of the barrier is obtained as
f (¿) =
¡ ln (H=S)
¾
p
¿3
n
¡
x1 ¡ ¾
p
¿
¢
(51)
with
A = F ¡ M (52)
B =
1
T
¢ Ln
µ
F ¢ Exp (¡r (T)) ¡ M
A
¶
C = Exp
³
¡¯¾
p
¢t
´
H = (F ¡ M) ¢ C
¹ =
µ
r + B ¡
1
2
¾2
¶
x1 =
£
ln (S=H) +
¡
¹ + ¾2
¢
(¿)
¤
=
¡
¾
p
¿
¢
The level of St at time ¿ is equal to the barrier at this time.
S¿ = Exp
³
¡¯¾
p
¢t
´
¢
¡
F ¢ e¡r¢(T ¡¿)
¡ M
¢
(53)
Replacing the value S¿ in the payo¤ equation by the above expression and integrating
over ¿ yields the following
CCR =
Z T¡t0
0
e¡r¢¿
à ¡
F ¢ e¡r¢(T¡¿)
¡ M
¢
¢³
1 ¡ Exp
³
¡¯¾
p
¢t
´´
!+
f (¿) d¿ (54)
=
Z T¡t0
0
¡
F ¢ e¡r¢T
¡ M ¢ e¡r¢¿
¢ ³
1 ¡ Exp
³
¡¯¾
p
¢t
´´
f (¿) d¿
30

31. As long as M is smaller than Fe¡rT
we can integrate from 0 to T-t0. If this
condition would not be satis…ed we would need to calculate a critical t, that would
guarantee that the expression in the braquets is >0. In a more complicated situation
involving several barriers we would have to work with the above integral expression.
In this very simple case we can derive the following closed form solution:
CCR =
Z T¡t0
0
F ¢ e¡r¢T
¢ X ¢ f (¿) d¿ ¡ M ¢ X ¢
Z T ¡t0
0
e¡r¢¿
f (¿) d¿
CCR = F ¢ X ¢ e¡r¢T
0
@
(H=S)m+n
¢ N
³
ln(H=S)+n¾2(T ¡t0)
¾
p
T ¡t0
´
+ (H=S)m¡n
¢ N
³
ln(H=S)¡n¾2(T ¡t0)
¾
p
T ¡t0
´
1
A
¡M ¢ X ¢
0
@
(H=S)m
¢ N
³
ln(H=S)+¹(T¡t0)
¾
p
T ¡t0
´
+N
³
ln(H=S)¡¹(T¡t0)
¾
p
T¡t0
´
1
A
where
m =
¹
¾2
(55)
n =
p
¹2 + 2r¾2
¾2
X = 1 ¡ Exp
³
¡¯¾
p
¢t
´
(56)
A margin requirement lower than 50% of the maximal possible loss would give
rise to some additional barriers. The valuation of the collateralized credit risk could
proceed along the same lines. The result would take the form of some nested integrals
with some restricted ranges of integration.
Figure 7 shows the e¤ect of variations in the two most important variables on the
collateralized credit risk in this situation.
31

32. 0.1
0.2
0.3
0.4
50
100
150
0
0.2
0.4
0.6
0.8
1
0.1
0.2
0.3
0.4
0
0.2
0.4
0.6
0.8
1
# of monitoring times
CCR Increasing frequency
of monitoring
Figure 7: Collateralized credit risk and marking-to-market frequency.
This …gure gives the values of collateralized credit risk (called CCR) for varying
levels of ¾ and numbers of marking-to-markets. The values of the …xed parameters
are: S0 = 100; F = 110:517; M = 50; r0 = 0:1; T ¡ t = 1:
The collateralized credit risk increases with the forward price. This is due to
the fact that with a higher forward price the possible loss increases. Default enables
the client to avoid that loss, hence its value increases the higher this possible loss
is. Obviously the credit risk decreases with a rising amount given as collateral. The
higher the volatility of the underlying the higher the value of credit risk. The intuition
behind this is, that with a higher volatility the chance that the underlying price
reaches the barrier between two marking-to-market instants increases. The more
frequent the forward contract is monitored the lower is the value of the implicit option.
This is captured by the term ¢t. The credit risk exposure in this framework with
dynamic marking-to-market and collateral management comes solely from the fact
that the barrier can be reached between the marking-to-market times. The relative
distance of the underlying price from the barrier a¤ects the credit risk only over the
density function. Intuitively this means that the underlying price today will in‡uence
the probability that we reach the barrier. The payo¤ however (conditional on having
reached the barrier) does not depend directly on the underlying price anymore. This
conclusion is quite intuitive when we recall the results of the preceding two sections.
The two means by which credit risk is managed are marking-to-market and the
requirement of collateral. Both will be associated with some costs. The bank might
have some preferences to use one of the two instruments more extensively then the
other. In order to determine how the marking-to-market frequency must be changed
to o¤set a lower collateral requirement, we need to check that marking-to-market and
32

33. collateralization are substitutes. The following …gure shows the collateralized credit
risk for various values of collateral and di¤erent marking-to-market frequencies.
40
45
50
55
60
50
100
150
0
0.2
0.4
0.6
0.8
1
40
45
50
55
60
# of monitoring times
CCR
Collateral
Increasing frequency
of monitoring
Figure 8: Collateralized credit risk for di¤erent levels of marking-to-
market frequency and collateral. This …gure gives the values of collateralized
credit risk (called CCR) for varying levels of collateral and numbers of marking-to-
markets. The values of the …xed parameters are: S0 = 100; F = 110:517; ¾ = 0:3;
r0 = 0:1; T ¡ t = 1:
40 45 50 55 60
20
40
60
80
100
120
140
#ofmonitoringtimes
Collateral
Figure 9: Collateralized credit risk for di¤erent levels of marking-to-
market frequency and collateral. This …gure gives the values of collateralized
credit risk (called CCR) for varying levels of collateral and numbers of marking-to-
markets. The lighter shading represents higher values of credit risk. Combinations of
collateral and marking-to-market frequency on the same line represent equal credit
33

34. risk levels. The values of the …xed parameters are: S0 = 100; F = 110:517; ¾ = 0:3;
r0 = 0:1; T ¡ t = 1:
60 80 100 120
0.04
0.06
0.08
0.1
0.12
0.14
45 50 55 60
0.04
0.05
0.06
0.07
MRS
Collateral
n = 90
n = 100
C = 60
MRS
C = 40
C = 50
# of m onitoring times
n = 110
Figure 10: MRS di¤erent levels of marking-to-market frequency and
collateral. This …gure gives the values of the marginal rate of substitution (MRS)
between collateral and marking-to-market frequency. The values of the …xed param-
eters are: S0 = 100; F = 110:517; ¾ = 0:3; r0 = 0:1; T ¡ t = 1:
Figures 8 to 10 illustrate that marking-to-market and collateralization are substi-
tutes. The lines of equal credit risk in …gure 9 are similar to indi¤erence curves of
two assets, which are substitutes to one another. Therefore the bank can achieve the
same level of credit-risk exposure by either requiring collateral or marking-to-market.
The bank can thus choose the mix of the two instruments, which turns out to be
optimal in the light of costs or regulatory requirements.
The valuation equation lets us draw some conclusions on the optimal strategy
of the bank. It can calculate the incremental bene…t of an additional marking-to-
market time and compare it with the marginal costs. Hence the optimal marking-to-
market frequency could be determined conditional on the collateral and the marking-
to-market costs. The optimal marking-to-market frequency will be an increasing
function of the volatility of the underlying. The value of the option to default will
be higher during turbulent times. One way of achieving a lower value of the option
to default would be to adopt a strategy with non constant time intervals between
the di¤erent marking-to-market times. The frequency would obviously have to be an
increasing function of the volatility.
34

35. VII. Renegotiation
In this section we want to examine under which circumstances there might be
some room for renegotiation in a situation with a credit risky forward contract. We
will analyze the optimal decision of the bank depending on the level of cost of default
and the bargaining power of the parties involved. We investigate the decisions at
some extreme values in order to emphasize the structure of the decision that the
parties face.
A. Renegotiation in the presence of costs
In this section we will analyze a situation where the bank faces some cost in seizing
the collateral. The …rst section will be a two period model. The bank incurs a cost
proportional to the value of the collateral. Hence the bank looses a fraction of c, when
it takes possession of the collateral. The gap in the value of the option to default to
the lender and the bank leaves some room for renegotiation.
The bank and the client have engaged in the standard forward contract at time 0,
the client has provided some collateral I. I is the abbreviation for initial margin. In the
following we will take into consideration that the bank will work with a maintenance
margin which is di¤erent from the initial margin. At time T the contract matures.
The clients payo¤ can be decomposed in a standard forward contract and put option.
The clients payo¤ from the put option is:
(¡I ¡ (ST ¡ F))+
= (F ¡ I ¡ ST )+
(57)
This is the payo¤ from a put option, hence it’s value at time 0 is
Put(ST ; F ¡ I) (58)
The banks payo¤ in the case of default will be given by
¡ (¡I ¢ (1 ¡ c) ¡ (ST ¡ F)) (59)
Hence the bank is short a put option with a strike F-M¢(1¡c). Note that this put
option is not a standard put option as the relevant range of integration is determined
by the clients optimal decision. It will be worth for the bank to make a payment of
the amount Y to the client in order to prevent him from defaulting. Y will be in the
range between
(F ¡ I ¡ ST )+
| {z }
Ymin
< Y <
(F ¡ I ¢ (1 ¡ c) ¡ ST )+
| {z }
Ymax
(60)
The possibility to renegotiate enables the bank to reduce the cost of default from
Ymax down to Ymin.The actual size of the surplus to the bank will be determined by
the bargaining power of the two parties. The client would chose to default if
35

36. F ¡ I ¡ ST > 0 (61)
Solving the inequality number 61 gives us the critical value of ST denoted by
Scrit1.
The relative bargaining power will determine the share of the surplus received by
the bank respectively by the client. The bargaining power of the bank is represented
by the parameter µ: This parameter can have a value between 0 (no bargaining power
for the bank) to 1 (all the bargaining power). The value of the risky forward can be
decomposed in the following way1
risky forward = riskless forward (62)
¡
Z Scrit1
0
(µ ¢ (F ¡ I ¡ ST ) + (1 ¡ µ) ¢ (F ¡ I ¢ (1 ¡ c) ¡ ST )) f (ST ) dST
= riskless forward ¡ µ ¢ Put (ST ; F ¡ I) ¡ (1 ¡ µ) ¢ Put (ST ; F ¡ I (1 ¡ c))
A numerical example of a more general model will be provided at the end of the
following section. The potential for renegotiation hinges on the fact, that the implicit
option of the client is di¤erent from the implicit option of the bank. In some sense
the contract bought by the client is not the same as the one sold by the bank.
B. The three period problem
We want to consider now the three period problem. First we will look at the
situation where the bank o¤ers a side payment in order to induce the client not to
default at the second period. The bank observes the value of the underlying and
in the case of possible default, it o¤ers the client a payment in the next period in
order to induce him not to default. In this section we take into account the di¤erence
between the initial margin and the maintenance margin. Technically this implies that
the bank will call for new collateral earlier as it requires the loss not to exceed the
maintenance level. The initial margin is denoted by I the maintenance margin by
M. The bank will only o¤er the client a payment Y at time T if he would default
otherwise. This setup encompasses a variety of concessions the bank might consider
in order to make the contact more attractive. Amongst these are a lowering of the
required margin, lowering of the forward price or a direct payment. This payment
will only be received by the client if he does not default at time T. The client will be
marked to market at time T/2 if
Scrit1 > F ¢ e¡r¢T=2
¡ (I ¡ M) (63)
he will choose to default without a side payment if
¡I ¡
¡
ST=2 ¡ F ¢ e¡r¢T=2
¢
¡ Put
¡
ST ; F ¡ I +
¡
ST=2 ¡ F ¢ e¡r¢T=2
¢¢
= 0 (64)
36

37. ST/2>Scrit1
No default
No default
Scrit1>ST/2>Scrit2
No default w ith
reincentivization
Scrit2>ST/2>Scrit3
Scrit3>ST/2
Default
No margin call at T/2
Margin call at T/2
Put2(F-I)
Put3(F-I+(ST/2-Fe-rT/2
))
Put4(F-I+(ST/2-Fe-rT/2
)-Y)
+Payment of Y
Put2(F-I(1-c))
Put3(F-(I-(ST/2-Fe-rT/2
))(1-c))
Put4(F-(I-(ST/2-Fe-rT/2))(1-c)-Y)
+Payment of Y
Put1(F-I)
Put1(F-I(1-c))
TT/2T=0
Value to the client
Cost of the bank
Figure 1:
Solving the equation 64 yields Scrit22
.
If the value of ST=2 happens to be lower than Scrit2 the bank will o¤er the agent
a side payment of Y, which he will receive if he does not default at time T. The client
will only default if
¡I ¡
¡
ST=2 ¡ F ¢ e¡r¢T=2
¢
¡ Put
¡
ST ; F ¡ I +
¡
ST=2 ¡ F ¢ e¡r¢T=2
¢
¡ Y
¢
¡ Y = 0
(65)
Solving the equation 65 for ST=2 yields Scrit3.
Figure 11: Ranges of ST=2 and implicit options in the three period
model. The gray shaded expressions are the implicit options seen from the banks
perspective.
The equilibrium strategies of the bank and the client are the following. All the
ranges refer to values of S at time T/2. In the range from Scrit1 to 1 the bank sticks
to the original contract and the client does not default at T/2. He has the implicit
put option number 2 at time T. For values of ST=2 from Scrit2 to Scrit1 the bank
does nothing and the client does not default. The client has the put option number
3 at time T. In the region from Scrit3 to Scrit2 the bank o¤ers the side payment Y
and the client chooses not to default at T/2. At time T he receives the put option
number 3 and the payment Y. For values of ST=2 lower then Scrit3 the bank o¤ers the
37

38. payment Y, but the client …nds it optimal to default. The clients payo¤ corresponds
to the implicit option number 1 at time T/2. There are no more payments at time T
in this case.
The bank will choose an Y, which minimizes its cost. This cost is represented by
the grey shaded expressions in …gure 11. A numerical example will be provided at
the end of the next section.
The collateralized credit risk for the bank given that it has no bargaining power
(hence µ = 0) is the sum of the following options:
Put1 = e¡rT=2
Z Scrit3
0
(¡I ¤ (1 ¡ c) ¡ ST=2 + F ¢ e¡r¢T=2
f(ST=2)dST=2 (66)
Put 2 = e¡rT=2
Z 1
Scrit1
Put (ST ; F ¡ I ¢ (1 ¡ c)) f
¡
ST=2
¢
dST=2 (67)
Put 3 = e¡rT=2
Z Scrit1
Scrit2
Put
¡
ST ; F + (1 ¡ c)
¡
¡I +
¡
ST=2 ¡ F ¢ e¡r¢T=2
¢¢¢
f
¡
ST=2
¢
dST=2 (68)
Put 4 = e¡rT=2
Z Scrit2
Scrit3
Put
¡
ST ; F + (1 ¡ c)
¡
¡I +
¡
ST=2 ¡ F ¢ e¡r¢T=2
¢
¡ Y
¢¢
f
¡
ST=2
¢
dST=2
3
(69)
PV (Y ) = e¡rT
Z Scrit2
Scrit3
Y ¢ f
¡
ST=2
¢
dST=2 (70)
C. Three periods with side-payment and renegotiation
In this section we take all the strategies together. The bank monitors the value
of the underlying. It will o¤er the client the same kind of deal as in the preceding
section. If the client still wants to default, the bank will bargain again with him
over the bene…t generated by saving the cost of seizing the collateral. Note that all
the critical values of ST=2 are the same as in the preceding section. The equilibrium
strategies are the same as in the last section except for the fact that in all the default
states the bank will o¤er a contemporaneous payment. The size of the payment
38

39. will again depend on the bargaining power of the two agents represented by µ. The
payments in the default state are again weighted sums of options, where the weight
is equal to µ:
The options are the same as in the preceding except that the costs c are set equal
to zero (represented by Put1;c=0) when the bank has all the bargaining power.
The implicit options are the following
Put1 = µ ¢ Put1;c=0 + (1 ¡ µ) ¢ Put1 (71)
Put2 = µ ¢ Put2;c=0 + (1 ¡ µ) ¢ Put2 (72)
Put3 = µ ¢ Put3;c=0 + (1 ¡ µ) ¢ Put3 (73)
Put4 = µ ¢ Put4;c=0 + (1 ¡ µ) ¢ Put4 (74)
PV (Y ) unchanged (75)
The model of the previous section is obtained by setting µ = 0, hence giving all
the market power to the client. In order to have the pure bargaining model (without
the payment of Y) the parameter Y must be set equal to zero. In this case the option
number 4 has a value of zero.
D. Results
The value of renegotiation comes from the fact that there are some costs associated
with obtaining the collateral. If the costs are equal to zero the optimal payment Y is
also equal to zero.
The following …gure shows the values of the implicit options for varying levels of
39

40. Y and cost c = 0.
5 10 15 20
0.2
0.4
0.6
0.8
1
Put 1
Y
5 10 15 20
0.002
0.004
0.006
0.008
0.01
0.012
Put 4
Y
5 10 15 20
0.5
1
1.5
2
2.5
PV(Y)
YY
CCR
5 10 15 20
2.6
2.8
3.2
3.4
3.6
3.8
4
Figure 12: CCR and values of the implicit options for various levels
of the side-payment Y. The above …gures show the values of the various implicit
options for di¤erent levels of Y. The values of options number 2 and 3 are not shown
as they are just horizontal lines. The values of the …xed parameters are: S = 100, F
= 102.5, I=20, M=1, ¾ = 0.3, r = 5%, T = 1, c = 0, µ =0.
The e¤ect in‡uence of Y >0 on the value of options 1 and 3 goes in the expect
direction. By making the no default state at time T more attractive the bank lowers
the value of these options. Hence it lowers the incentive of the client to default at
time T/2. Thus it re-incentivizes the client to keep the contract alive. Technically
this is re‡ected in a smaller range of integration of option number 1. However the
cost of doing this, which is equal to PV(Y) is much larger then the bene…ts. Thus if
the bank can take over the collateral without any costs, the optimal Y is equal to 0.
Hence the bank will not o¤er the deal. For values c >0 the optimal Y is also greater
40

41. then zero.
5 10 15 20
4.5
4.6
4.7
CCR
Y
C = 0.4
5 10 15 20
3.25
3.5
3.75
4
4.25
4.5
4.75
5
CCR
Y
C = 0.1
C = 0.2
C = 0.3
C = 0.4
C = 0.5
Figure 13 : Collateralized credit risk for di¤erent values of costs and
side payment Y. The left …gure shows the value of CCR for c = 0.4. The optimal
value of Y is given by 9.75. The right …gure shows the CCR for di¤erent levels of c.
The values of the …xed parameters are: S = 100, F = 102.5, I=20, M=1, ¾ = 0.3, r
= 5%, T = 1, µ = 0.
The most important feature of …gure 13 is that the CCR has a minimum for a
Y>0. The optimal Y can be calculated easily for various values of c. For values of
c up to 50% it is increasing with c. For very extreme values of c it decreases again
slightly.
The in‡uence of the bargaining power (measured by µ) is straightforward. The
collateralized credit risk is a weighted average of the two options. Due to the fact
that the …rst option is always worth less than the second, the collateralized credit
risk is decreasing in µ. The following graph shows the CCR for various values of c, Y
and µ. One sees clearly that for c>0 there exists an optimal Y >0. The in‡uence of
µ can be seen from the three di¤erent layers of CCR values.
41

42. 0
5
10
15
20
0.1
0.2
0.3
0.4
0.5
3
3.5
4
0
5
10
15
20
CC R
Y
= 1
C
= 0.5
= 0
Figure 14: Collateralized credit risk for di¤erent levels of c,Y and µ.
The above …gure shows the values of CCR for di¤erent combinations of C,Y and µ.
The three value of µ give rise to the three layers of CCR. The values of the …xed
parameters are: S = 100, F = 102.5, I=20, M=1, ¾ = 0.3, r = 5%, T = 1.
42

43. VIII. Conclusions
We have shown the in‡uence of collateral on the valuation of credit risk in some
stylized situations. We have explored the e¤ect of di¤erent asset classes serving
as collateral. The analysis underlines the necessity to think of collateral valuation
jointly with the source of risk. Correlation matters. Collateralized credit risk depends
obviously on all the characteristics of the assets involved. Therefore the decision on
margin requirements and haircuts needs to take all of them into account at the same
time. The models proposed give a technical solution to the problem of determining
haircut levels across di¤erent classes of assets.
We have analyzed the structure of the collateralized credit risk when a simple
marking-to-market rule is used. We have demonstrated the way in which marking-
to-market reduces collateralized credit risk. We have shown how collateral can be
substituted by a higher marking-to-market frequency. In the last part we have ex-
plored the e¤ect of dynamic collateral management. We have shown the sources
of credit risk in this context. The results have practical implications on optimal
marking-to-market strategies. The aim of this work was to put economic intuition
on collateral on a sound basis. Moreover it is a starting point to include collateral in
more general credit risk frameworks.
Future work could focus on more complex marking-to-market mechanisms. It
could introduce the possibility of double default with stochastic collateral as well as
in the dynamic setting. The dynamic management setting itself could be extended to
the case of stochastic collateral. More importantly, the implications of the framework
on credit portfolios could be further examined.
43

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46

47. Footnotes
1
The solution has been obtained by using the standard Martingale approach. The
resulting equation can be found for example in Zhang (1998).
2
Please note that this is not a standard put option. The range of integration is
determined by the clients’s optimality condition not by the one of the bank.
3
Please note that in the range from Scrit1 up to Scrit2, the agent does not default
even without side payment.
4
Alternatively one could not substract the Y payment in the payo¤ function. In
this case the payment Y would be conditional on no default at time T. This would
result in a double integral expression for Y.
47