An Analysis of Credit RiskWith RiskyCollateral:

A Methodology for Haircut Determination

Didier Cossin¤ & Tomas Hricko¤¤

March 2001

Comments Welcome

¤HEC, University of Lausanne, CH-1015 Lausanne, SwitzerlandTel: 41 21 692 34 69Fax: 41 21 692 33 05Email: 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

An Analysis of Credit RiskWith RiskyCollateral:

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

Abstract

Although many credit risk pricing models exist in the academic literature, verylittle attention has been paid to the impact of risky collateral on credit risk. It isnonetheless well known that practitioners often mitigate credit risk with collateral,using so-called haircuts for collateral level determination. The presence of collateralhas 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 andmargin calls. The models con…rm many practical intuitions, such as the impact onthe haircut level required of the risks of the collateral asset and of the underlyingasset to the forward as well as the impact of their correlation. Moreover the modelsupports the intuition that the frequency of marking-to-market and collateral aresubstitutes. The models also stress the possibly unexpected magnitude of these fac-tors. More importantly, they give actual solutions to determining the value of thecredit 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 tounderstand the portfolio e¤ect of collateral management. Finally they illustrate howdi¤erences in prices may arise from pure di¤erences of credit risk management, asillustrated here in the case of futures and forwards.

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I. IntroductionCredit 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 oneof the major elements of …nancial markets risks. Financial crises keep remindingbanks and regulators of the importance of credit risk. For example, in a recent ISDA(International Swap Dealer Association) paper, the self regulating body puts underscrutiny (and under question) the Basle Committee on Banking Supervision’s ruleson credit risk capital, which still date back from 1988 and are outdated in manydimensions. The paper calls for major reform of the credit risk guidelines, notably asfar 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 thanstandard loans), the apparition of credit risk derivatives, the widespread use of col-lateral by …nancial intermediaries and the use of risky securities such as corporatebonds or stocks as collateral, all have dramatically changed the face of the credit riskexposure 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 exchangetraded derivatives do not present credit risk, OTC derivatives do. With an estimateof the OTC market at $65 trillion (Source : Risk Magazine, July 1998), it becomesobvious that the issue of credit risk management, either through pricing or throughother 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 putsinto question the academic models of credit risk pricing that tend not to incorporatecollateralization.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 goodtheory 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 thumbrather than to advanced analysis. The goal of this research is to participate in …llingthat 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 fewyears, such as, amongst others, Longsta¤ and Schwartz (1995), Jarrow and Turn-bull (1995), Anderson and Sundaresan (1996), Leland and Toft (1996), Du¢e andHuang (1996), Du¢e and Singleton (1997), Mella-Barral and Perraudin (1997), Jar-row, Lando and Turnbull (1997)). Many of these papers approach the credit riskissue from the pricing standpoint. It has been a classical way to approach the prob-

4

lem since the seminal Merton paper of 1974 that derives a simple, option-based modelof a zero-coupon bond pricing with credit risk. This elegant model prices credit riskendogenously 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 directionof 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 itselfand considers it as an exogenous process. From a theoretical point of view, this isnot a welcome concession. On the other hand, it allows for an easier treatment ofpractical cases (with the weakness of ignoring the …nancial economics behind thedetermination of the bankruptcy process). Many papers have recently appeared thatfollow 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 awidely 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 approachDu¤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 riskyinstrument. Notice that the issue of pricing an instrument that is collateralized withanother risky instrument is not trivial and becomes complex when marking-to-marketor margin calls are considered. Margrabe(1978) has mentioned the analogy betweenan exchange option and a margin account and provides the pricing for a very simpleframework with no marking-to-market. Stulz and Johnson (1985) have priced secureddebt using contingent claim analysis and study the use of collateralization in a cor-porate …nance framework, analysing the impact of collateralization on the value ofthe …rm. The rest of the economic literature has addressed the rationale behind theuse of collateral in debt contracts and is an extension of the questions arising in thetheory of debt (see Benjamin (1978), Plaut (1985), Bester (1994)) but has not beenconcerned with pricing the credit risk with collateral or with evaluating the impactof haircut levels on the credit risk value. Our goal of pricing credit risky instrumentswith risky collateral is thus fundamentally di¤erent and has not been fully addressedyet.While the …nance academic world has focused on the issue of pricing credit risk,

practitioners and regulators have used collateralization very commonly rather thanpricing to manage credit risk exposures. Collateralization is an elegant way to trans-

5

form a credit risk issue into a market risk one. For example, it is well known inpractice that pricing of swap contracts for example bears little or no dimension ofcredit risk (a problem examined in Cossin and Pirotte (1997)), notably because col-lateralization is today used extensively. Many securities can in practice be used forcollateralization of risky contracts. The amount of the collateral required typicallydi¤ers depending on the risk of the security being used in the collateral. This is whypractitioners traditionally use so-called haircuts, that determine how much collateralis required depending on the type of security used as collateral, a phenomenon notwell incorporated in current academic research (nor in current regulation). We aimhere at providing with this research a framework to analyze haircut determinationand the impact of risky collateral on credit risk and look precisely at the case of riskyforwards, 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 inthis case. We then present an analysis of collateralizing an instrument with stochasticequity when there is no marking-to-market, followed by an analysis of collateralizingwith bonds when interest rates are stochastic. Finally, we approach the problem ofpricing a credit risky instrument with collateral when there are marking-to-marketand margin calls.

6

II.Model with dual default and non-stochasticcollateral

A. Assumptions

In this very simple set-up, and before studying the more pertinent situation ofrisky collateral, we analyze the value of the collateralized credit risk (CCR) for twocredit-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 shortside). In order to manage their respective credit risk exposure both agents requirethe counterparty to deposit some collateral either with their counterparty or with aneutral institution, for example a clearing house. There is no intermediate marking-to-market between initiation of the contract and expiration. Hence the only timewhen default can occur is at maturity. The collateral deposited by both agents iscash (and thus is not risky itself).We assume that the decision to default dependsonly on the value of the original contract and the value of the collateral. We thusconsider default to be endogenous. Extension to exogenous default could be thework of future research but the assumption of endogenous default may be strongerthat it appears to be …rst. Indeed, we assume here that an agent defaults if the lossof the collateral and the underlying contract is worth less than the future paymentss/he had committed to make. One might object, that a company or an individualmight default for a wide variety of reasons. Some of them might be totally unrelatedto the speci…c contract considered. For example a company might have to …le forbankruptcy for some liquidity reasons. In the case of bankruptcy though, creditorswill take over control and behave optimally, by maximizing the contract value, asdescribed in our model. Therefore they will take the same decisions as the companywould have taken, namely they will not default on the speci…c contract if default iscostlier than honoring the contract. We also assume no external costs to default (forsimpli…cation), which means that both agents will exercise their options if they arein 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 betraded 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 atthe forward price at expiration (hereafter called time T), while agent 2 must sell theasset at the same time. The forward price is set at time 0 so as to make the value ofthe forward contract equal to zero. We assume that the storage cost for this asset canbe 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 forwardprice that yields a forward contract value of zero at time 0 is given by:

H = S0 ¢ ert (1)

7

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 0ST Value of the underlying of the forward contract at time TH Forward priceM1 Cash amount that agent 1 has to give as collateralM2 Cash amount that agent 2 has to give as collateral

We assume that the price of the underlying of the forward contract follows ageometric Brownian motion process. The interest rate is assumed to be constant forthe time being.

B. The basic modelUnder 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 STriskless forward ST ¡H ST ¡H ST ¡Hrisky 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 ofthe underlying of the forward contract at maturity.

The value of the forward contract with two-sided default risk can be smaller, largeror equal to the situation without the possibility of default. Agent 1 with the longposition gains because he has the possibility to default if the price of the underlyingof the forward contract drops. On the other hand he looses because the counterpartymight not honor its obligation if the contract evolves in agent 1’s favor. He looses dueto the presence of the implicit call option. The resulting value of the long position inthe credit risky forward contract can be bigger or smaller than the risk free forwardcontract.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 attime 0, the value of both options must be the same. As a consequence of the limitedliability assumption the price of the underlying of the forward contract can not be

8

negative. This implies that agent 1’s loss is bounded. On the opposite side, the lossof agent two with the short position is unbounded. Hence the possibility to defaultis of greater value to him. In order to obtain the same value for both options agent2 must deposit a higher amount.

The following example illustrates the decomposition:S0 = 100H = 100 ¢ exp (r ¤ (T ¡ t)) = 102:532M1 = 20:5063 (20% of H)M2 = 20:5063r = 0:1¾ = 0:3T-t = 0:25

If the forward price is chosen in a way as to make the value of the forward contractat time zero equal to zero, the value of the risky forward contract at time 0 is justthe 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 atinitiation, i.e. to give both options the same value, the party with the short positionin the forward contract would need to deposit a higher margin. In the above examplethe necessary value of M2 for a 0 contract value at initiation is equal to 27% of theforward price (versus 20% for the long position).

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III. Model with stochastic collateralA. Assumptions

We now generalize our model to the more realistic situation of one credit riskyagent 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 assumptionfor 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 ofdefaulting 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, thebuyer of a forward on an exchange rate can post a collateral of a certain amount ofa 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 thatmoment. We assume that the price of the underlying asset to the forward contractand the price of the asset given as collateral follow two separate Geometric Brownianmotion processes. The two processes are correlated. There is no marking-to-marketand 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 theforward contract. M stands for the value of the asset given as collateral, St standsfor 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 Sp¿ du

dm = M r dt+ ¾M Mp¿ dv

½ = Correlation between dv and du

The collateralized credit risk can be compared to an exchange option. The mostsimple situation involving the possibility to exchange one stochastic asset for another

10

stochastic asset was analyzed by Margrabe (1978). He mentioned the situation of amargin 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 heexchanges the di¤erence of the forward and the spot price for the collateral. Hencethe implicit option is a spread option. The value of the collateralized credit riskoption is:

CCR option =

Z +up(v)

¡1

Z ¡d¡Á(v)

¡1

"H ¡ Se(¹s¡ 1

2¾2s)¿+¾s

p¿u

¡Me(¹M¡ 12¾2M)¿+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+½¾Sp¿ )

¡1f (v)N

ád¡ Á (v + ½¾Sp¿ )¡ ½v ¡ ¾Sp¿p

1¡ ½2

!dv (8)

A2 =

Z +up(v+¾Mp¿)

¡1f (v)N

ád¡ Á (v + ¾Mp¿)¡ ½v ¡ ½¾Mp¿p

1¡ ½2

!dv

A3 =

Z +up(v)

¡1f (v)N

ád¡ Á (v)¡ ½vp

1¡ ½2

!dv

Assume H is the risk free forward price. We will compare di¤erent collateralsas well as di¤erent levels of collateral demanded and analyze the in‡uence of thevarious parameters on the option value and interpret the di¤erent parameters in thecontext of collaterals and the setting of optimal haircuts. The benchmark is the useof non-stochastic collateral.Figures 1 and 2 illustrate the in‡uence of changes in the various parameters.

11

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

1000

0.01

0.02

0.03

0.04

S M

Increas ing va lueso f M

Increas ing va lueso f 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 ofthe values of S (the underlying asset of the forward) and M (the collateral) at time0. The values for the …xed parameters are: r = 0:1, ¾1 = 0:3, ¾2 = 0:3, ½ = 0:2 andT-t = 0:5.(called CCR)

0.2

0.25

0.3

0.35

0.4

0.2

0.3

0.4

0.50

0.1

0.2

0.2

0.25

0.3

0.35

0.40

0.1

0.2

S M

CCR

= -0.2

= 0

= 0.2

.

Figure 2: Collateralized credit risk and the correlation of collateral andunderlying. This graph shows the collateralized credit risk (called CCR) for di¤erentlevels of ¾S (the volatility of the underlying asset of the forward), ¾M (the volatilityof the collateral) and ½ (the correlation of the underlying asset and collateral). Thevalues 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

exchange option. An intuitive result is that the value of the credit risk increasesif the amount of collateral that is requested decreases. The e¤ect of the value ofthe underlying security to the forward may seem less obvious. One would expectthat 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 theoption would be …xed. In the case of the forward contract the bank will increasethe strike price with a rising level of the underlying, by keeping in our example aconstant proportional value of the forward as collateral (i.e., proportional haircuts).The value of the collateralized credit risk increases if the maximum possible loss fromthe underlying contract increases. This e¤ect dominates if the bank uses the cashand carry arbitrage price in setting the forward price (resp. the strike price of theoption). Figure 2 shows the credit risk for di¤erent levels of ¾S, ¾M and ½: The riskierthe underlying contract, the higher is the value of the possibility to default. This alsoimplies that regulations on margin requirements have to be di¤erent for various riskyassets underlying the original contract. The riskier the asset taken as collateral thehigher 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 importantto stress the fact that the in‡uence of the volatility of the underlying seems to be atleast as important as the in‡uence of the volatility of the collateral assets. This factis highlighted with the following …gure.

M

S

0.2 0.22 0.24 0.26 0.28 0.30.2

0.22

0.24

0.26

0.28

0.3

Figure 3: Collateralized credit risk for various values of volatilities ofthe underlying and the collateral. This graph shows the collateralized creditrisk (called CCR) for di¤erent levels of ¾S (the volatility of the underlying assetof the forward),and ¾M (the volatility of the collateral) and (the correlation of theunderlying 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

The lines and degrees of shading represent constant levels of credit risk. We seethat there is a nearly linear relationship between the two volatilities. If one wouldbe 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 thecharacteristics of the collateral, one is holding implicitely constant the volatility ofthe underlying. It is often the case in practice that the volatility of the collateral assetseems to be considered of a greater importance than the volatility of the underlyingcontract 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 andcan be highly sensitive to the correlation. This underlines the necessity to considercorrelation 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 riskfreecollateral of the same size. Collateralized credit risk is lower (in our example) in thecase of stochastic collateral than in the case of non stochastic collateral with negativecorrelations of about -0.2 and a volatility of 15% of the collateral asset. This meansthat 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-stochasticcollateral

M

CCR���������������������������������������������������������

Area of negative haircuts

Figure 4: Comparison of riskless and risky collateral. This graph showsthe collateralized credit risk (called CCR) for di¤erent collaterals (various values ofcorrelation 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 comparedto riskless collateral in the gray shaded area. For points in this region one obtainsnegative haircuts if riskless collateral is used as a benchmark. The values for the …xedparameters 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

The in‡uence of ¾S and ½ suggests that it is important to take into account equallythe source of risk in the underlying asset as well as its relationship to the risk of thecollateral 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 willnot have only one contract with the bank. On the other hand the collateral will be aportfolio of assets rather than a single asset. How should the bank structure collateraluse when asset portfolios are concerned? Should one asset serve as collateral for someuse or should all the assets of the collateral portfolio serve jointly as collateral for allthe liabilities? In the context of our framework the answer would depend on the valueof the implicit options. We have seen that the collateralized credit risk is in‡uencedby the volatility of both the underlying contracts and the collateral portfolio. Due todiversi…cation e¤ects, the variance of both sides will be reduced when the portfolioview is adapted. The e¤ect on the correlation can not be unambiguously given asign. Therefore the optional structure has to be designed case by case. The presentframework is well suited to perform this analysis.

15

IV. Bonds as collateralIt 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 thisphenomenon, the asset class which is used most often as collateral are governmentbonds. The preceding set up is not well adapted to pricing collateralized credit riskwhen bonds are used as collateral as we considered non stochastic interest rates anda 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 abond of maturity U, where U > T. We want to present in this section the valuationequation for the collateralized credit risk in order to show that the structure of thesolution remains the same. The valuation methodology is based on results by Gemanet alii (1995) for the change of numeraire and Harrison and Pliska (1981,1983) for themartingale pricing.

A. AssumptionsWe 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 processwith two sources of randomness

dStSt

= r ¢ dt+ ½ ¢ ¾s ¢ dWr +p1¡ 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

B. Valuation of the collateralized credit risk option

The gain from defaulting corresponds to

CCR (t) = ¯ (t) IEt

"µ1

¯ (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. Mrepresents the number of zero-coupon bonds. We split the above expression intothree parts and evaluate them separately.

Part1 = ¯ (t) IE

·µ1

¯ (T )F ¢ 1IA

¶¸(16)

Part2 = ¯ (t) IEt

·µ1

¯ (T )ST ¢ 1IA

¶¸(17)

Part3 = ¯ (t) IEt

·µ1

¯ (T )M ¢B (T; U) ¢ 1IA

¶¸(18)

By using the forward measure approach (change of numeraire) and some ordinarychanges of measure we derive the above value as

CCR (t) = ¯ (t) IEt

"µ1

¯ (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) ¢ IN3where 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 collateralizedcredit risk is derived in a classical contingent claims setting.

There is no fundamental di¤erence in the structure of the above result from theone in the previous section. The value of the option would certainly be lower for acomparable set of parameters. However the values obtained can not be compareddirectly as the model in this section works with stochastic interest rates whereasthe previous results assumed constant interest rates. This result allows us to extendall the conclusions of the preceding sections to situations when bonds are used ascollateral. The only major di¤erence from the previous setup lies in the in‡uence ofthe correlation between the interest rate and the underlying of the forward contract

17

on the collateralized credit risk. Figure 5 shows the collateralized credit risk fordi¤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 graphshows the collateralized credit risk under stochastic interest rates for di¤erent valuesof 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

Figure 6. Collateralized credit risk for di¤erent levels of ½. This graph showsthe 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 interestrate in‡uences the value of the collateral and the stochastic discount factor. Thebond values are negatively correlated with the interest rate. Hence an increase of theinterest 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 optionto 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 oppositedirections. The bond e¤ect is stronger for longer maturity bonds relative to thematurity of the underlying contract. If the collateral bond matures shortly afterthe expiration of the underlying contract the e¤ect related to the discount factordominates. Hence the value of CCR can be a decreasing function of the correlation. Ifthe 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 functionof the correlation. For some values in between, the two e¤ects o¤set each otherand we observe the relationship depicted in …gure..This leads to the result that thecollateralized credit risk can be an increasing or decreasing function of the correlationof the interest rate and the underlying of the forward contract, depending on thematurity of the collateral bond.

19

V. On forward and futures contractsUp 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 collateraland by introducing marking-to-market and margin calls. The rest of the paper isdevoted 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 beenmost often seen as merely replacing the multi-period decisions by a series of oneperiod decisions. Within this series all the decisions have been considered as beingidentical. It will become clear that this view is not correct. It is true that one ofthe bene…ts of marking-to-market and margin calls is to replace the option to defaultby the sum of options of shorter maturity. Therefore the bene…t of introducing amarking-to-market procedure stems from the fact that the period during which thecredit risk exposure builds up is shorter. This is however not the only bene…t. Agentsare not myopic. Therefore they consider all the e¤ects of their actions. When takingthe decision whether to default or not the agent will take into account the fact thats/he will loose the contract and all the possibilities to default in the future. Thiswill cause the probability of default to be lower than in the myopic case. Technicallyspeaking 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 hasbeen shown by Cox,Ingersoll and Ross(CIR) (1981) that without the possibility ofdefault under non-stochastic interest-rates the forward price is the same as the futuresprice. We will show that this is only true in a setting without credit risk. Futurescontracts have been invented especially to deal with credit risk. An analysis whichis conducted in a framework without the possibility to default is lacking the mostimportant feature of the contract. This section is organized as follows. We will …rstrepeat the classical argument of CIR. Afterwards we will show that the payo¤ ofthe two strategies contracts is not the same, even in a setting without stochasticinterest-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 clienthas to provide a cash amount M. If the client increases her position she will haveto provide additional collateral. Hence a strategy with one change in the size of thefutures position will involve one marking-to-.market.

The following variables will be used:Si Price of the underlying of the forward contract at time ir continuously compounded interest rate at time iFi Value of the futures contract maturing at T at time iGi Value of the forward contract maturing at T at time i

A. The CIR argument without credit risk

20

The CIR argument is based on the following strategy:

1.At time 0, take a long position of Exp¡T2

¢(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 TFutures price F0 FT=2 FTFutures position e(T=2)r e(T )r 0Gain/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 spotprice of the underlying at time T. If you combine this strategy with an investment ofF0 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 toimplement 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 andshow the structure of the implicit options. We assume, that the amount that needs tobe given as collateral is proportional to the size of the contract. Doubling the numberof forward contracts thus leads to a doubling of the required collateral. In the case ofthe futures contract the client can default at the intermediate marking-to-market at

21

time T/2 and at maturity, while default is only possible at maturity for the forwardcontract.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 )IEh¡¡M ¡ ¡STe(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 astrike price equal to G0e(T )r ¡M:

Now we turn to the futures strategy. We will again assume, that the client has togive an amount of cash equal to M as collateral. At time T/2 he faces the followingdecision: He can exchange the payment that he is supposed to make (which is givenby¡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 secondfutures position has a value of zero. The client will choose to default, if the moneyshe 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 defaultinglater. It is important to note that the strategy requires us to increase the number offutures contracts and therefore also the amount of collateral. The gain from defaultingat time T is ¡¡e(T )r=2M ¡ ¡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)rM¢ (27)

The range of integration for ST=2 is obtained by …nding the value of ST=2 thatmakes inequality 25 a strict equality. The gain from defaulting at time T/2 doesnot include the value of the default at time T. The distinction of the equation thatdetermines the relevant range and the payo¤ of the option is quite important. Thisis the e¤ect of the non-myopic behavior of the agent. When taking the decision todefault he takes into account that he will lose the future cash‡ows. This implies thathe will not default in some range even if the value of the option at time T/2 is in themoney.

22

The value of the option to default at time T/2 is a function of ST=2: This pricehowever 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=2At 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¡STe

(T )r; FT=2e(T )r ¡ e((T )=2)rM¢ (29)

¢1INo def at T=2f¡ST=2

¢dST=2

= e¡r((T )=2)Z 1

Scrit

Put¡STe

(T )r; FT=2e(T )r ¡ e((T )=2)rM¢ f ¡ST=2¢ dST=2

In order to compare the result to the one obtained without default risk we lookat the future value of the strategy.

Futures Position =¡FT=2 ¡ F0

¢e(T )r + PutT=2e

(T )r + (30)¡FT ¡ FT=2

¢e(T )r1INo def at T=2

¡Put2¡FT ; FT=2 ¡ e((T )=2)rM

¢e(T )r + F0e

(T )r

We can do the following simpli…cations: FT is again equal to ST :We know that thefutures 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 istherefore

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=2M¢ 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 ifthe forward price is equal to the futures price. In order to determine explicitly thevalue of the two strategies at time 0, we will take the expectation of the above ‡owsunder the riskneutral probability and calculate its present value.

23

The discounted expected values of the various positions in the futures strategyare 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

¡FTe

(T )r; FT=2e(T )r ¡ e(T )r=2M¢¤ = (37)

e¡r((T )=2)Z 1

uc

Put¡STe

(T )r; FT=2e(T )r ¡ e(T )r=2M¢ 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=2M¢¤+ 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=2M¢+G0 (41)

The value of the two positions will not be the same anymore if forward and futuresprices 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 = 106:318Collateralized credit risk = 1:191V alue forward position = 106:717Collateralized credit risk = 1:589

Table III. Values of the forward and futures positions. The parametersused 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¤erencein managing the credit risk exposures. The credit risk implicit in the forward strategyis higher compared to the credit risk in the futures strategy, despite the amount of

24

collateral set at the beginning being the same. We would assume that the amount ofcollateral required will be lower in a situation with marking-to-market. In order toobtain the same value of the option to default, the required margin for the futurescontract needs to be nearly 9% lower then for the forward contract. This showsclearly 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 decisionof the agent by taking explicitly into account that s/he will consider the loss of thesecond option when s/he decides whether to default or not. From this considerationit follows that it is incorrect to model the situation with marking-to-market as aseries of independent put options. The e¤ect of neglecting the more restricted rangesof the underlying variable is to overestimate the value of the default option undermarking-to-market. The value of the marking-to-market procedure comes from thefact that it splits the longer maturity option into options of shorter maturities andthat it introduces the dependence among the successive decisions to default.

25

VI. Dynamic collateral managementIn 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. Theresulting collateralized credit risk values have a structure which is comparable tonon-standard compound options. In order to generalize the results one could use thesame kind of setup and increase the number of marking-to-market times, but pricingbecomes cumbersome quickly as option numbers increase exponentially. In this sec-tion we want to take a di¤erent viewpoint starting from a situation with continuousmarking-to-market. This will have a deep impact on the structure of the result. Weshow how the time dimension in the continuous framework becomes less pertinentthan the underlying value dimension, an intuition that leads us to a simpler pricingmethodology 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 creditrisk when the bank can monitor the value of the contract continuously. The bank isassumed to be able to issue a margin call whenever it considers it to be necessary. Wewill show that there is no credit risk in this setting, an intuitive result. The settingprovides a starting point for the next section where we will show that collateralizedcredit risk arises from the fact that the bank will not monitor the value of the contractcontinuously 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 canmonitor the value of the forward contract continuously. It will issue a margin callwhen the value of the contract has decreased by the amount given as collateral. Thebank will require a cash amount M as collateral. In order to clarify the setup we willfurther assume that the client can only default at maturity. This means that s/hewill always make the required margin payments until time T. The di¤erence to themodel without marking-to-market is that the collateralized credit risk is comparableto a put option with a strike price that is strongly path-dependent. This implicitoption corresponds to a number of barrier options. The variables used in this sectionare:

St Price of the underlying of the forwardF Forward pricer Interest rateLit ith barrierM Collateral (cash amount)

26

A.II. The model

The value of the credit risky contract can be modelled as a riskless forward contractplus 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 putoptions and one down-and-in put option. The couples of long and short positions ofoptions 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 margincall if the loss is not covered anymore by the collateral. Therefore the out-barrier ofthe …rst option will be set at the level of St, which makes the value of the forwardcontract equal to the negative value of the collateral. Hence the out-barrier of thisoption 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 isnow given by 2M. The next option is a long position in a down-and-in put optionwith a strike price equal to F- 2¢M and the same in-barrier as the out-barrier of thedown-and-out put option. The next option is a short position in a down-and-in putoption 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 barrierequal to the out-barrier of the preceding out option. The next position is a shortposition in a down-and-in put option with a strike equal to F-3¢M and a barriergiven 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 ¢Mlower 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 greateror equal to zero the barrier can never be lower than zero. The couple of options thatwould have a lower barrier (resp. the in barrier of the short position would be lowerthan zero) is just a down-and-in put option. There is no o¤setting short position ina 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

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 presentin 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 atmaturity. The bank will issue a margin call, if the value of the underlying of theforward contract is smaller than the negative value of the margin payment M. Hencethe bank will ask for additional collateral if the potential loss arising from credit riskis bigger than the collateral to cover it. The critical level of St at which a maintenancecall 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 thebarrier is hit, the client can choose if he wants to provide additional collateral andthus keep the contract alive or default. The gain resp. loss for the client dependingon 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 laterif s/he chooses not to default at this point. In the case of no default the clienthas to provide new collateral M. By doing this he keeps the contract alive with thepossibility to default later. He then has a second option which is conditional on thefact that he didn’t default when the …rst barrier has been breached. The level of thesecond barrier is the same as in the previous section. Without dwelling to much onthe 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 valuein the default condition yields

gain=loss = ¡M ¡ ¡¡F ¢ e:r(T¡t) ¡M¢¡ F ¢ e¡r(T¡¿ )¢ (47)

¡future cash flowsgain=loss = ¡future cash flows < 0 (48)

This implies that the agent will never …nd it advantageous to default or will atleast be indi¤erent between defaulting or not if the value of the future cash ‡ows willbe zero.

28

This result holds for all the options forming the barrier option structure. None ofthe option has ever a greater value than zero if the bank can issue a margin call atany 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 contractduring the time between the margin call and the time of delivery of the additionalcollateral is managed by adopting a double trigger strategy. When the …rst barrieris reached the margin call is issued. The client has a prespeci…ed amount of timeto react. However if in the meantime the second (lower) barrier is reached the sameprocedure is applied as in the case of default, meaning the clients position is closedout.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 toinvestigate 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 levelat which to issue a margin call. The same kind of problem arises in the valuation ofordinary barrier options. Broadie et al. (1997) have developed an approximation forthe case of ordinary barrier options. The basic idea is that the value of a discretelymonitored barrier option corresponds to a continuously monitored barrier option witha lower (for a down option) barrier. Therefore the uncertainty about the price at thenext marking-to-market instant is translated into a lower barrier. The methodologyfor pricing barrier options is well explained in Rich (1994). The valuation of curvedboundaries is due to Kunitomo et al. (1992). The actual valuation obtained then isoriginal.

B.II. Valuation using a correction of the barrier for lowermarking-to-market frequency

In this section we calculate the value of the option to default under the abovedescribed rules for collateral management. The amount of collateral taken at theinitiation of the contract is assumed to be at least 50% of the initial maximum possibleloss. If the next barrier is hit, the bank will require again the same amout. Thereforeit is fully hedged against a loss arrising from this contract if the client provides thecollateral. This implies that there is only one barrier option as the barrier optionwith 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

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 isat least 50% of F. This implies that the option the agent receives if he decides tocontinue 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 thefactor

Exp³¡¯¾

p¢t´

(50)

as shown in Broadie et al. (1997). The parameter ¯ is a constant and ¾ is thevolatility of the underlying of the forward contract. The corrected density functionof the barrier is obtained as

f (¿ ) =¡ ln (H=S)¾p¿ 3

n¡x1 ¡ ¾

p¿¢

(51)

with

A = F ¡M (52)

B =1

T¢ Ln

µF ¢ Exp (¡r (T ))¡M

A

¶C = Exp

³¡¯¾

p¢t´

H = (F ¡M) ¢ C¹ =

µr +B ¡ 1

2¾2¶

x1 =£ln (S=H) +

¡¹+ ¾2

¢(¿ )¤=¡¾p¿¢

The level of St at time ¿ is equal to the barrier at this time.

S¿ = Exp³¡¯¾

p¢t´¢ ¡F ¢ e¡r¢(T¡¿ ) ¡M¢ (53)

Replacing the value S¿ in the payo¤ equation by the above expression and integratingover ¿ yields the following

CCR =

Z T¡t0

0

e¡r¢¿Ã ¡

F ¢ e¡r¢(T¡¿ ) ¡M¢ ¢³1¡ Exp

³¡¯¾p¢t

´´ !+ f (¿) d¿ (54)

=

Z T¡t0

0

¡F ¢ e¡r¢T ¡M ¢ e¡r¢¿¢ ³1¡ Exp³¡¯¾p¢t´´ f (¿ ) d¿

30

As long as M is smaller than Fe¡rT we can integrate from 0 to T-t0. If thiscondition would not be satis…ed we would need to calculate a critical t, that wouldguarantee that the expression in the braquets is >0. In a more complicated situationinvolving 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¢T0@ (H=S)m+n ¢N

³ln(H=S)+n¾2(T¡t0)

¾pT¡t0

´+ (H=S)m¡n ¢N

³ln(H=S)¡n¾2(T¡t0)

¾pT¡t0

´ 1A¡M ¢X ¢

0@ (H=S)m ¢N³ln(H=S)+¹(T¡t0)

¾pT¡t0

´+N

³ln(H=S)¡¹(T¡t0)

¾pT¡t0

´ 1Awhere

m =¹

¾2(55)

n =

p¹2 + 2r¾2

¾2

X = 1¡ Exp³¡¯¾

p¢t´

(56)

A margin requirement lower than 50% of the maximal possible loss would giverise to some additional barriers. The valuation of the collateralized credit risk couldproceed along the same lines. The result would take the form of some nested integralswith 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

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 tim es

CCR Increasing frequencyof monitoring

Figure 7: Collateralized credit risk and marking-to-market frequency.This …gure gives the values of collateralized credit risk (called CCR) for varyinglevels of ¾ and numbers of marking-to-markets. The values of the …xed parametersare: 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 tothe fact that with a higher forward price the possible loss increases. Default enablesthe client to avoid that loss, hence its value increases the higher this possible lossis. Obviously the credit risk decreases with a rising amount given as collateral. Thehigher the volatility of the underlying the higher the value of credit risk. The intuitionbehind this is, that with a higher volatility the chance that the underlying pricereaches the barrier between two marking-to-market instants increases. The morefrequent 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 withdynamic marking-to-market and collateral management comes solely from the factthat the barrier can be reached between the marking-to-market times. The relativedistance of the underlying price from the barrier a¤ects the credit risk only over thedensity function. Intuitively this means that the underlying price today will in‡uencethe probability that we reach the barrier. The payo¤ however (conditional on havingreached the barrier) does not depend directly on the underlying price anymore. Thisconclusion 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 mighthave some preferences to use one of the two instruments more extensively then theother. In order to determine how the marking-to-market frequency must be changedto o¤set a lower collateral requirement, we need to check that marking-to-market and

32

collateralization are substitutes. The following …gure shows the collateralized creditrisk 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

In creasing frequen cyof m onito ring

Figure 8: Collateralized credit risk for di¤erent levels of marking-to-market frequency and collateral. This …gure gives the values of collateralizedcredit 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

# of

mon

itori

ng ti

mes

Collateral

Figure 9: Collateralized credit risk for di¤erent levels of marking-to-market frequency and collateral. This …gure gives the values of collateralizedcredit risk (called CCR) for varying levels of collateral and numbers of marking-to-markets. The lighter shading represents higher values of credit risk. Combinations ofcollateral and marking-to-market frequency on the same line represent equal credit

33

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

M RS

Collateral

n = 90

n = 100

C = 60

M RS C = 40

C = 50

# of m onitoring tim es

n = 110

Figure 10: MRS di¤erent levels of marking-to-market frequency andcollateral. 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 oftwo assets, which are substitutes to one another. Therefore the bank can achieve thesame 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 beoptimal 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 increasingfunction of the volatility of the underlying. The value of the option to default willbe higher during turbulent times. One way of achieving a lower value of the optionto default would be to adopt a strategy with non constant time intervals betweenthe di¤erent marking-to-market times. The frequency would obviously have to be anincreasing function of the volatility.

34

VII. RenegotiationIn 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. Wewill analyze the optimal decision of the bank depending on the level of cost of defaultand the bargaining power of the parties involved. We investigate the decisions atsome extreme values in order to emphasize the structure of the decision that theparties face.

A. Renegotiation in the presence of costs

In this section we will analyze a situation where the bank faces some cost in seizingthe collateral. The …rst section will be a two period model. The bank incurs a costproportional to the value of the collateral. Hence the bank looses a fraction of c, whenit takes possession of the collateral. The gap in the value of the option to default tothe 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 thefollowing we will take into consideration that the bank will work with a maintenancemargin 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 putoption is not a standard put option as the relevant range of integration is determinedby the clients optimal decision. It will be worth for the bank to make a payment ofthe amount Y to the client in order to prevent him from defaulting. Y will be in therange 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 fromYmax down to Ymin.The actual size of the surplus to the bank will be determined bythe bargaining power of the two parties. The client would chose to default if

35

F ¡ I ¡ ST > 0 (61)

Solving the inequality number 61 gives us the critical value of ST denoted byScrit1.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 representedby the parameter µ: This parameter can have a value between 0 (no bargaining powerfor the bank) to 1 (all the bargaining power). The value of the risky forward can bedecomposed 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 thefollowing section. The potential for renegotiation hinges on the fact, that the implicitoption of the client is di¤erent from the implicit option of the bank. In some sensethe 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 thesituation where the bank o¤ers a side payment in order to induce the client not todefault at the second period. The bank observes the value of the underlying andin the case of possible default, it o¤ers the client a payment in the next period inorder to induce him not to default. In this section we take into account the di¤erencebetween the initial margin and the maintenance margin. Technically this implies thatthe bank will call for new collateral earlier as it requires the loss not to exceed themaintenance level. The initial margin is denoted by I the maintenance margin byM. The bank will only o¤er the client a payment Y at time T if he would defaultotherwise. This setup encompasses a variety of concessions the bank might considerin order to make the contact more attractive. Amongst these are a lowering of therequired margin, lowering of the forward price or a direct payment. This paymentwill only be received by the client if he does not default at time T. The client will bemarked 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

ST/2>Scrit1

No default

No default

Scrit1>ST/2>Scrit2

No default w ithreincentivization

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 clientwill 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 periodmodel. The gray shaded expressions are the implicit options seen from the banksperspective.

The equilibrium strategies of the bank and the client are the following. All theranges refer to values of S at time T/2. In the range from Scrit1 to1 the bank sticksto the original contract and the client does not default at T/2. He has the implicitput option number 2 at time T. For values of ST=2 from Scrit2 to Scrit1 the bankdoes nothing and the client does not default. The client has the put option number3 at time T. In the region from Scrit3 to Scrit2 the bank o¤ers the side payment Yand the client chooses not to default at T/2. At time T he receives the put optionnumber 3 and the payment Y. For values of ST=2 lower then Scrit3 the bank o¤ers the

37

payment Y, but the client …nds it optimal to default. The clients payo¤ correspondsto the implicit option number 1 at time T/2. There are no more payments at time Tin 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 atthe 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=2Z Scrit3

0

(¡I ¤ (1¡ c)¡ ST=2 + F ¢ e¡r¢T=2

f (ST=2)dST=2 (66)

Put 2 = e¡rT=2Z 1

Scrit1

Put (ST ; F ¡ I ¢ (1¡ c)) f¡ST=2

¢dST=2 (67)

Put 3 = e¡rT=2Z 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=2Z 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¡rTZ 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 valueof the underlying. It will o¤er the client the same kind of deal as in the precedingsection. If the client still wants to default, the bank will bargain again with himover the bene…t generated by saving the cost of seizing the collateral. Note that allthe critical values of ST=2 are the same as in the preceding section. The equilibriumstrategies are the same as in the last section except for the fact that in all the defaultstates the bank will o¤er a contemporaneous payment. The size of the payment

38

will again depend on the bargaining power of the two agents represented by µ. Thepayments in the default state are again weighted sums of options, where the weightis 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 allthe market power to the client. In order to have the pure bargaining model (withoutthe payment of Y) the parameter Y must be set equal to zero. In this case the optionnumber 4 has a value of zero.

D. Results

The value of renegotiation comes from the fact that there are some costs associatedwith obtaining the collateral. If the costs are equal to zero the optimal payment Y isalso equal to zero.

The following …gure shows the values of the implicit options for varying levels of

39

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

C CR

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 levelsof the side-payment Y. The above …gures show the values of the various implicitoptions for di¤erent levels of Y. The values of options number 2 and 3 are not shownas 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 expectdirection. By making the no default state at time T more attractive the bank lowersthe value of these options. Hence it lowers the incentive of the client to default attime T/2. Thus it re-incentivizes the client to keep the contract alive. Technicallythis is re‡ected in a smaller range of integration of option number 1. However thecost of doing this, which is equal to PV(Y) is much larger then the bene…ts. Thus ifthe 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

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 andside payment Y. The left …gure shows the value of CCR for c = 0.4. The optimalvalue 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 aY>0. The optimal Y can be calculated easily for various values of c. For values ofc up to 50% it is increasing with c. For very extreme values of c it decreases againslightly.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 factthat the …rst option is always worth less than the second, the collateralized creditrisk is decreasing in µ. The following graph shows the CCR for various values of c, Yand µ. One sees clearly that for c>0 there exists an optimal Y >0. The in‡uence ofµ can be seen from the three di¤erent layers of CCR values.

41

05

1015

200.1

0.2

0.3

0.4

0.5

33.54

05

1015

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 …xedparameters are: S = 100, F = 102.5, I=20, M=1, ¾ = 0.3, r = 5%, T = 1.

42

VIII. ConclusionsWe 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 servingas collateral. The analysis underlines the necessity to think of collateral valuationjointly with the source of risk. Correlation matters. Collateralized credit risk dependsobviously on all the characteristics of the assets involved. Therefore the decision onmargin requirements and haircuts needs to take all of them into account at the sametime. The models proposed give a technical solution to the problem of determininghaircut 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 besubstituted 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 sourcesof credit risk in this context. The results have practical implications on optimalmarking-to-market strategies. The aim of this work was to put economic intuitionon collateral on a sound basis. Moreover it is a starting point to include collateral inmore 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 asin the dynamic setting. The dynamic management setting itself could be extended tothe case of stochastic collateral. More importantly, the implications of the frameworkon credit portfolios could be further examined.

43

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46

Footnotes

1The solution has been obtained by using the standard Martingale approach. Theresulting equation can be found for example in Zhang (1998).

2Please note that this is not a standard put option. The range of integration isdetermined by the clients’s optimality condition not by the one of the bank.

3Please note that in the range from Scrit1 up to Scrit2, the agent does not defaulteven without side payment.

4Alternatively one could not substract the Y payment in the payo¤ function. Inthis case the payment Y would be conditional on no default at time T. This wouldresult in a double integral expression for Y.

47