A Bond Consistent Derivative Fair Value
AA Bond Consistent Derivative Fair Value
C. Johan Gunnesson ∗ Alberto Fern´andez Mu˜noz de Morales † BBVA - Risk Methodologies ‡ September 22, 2014
Abstract
In this paper we present a rigorously motivated pricing equation for derivatives,including cash collateralization schemes, which is consistent with quoted market bondprices. Traditionally, there have been differences in how instruments with similarcash flow structures have been priced if their definition falls under that of a financialderivative versus if they correspond to bonds, leading to possibilities such as fundingthrough derivatives transactions. Furthermore, the problem has not been solved withthe recent introduction of Funding Valuation Adjustments in derivatives pricing, andin some cases has even been made worse.In contrast, our proposed equation is not only consistent with fixed income assetsand liabilities, but is also symmetric, implying a well-defined exit price, independentof the entity performing the valuation. Also, we provide some practical proxies, suchas first-order approximations or basing calculations of CVA and DVA on bond curves,rather than Credit Default Swaps. ∗ [email protected] † [email protected] ‡ The opinions of this article are those of the authors and do not reflect in any way the views or businessof their employer. a r X i v : . [ q -f i n . P R ] S e p Introduction and Final Pricing Formula
Ever since Black, Scholes and Merton’s seminal works [1, 2], and until recently, financialderivatives products have been priced without taking into consideration credit- or fundingspreads of either counterparty in the transaction. A frequently stated reason for thisapproach (eg, [3]) was that financial institutions could, in the pre-financial crisis world,borrow funds at the prevailing Libor rate, and any funding considerations could thereforebe taken into account by discounting cash flows accordingly.This did, however, not correctly reflect the counterparty credit risk inherent in anygiven derivative. A digital option which is far in the money behaves similarly to a zero-coupon bond, yet traditionally its cash flows were discounted at Libor, or a similar rate,instead of applying the corresponding bond curve. The inconsistency in the way thesetwo, functionally similar, deals were treated led some market participants to securingfunding through derivatives transactions. Derivatives desks are of course aware of cur-rent bond prices, and it is therefore expected that they should charge the counterpartiesaccordingly, reflecting the value of the liquidity provided to the counterparty. But theaccounting mismatch still provided incentives for closing deals entailing funding, thus ren-dering an upfront profit for the dealer, while at the same time reducing funding costs forthe counterparty. It is true that, from the risk management side, some sophisticated bankswere provisioning expected counterparty credit losses based on market-implied estimates.However, many were basing such provisions on historical data.A step in the direction of reconciling bond- and derivative valuations was taken withthe entry into force of the accounting standard IFRS 13. This standard defines fair valueas an exit price, further stressing the use of market-implied (or at least market-adjusted)valuations, and including a bank’s own non-performance risk, ie, the possibility that thebank may not fulfill all of its obligations. The fair value should not be entity-specific,in the sense that other market participants should arrive at the same valuation. Thestandard interpretation of IFRS 13 is to include in the derivative price a Credit Valua-tion Adjustment (CVA), representing the market value of the deal’s counterparty creditrisk , together with a Debit Valuation Adjustment (DVA), representing the bank’s non-performance risk and based on its credit spread. The fair value obtained in this way issymmetric, in the sense that two counterparties will arrive at the same value if they usethe same calculation methodology and market inputs. For a technical account of thesesubjects see [4, 5], or the textbooks [6, 7, 8].After including CVA and DVA, deals with risky counterparties that once seemed ar-tificially appealing will not produce as large an accounting profit upfront, thus reflectingthe true nature of these transactions. The most frequent way to quantify CVA and DVAis to estimate market-implied default probabilities using Credit Default Swaps (CDS), in- The market value of a given risk can be defined as the cost of buying protection against it in the market,ie, the cost of hedging it. It is true that CVA is often expressed as the expected value of discounted lossesdue to counterparty defaults, but it should be born in mind that these expectations are based on marketinputs, and not on historical (or real-world) losses due to counterparty defaults. plus the CDS spread.Nevertheless, a bond and a CDS on the same reference basically refer to the same type ofrisk, so in a frictionless market, arguments of arbitrage should end up driving such differ-ence, called the bond-CDS basis, to zero. There are a number of reasons that explain whythis gap fails to disappear completely, besides the classical argument of capital constraintspreventing arbitrage opportunities (see [9]). Certain frictions, like the Cheapest-to-deliveroption embedded in CDS contracts ([10, 11]) or haircuts that the arbitrageur encounterswhen financing bond purchases in the repo market ([12]), explain the rationale behind thebasis. For further details on the origin of the basis see, for example, [13]. For simplicity,the many reasons underlying the basis are commonly referred to as liquidity risk.In the past couple of years, an additional step has been taken by some sophisticatedbanks, with the inclusion of a Funding Valuation Adjustment (FVA). The aim of such anadjustment is to take into consideration the funding costs associated to the ”production” ofa derivative’s transaction, ie, the cost of funding the hedging of its risks during the lifetimeof the deal. This introduces the bank’s complete bond spread into the derivatives price.However, rather than solving the discrepancy between bond- and derivatives prices, manyapproaches to FVA actually make it larger. Consider, for example, the in-the-money digitaloption mentioned above, and suppose that the bank has bought the option, analogously toa bond purchase. Any approach to pricing it consistently with bonds should thus containa CVA, reflecting the counterparty’s credit risk, together with an additional term governedby the counterparty’s bond-CDS basis to reflect the bond’s liquidity premium. In contrast,one approach to FVA adds the bank’s complete funding cost (proportional to its fundingspread) to the calculated CVA. The counterparty’s bond-CDS basis is therefore not partof the price and furthermore, as we will explain later, its CVA contributes implicitly tothe bank’s funding spread, and is therefore double counted. Needless to say, the obtainedvaluation will not be symmetric, and the counterparty will calculate a different derivativesprice. In Section 2 we will explain the limitations of current FVA frameworks in moredetail.In this paper we provide a solution in the form of a rigorously motivated derivativepricing equation that is completely consistent with market bond prices. We will be con-cerned here with uncollateralized derivatives, since the direct exposure that they generateto the counterparty is analogous to bond exposure, although we will briefly comment onthe partially collateralized case in Section 4. At a given time t , the pricing equation takes It is, of course, doubtful that any truly risk-free interest rate can be said to exist, but for practical (andtheoretical, as will be discussed below) purposes an Overnight Indexed Swap (OIS) rate is often employed(see [3]). It has become a standard to pay such rates for held collateral, and they are therefore oftenreferred to as collateral rates. See, for example, the aforementioned textbooks or [14, 15, 16]. V t = V ct − CV A t + DV A t + BF V A t , (1)where V t is the fair value at t , and its components are • V ct : the fair value that would be obtained at t if the derivative were perfectly col-lateralized, meaning that collateral is posted in a continuous fashion by the bank orcounterparty in response to changes in the derivative valuation. In [17] it was shownthat in such idealized cases the derivative value is simply obtained by discountingall future cash flows using the rate paid on the collateral accounts. • CV A t : The CVA calculated to adjust for the counterparty’s credit risk. • DV A t : The DVA reflecting own credit risk, and which equals the CVA that wouldbe calculated by the counterparty. • BF V A t : The new term in our approach, which we call a Bilateral FundingValuation Adjustment , and that incorporates the effects of both the bank’s andcounterparty’s bond-CDS bases. In turn, we separate it as
BF V A t = − CF V A t + DF V A t , (2)where CF V A t stands for Credit Funding Valuation Adjustment , and is gov-erned by the counterparty’s bond-CDS basis, while
DF V A t means Debit FundingValuation Adjustment , and depends on the bank’s bond-CDS basis. They canbe thought of as correcting CVA and DVA, respectively, extending them to a fullfunding adjustment. In particular, positive exposure to the counterparty, given by V + t ≡ max( V t , , (3)and which arises when the derivative can be considered an asset, generates CF V A t ,while negative exposure (the derivative is a liability) V − t ≡ max( − V t ,
0) (4)gives rise to a
DF V A t . In more detail, CF V A t = E (cid:20)(cid:90) Tt alive ( s ) D ( t, s ) γ Cs V + s ds (cid:21) , (5)where E [ · ] stands for ”Expected Value ”, t is the current valuation time, T is ma-turity of the deal, s is an integration variable representing all intermediate timesbetween the present ( t ) and maturity ( T ), 1 alive ( s ) means that the deal should bealive at s (it is a variable that is equal to 1 if the deal is alive at s , and zero otherwise), D ( t, s ) is the discount factor between t and s , γ Cs is the counterparty’s bond-CDSbasis at s , and V + s is the aforementioned positive exposure. In the same way, DF V A t = E (cid:20)(cid:90) Tt alive ( s ) D ( t, s ) γ Bs V − s ds (cid:21) , (6)where γ Bs is the bank’s bond-CDS basis. Under the risk-neutral, or market-implied, measure.
4t should be noted that the pricing equation (1) is symmetric, and as a consequenceprice agreement between two counterparties using it will always be possible. The proposalis therefore especially suitable in an accounting framework, since price agreement impliesthat it should always be possible to exit the deal at that price .In the case that the derivative is partially collateralized, the resulting expressionsremain relatively simple, assuming cash-collateralization. If we name C ( t ) the amount ofcollateral held by the bank at time t (which is defined as negative if the bank has postednet collateral), we get CF V A t = E (cid:20)(cid:90) Tt alive ( s ) D ( t, s ) γ Cs ( V s − C ( s )) + ds (cid:21) , (7)and DF V A t = E (cid:20)(cid:90) Tt alive ( s ) D ( t, s ) γ Bs ( V s − C ( s )) − ds (cid:21) . (8)In the next section we will bring into focus the limitations of current approaches toderivative pricing and FVA. We will then detail and motivate our proposal for a derivativefair value in Section 3 and discuss briefly the (partially) collateralized case in Section4. We finish with conclusions and a final discussion in Section 5. In the appendices,a mathematically rigorous derivation of the pricing formula is provided in Appendix A,while an analytical expression for bond prices, allowing a calibration to market prices, canbe found in C. Before continuing in Section 3 with our proposal, we will now highlight some of the currentapproaches to derivative pricing and FVA, and in particular focus on the discrepancybetween bond- and derivative valuations. This section is independent of the rest of thepaper, but should shed some light on the validity of the pricing equation (1).The alternative approaches that we will discuss are
F V A = 0, own bond-CDS basedFVA, transfer cost-based FVA, CVA and full FVA with no DVA, asymmetric FVA withonly a funding cost, and CVA calculated using bond curves. In many cases, the inclusionof FVA will make the discrepancy with bond prices worse, entailing for example doublecounting of Credit- and Funding Valuation Adjustments. Recently, a similar result has been obtained in [18], starting from rather different assumptions. Inthis paper, the author posits that counterparties, rather than remaining in an uncollateralized setup, haveeconomic incentives to willingly enter into collateral agreements in which the rate paid on received collateralis their funding rate. The different rate paid on collateral then leads to funding adjustments similar to our(5) and (6). .1 Should FVA be zero? In [19] the controversial claim was made that derivative pricing should include CVA andDVA, but not FVA. In [20] and [21] the same authors admit that it is only defensible tobase an FVA on the bank’s bond-CDS basis, but they also argue that it does not seemcompatible with IFRS 13 to include such an FVA in the accounting fair-value, since itwould consist of an entity-specific valuation. Arguably, different counterparties will arriveat different prices.Others have recognized that it seems to make economic sense to calculate even asubstantial FVA, reflecting the costs of funding the derivative’s hedges, and take it intoconsideration in decision-making processes, but that it should not be used for accountingpurposes [22]. The reason given is that the market price is not fixed by a given entity’sfunding costs, in the same way that the market price of a given commodity is not adirect function of a given producer’s extraction costs. Yet another interpretation of thisobservation is that a ”market” FVA should be calculated, based on the average fundingcosts of different market actors.However, it is not correct that IFRS 13 implies that valuations cannot be based onparameters depending on the own entity, as the inclusion of DVA exemplifies. Instead, theparameters should be such that agents other than the entity will arrive at the same values.In other words, they should be based on objective, market-based information. Consistencywith bond prices requires the introduction of an FVA, but as long as it is calculated usingmarket information, there is no problem in including it in an accounting fair value.In [23, 24] a zero FVA is also obtained under the assumption that the bank can freelytrade in own bonds of different seniority. In fact, the methodology that we use in thispaper, that of replication, follows the same principles as that of these papers. The resultof this pricing methodology depends on the risk factors on which the product value isassumed to depend, and if a dependence on the full funding spread is required, as it mustif consistency with bond prices is required, the bond-CDS basis will emerge from thereplication.
In the previous case we mentioned that [20] and [21] allowed for an FVA depending onthe bank’s bond-CDS basis, although the authors expressed doubts whether such an FVAis compatible with accounting standards. Such a dependence was first proposed in [14],and in [25] we arrived at a similar result by different means, although we do not believethat the accounting mismatch is irreconcilable. The only problem with these approaches,in our view, is that they do not incorporate the counterparty’s funding spread, and cantherefore not reproduce the market prices for the counterparty’s bonds.6 .3 Transfer cost-based FVA
From the perspective of managing a derivative desk it seems appealing to base FVA onthe bank’s internal funds transfer rates, since they determine the actual financing costsexperienced by the desk. In Proposition 3 of the report [26], it is stated that it is desirableto have a close alignment between FVA calculations and funds transfer pricing rules,in order to avoid arbitrage opportunities for external liquidity takers. A client shouldnot be able to benefit from entering an in-the-money derivative as compared to taking atraditional loan, it is argued.From the accounting perspective it makes less sense to base FVA on funds transferrates. Clearly, doing so would constitute an entity-specific calculation, and not be directlymarket based. And the problem goes deeper as banks that can freely alter their transferrates could directly manipulate their obtained accounting FVA. It should instead be thetransfer costs that should adapt to the external conditions, with FVA calculated indepen-dently. Finally, if the goal is to avoid arbitrage opportunities for external liquidity takersit becomes even more relevant to impose consistency between funding through derivativesand bonds.
As summarized in Proposition 8 of the report [26], DVA is difficult to monetize, impossibleto completely hedge, and from the point of view of the derivatives desk leads to a negativecarry ie, a loss of value with the passage of time. This has led to proposals excludingDVA from pricing, and instead including a full FVA, that is, an FVA based on the bank’sfull funding spread. In Section 14.3.6 of [6], a first-order approach in this spirit is pre-sented, denoted as CVA+FCA+FBA. Furthermore, in [27] such a price was obtained viareplication, including higher order corrections, by not considering the bank’s default as arisk-factor.In the case of liabilities this approach will yield a numerically similar result as ourproposal (1), since an FVA based on the complete funding spread is at first order equalto DVA plus an FVA based on the bond-CDS basis, but there will be large differences forassets. The reason for this is that in the CVA + full FVA setup there is an importantdouble counting of CVA. Following the example given in Proposition 6 of [26], if the bankwere to deal with a single counterparty, the bank’s riskiness, and therefore its fundingspread, should be entirely determined by the riskiness of the counterparty. Implicit inthe bank’s funding spread is therefore the counterparty’s CVA and adding the two willproduce a full double counting. Proponents of this approach argue that both the CVAhedge and the liquidity provided to the counterparty must be financed independently.However, if the bank were to hedge its exposure to the counterparty, what is left is apurely riskless instrument, and the bank’s funding spread should adjust accordingly, sincewhat remains will be a riskless balance sheet. In other words, hedging CVA does not have7o be considered a cost, but can instead be thought of as an investment in risk reduction,accompanied by the associated funding benefit.It is true that funding costs will not adapt instantly to variations in risk profiles.However, as discussed in [25], large portions of the bank’s derivatives portfolio are con-tinuously renewing, with similar deals replacing maturing ones. For such a steady statebalance sheet, the funding spread should have adapted to the riskiness of the portfolio.In the case of an expanding or contracting business the situation is more complex, butthere will in general still be a significant overlap between the portfolio compositions in,for example, consecutive months, implying that funding costs should reflect the assumedrisks.The use of CVA + full FVA means that credit given to the counterparty throughderivatives transactions will be charged a rate greater than the market rate. Of course,an institution should always try to maximize profits, and if a counterparty is preparedto accept such a price the bank should agree, but we would argue that such a scenarioshould lead to an upfront accounting gain. The risk of using CVA + full FVA is insteadthat deals that are actually profitable may not appear as such, leading to lost businessopportunities. Furthermore, the negative carry of DVA is actually not real since DVAcan be identified with a funding benefit. If this benefit is compensated for internally, thenegative carry will go away. Yet another possibility, presented in Section 14.3.6 of [6], is to calculate a CVA + DVA+ FCA, where FCA (the funding cost) is the part of a full FVA that is based on positiveexposures. This has the advantage, from the management point of view, that FCA, beingbased on the bank’s complete funding spread, constitutes a natural hedge for the DVAterm. However, numerically the approach is similar to CVA + full FVA, and therefore hasthe same problems. There will be a mismatch between credit provided to the counterpartythrough derivatives versus by other means.
In this section we have taken a critical position regarding common FVA frameworks, butlet us end it on a positive note. Some financial entities have reported calculating a DVAterm based upon the default probabilities extracted from their own issued bonds. Bydoing so, they argue, they do not need to calculate any FVA term, since such a DVA alsoencompasses their funding capabilities. Ordinarily, CVA and DVA depend on the creditcomponent of the counterparty’s and bank’s respective funding spreads, and the questionis therefore whether one can incorporate funding considerations by extending them to the This is at least the case for a steady state balance sheet.
In this section we will motivate our proposal. For a more rigorous derivation see AppendixA. We will start with the so-called perfectly collateralized case in Section 3.1, for whichthere is neither CVA, DVA nor FVA, and we will then explain how the general case arisesby modeling the financing of the absence of collateral in terms of bonds.
The point of departure for the general case will be a perfectly collateralized trade, atheoretical construct in which there is a continuous exchange of cash collateral so thatat all times neither counterparty has any net exposure to the other. This implies thatthere will be no losses due to defaults, and therefore no CVA or DVA. What is perhapsless obvious is that there will not be any funding adjustments either. The reason is thatit is assumed that the trade can be hedged in the interbank market using (different)collateralized trades. The initial cost of setting up hedges and residual balances in cashaccounts should coincide with the premium paid by the counterparty. Furthermore, anyvariation of the MtM of the original deal implying, for example, that the bank must postadditional collateral to the counterparty will be offset by collateral posted to the bank bythe counterparties of the hedges.As shown in [17], the value of such a perfectly collateralized derivative is obtained bydiscounting all future cash flows by the rate paid on the held collateral, which in typicalcollateral agreements is taken to be an OIS rate c based on overnight interbank lending.To understand why this discount rate should be used, consider a deal in which the bankreceives a single fixed cash flow N . An instant before maturity, the deal value will be equalto N , since any discount can be ignored, and the cash held in collateral accounts will alsoequal N . At maturity, the deal is closed, but there is no net exchange of cash betweenthe bank and the counterparty since the final cash flow N will cancel the return of thecollateral. Now, at previous times, the bank must pay the OIS rate c on the collateralaccount balance, but since there will be no net cash exchange at maturity it will onlyagree to paying this rate if at the same time the collateral balance increases by the same9mount, compensating this outflow (so that there are no net cash interchanges at previoustimes either).The end result is that the amount of collateral, and therefore the value of the deal,grows at the rate c . In other words, previous values of the deal are obtained by discountingusing precisely c . If we denote the deal value at time t by V ct , maturity by T , and takinginto account the continuous accruing of the collateral account, we therefore have V ct = exp − ( T − t ) · c N .
Allowing for the possibility that c may vary in time (although in a way known beforehand)as c t , this becomes V ct = exp − (cid:82) Tt c s ds N .
Of course, in reality the values of c s at times later than t are not known at t , so weneed to take expectations over the different paths that the collateral rate can take : V ct = E (cid:104) exp − (cid:82) Tt c s ds (cid:105) N .
Following, for example, [28], we introduce the stochastic discount factor D ( t, t (cid:48) ) ≡ exp (cid:16) − (cid:90) t (cid:48) t c s ds (cid:17) . (9)In general, since we are dealing with derivatives, we should expect the final cash flowto be an unknown payoff, following some statistical distribution, possibly correlated withthe discount factor. This payoff, occurring at maturity, can be written simply as V cT , andwe therefore have V ct = E (cid:2) D ( t, t (cid:48) ) V cT (cid:3) . (10) Let us now turn to the case of main interest, in which the derivative is uncollateralized,leaving a discussion of the partially collateralized case for Section 4. Let t = t, t , . . . , t n = T be times between which the collateral rate c is fixed, and which definethe periods ∆ t i ≡ t i +1 − t i upon which the collateral account interest rate payments are based (typicallydaily periods). Ignoring subtleties regarding day-count conventions, the collateral account will then growbetween t i and t i +1 by a factor (1 + c t i · ∆ t i ). Using that ∆ t i is small for frequent collateral margining, thisfactor can be substituted for exp ( c t i · ∆ t i ), since the former expression is the first order approximationof the latter. Multiplying the factors stemming from all periods gives V cT = (cid:81) n − i exp ( c t i · ∆ t i ) V ct =exp (cid:0)(cid:80) n − i c t i · ∆ t i (cid:1) V ct . Substituting the sum inside the exponential for an integral (exact in the limitwhen ∆ t i →
0) gives the result. The exact measure needed to take this expectation is obtained from a more careful analysis, see [17],or Appendix A.
10f the deal is not collateralized there will be two consequences: Firstly, there will belosses upon defaults of either the bank or counterparty, depending on the sign of the valueof the trade at that moment. Secondly, the lack of posted collateral has to be financedto compensate for the collateral movements of any market hedges, and also the fundingimplicit in the trades net value. Funding the trades NPV, applying the bank’s fundingcurve, transfer costs, etc, is what has led to previous proposals on FVA. Our main pointis that we recognize that in the case of positive exposure to the counterparty, we arefinancing the counterparty for the amount that would have been transferred as collateralin the perfectly collateralized case. The corresponding funding adjustment thus dependson the accounting value of financing the counterparty, and not the cost of obtaining bankfunds.Any derivatives trade can be thought of as composed of its perfectly collateralizedcounterpart together with a financing of any deficit or excess of collateral. If the exposure ispositive V + t >
0, the counterparty would have transferred posted collateral correspondingto this exposure in the perfectly collateralized case. In allowing it not to do so, the bankis implicitly lending the counterparty funds, or equivalently, buying the counterparty’sbonds, whose value we will write as B Ci ( t ), with the label ’ i ’ specifying the individualbonds. Let ω Ci be the quantity held of bond ’ i ’. The total value of these bonds will beprecisely V + t , so we have the Credit Constraint (cid:88) i ω Ci B Ci ( t ) = V + t , (11)which holds at all times. The bank can hedge this bond component of the derivative bytaking a short position in the counterparty’s bonds.But why is more than one bond needed in (11)? Could we not simply include asingle bond with the same maturity as the derivative? No, unfortunately, the uncertainand possibly bilateral nature of derivative cash flows complicates matters. Consider, forinstance, an interest rate swap with a value of 0 at inception, implying a vanishing positiveexposure. Does this, together with (11), mean that the bond-component of the derivativeis zero as well? Clearly, this cannot be the case, because there is a positive probabilitythat the derivative value in the future will move in favor of the bank, generating a non-zero positive exposure and thereby producing a position in the counterparty’s bonds.The possibility of this forward position must be included in the initial valuation, and ifthe counterparty’s bond yield rises, the fair value, from the point of view of the bank,should decrease . Consider, for example, the case of two bonds, and V + t = 0. The creditconstraint then becomes ω C B C ( t ) + ω C B C ( t ) = 0 . If the duration of, for instance, the second bond is greater than the first and ω C > ω C < Since no additional cash flows are interchanged as a consequence of the change in bond yield, the bankwill be lending funds to the counterparty at a rate cheaper than the new market rate, entailing a loss invalue.
11f the yield curve. The exact amount of bonds would then be determined by requiring amatching of the bond and derivative sensitivities to the yield curve. Due to the constraint,in order to capture n points of the yield curve, at least n + 1 bonds must be included.However, in order to correctly represent the dependence of the derivative on the coun-terparty’s credit risk, besides matching derivative- and bond curve sensitivities, the jump-to-default components ie, the change in value as a consequence of the counterparty’s defaultevent, must be captured as well. One way to achieve this would be to include additionalbonds in (11) (in, for example [24] the jump-to-default component was replicated usingbonds of different seniority), and simultaneously choose the quantities ω Ci so that bothbond curve and jump-to-default dependencies take the correct values. Another possibilityis to include Credit Default Swaps in the mix, as we do in our derivation in AppendixA, greatly simplifying the calculation. Assuming that such CDS can be entered withouta relevant upfront payment, they can be added without changing the amount of creditgiven to the counterparty, and therefore without affecting (11). The choice of whether toreplicate the jump-to-default component using additional bonds or CDS should not alterthe obtained result.We repeat the steps outlined above for negative exposure, V − t >
0. In this case, thebank has transferred less collateral to the counterparty, as compared with the perfectlycollateralized case. In analogy with the positive exposure case, the counterparty is financ-ing the bank, and it is therefore as if the bank had issued bonds B Bi ( t ), of total value − V − t (the value is negative for the bank since they are a liability), which were bought bythe counterparty. We therefore have the Debit Constraint (cid:88) i ω Bi B Bi ( t ) = − V − t . (12)This component can of course be hedged by trading in the entity’s bonds.The issue of the bank’s jump-to-default is more contentious. The reason is that itis difficult for a bank to replicate its own jump-to-default using real instruments , andhedging it can prove to be pernicious (see for example [29]). As a case in point, considerthat no bank can sell its own CDS (who would buy protection from a seller on the seller’sown default?). However, in [25] it was explained that, even in a hedging context, it makessense to model the existing unhedged jump-to-default component using a fictitious positionon a CDS written on the bank, which would be prepared to pay or receive precisely themarket spread for such a CDS, even though it could not enter into it in practice. The reasonis that cash-flows occurring at default will affect the bank’s recovery, which should alter itsfunding costs. The additional funding costs or benefits obtained in this way compensatefor the CDS premium. Furthermore, from an accounting perspective it is important tofully include own credit spreads, since they should be taken into account in any exit price. For example, in [24] the jump-to-default component is replicated using two bonds of different seniority.However, the authors recognize that there might be a mismatch between the post-default value of the bondportfolio and the derivative due to a lack of control over bond recoveries. At least in the case of a steady state balance sheet, in which new deals replace other deals on acontinuous basis. . (13)The way to proceed is to calculate the amount of bonds and CDS, at current and futuretimes, implied by this decomposition, and derive a pricing equation, as is done in theappendix. Here, we will give a heuristic motivation of the result.Firstly, any cash flows occurring at default of either party are precisely those underly-ing calculations of CVA and DVA. By calculating CVA and DVA we are therefore takinginto consideration the deal’s credit risk. CVA is contingent, meaning that it is based oncounterparty defaults taking place prior to any default by the bank itself. The same hap-pens with DVA. The rationale behind this contingency is that the derivative is liquidatedupon the first default, by either counterparty, and so its value will not be affected bydefaults beyond the first. The expressions for CVA and DVA are therefore CV A t = E (cid:104) { τ C <τ B } D ( t, τ C ) (cid:0) − R C ( τ C ) (cid:1) (cid:0) V cτ C (cid:1) + (cid:105) (14)and DV A t = E (cid:104) { τ B <τ C } D ( t, τ B ) (cid:0) − R B ( τ B ) (cid:1) (cid:0) V cτ B (cid:1) − (cid:105) , (15)where τ C and τ B are times at which the defaults occur for the counterparty and bank,respectively, R C ( t ) and R B ( t ) are their recovery fractions, V ct is the value of the perfectlycollateralized equivalent of the derivative , D ( t, · ) is the stochastic discount factor (9),and 1 ··· is an indicator function requiring the stated condition to be satisfied. Basically,we are taking the expected value of all future losses due to defaults, given by the perfectlycollateralized equivalent exposures at the time of default discounted to the present time.The indicators imply that only losses corresponding to the first default will contribute,reflecting the contingency. Usually, we will assume constant recoveries, and so R C ( t ) ≡ R C and R B ( t ) ≡ R B .These terms include the credit risk of the bonds satisfying (11) and (12), so the onlyadditional contribution to the derivative value should depend on the fraction of the bondyield not explained by credit risk. If we decompose a bond yield f Xt , where X is eitherequal to C , for the counterparty, or B for the bank, as f Xt = c t + π Xt + γ Xt , We adopt the riskless close-out convention, leading to perfectly collateralized exposures being used inthe expressions for CVA and DVA. An important point, though, is that the value of future defaults may beincorporated into the close-out claim, put forth upon default, if a so-called risky close-out is performed (see[30, 31]). However, for unsophisticated counterparties, such as the ones that tend to not have collateralagreements with the bank, we would expect a riskless closeout, in which the claim is based on the risklessvalue, since such counterparties are not likely to have the ability to calculate CVA and DVA. On the otherhand, for sophisticated counterparties it is frequent to have a collateral agreement with daily margin, basedon the perfectly collateralized value, so it is natural that a close-out should also be based upon this value. c t is the OIS rate (often considered as a proxy for a ”risk-free” rate), π Xt is theCDS spread, encoding credit risk, and γ Xt the bond-CDS basis (by definition), we arethus saying that the remaining terms contributing to the derivative valuation should begoverned by the bond-CDS basis.In effect, on the credit side we are charging the counterparty for the additional liquiditypremium implicit in the funds provided to the counterparty due to the lack of (expected)posted collateral. We will call this term a Credit Funding Valuation Adjustment (CFVA), and not a Funding Cost Adjustment, a common term in the FVA literature,since it should not be confused with the bank’s own funding costs. The properties thatwe expect that this CFVA should have are: • Contributions to the adjustment from any future time s ≥ t should be proportionalto γ Cs · V + s , since by (11) the net amount of bonds held are V + s , and the liquiditypremium charged for these bonds is γ Cs . • Only future scenarios in which the deal is alive should contribute. If either counter-party has defaulted previously, the deal will have been liquidated, and no fundingadjustment should be made. This condition can be incorporated by the indicators1 { τ C >s } { τ B >s } . • Future contributions to the adjustment should be discounted to the present timeusing the stochastic discount factor D ( t, s ). • We should sum over the contributions to
CF V A t from all future times s , implyingan integration over the lifetime of the deal. • Since the previous discussed factors are stochastic, in general, we should take theexpectation of their product (in the appropriate measure) in order to obtain the finaladjustment.In total this gives
CF V A t = E (cid:20)(cid:90) Tt { τ C >s } { τ B >s } D ( t, s ) γ Cs V + s ds (cid:21) , (16)and the fair value will be reduced by this amount.In the same way, we obtain for the debit side the Debit Funding Valuation Ad-justment (DFVA), given by
DF V A t = E (cid:20)(cid:90) Tt { τ C >s } { τ B >s } D ( t, s ) γ Bs V − s ds (cid:21) , (17)which enters with a positive sign in the fair value.A nice property of (16) and (17) is that they are symmetric: the CFVA calculated bythe bank coincides with the DFVA calculated by the counterparty, and vice versa. This14rompts us to define a Bilateral Funding Valuation Adjustment
BFVA, as
BF V A t = − CF V A t + DF V A t , (18)analogously to the Bilateral Credit Valuation Adjustment (introduced in [5]).Adding up the different contributions, our complete proposal for the fair value V t ofthe derivatives transaction is thus V t = V ct − CV A t + DV A t + BF V A t , (19)with BF V A t defined by equations (16)-(18). It should be noted that (19) is recursive,since V t will depend, through the funding adjustment BF V A t , on V + s and V − s at latertimes. This makes the equation difficult and costly to implement for the entire derivativesportfolio, but as we show in Appendix B, and have discussed in Section 2.6, reasonableand practical approximation are obtained by either omitting the recursive behaviour ofthe F V A terms, or by dropping the
BF V A t term altogether and instead incorporatingfunding considerations into CV A t and DV A t by basing them on bond-implied defaultprobabilities.We have constructed the pricing equation (19) taking into consideration bond-CDSbases in order to be consistent with both the counterparty’s and the bank’s bonds. Butcan we actually reproduce observed bond prices in the current framework? Yes, butthere are some subtleties when applying so-called recovery conventions. In Appendix Cwe attempt to shed light on this issue, deriving closed expressions for bond prices underdifferent conventions. The resulting formulae can be used to calibrate the bond-CDS basesto bond market prices. Even though we will not give a rigorous motivation here, we will dedicate a few linesto explain how the pricing equation can be extended to include partially collateralizedderivatives, meaning derivatives subject to a Credit Support Annex (CSA), but cannot beconsidered perfectly collateralized (for which the value is simply V ct ). Depending on thedegree of accuracy required, this may encompass practically all deals, since in reality it isnot possible to obtain the ideal of perfect collateralization. We will limit ourselves to cashcollateral, noting that non-standard collateral, such as bonds denominated in domesticor foreign currencies, commodities, equities, etc, can be incorporated at the expense ofconsiderable complexity in the pricing (affecting the relevant pricing measures, etc). Foran example of the treatment of collateralization in a funding framework, see [15, 16].Two ways in which counterparty exposure can arise when cash collateralization is notperfect are 15. The existence of thresholds, minimum-transfer-amounts, independent amounts andnon-continuous remargining will imply that, at any given moment t , the amount ofheld collateral C ( t ) is not exactly equal to the derivatives value V t .2. At default of either party, there is typically a period of uncertainty ∆, a so-calledcure period (or margin period of risk), in which no further collateral is posted bythe defaulting party, but it is not yet legally clear that the default has actuallytaken place and/or the surviving party has not yet closed out all of the markethedges corresponding to the derivatives trade in question. This incremental lackof collateral at default implies an additional exposure to the defaulting party, andthereby a potentially greater loss.In order to incorporate partial collateralization the first thing that must be done isto model the collateral function C ( t ). In general, it can be a rather complex functiondependent on the path taken by the derivatives price up to t . In many circumstances itis sufficient, however, to use an approximate form. For example, if the collateralization isconsidered perfect, with the exception of the existence of a bilateral threshold H , we have C ( t ) = sign( V ct ) max ( | V ct | − H, , where the amount of collateral depends on the perfectly collateralized value, as is commonin collateral agreements. Another special case, which we have already seen, is of coursethe uncollateralized case in which C ( t ) = 0. Further examples can be obtained in [32] or[33].Introducing collateral in the framework is then straightforward. To begin with, it iswell-known how to incorporate held collateral into CVA and DVA calculations. For CVA,we have: CV A t = E (cid:104) { τ C <τ B } D ( t, τ C ) (cid:0) − R C ( τ C ) (cid:1) (cid:0) V cτ C − C ( τ C ) (cid:1) + (cid:105) (20)and analogously for DVA. This formula is based on a reduced exposure to the counterpartyat the time of default by C ( τ ).Now, any amount of held collateral C ( t ) obviously reduces the collateral gap that mustbe financed, so the Credit and Debit Constraints become (cid:88) i ω Ci B Ci ( t ) = ( V t − C ( t )) + (21)and (cid:88) i ω Bi B Bi ( t ) = − ( V t − C ( t )) − . (22)It is therefore not surprising that (and as can be shown performing a more rigorous anal-ysis), the modified Credit- and Debit Funding Valuation Adjustments will be CF V A t = E (cid:20)(cid:90) Tt { τ C >s } { τ B >s } D ( t, s ) γ Cs ( V t − C ( t )) + ds (cid:21) (23)16nd DF V A t = E (cid:20)(cid:90) Tt { τ C >s } { τ B >s } D ( t, s ) γ Bs ( V t − C ( t )) − ds (cid:21) , (24)In these equations it is implicitly contained that the collateral itself can generate expo-sure to the counterparty. For example, if C ( t ) is negative, due to some large IndependentAmount of collateral, while V t ≈
0, the positive exposure will then be( V t − C ( t )) + ≈ | C ( t ) | . The second effect that must be taken into account is that of the cure period ∆. Ifthe period is sufficiently short, in the sense that the potential variation in the derivativesvalue over the period is small compared to the exposures ( V t − C ( t )) ± , this effect can beignored. And in general, we would typically ignore it in the funding terms, since they aregiven as integrals over time, and the additional funding required during the cure periodis therefore relatively small. However, for a sufficiently well-collateralized deal it maybecome the main contribution to the value of CVA and DVA. In order to incorporate it,we set CV A t = E (cid:104) { τ C <τ B } D ( t, τ C ) (cid:0) − R C ( τ C ) (cid:1) (cid:0) V cτ C +∆ − C ( τ C ) (cid:1) + (cid:105) (25)with the understanding that if τ C + ∆ > T we redefine V cτ C +∆ as V cT . The rationale issimply that we take the default time to be the moment in which collateral is no longerposted, while we are still exposed to the counterparty during the cure period.In practice, it can be difficult to solve the partially collateralized pricing equationexactly. Consider for example taking into consideration Minimum Transfer Amounts inthe collateral function C ( t ). This will introduce a path-dependent derivatives price, whichis difficult to combine with the recursive nature of the pricing equation (recall that equation(19) is recursive since V t will depend on future values V s , with s ≥ t through the fundingterms), which is simplest to treat if the equation can be solved from maturity backwardsto the present.For the uncollateralized case we mentioned the use of bond curves in the calculationof CVA and DVA as an approximation to the full CFVA and DFVA calculations. Somecare must be taken when extending this approach to the partially collateralized case,however. The issue is that the cure period ∆ enters into the CVA term but not CFVA.Calculating a bond-based CVA would thus imply financing an exposure that only appearsat default during the entire derivative lifetime. The solution is to split the exposureunderlying CVA into two pieces, one which is an exposure during life, and one which isan incremental Exposure At Default (which would contain effects stemming from the cureperiod), defined as the difference between the exposure entering (25) and that of (23).We can then calculate two CVA terms, one based on the exposure during life and the fullbond curve, and one based on the incremental EAD and credit spreads. The symmetrictreatment should of course be given to DVA. This ignores the discount factor between τ C and τ C +∆, which for practical purposes can be neglected. Conclusions and Discussion
The aim of this paper has been to define a fair value for a financial derivatives transaction,entered into by a bank and its counterparty, completely consistent with all available marketinformation, including bond prices. After summarizing the steps taken, we will discusshow the approach is • Compatible with the notion of exit price, underlying IFRS 13. • Consistent with market bond prices. • Not entity specific. • Not based on future costs. We will explain why cost-based derivative valuations arenot desirable from an accounting point of view. • Free of double-counting between CVA and funding costs.In summary, taking the well-defined case of a perfectly collateralized derivative as apoint of departure, we obtain the uncollateralized and partially collateralized cases bymodeling the financing of the lack or excess of collateral with respect to this case. Forexample, if the counterparty should have posted more collateral in the perfectly collater-alized case, we recognize that we are lending the missing collateral to the counterparty,which implicitly means that we are holding bonds issued by the counterparty for the sameamount. Modeling the bonds in a way consistent with the derivative’s dependence on thebond curve, and its behavior in default, provides the pricing equation (19), containing theperfectly collateralized value, contingent CVA and DVA and a Bilateral Funding Valua-tion Adjustment. The funding terms consist of a Credit Funding Valuation Adjustment,depending on the counterparty’s bond-CDS basis, and a Debit Funding Valuation Adjust-ment, governed by the bank’s bond-CDS basis. Altogether we obtain a symmetric pricingformula implying that both parties will obtain the same valuation, which we have shownto be consistent with bond prices . Due to its recursive nature, the pricing formulamay prove difficult to implement, but can be approximated by for instance incorporat-ing the calculation of the FVA terms into CVA and DVA based on bond-implied defaultprobabilities.From the management point of view it might not be desirable for a deep in-the-moneyderivative to behave like one of the counterparty’s bonds, since the cost of funding of col-lateral depends on the bank’s funding spread. However, the counterparty bond componentof the derivative can in principle be hedged by taking a short position in the same bonds.It is true that selling bonds short often entails additional costs, but this is not always soif existing long positions can be reduced, and in any case it does not seem defensible toinclude such costs in accounting fair value. The present proposal may present challengesif applied to derivatives management, since the bond-CDS bases are frequently difficultto estimate, introduce additional volatility in the fair value, and can become negative,18roducing counterintuitive results such as negative CFVA. However, if such issues areconsidered problematic, a different framework can be used for determining managementP&L and other performance metrics .Our ultimate goal is to be compliant with IFRS 13 , which defines fair value as a non-entity-specific, market based, exit price . In turn, exit price is defined as the price obtainedor paid when transferring an asset or liability, respectively, with an emphasis on thevaluations performed by other market participants. The problem is that, strictly speaking,a derivative cannot be transferred. Its very nature depends on the two counterpartiesinvolved, and a transferred derivative is no longer the same derivative, exhibiting a differentdistribution of cash flows (since defaults alter the cash flow structure). Also, the bankcan not freely transfer its side of the derivative to a different market participant. Insteadthe counterparty must agree to cancel the deal, and enter a new one with the marketparticipant.One solution then is to define exit price as the counterparty’s replacement cost, shiftingthe issue of transfer to the counterparty. For an asymmetric derivative price, such as whatis obtained with many approaches to FVA, this becomes awkward, resulting in an exitprice depending on the counterparty’s funding spread, but not the bank’s, unless it isassumed that the third party involved (which takes over the bank’s side of the derivative)has an identical funding spread as the bank. A symmetric formulation, such as the onedefined here, does not have this problem, since price agreement is, in principal, alwayspossible.Furthermore, our formulation is not entity-specific . It is true that the proposed fairvalue depends on the bank’s bond-CDS basis, but this is fundamentally not different fromthe dependence of DVA on its credit spreads. What we are doing is simply extending thecredit component already present in DVA to encompass the full bond price, which is fixedby the market. What IFRS 13 actually means by a non-entity-specific valuation is that theexit price should not depend on such factors as the intention to hold the deal to maturity,etc. Another way of looking at this issue is to recognize that we are decomposing a giventrade into a perfectly collateralized one, some bonds, and possibly CDS, all of which havewell defined market values, determined by market participants independent of the entity.A similar FVA term compliant with the existing accounting framework has also beenobtained in [18], where a bilateral funding valuation adjustment is achieved, also depend-ing on the bond-CDS bases of both counterparties. The point of departure, however, israther different from the one shown here. While in this paper the value of an uncollater-alized derivative has been decomposed into a perfectly collateralized portfolio plus bondsissued by both participants, finally leading to terms 16 and 17, in [18] it is argued thatcounterparties agreeing on a derivative transaction should be indifferent between enter-ing its uncollateralized version or the collateralized equivalent with the collateral account In our view, a CVA desk should only be affected by risks that it is expected to manage. If it suffersfrom, for example, DVA volatility, it will have incentives to hedge DVA, while such hedging might not bedesirable from a global balance sheet perspective: it introduces additional costs and risks with no clearassociated benefit, while an unhedged DVA acts as a natural hedge for global earnings volatility. fu-ture costs associated to a given derivatives transaction are not included in the valuation,the derivative will almost certainly lead to a loss over time. There are two problems withthis statement. Firstly, the costs may not have been correctly estimated, such as is thecase with approaches where a full FVA, based on the bank’s full funding spread, is addedto the CVA . Secondly, such an approach will behave in unpredictable ways when dealsare terminated early. A counterparty should be charged upfront for any future estimatedcosts, by increasing the gross margin to anticipate them, but it would be paradoxical tocharge the counterparty even more when closing out the deal, citing variations in expectedfuture costs.To illustrate why this could happen let us consider, tongue-in-cheek, that besidesan FVA, a manager with foresight decides to include an EVA, an Electricity ValuationAdjustment, in his derivative pricing. The operation of the derivatives desk will contributeto the electricity consumption of the bank, introducing a dependence on the electric spotprice, he argues. Fortunately it can be hedged by entering into an electricity swap, andthe cost of hedging can easily be included in the valuation. Counterparties start payingupfront in accordance with this new pricing, and time passes. Some time later, electricityprices suddenly plummet, but due to the hedges there is no impact in P& L, and all iswell. Until one day, when one of the largest counterparties asks to undo a large positionwith the bank.-’Alright’, the bank manager says, ’but you will have to compensate me for the variationin my future electricity costs.’-’What? Electricity prices have fallen! And why should I have to compensate you for coststhat you will not end up incurring, once the transaction has been closed?’-’I have hedged my electricity exposure, so closing out my hedges will imply an additionalcost.’ As explained in Section 2.4, this does not take into consideration the overlap between CVA andthe bank’s funding costs . Trading with counterparties with a better credit quality than the bank itselfwill produce a funding benefit over time, and in the case of a balance sheet in equilibrium (with new dealsreplacing similar old deals on a continuous basis) this funding benefit is already incorporated into thecurrent funding spread.
Acknowledgements
We would like to thank Juan Antonio de Juan Herrero, Jos´e Manuel L´opez P´erez andRobert Dargavel Smith for helpful comments and suggestions.21
A formal derivation of the pricing formula
Let us now derive the pricing formula shown above with some analytical rigor. To do so, wewill study a derivatives transaction and its replication building upon the setup described in[27] and [25], but with the important new ingredient being the adoption of the Credit- andDebit Constraints (11) and (12), respectively, which enforce the consistency of derivative-and bond valuations. The setup has the properties : • The derivative depends on a single underlying market factor S t and has only a singlepayoff at maturity date T . • The parties involved in the transaction are separated into a Bank (B) and a Coun-terparty (C). The sign of its value is as seen by the bank. • Credit spreads are allowed to be stochastic. • The portfolio should be self-financing, with no additional funding obtained. • We assume a riskless close-out in the event that either bank or counterparty defaults,implying that claims put forth by either the surviving party or the liquidators ofthe defaulted party will be based upon the MtM of the corresponding perfectlycollateralized transaction. No further defaults are thus considered when determiningthe close-out amounts. • Risk factors are, for simplicity, taken to be driven by single factor models, such as asingle credit spread factor, and interest rates are deterministic.In this setup, the bank will construct a replicating portfolio in a similar fashion as in [27]and [25]. We will skip some steps in the derivation of the strategy. The interested reader isreferred to these works for more detail. We must note that the approach followed here willbe that of replication, which is not necessarily the same thing as hedging. Although thedifference between both concepts may appear subtle, it does plays a role in this context.Let us comment a bit on this before continuing with the derivation.We use the term ’replication’ in the sense of decomposition, so that the price of a givendeal is simply equated to the price of a portfolio of simpler instruments (the replicatingportfolio) which in aggregate produce the same cashflows and sensitivities to market vari-ables as the original deal. By contrast, pricing by hedging would entail assigning a valueto a given derivative based on the hedging transactions that the corresponding bank orfinancial institution is able (or willing) to perform. The rationale behind this approachstates that a bank who intends to produce and sell a derivative will hedge its position inthe market. The price to be charged to the customer, that could be regarded as the valueof the transaction, would be that of the hedging portfolio. These conditions can be easily relaxed, which would, with the exeption of the close-out assumptions,not affect the final result. .The replicating portfolio will consist of a collateralized derivative H t , used for capturingmarket risk, bonds issued by both the counterparty and the bank, a short-term CDS written on the Counterparty, CDS C ( t, t + dt ), for eliminating counterparty’s jump-to-default risk, plus another short-term CDS written on the Bank, CDS B ( t, t + dt ), foreliminating bank’s jump-to-default risk .Moreover, since credit spreads are taken to be stochastic, the bank will also need toeliminate these sources of risk. To do so, it will trade in bonds issued by the counterpartyand itself, taking care that the portfolio remains self-financing, which can achieved in thefollowing way: • To replicate its own credit spread risk, the bank will be trading in bonds of itsown of different maturities satisfying the debit constraint (12). Since here we willonly be concerned with a single-factor model, just two bonds will be sufficient. Forsimplicity, we will take one of them, B B ( t, t + dt ), to be infinitesimally short-termed,while the other, B B ( t, T ), is of finite maturity. In this case, the debit constrainttakes the form As an example, IFRS 13 states that the fair value of a liability must reflect non-performance risk,that is, ”the risk that the entity will not fulfill an obligation”. Such a statement gives rise to the usualDVA term in annual reports of financial institutions. However, DVA is difficult to motivate in terms ofhedging since no institution is able to completely hedge its own credit risk, including the jump-to-defaultcomponent. But this component does exist and, as shown in [25], has a well-defined value even for theinstitution considered as a going concern. Furthermore, IFRS 13 emphasizes that fair value must be amarket-based measurement, and ”not an entity-specific measurement”, ie, the value of a derivative shouldnot depend on internal decisions taken by the entity, such as the intention to hold the asset, to settle orto partially or totally hedge it. The short-term (infinitesimal) CDSs are a theoretical construct (see [27] for more details) used herefor simplicity, but in practice, the same hedging can be carried out by trading in two contracts of finite,but different, maturity. Although the bank does not have free access to its own CDS, as shown in [25], this instrument has awell-defined value for it and lies implicitly in the derivative. Therefore, by including it in the replicationportfolio, we are simply expliciting an existing component of the payoff. Alternatively, as in [24], twobonds of different seniority could have been used. V − t = − max {− V t , } = Ω Bt B B ( t, t + dt ) + ω Bt B B ( t, T ) , (26)where Ω Bt and ω Bt are the quantities held of each bond. The bank could hedge thiscomponent by buying back its own debt of an amount precisely equal to − V − t . • To eliminate the counterparty’s credit spread risk, the bank will trade in bondsissued by the counterparty satisfying the credit constraint (11). As before, we willuse just two bonds, B C ( t, t + dt ) and B C ( t, T ). The credit constraint becomes: V + t = max { V t , } = Ω Ct B C ( t, t + dt ) + ω Ct B C ( t, T ) , (27)with Ω Ct and ω Ct defined analogously as Ω Bt and ω Bt .Finally, we will have some cash in a collateral account, described using a unit-of-account C t , of constant value 1, which generates an annualized interest of c t , given byprecisely the rate paid on collateral.Putting all pieces together, the replicating portfolio will be V t = α t H t + β t C t + (cid:15) t CDS C ( t, t + dt ) + η t CDS B ( t, t + dt )+Ω Ct B C ( t, t + dt ) + ω Ct B C ( t, T ) + Ω Bt B B ( t, t + dt ) + ω Bt B B ( t, T ) , (28)where the Greek letters represent the amounts held of each instrument in the portfolio.We assume that the evolution of the relevant market variables under the real measure P is described as dS t = µ St S t dt + σ St dW S, P t dπ Ct = µ Ct dt + σ Ct dW C, P t dπ Bt = µ Bt dt + σ Bt dW B, P t (29)where S t represents the price of the derivative’s underlying asset at time t , while π Ct and π Bt are the short term CDS spread of the counterparty and bank, respectively. Thesespreads are defined so that CDS k ( t, t + dt ) = 0 , k ∈ { C, B } . µ St , µ Ct and µ Bt are the realworld drifts of these processes, while σ St ( t, S t ), σ Ct ( t, π Ct ), σ Bt ( t, π Bt ) are their volatilities.Interest rates will be taken to be deterministic.The three processes will be correlated with time dependent correlations: ρ S,Ct dt = dW S, P t dW C, P t ρ B,Ct dt = dW B, P t dW C, P t ρ S,Bt dt = dW S, P t dW B, P t (30)Two additional sources of uncertainty are described by the default indicator processes N C, P t = 1 { τ C ≤ t } and N B, P t = 1 { τ B ≤ t } , with real world intensities λ C, P t and λ B, P t , with τ C and τ B being the default times of the counterparty and the bank, respectively.24o do the replication, we will proceed in the standard way, equating the differentialof (28), assuming a self-financing strategy, with the expression obtained by expanding dV t using Itˆo’s Lemma, and choosing the available coefficients so that the stochastic termscancel. The remaining, deterministic terms then imply a differential equation for V t . Sincethis procedure is fairly standard, for the sake of brevity, we will omit some steps, and notspell out explicitly certain terms.Conditional on both the counterparty and the bank being alive at time t , the changein V t will be given by (applying Itˆo’s Lemma for jump diffusion processes) dV t = L SCB V t dt + ∂V t ∂S t S t σ St dW S, P t + ∂V t ∂π Ct σ Ct dW C, P t + ∂V t ∂π Bt σ Bt dW B, P t +∆ V Ct dN C, P t + ∆ V Bt dN B, P t , (31)where L SCB V t groups together deterministic terms.On the other hand, by assuming a self-replicating trading strategy, taking the differ-ential of (28) gives dV t = α t dH t + β t dC t + (cid:15) t dCDS C ( t, t + dt ) + η t dCDS B ( t, t + dt )+Ω Ct dB C ( t, t + dt ) + ω Ct dB C ( t, T ) + Ω Bt dB B ( t, t + dt ) + ω Bt dB B ( t, T ) (32)We write down the differentials of the short- and long term bonds, including termscorresponding to their jump-to-default, as well as the instantaneous CDS and the collateralaccount : dB k ( t, t + dt ) = f kt B k ( t, t + dt ) dt + ( R k − B k ( t, t + dt ) dN k, P t dB k ( t, T ) = L k B k ( t, T ) dt + ∂B k ( t,T ) ∂π kt σ kt dW k, P t + ∆ B k ( t, T ) dN k, P t dCDS k ( t, t + dt ) = π kt dt − (1 − R k ) dN k, P t dC t = c t dt (33)for k ∈ { C, B } , where f kt = c t + ¯ f kt represents the short term funding rate, ¯ f kt is the shortterm funding spread over the OIS rate c t , R k is the recovery rate upon default, and L k B k ( t, T ) = ∂B k ( t,T ) ∂t + µ kt ∂B k ( t,T ) ∂π kt + ( σ kt ) ∂ B k ( t,T ) ∂ π kt If we substitute all the differential terms in (32), we reach a replicating equation inwhich we eliminate the stochastic terms driven by dW k, P t , k ∈ { C, B, S } and dN C, P t , dN B, P t by taking Actually, as explained in [35], this formulation of the self-financing condition is an abuse of notation.For example, we have stated that C t is a unit-of-account of constant value 1, and should therefore obey dC t = 0. However, it should be understood implicitly, when reading (32), that the differentials of thedifferent components refer to the gain processes , including generated dividends. t = ∂Vt∂St∂Ht∂St ω Ct = ∂Vt∂πCt∂BC ( t,T ) ∂πCt ω Bt = ∂Vt∂πBt∂BB ( t,T ) ∂πBt (cid:15) t = − V + t − ∆ V Ct − R C η t = V − t − ∆ V Bt − R B (34)while Ω kt , k ∈ { C, B } , are fixed by the constraints (26) and (27).Making use of the PDEs for H t , B C ( t, T ) and B B ( t, T ), we obtain the final PDE:ˆ L SCB V t + π Ct − R C ∆ V Ct + π Bt − R B ∆ V Bt = ( f Ct − π Ct ) V + t − ( f Bt − π Bt ) V − t , (35)where ˆ L SCB V t = ∂V t ∂t + ( r t − q t ) S t ∂V t ∂S t + ( µ Bt − M Bt σ Bt ) ∂V t ∂π Bt + ( µ Ct − M Ct σ Ct ) ∂V t ∂π Ct + ∂ V t ∂S t S t ( σ St ) + ∂ V t ∂π B t ( σ Bt ) + ∂ V t ∂π C t ( σ Ct ) + ∂ V t ∂S t ∂π Bt S t σ St σ Bt ρ S,Bt + ∂ V t ∂S t ∂π Ct S t σ St σ Ct ρ S,Ct + ∂ V t ∂π Ct ∂π Bt σ Ct σ Bt ρ C,Bt ,M Ct and M Bt are the market price of credit risk of the counterparty and bank, respectively,that is, the expected excess return of a credit derivative on each of them over the collateralrate divided by the derivatives’ volatility.Once we have arrived to the PDE depicted in (35), we can follow the well-known stepsleading to the Feynman-Kac formula (shown in, for example, [36]). To do so, we definethe process: X t = V t exp (cid:16) − (cid:90) ts =0 c s ds (cid:17) { τ C >t } { τ B >t } (36)We then place ourselves in the risk-neutral measure Q in which the drifts of S t , π Bt and π Ct are given by ( r t − q t ) S t , µ Bt − M Bt σ Bt and µ Ct − M Ct σ Ct , respectively. Furthermore,default intensities of the counterparty and bank are given by: λ C, Q t = π Ct − R C λ B, Q t = π Bt − R B We apply Itˆo’s Lemma for jump diffusion processes to X t in Q , using the stochasticdiscount factor notation already introduced in (9) dX t = D (0 , t ) (cid:104) { τ C >t } { τ B >t } (cid:16) − c t V t dt + ˆ L SCB V t dt + ∂V t ∂S t S t σ St dW St + ∂V t ∂π Ct σ Ct dW Ct + ∂V t ∂π Bt σ Bt dW Bt (cid:17) − { τ C >t } V t dN B, Q t − { τ B >t } V t dN C, Q t (cid:105) , L SCB V t = ( f Ct − π Ct ) V + t − ( f Bt − π Bt ) V − t − λ C, Q t ∆ V Ct − λ B, Q t ∆ V Bt = ( c t + ¯ f Ct − π Ct ) V + t − ( c t + ¯ f Bt − π Bt ) V − t − λ C, Q t ∆ V Ct − λ B, Q t ∆ V Bt = c t V t + ( ¯ f Ct − π Ct ) V + t − ( ¯ f Bt − π Bt ) V − t − λ C, Q t ∆ V Ct − λ B, Q t ∆ V Bt so that, naming γ kt = ¯ f kt − π kt , k ∈ { C, B } , we get dX t = D (0 , t ) (cid:104) { τ C >t } { τ B >t } (cid:16) ( ¯ f Ct − π Ct ) V + t dt − ( ¯ f Bt − π Bt ) V − t dt − λ C, Q t ∆ V Ct dt − λ B, Q t ∆ V Bt dt + ∂V t ∂S t S t σ St dW St + ∂V t ∂π Ct σ Ct dW Ct + ∂V t ∂π Bt σ Bt dW Bt (cid:17) − { τ C >t } V t dN B, Q t − { τ B >t } V t dN C, Q t (cid:105) , Next, we can integrate between t and T and take expectations conditional on F t . Termsin dW disappear since they are expected values of Itˆo integrals. Furthermore, terms in λ also vanish due to the definition of default intensity, allowing us to write: V t = E Q (cid:104) V T D ( t, T )1 { τ C >T } { τ B >T } |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ B >s } D ( t, s )( V s + ∆ V Cs ) dN C, Q s |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ C >s } D ( t, s )( V s + ∆ V Bs ) dN B, Q s |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Cs V + s ds |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Bs V − s ds |F t (cid:105) (37)We let V ct be the time t value that the derivative would have if it were perfectlycollateralized. Now, if we assume, in accordance with a riskless close-out, that after thecounterparty’s default, V s jumps to R C V cs if V cs ≥ V cs if V cs <
0, and that afterthe bank’s default V s jumps to V cs if V cs ≥ R B V cs if V cs <
0, then, after someindicator manipulations (see, for example, Appendix A of [25] for details), we arrive at V t = E Q (cid:104) V T D ( t, T ) |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t (cid:82) + ∞ u = s D ( t, s )(1 − R C )( V cs ) + dN B, Q u dN C, Q s |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t (cid:82) + ∞ u = s D ( t, s )(1 − R B )( V cs ) − dN C, Q u dN B, Q s |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Cs V + s ds |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Bs V − s ds |F t (cid:105) (38)Intermediate cash flows can easily be incorporated by substituting the first term onthe RHS of (38) for the complete perfectly collateralized price V ct , which discounts all cashflows to present time using OIS rate c t . 27 Practical approximations
The obtained pricing equation (38) may be difficult to evaluate directly, due to its recursivenature. For this reason we will now provide a couple of approximations that can be used inpractice. We begin by suggesting the omission of the recursivity in the CFVA and DFVAterms, providing a straightforward option. We then proceed to study whether it would bevalid to drop both FVA terms and instead calculate contingent debit and credit valuationadjustments based on the default probabilities extracted from bonds issued by the bankand its counterparty, respectively, and thereby incorporate funding considerations via theCVA and DVA terms. As we will see, the resulting expression equals the pricing equationto a first-order approximation.Let us first rewrite (38) using the definition of default intensity. V t = E Q (cid:104) V T D ( t, T ) |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t λ Cs { τ C >s } { τ B >s } D ( t, s )(1 − R C )( V cs ) + ds |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t λ Bs { τ C >s } { τ B >s } D ( t, s )(1 − R B )( V cs ) − ds |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Cs V + s ds |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Bs V − s ds |F t (cid:105) (39)If we iterate the integral equation, the future derivative exposures V ± s entering into theFVA terms become ( V cs − CV A s + DV A s − CF V A s + DF V A s ) ± . Since the FVA termswill be O ( γ ), they can be dropped in a first-order approximation in the bond-CDS bases(which tend to be small). Also, in most cases CV A s and DV A s will be small comparedto V cs , and we can drop their feedback into the FVA terms as well. The upshot is that inthe FVA terms the full exposure V s can be approximated by its collateralized equivalent V cs . Recalling that π ks = λ ks (1 − R k ), k ∈ { B, C } we therefore get V t = E Q (cid:104) V T D ( t, T ) |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s )( π Cs + γ Cs )( V cs ) + ds |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s )( π Bs + γ Bs )( V cs ) − ds |F t (cid:105) (40)We recall that π ks + γ ks = ¯ f ks is the short term funding spread over the OIS rate c t . Thisequation is much simpler to evaluate than (39), and in fact the second and third termsare structurally equivalent to the Funding Cost Adjustment (FCA) and Funding BenefitAdjustment (FBA) found in, for example, [6]. The difference is that in [6] the FCA termis governed by the bank’s, and not the counterparty’s, funding spread, and the overlapwith CVA is also not corrected for. Here, a separate term for CVA is not obtained sinceit is fully contained in the FCA term. Keeping CVA and DVA separate, we thus have the28pproximate expressions CF V A t = E Q (cid:104) (cid:90) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Cs ( V cs ) + ds |F t (cid:105) , (41)and DF V A t = E Q (cid:104) (cid:90) Ts = t { τ C >s } { τ B >s } D ( t, s ) γ Bs ( V cs ) − ds |F t (cid:105) . (42)Now, returning to equation (40): if we bootstrapped default probabilities from a bond,the funding spread ¯ f kt can be approximated as the instantaneous default probability ex-tracted from the bond, ¯ λ kt , times the loss-given-default . Thus, the resulting equationis: V t = E Q (cid:104) V T D ( t, T ) |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s )¯ λ Cs (1 − R C )( V cs ) + ds |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t { τ C >s } { τ B >s } D ( t, s )¯ λ Bs (1 − R B )( V cs ) − ds |F t (cid:105) (43)Finally, the distributions of the default indicator functions 1 { τ k >s } depend on the de-fault probabilities λ ks = ¯ λ ks − γ ks / (1 − R k ) = ¯ λ ks + O ( γ k ). Thus, at first-order approximationin γ k , and back to the notation of (38), we obtain the following expression: V t = E Q (cid:104) V T D ( t, T ) |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t (cid:82) + ∞ u = s D ( t, s )(1 − R C )( V cs ) + d ¯ N B, Q u d ¯ N C, Q s |F t (cid:105) + E Q (cid:104) (cid:82) Ts = t (cid:82) + ∞ u = s D ( t, s )(1 − R B )( V cs ) − d ¯ N C, Q u d ¯ N B, Q s |F t (cid:105) (44)where the d ¯ N k, Q s are default processes inferred from the bond curves. We have thus alsoshown that the CFVA and DFVA terms can be absorbed, to first order, into the CVA andDVA adjustments. C Consistency with Bond Prices
We have constructed the pricing equation (19) taking into consideration bond-CDS basesin order to be consistent with both the counterparty’s and the bank’s bonds. But can weactually reproduce observed bond prices in the current framework? Yes, but we have to Imagine a one-period zero coupon bond that gave us 1 at time 1 with probability 1 − p and R withprobability p , being c the constant risk-free rate between t = 0 and t = 1. Using standard valuationtheory, the price of this bond at time 0 is B (0) = (1 − p ) e − c + pRe − c = e − c (1 − (1 − R ) p ), which canbe approximated as B (0) = e − ( c +(1 − R ) p ) . Therefore, the funding spread needed to match the price of thebond equals its probability of default times 1 − R .
29e careful when doing so. The difference between bonds and derivatives is that the latterwill be liquidated upon default of either counterparty. A bond, by contrast, is traded inthe market, and the market itself cannot in general cause the bond to be liquidated. Sothe first step in reproducing bond prices is to make the bond buyer default-free, based onthe understanding that if the buyer of the bond were to default, the bond would simplybe sold in the market with no change in value .Assuming that the bank is the buyer (the bond is an asset), the pricing equation forthe bond would then be V t = V ct − CV A t − CF V A t , (45)where there is no DVA since the bank’s default is irrelevant to the bond price, and noDFVA since the exposure is always positive for an asset.Equation 38 then becomes V t = V ct − E Q (cid:104) (cid:82) Ts = t D ( t, s )(1 − R C ) V cs dN C, Q s |F t (cid:105) − E Q (cid:104) (cid:82) Ts = t { τ C >s } D ( t, s ) γ Cs V s ds |F t (cid:105) (46)where positive exposures V + s have been substituted for V s since the valuation will alwaysbe positive.Let us now check what this equation gives for the simple case of a default-free issuer anda zero-coupon bond of notional N and maturity T , and under the simplified assumptionthat the discount rate c , as well as the bond-CDS basis γ , are deterministic and constant.Then there will be no CVA, the deal will always be alive, and CF V A t = (cid:90) Tt D ( t, s ) γ V s ds . Using that V ct = e − c ( T − t ) N The full pricing equation will therefore be V t = e − c ( T − t ) N − (cid:90) Tt e − c ( s − t ) γ V s ds . It can easily be checked that the solution to this equation is V t = e − ( c + γ )( T − t ) , and so we recover the bond price corresponding to a default-free bond.The issue gets more complicated when credit risk is involved, and solving the equationdoes not directly produce a simple exponential expression for the bond price. The reason issubtle, and related to the different recovery conventions used when modeling the defaultsof derivatives and bonds, respectively. For derivatives we are, as explained in Section 3,assuming a riskless close-out, which means that the post-default value of a derivative is An exception of course is if the buyer is a systemic entity, correlated with the bond issuer. Suchsystemic dependence should already be contained in the issuer’s bond curve, however, and so when pricingwe can assume that the buyer is a small, with no systemic impact R of the riskless (or perfectly collateralized) value. By contrast, formathematical simplicity we have modeled, in Appendix A, the recovery of the bonds inthe replicating portfolio as relative, meaning that they denote the fraction of the full pre-default value that is recovered after default. We can easily accommodate such a recoveryconvention by performing the substitution R → R V t V ct in equation (46). By doing so, and solving the equation, one obtains the price of the zerocoupon bond as V t = e − ( c + π + γ )( T − t ) N , (47)where π is the CDS spread (again assuming that all parameters are constant in time).The notional amount is therefore simply discounted by the bond curve.Yet a third approach is the common standard consisting of defining the recovery frac-tion as the proportion of the notional amount that is recovered. In general, holding allother parameters constant, different bond prices will be obtained for different recovery con-ventions. Market prices are unique, though, which implies that bond-CDS bases directlyobtained from bond curves will differ depending on the recovery convention adopted, insuch a way as to compensate for the differences. The bond-CDS basis we use, is that of arelative recovery convention, so if such a convention is adopted, the bond-CDS basis caneasily be obtained by inferring the bond curve used to discount bond cash flows. Otherrecovery conventions are not on the same footing as the relative one, but the difference inbond-CDS bases should not be large.In order to be able to obtain greater precision, we will derive expressions for bond pricesfrom (46), which can be used to calibrate the bond-CDS basis parameters γ Xt , X ∈ { C, B } to market prices. An alternative is of course to use observed bond-CDS bases directly,but it should be born in mind that strictly speaking the bases introduced in Appendix Acorrespond to bonds using a relative recovery convention. In Section C.1 we derive thevalue for a bond under a riskless recovery assumption (the default value of the bond is therecovery fraction times the bond valuation obtained by discounting using the OIS curve),while we obtain the price corresponding to an absolute recovery convention (in which thedefault value is a constant recovery fraction of the notional) in Section C.2. C.1 Bond with Riskless Recovery
Let us assume a bond issued by the counterparty with notional N , generating a seriesof cash flows c i at payment dates I c = { t Pi } . As stated above, the bond’s holder can beassumed not to affect its price, so we can suppose that the bank cannot default, that is, λ B = 0, so we will only care for those variables related to the counterparty (the bank’sbond-CDS basis is also irrelevant since V − t = 0). For simplicity, we will assume that recov-eries and interest rates (and therefore, discount factors) are deterministic. Furthermore,hazard rates λ j and bond-CDS basis γ k are deterministic and display a piecewise-constant31tructure, with parameters remaining constant between dates { t λj } and { t γk } , respectively.Let us define { t A } = { t Pi } ∪ { t λj } ∪ { t γk } .Let us first consider the case in which we only have one period: [ t , t ]. The bond onlypays at date t a cash flow equal to c (which includes the notional). With this in mind,substituting in (46) for t ∈ [ t , t ] yields: V t = D ( t, t ) c − γ (cid:82) t s = t e − λ ( s − t ) D ( t, s ) V s ds − (cid:82) t s = t λ (1 − R ) D ( s, t ) c e − λ ( s − t ) D ( t, s ) ds (48)where we have used that ( V cs ) + = D ( s, t ) c . If we rewrite the expression, evaluate thelatter integral and multiply everything by D − ( t, t ) /c , we get: D − ( t, t ) V t c = 1 − γ (cid:82) t s = t e − λ ( s − t ) D − ( s, t ) V s c ds − (1 − R )(1 − e − λ ( t − t ) ) (49)Let us define V ∗ t = D − ( t, t ) V t c . Then we have V ∗ t = 1 − γ (cid:82) t s = t e − λ ( s − t ) V ∗ s ds − (1 − R )(1 − e − λ ( t − t ) ) (50)Performing a change of variables by defining ˜ t = λ ( t − t ) and ˜ s = λ ( t − s ), togetherwith the notation ˜ V ˜ t = V ∗ t , we get˜ V ˜ t = 1 − γ λ e − ˜ t (cid:82) ˜ t ˜ s =0 e ˜ s ˜ V ˜ s ds − (1 − R )(1 − e − ˜ t ) (51)If we now multiply all terms by e ˜ t and differentiate with respect to ˜ t , we have trans-formed the integral equation into the ordinary differential equation˜ V (cid:48) ˜ t + (cid:16) γ λ (cid:17) ˜ V ˜ t = R (52)Solving this equation and undoing previous changes of variables, we find: D − ( t, t ) V t c = Rλ λ + γ + ¯ Ke − ( λ + γ )( t − t ) (53)where ¯ K is an integration constant. Imposing the terminal condition V t = c , gives V t = c D ( t, t ) (cid:104) − (1 − R ) λ + γ λ + γ (cid:16) − e − ( λ + γ )( t − t ) (cid:17)(cid:105) (54)Next we consider the case with two periods: [ t , t ] and [ t , t ], with cash flows c at t and c at t . First, we can restrict ourselves to the period [ t , t ]. There, we can apply32he solution (54). If we name V t +1 as the value of the bond at date t +1 , that is, immediatelyafter cash flow c at t is paid, we have: V t +1 = c D ( t , t ) (cid:104) − (1 − R ) λ + γ λ + γ (cid:16) − e − ( λ + γ )( t − t ) (cid:17)(cid:105) (55)Furthermore, we can also define V t − as the value of the bond at date t − , immediatelybefore cash flow c at t is paid, so: V t − = c + V t +1 ≡ A (56)Finally, we also consider the value of the collateralized equivalent transaction at date t − , defined as V ct − = c + c D ( t , t ) ≡ B (57)When we consider t ∈ [ t , t ] and substitute V ct for B in (46), the equation we obtainis very similar to the one that we saw in the single-period example. The same argumentsoutlined above lead us to the following expression: D − ( t, t ) V t B = Rλ λ + γ + ¯ Ke − ( λ + γ )( t − t ) (58)We then apply the terminal condition V t = A to determine ¯ K . Putting all piecestogether, we arrive at the final expression: V t = c D ( t, t ) (cid:104) − (1 − R ) λ + γ λ + γ (cid:16) − e − ( λ + γ )( t − t ) (cid:17)(cid:105) + c D ( t, t ) (cid:104) − (1 − R ) λ + γ λ + γ (cid:16) − e − ( λ + γ )( t − t ) (cid:17) − (1 − R ) λ + γ λ + γ (cid:16) − e − ( λ + γ )( t − t ) (cid:17) e − ( λ + γ )( t − t ) (cid:105) (59)By applying sequentially the corresponding terminal conditions, we arrive to the ex-pression for an arbitrary number of periods: V t = (cid:80) p ∈ I c D ( t, t p ) c p (cid:104) − (cid:80) ni =1 (1 − R ) λ i + γ i λ i + γ i (cid:16) − e − ( λ i + γ i )( t i − t i − ) (cid:17) × (cid:81) i − j =1 e − ( λ j + γ j )( t j − t j − ) (cid:105) (60)If we make the intervals defining the piecewise-constant functions tend to zero, λ i → λ ( s ) and γ i → γ ( s ), and we have the alternative expression valid for any deterministicfunctions: 33 t = (cid:80) p ∈ I c D ( t, t p ) c p (cid:104) − (cid:82) t p t ((1 − R ) λ ( s ) + γ ( s )) exp (cid:16) − (cid:82) st ( λ ( u ) + γ ( u )) du (cid:17) ds (cid:105) (61)If we define the liquidity-adjusted survival probability P L ( t, s ) ≡ exp (cid:16) − (cid:82) st ( λ ( u ) + γ ( u )) du (cid:17) , (62)the equation simply becomes V t = (cid:80) p ∈ I c D ( t, t p ) c p (cid:104) − (cid:82) t p t ((1 − R ) λ ( s ) + γ ( s )) P L ( t, s ) ds (cid:105) (63) C.2 Bond with Absolute Recovery
In the case of a bond with absolute recovery, the payoff upon default is not a recov-ery fraction of the collateralized equivalent of the bond, but recovery times the notionalamount.We place ourselves in the same framework as in C.1. However, besides being determin-istic, we now will assume that interest rates r j also display a piecewise-constant structure,with parameters remaining constant between dates { t rj } . Again, we first consider the casein which we only have one period: [ t , t ]. The situation is the same as in (48), where,using the new recovery convention, we substitute R → R NV ct . This yields: V t = D ( t, t ) c − γ (cid:82) t s = t e − λ ( s − t ) D ( t, s ) V s ds − (cid:82) t s = t λ ( D ( s, t ) c − RN ) e − λ ( s − t ) D ( t, s ) ds (64)Proceeding as in C.1 and imposing the terminal condition V t = c , it needs to be V t = c D ( t, t ) e − ( λ + γ )( t − t ) + λ RNr + λ + γ (cid:16) − D ( t, t ) e − ( λ + γ )( t − t ) (cid:17) (65)In general, for an arbitrary number of periods, the expression would be V t = (cid:80) p ∈ I c c p D ( t, t p ) exp (cid:16) (cid:80) pi =1 − ( λ i + γ i )( t i − t i − ) (cid:17) + RN (cid:80) pi =1 λ i r i + λ i + γ i (cid:16) − D ( t i − , t i ) e − ( λ i + γ i )( t i − t i − ) (cid:17) (cid:81) i − k =1 D ( t k − , t k ) e − ( λ k + γ k )( t k − t k − ) (66)Again, we make the intervals defining the piecewise-constant functions tend to zero, r i → r ( s ), λ i → λ ( s ) and γ i → ( s ), and, making use of the definition (62), we have thealternative expression: 34 t = (cid:80) p ∈ I c (cid:104) c p D ( t, t p ) P L ( t, t p ) + RN (cid:82) t p t λ ( s ) D ( t, s ) P L ( t, s ) ds (cid:105) . (67)35 eferences [1] Black, Fischer and Myron Scholes (1973), ”The Pricing of Options and CorporateLiabilities.” Journal of Political Economy , Vol. 81, No. 3, May-June, pp. 637-654[2] Merton, Robert (1973), ”Theory of Rational Option Pricing.”
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