An analysis of credit risk with risky collateral a methodology for haircut determination

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An analysis of credit risk with risky collateral a methodology for haircut determination

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  44. 44. References - Altman, E. and D.L. Kao, 1992, ” Rating Drift in High-Yield Bonds ”, Journal of Fixed Income, March 1992. - Anderson, R. W., and S. M. Sundaresan, 1996, ” The Design and Valuation of Debt Contracts ”, Review of Financial Studies, 9, (1996), pp: 37-68. - Benjamin, Daniel K, 1978, ”The Use of Collateral to Enforce Debt Contracts.” Economic Inquiry. Vol. 16 (3). p 333-59. July 1978. - Bester, Helmut, 1994, ”The Role of Collateral in a Model of Debt Renegotia- tion.”, Journal of Money, Credit & Banking. Vol. 26 (1). p 72-86. February 1994. - Broadie, M.; Glasserman, P.; Kou, S., 1997, ”A Contimuity Correction For Dis- crete Barrier Options”; Mathematical Finance, Vol. 7 (4). p 325-49. October 1997. - Cossin, Didier; Pirotte, Hugues, 1997. ”Swap Credit Risk: An Empirical Inves- tigation on Transaction Data.”. Journal of Banking & Finance. Vol. 21 (10). p1351-73. October 1997. - Cossin, D., and H. Pirotte, 1998, ” How Well Do Classical Credit Risk Models Fit Swap Transaction Data ? ”, European Financial Management Journal, vol. 4, no 1, March 1998, p.65-78.. - Du¤ee, G.R., 1995a, ” On Measuring Credit Risks of Derivative Instruments ”, Working paper, Federal Reserve Board, February 1995. - Du¤ee, G.R., 1995b, ” The variation of default risk with Treasury yields ”, Working paper, Federal Reserve Board, January 1995. - Du¢e, Darrel and Ming Huang, 1996, ”Swap Rates and Credit Quality”, Jour- nal of Finance, 51(3), July 1996, 921-949. - Du¢e, Darrell; Singleton, Kenneth J., 1997, ”An Econometric Model of the Term Structure of Interest-Rate Swap Yields.”. Journal of Finance. Vol. 52 (4). P. 1287-1321. September 1997. - France,V.G.; Baer, H.L.; Moser, J.T., 1996,”Opportunity Cost and Prudential- ity: An Analysis of Futures Clearinghouse Behavior”, OFOR working paper 96-01. - Geman, H.; El Karoui, N., Rochet, J.C., 1995, ”Changes of Numeraire, Changes of Probability Measure and Pricing of Options ”, J. Appl. Probab 32, 1995 , 443-458 44
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  47. 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

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