Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA Samim Ghamami and Lisa R. Goldberg 2014-54 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA ∗ Samim Ghamami† and Lisa R. Goldberg‡ July 30, 2014 Abstract Wrong way risk can be incorporated in Credit Value Adjustment (CVA) calculations in a reduced form model. Hull and White [2012] introduced a CVA model that captures wrong way risk by expressing the stochastic intensity of a counterparty’s default time in terms of thefinancialinstitution’screditexposuretothecounterparty. Weconsideraclassofreduced formCVAmodelsthatincludestheformulationofHullandWhiteandshowthatwrongway CVA need not exceed independent CVA. This result is based on some general properties of themodelcalibrationschemeandaformulathatwederiveforintensitymodelsofdependent CVA (wrong or right way). We support our result with a stylized analytical example as well asmorerealisticnumericalexamplesbasedontheHullandWhitemodel. Weconcludewith a discussion of the implications of our findings for Basel III CVA capital charges, which are predicated on the assumption that wrong way risk increases CVA. 1 Introduction and Summary Since the 2007–2009 credit crisis, the emphasis on counterparty credit risk by both global and US regulators has increased dramatically. Credit Value Adjustment (CVA) is one of the most important counterparty credit risk measures; according to Basel III,1 banks are required to hold regulatory capital based on CVA charges against each of their counterparties, (see, for instance, Bohme et al. [2011]). Consider a portfolio of derivative contracts that a financial institution, such as a dealer, holds with a counterparty. CVA is the difference between the portfolio value before and after ∗We are grateful to Robert Anderson, Stephen Figlewski, and Travis Nesmith for helpful discussions. †Federal Reserve Board, Washington, D.C. 20551, USA, email: samim.ghamami@frb.gov, and Center for Risk Management Research, University of California, Berkeley, CA 94720-3880, USA, email: samim ghamami@berkeley.edu. ‡Department of Statistics and Center for Risk Management Research, University of California, Berkeley, CA 94720-3880, USA, email: lrg@stat.berkeley.edu. 1Basel III is a global regulatory standard on bank capital adequacy, stress testing and market liquidity risk agreed upon by the members of the Basel Committee on Banking Supervision in 2010-11, and scheduled to be introduced from 2013 until 2018. 1

adjustmentfortheriskthatthecounterpartymightdefault;itisthemarketpriceofcounterparty credit risk.2 CVA is expressed in terms of the dealer’s counterparty credit exposure, V, which is the maximum of zero and the future value of the portfolio. It also depends on the maturity, T, of the longest transaction in the portfolio and the default time, τ, of the counterparty. CVA can be expressed as a risk-neutral expected discounted loss: CVA = E[(1−R)D V 1{τ ≤ T}], (1) τ τ where D is the stochastic discount factor at time t, 1{·} is an indicator function, and R is t the financial institution’s recovery rate.3 Hereafter, for notational simplicity, we suppress the dependence of the CVA to the recovery rate, R. A widely adopted assumption is that credit exposure, V, andthecounterparty’sdefaulttime, τ, areindependent. Thisleadstoindependent CVA, denoted CVA , and it is expressed in terms of the density, f, of τ: I (cid:90) T (cid:90) T CVA = E[D V |τ = t]f(t)dt = E[D V ]f(t)dt, (2) I τ τ t t 0 0 where the last equality follows from the independence of τ and V. In practice, a counterparty’s default time distribution is approximated from counterparty credit spreads observed in the market. Monte Carlo simulation is then used to estimate independent CVA by estimating E[D V ] based on a discrete time grid. t t The efficacy of independent CVA is limited since there are important practical cases where credit exposure, V, and the counterparty’s default time, τ, are correlated, (see Chapter 8 of Gregory [2010]). When credit exposure is negatively correlated with a counterparty’s credit quality, the exposure and its associated risk measures are said to be wrong way. Wrong way CVA, denoted CVA , refers to CVA in the presence of wrong way risk. When the correlation is W positive, the exposure and its associated risk measures are said to be right way. To simplify the exposition, we concentrate on wrong way CVA. However, there are analogous results for right way CVA. A basic example of wrong way risk occurs when a derivatives dealer takes a long position in a put option on a stock of a company whose fortunes are highly positively correlated with those of its counterparty. A widely held view among practitioners is that wrong way risk decreases a counterparty’s credit quality, and this, in turn, increases CVA. This is also evident from Basel III CVA capital charges where CVA ≈ α×CVA with α > 1. It is difficult to build any practical intuition W I on the impact of wrong way risk on CVA in the absence of a mathematical model capturing the correlation between credit exposure, V, and the counterparty’s default time, τ, with a well definedcalibrationschemeusinghistoricaldatatoestimatethemodelparameters. Inthispaper, working within the widely used reduced form modeling framework, we show that wrong way risk does not necessarily increase CVA, i.e. CVA could exceed CVA . I W OurstartingpointisthemodelintroducedbyHullandWhite[2012]summarizedinSection2. In that model, the logarithm of the counterparty’s default time intensity is an affine function of 2Throughout this paper, we consider the unilateral CVA. See Chapter 7 of Gregory [2010] for discussions on Bilateral CVA. 3A derivation of Formula (1) is in Chapter 7 of Gregory [2010]. 2

the dealer’s exposure to the counterparty. In Section 3, we consider a class of intensity models of CVA that includes the formulation of Hull and White [2012]. We show that the calibration scheme of intensity models imply that the model implied credit quality is to match the market implied credit quality. This holds regardless of how the exogenous relationship between V and τ is specified. Let λ denote the counterparty’s default time stochastic intensity. As shown in Sections3.1–3.3,thisimportantimplicationofthecalibrationschemegivesusausefulexpression for CVA : I CVA I = (cid:90) T E[D t V t ]E (cid:104) λ t e− (cid:82) 0 tλudu (cid:105) dt. 0 Deriving the following formula for CVA in the presence of wrong way risk, CVA W = (cid:90) T E (cid:104) D t V t λ t e− (cid:82) 0 tλudu (cid:105) dt, 0 enables us to directly compare CVA and CVA and conclude that wrong way CVA need not W I exceed independent CVA. That is, using reduced form modeling, we derive a formula for CVA W and a calibration-implied formula for CVA so that CVA and CVA become comparable. We I W I shall emphasize that in the absence of such a framework, i.e., a dependent CVA model with a well defined calibration scheme, no practical comparison can be made between wrong way CVA andindependentCVA.InSection4weprovidenumericalexamplesbasedontheHullandWhite model showing that CVA can exceed CVA .4 We discuss the regulatory implications of our I W result in Section 5.5 2 The Hull and White Stochastic Intensity Model of CVA Hull and White [2012] incorporate wrong way risk in a CVA model by formulating a counterparty’s default intensity in terms of a dealer’s credit exposure to the counterparty. They assume that the stochastic intensity of a counterparty’s default time, τ, denoted by λ, is given by: λ = ebVt+at (3) t where b is a constant and a is a deterministic function of time. The parameter b governs the t typeandlevelofdependentrisk,anditiscalibratedby“subjectivejudgment”inHullandWhite [2012]. A positive value for b indicates wrong way risk and a negative value indicates right way risk.6 Let s denotes the counterparty’s maturity-t credit spread and let R denotes the recovery t 4Inpracticedealerportfoliosarecomplex,andtherearealmostalwayscollateralandnettingagreementsassociatedwithpositions. However,inordertoeffectivelycommunicateourmainresults,weconsideruncollateralized contract-level exposure in our numerical examples. 5Weusethefollowingtermsinterchangeablyinthesequel: counterpartycreditexposuresandcreditexposures; also, stochastic default intensity models, intensity models, and reduced form models. 6Hull and White [2012] discuss, but do not implement, an estimation scheme based on historical observations of the exposure V and credit spread of the counterparty. 3

rate.7 Given b, the piecewise constant a are sequentially chosen to satisfy: t e− 1 t − st R = E (cid:104) e− (cid:82) 0 tλudu (cid:105) , (4) as closely as possible. Hull and White [2012] use the left side of (4) as an approximation of the counterparty’s survival probability up to time t > 0, i.e., P(τ > t). The Appendix of Hull and White [2012] details how a is sequentially specified using Formula (4); the expectation on the t right side is estimated by Monte Carlo simulation after time discretization of the integral of the intensity process, λ. 3 Stochastic Intensity Models of CVA Motivated by the Hull and White model, we consider intensity models of CVA in which a counterparty’s default intensity, λ, is driven by a single risk factor, V. The real-valued process {V } is defined on a filtered probability space (Ω,F,{G } ,P), where {G } denotes the t t≥0 t t≥0 t t≥0 filtration generated by V. To incorporate wrong way risk, the intensity, λ, is defined as an increasing function of exposure, V. In this setting the default time, τ, admits a stochastic intensity, λ. A consequence of this is an expression for survival probabilities, (under technical conditions summarized in Appendix A): P(τ > t) = E (cid:104) e− (cid:82) 0 tλudu (cid:105) , (5) and conditional survival probabilities: P(τ > t|τ > s) = E s [e− (cid:82) s tλudu], (6) where 0 < s < t and E denotes expectation conditional on all available information at time s.8 s Also, (5) implies that the density of the default time τ is given by: f τ (t) = E (cid:104) λ t e− (cid:82) 0 tλudu (cid:105) . (7) Remark 1 The results of this paper hold when λ is driven by more than one risk factor. This becomes evident from Remark 2 in Appendix A and a common implication of calibration schemes in reduced form models as discussed in Sections 3.1 and 3.3. Defining λ as a function of a single risk factor, V, merely facilitates the communication of our results; it simplifies the notation and resembles the Hull and White model. 7This is the recovery rate associated with the credit default swap contract “on” the counterparty, and it may ormaynotbeequaltotherecoveryratethatappearsintheCVAformula. TherecoveryrateintheCVAformula referstothefractionoflossthatisrecoveredbythefinancialinstitution(aderivativesdealer)ifthecounterparty defaults. 8Clearly, (5) follows from (6) by taking s=0. We have presented them separately to simplify the exposition as each expression is used to calibrate an intensity model to different types of historical data. We discuss this in Section 3.1 and Appendix B. 4

3.1 Calibration of Stochastic Intensity Models Many of the reduced form models in the credit literature benefit from the the computational convenience of affine intensity modeling by assuming that λ is an affine function of a given Markov process X, such that the conditional expectation in (6) can be written as: P(τ > t|τ > s) = E s (cid:104) e− (cid:82) s tλ(Xu)du (cid:105) = eα(s,t)+β(s,t).Xs, (8) where coefficients α and β depend only on s and t, 0 < s < t, (see Duffie and Singleton [2003] and Duffie et al. [2000]). The Markov process X can be multidimensional. However, here, for simplicity, we think of X as a 1-dimensional process, e.g., a square-root diffusion. Suppose that the conditional survival probabilities on the left side of Formula (8) are market implied. For instance, they may be approximated from corporate bond spreads. Given the convenient form of the conditional expectation in (8) and given that X has usually well known distributional properties, statistical estimates of the parameters of X and λ are often based on (approximate) maximumlikelihoodestimationmethodsortheKalmanfilter. (SeeDuffieetal.[2000],Appendix BofDuffieandSingleton[2003],andLando[2004]. Also,Duffieetal.[2003]andDuffee[1999]are examples of papers using an approximate maximum likelihood estimation method and Kalman filter, respectively.) In CVA stochastic intensity modeling, the unknown parameters of λ are also to be estimated via(5)or(6)assumingthatsurvivalprobabilitiesorconditionalsurvivalprobabilitiesaremarket implied. Hull and White [2012] use (5) and approximate survival probabilities based on CDS spreads. Corporate bond spreads can be used to approximate conditional survival probabilities, (see Appendix B). That is, (6) can also be used for the calibration of an intensity model of CVA. In CVA intensity models considered in this paper, λ is a function of the credit exposure process V, which is the maximum of zero and the value of a derivatives portfolio consisting of possibly thousands of derivatives contracts. So, the stochastic process governing the dynamics of V cannot be assumed as given a priori, and affine intensity modeling cannot be applied here. That is, when the distributional properties of V are not given a priori, the parameters of λ cannot be specified by benefitting from convenient expressions similar to the one on the right side of (8) and using well-known statistical parameter estimation methods. In this sense the term “calibration” as opposed to “statistical estimation” is more suitable for CVA intensity modeling. We shall emphasize that regardless of the sophistication and the mechanics of statistical estimation or calibration schemes, the parameters of λ are to be estimated or approximated such that the model implied survival probabilities: E (cid:104) e− (cid:82) 0 tλudu (cid:105) match the market implied survival probabilities, or, similarly, the model implied conditional survival probabilities: E s (cid:104) e− (cid:82) s tλudu (cid:105) 5

match the market implied conditional survival probabilities, where 0 < s < t. That is: The statistical estimation or calibration scheme of stochastic default intensity models is to ensure that model generated (conditional) survival probabilities match market implied (conditional) survival probabilities. Hereafter, for simplicity we focus on (5) and survival probabilities. The above observation has important implications for CVA calculations in the presence of wrong way-right way risk. In what follows we further elaborate on this by revisiting the Hull and White calibration scheme. Consider the Hull and White model again, where λ t = ebVt+at. Let 0 ≡ t 0 < t 1 < ... < t n ≡ T denote a discrete time grid and set P(τ > t ) ≡ p , i = 1,2,...,n. Suppose that n market i i implied survival probabilities p ,...,p , approximated based on maturity-t CDS spreads with 1 n i e− t 1 1 − st R 1 ,...,e− t 1 n − st R n , are given. Suppose that b is given and the model’s unknown parameters are a ,...,a , on the above-mentioned time grid; a ≡ a . The Hull and White calibration scheme 1 n i ti sequentially estimates a ’s by estimating: i (cid:20) (cid:21) E e − (cid:82) t t i i −1 λudu withMonteCarlosimulationandmakingtheseMonteCarloestimatesequaltop ,fori = 1,...,n. i For instance, given b and p , the calibration scheme uses: 1 p 1 = E (cid:104) e− (cid:82) 0 t1ebVu+a1du (cid:105) , atitsfirststeptospecifya . ThisisdonebyreplacingtheexpectationabovewithitsMonteCarlo 1 estimate based on sampling from V and then numerically solving for a . That is, the calibration 1 scheme approximates a ’s sequentially by making the Hull and White model generated survival i probabilities equal to market implied survival probabilities, p ,...,p . 1 n 3.2 Model Implied Counterparty Credit Quality Suppose that a counterparty’s survival probabilities P(τ > t), for t > 0, are considered to be a measure of its credit quality. Wrong way exposures are defined by Canabarro and Duffie [2003] as “credit exposures that are negatively correlated with the credit quality of the counterparty.” In what follows, we show that stochastic intensity models of CVA capture this basic definition. However, reiterating the result of the previous section: the calibration scheme equates a counterparty’s model implied credit quality to the counterparty’s market implied credit quality. In other words, wrong way risk does not affect a counterparty’s credit quality. In the presence of wrong way risk, the stochastic intensity, λ, of a counterparty’s default time, τ, is defined as an increasing function of the credit exposure V. Conditional on a given sample path of the credit exposure process in [0,t], we can write: P(τ > t|G t ) = e− (cid:82) 0 tλGt (Vu)du. (9) 6

Hereafter,whenconditioningonagivensamplepathoftheexposureprocessin[0,t],wesuppress the dependence of λ on G , and we refer to the survival probabilities on the left side of (9) Gt t (k) as path dependent survival probabilities. Consider two given sample paths, {V ; u ≤ t}, u k = 1,2, for which: (cid:90) t (cid:90) t λ(V(1))du < λ(V(2))du. u u 0 0 This implies that the counterparty’s credit quality is lower along the second sample path, i.e. counterparty’s path dependent survival probability is lower along the second sample path: e− (cid:82) 0 tλ(Vu (2))du < e− (cid:82) 0 tλ(Vu (1))du. In other words, wrong way risk affects a counterparty’s credit quality on a path-wise basis, i.e., it lowers the credit quality along some paths. However, the calibration strategy that uses (5) equates the average of path dependent survival probabilities with the market implied survival probabilities: P(τ > t) = E[P(τ > t|G )] = Market Implied Time-t Survival Probability. t An analogous argument shows that right way risk does not affect the credit quality of the counterparty. 3.3 Wrong Way CVA Need Not Exceed Independent CVA In Lemma 1 of Appendix A, we derive the following formula for dependent CVA (right or wrong way), which assumes that the stochastic intensity of counterparty’s default time, τ, is a function of dealer’s credit exposure, V: CVA W = (cid:90) T E (cid:104) D t V t λ t e− (cid:82) 0 tλudu (cid:105) dt. (10) 0 Focusing on wrong way CVA, we now show that CVA need not exceed CVA in stochastic W I intensity models of CVA. Our result is based on comparing the wrong way CVA formula with a calibration-implied independent CVA formula introduced below. The calibration-implied expressionforindependentCVAholdsforallintensitymodelswhosecalibrationschemeuses(5)or (6). We further support our result by constructing a stylized example at the end of this section and our numerical examples of Section 4. Recall that to calculate CVA , I (cid:90) T CVA = E[D V ]f(t)dt, I t t 0 the probability density function (p.d.f), f, of a counterparty’s default time is market implied and approximated from CDS or bond spreads. The calibration scheme of stochastic intensity models equates the market implied (conditional) survival probabilities to the model implied 7

(conditional) survival probabilities as suggested by Formula (5) and (6). This implies that the market implied p.d.f. of counterparty’s default time, f(t), is to match the model implied p.d.f.: E (cid:104) λ t e− (cid:82) 0 tλudu (cid:105) , for all t ∈ [0,T] as also suggested by Formula (7). This gives the useful calibration-implied expression for CVA : I CVA I = (cid:90) T E[D t V t ]E (cid:104) λ t e− (cid:82) 0 tλudu (cid:105) dt, (11) 0 which enables us to compare CVA and CVA directly, regardless of the mechanics and sophis- W I tication of the model calibration strategy. Hereafter, for simplicity, assume that the stochastic discount factor D is constant or independent of λ and V. A comparison of the calibrationimplied CVA (right hand side of Formula (11)) and CVA (right hand side of Formula (10)) I W suggests that wrong way CVA need not exceed independent CVA. Note that since the stochastic default intensity process is defined as an increasing function of the credit exposure process, λ, and V are positively correlated. That is, E[D V λ ] ≥ E[D V ]E[λ ]. t t t t t t However, this has no implication for the pair of terms: E (cid:104) D t V t λ t e− (cid:82) 0 tλudu (cid:105) and E[D t V t ]E (cid:104) λ t e− (cid:82) 0 tλudu (cid:105) , (12) or for the time integrals of those terms. We end this section by constructing a stylized example for which we analytically prove that CVA ≥ CVA in some parts of the parameter space. In Section 4, we give more realistic I W numerical examples for which CVA > CVA in the Hull and White model. I W Example 1 Let X denote a [0,1] uniform random variable. Define the exposure V in the interval [0,T] based on X as follows: (cid:40) X 0 < t ≤ t ≡ T/2 1 V = t nX t < t ≤ t ≡ T, 1 2 where n is a positive constant. Let λ be the stochastic intensity of a counterparty’s default time, τ, and suppose: (cid:90) ti λ du = bK +a , (13) u i i 0 for i = 1,2 and K = X, K = nX. In Formula (13), b is a positive constant and the parameters 1 2 a and a are calibrated to market credit spreads. Note that since the time integral of the 1 2 stochastic intensity is an increasing function of the exposure, the definition of wrong way risk is 8

capturedinthisstylizedexample. Letp andp denotethemarketimpliedsurvivalprobabilities 1 2 ofthecounterpartybytimet andt ,respectively. ThecalibrationschemethatusesFormula(5) 1 2 specifies the unknown parameters a and a based on: 1 2 p = P(τ > t ) = E[e−bKi−ai], (14) i i for i = 1,2 and K = X and K = nX. We show that for large n: 1 2 CVA ≥ CVA . (15) I W The proof is in Appendix C. 3.4 Discussion of Our Results Our study challenges the premise that wrong way risk always increases CVA and shows that independent CVA can exceed wrong way CVA. Reduced form modeling enables the modeler to exogenously correlate credit exposures and the default time of a counterparty by making the default time’s intensity an increasing function of credit exposures. The calibration scheme of any intensity model equates the model implied counterparty’s credit quality with the market implied counterparty’s credit quality derived from, for instance, CDS prices. This statement has been rephrased by “in intensity models, wrong way risk does not affect a counterparty’s credit quality” in Section 3.2 to further emphasize this important implication of the calibration scheme. Using this, we derive a calibration-implied expression for the independent CVA formula to make it directly comparable with dependent CVA, whose formula is derived in the Appendix. See the right side of (11) and (10), respectively. Then, it follows that there is no reason that one should exceed the other. It is not the purpose of our paper to numerically experiment with a fixed model in order to attach financial interpretations to different parts of the parameter space to formulate a rule prescribing where CVA could exceed CVA . A different intensity model of CVA, i.e., different W I functionalrelationbetweenλandV, couldleadtodifferentnumericalresultsleadingtodifferent sets of financial rules and interpretations. It is the purpose of this paper to show that CVA I can exceed CVA for a broad class of reduced form models. Example 1 in the previous section W is a stylized setting in which we show that CVA exceeds CVA in some part of the parameter I W space. On the basis of our study, one could argue that the dependence of a counterparty’s credit quality on credit exposures is already reflected, for instance, in CDS prices, which are indicators of the credit quality. In fact, when CDS prices are believed to reflect all the information on counterparty’s credit quality, one could question the need for dependent CVA, which is then to be compared with independent CVA. After all, in a dependent CVA intensity model, after exogenously fixing a relation between a counterparty’s default intensity and credit exposures, one should fit the model to the market implied credit quality, which is also present in the independent CVA formula. 9

4 Numerical Examples This sectionis a summaryof ournumericalexamples based onthe Hulland White [2012]model. TheydemonstratethatindependentCVAcanexceedwrongwayCVA.Therearemanypractical instances where Monte Carlo estimates of CVA and CVA are close but the former exceeds I W the latter. We consider contract level exposures for forward type contracts and put options. In what follows we assume that the risk free rate, r, is constant. That is, the discount factor is D = e−rt and independent and wrong way CVA are: t CVA I = (cid:90) T D t E[V t ]f(t)dt and CVA W = (cid:90) T D t E (cid:104) V t λ t e− (cid:82) 0 tλudu (cid:105) dt, 0 0 where V denotes the time t ≥ 0 value of the derivative contract and T is the maturity of the t contract. Also, λ is the stochastic intensity proposed by Hull and White, i.e., λ = exp(bV +a ). t t t Assuming that b is given, the piece-wise constant deterministic function a is approximated t based on counterparty’s t-maturity credit spreads, s , and (4), (see the details in the Appendix t of Hull and White [2012]). The expected exposures, E[V ], are with respect to the physical measure in our numerical t examples. There is no consensus in counterparty credit risk around choices of measure for CVA calculations, (see Gregory [2009] and Chapters 7 and 9 of Gregory [2010] for discussions on the use of risk-neutral and physical measure in CVA calculations).9 Monte Carlo CVA Estimation Monte Carlo estimators of CVA and CVA , denoted θˆ I W I and θˆ , are defined as follows. Consider the time grid, 0 ≡ t < t < ... < t ≡ T, W 0 1 n n n (cid:88) (cid:88) θˆ = D V¯f(t )∆ , and θˆ = D ξ ∆ , I i i i i W i i i i=1 i=1 (cid:16) (cid:17) where ∆ ≡ t −t , and, V¯ = 1 (cid:80)m V , with V being the jth Monte Carlo realization of i i i−1 i m j=1 ij ij V i ≡ V ti . Similarly, ξ i is the m-simulation-run average of V i λ i e−(cid:80)i k=1 λ k ∆˜ k, with ∆˜ k = t˜ k −t˜ k−1 being defined based on a finer time grid, 0 ≡ t˜ < t˜ < ... < t˜ ≡ T, l > n. 0 1 l Let {S t } t≥0 denote a geometric Brownian motion, S t = S 0 eXt, where {X t } t≥0 is a Brownian motion with drift µ and volatility σ. We sample from the risk factor S based on the physical t measure. Then,giventheMonteCarlorealizationofS,thevaluationisbasedontherisk-neutral measure.10 This implies V = e−r(T−t)E[S | S ] = S for a forward type contract. For the put t T t t options, we simply set V = e−r(T−t)E[(K −S )+ | S ]. t T t The credit curve is assumed to be flat at s. So, in the independent case, the default time, τ, is an exponential random variable with mean 1/s. This leads to the following closed form 9Note that in the above setting {D V } is a martingale under the risk-neutral measure. We have chosen t t t≥0 the physical measure merely to avoid the trivial case, V =E[D V ] and so CVA =V P(τ ≤T), resulting from 0 t t I 0 {D V } being a martingale. t t t≥0 10This is the common and well known practice in risk management: sampling from the risk factors based on the physical measure and then risk neutral valuation, (see, for instance, Chapter 9 of Glasserman [2004]). 10

formula for independent CVA in the forward contract case, CVA = sS0(exp(αT) − 1) with I α α = µ+ σ2 −r−s. 2 NumericalResults CVAestimatesinthefollowingnumericalexamplesarebasedonm = 105 simulation runs.11 We assume a recovery rate of R = 0, a constant risk free rate of r = .01, and an annualized volatility of 25%. The credit quality of the counterparty is investment grade with a flat spread curve at 100 basis points. The family of forward contracts presented in Exhibit 1 and the family of in-the-money put options analyzed in Exhibit 2 are both examples where independent CVA and wrong way CVA are close, but CVA exceeds CVA at each maturity. The I W coefficient b = .02 in both Exhibits 1 and 2 indicates a relatively low dependence of stochastic intensity on exposure. Exhibit 3 presents another 20% in-the-money put option example where CVA exceeds CVA at each maturity; note that the difference is most pronounced for T = 1. W I The coefficient b = 1 in Exhibit 3 indicates a relatively higher dependence of intensity on exposure. T .1 .2 .4 .6 .8 1 CVA 2 4 8 12 16.1 20.1 I θˆ 1.9 3.7 7.5 10.5 15.5 19.6 W Exhibit 1: Forward contract: CVA numbers and estimates are of order 10−3, m = 105, b = .02, µ = 0, σ = .25, S = 2, spread = .01, ∆ = 5∆˜, ∆˜ = .01 for T = 1,.8,.6,.4, and ∆˜ = .001 for 0 T = .1,.2. T .1 .2 .4 .6 .8 1 θˆ 2 4 8.1 11.5 17.1 21.9 I θˆ 1.9 3.7 7.6 11.1 16.7 21.6 W Exhibit 2: Put option: CVA estimates are of order 10−3, m = 105, b = .02, µ = 0, σ = .25, S = 10,K = 12,spread = .01,∆ = 5∆˜,∆˜ = .01forT = 1,.8,.6,.4,and∆˜ = .001forT = .1,.2. 0 We also came across unrealistic cases of put options where CVA exceeds CVA in a more I W pronounced way. For instance, consider the case where credit spread is flat at 106 basis points, i.e.,s = 100. ThisgivesCVA = .0169andCVA = .0057forT = 1. Thatis,independentCVA I W 11We use MATLAB to produce the results. 11

T .1 .2 .4 .6 .8 1 θˆ 2 4 8.1 11.5 17.1 21.9 I θˆ 2.2 4.8 11.27 17.6 29.4 37.9 W Exhibit 3: Put option: CVA estimates are of order 10−3, m = 105, b = 1, µ = 0, σ = .25, S = 10, K = 12, spread = .01, ∆ = 5∆˜, ∆˜ = .01 for T = 1,.8,.6,.4, and ∆˜ = .001 for 0 T = .1,.2. is roughly 3 times larger than wrong way CVA.12 Note that θˆ and θˆ are biased estimators I W of CVA and CVA due to the time-discretization. Ideally, the mean square error of these I W estimators should be estimated. This is computationally extremely expensive in our setting. To get a feel for the statistical efficiency of our estimators, we note that for the forward contract example presented in Exhibit 1, CVA is analytically calculated, and Monte Carlo estimates of I CVA coincide with the exact values. Since Monte Carlo estimation of CVA is computationally I intensive, a valuable line of research is to develop efficient Monte Carlo estimators of CVA. (See Ghamami and Zhang [2013] for efficient Monte Carlo independent CVA estimation.) 5 Regulatory Treatment of Wrong Way Risk Basel III’s counterparty credit risk (CCR) regulatory capital charges consist of counterparty default risk (carried over from Basel II) and CVA capital charges for bilateral derivatives transactions (see BCBS [2011]). For centrally-cleared derivatives transactions, the Basel Committee on Banking Supervision (BCBS) has recently devised capital charges on banks for their central counterparty credit risk (see BCBS [2012]). In all these CCR regulatory capital charges, the BCBS assumes that wrong way risk increases different measures of CCR; CVA being one of them. It then approximates a wrong way CCR measure by increasing the independent CCR measure using the so-called α multiplier, which is often set to 1.4. That is, in the case of CVA, wrong way CVA is often approximated by the independent CVA times 1.4. It should be noted that capturing wrong way risk is not the only purpose of the BCBS’s α multipliers (see Section 4.2 of Pykhtin and Zhu [2006] on α multipliers and the references there). Similar to the view often held by practitioners in the financial industry, the BCBS’s premise in CVA calculations is that wrong way risk increases CVA. Our findings challenge this premise. Our results would be useful when reviewing the methodology underlying CCR capital charges that incorporate dependent risk (wrong or right way). Historically, BCBS has taken relatively simple and conservative approaches in areas where mathematicalmodelingbecomeschallenging–thealpha-multiplierapproachtowrongwayCVA estimation was to provide simple and conservative wrong way CVA estimates. Financial insti- 12The remaining parameters for this unrealistic example are σ = .3, b = 2, µ = 0, S = 1, K = 1.5. Also, 0 ∆˜ =.01 and ∆=.05 12

tutions that prove to be sufficiently sophisticated in terms of their quantitative capabilities are usually approved by regulators to use their own internally developed risk sensitive models. Our results would be informative for regulators when financial institutions’ CVA models are being evaluated to replace the the BCBS’s less risk sensitive proposed methods. 6 Conclusion Amathematicalmodelisrequiredtoincorporatethedependencybetweenacounterparty’scredit quality and credit exposures to compare independent CVA and dependent CVA (wrong or right way). The calibration scheme of the model plays a critical role in quantifying this comparison. In this paper, we focus on stochastic intensity models of CVA that include the formulation of Hull and White [2012]. We derive a formula for CVA and show that the general properties of calibration scheme, regardless of its level of sophistication, imply that dependent CVA may or may not exceed independent CVA. Using the Hull and White model, we generate numerical examples that confirm our result for wrong way and independent CVA. BCBS’s regulatory CCR capital charges assume that wrong way risk increases CVA, and CVA is approximated by W α×CVA , where α is often assumed to be 1.4. Our results would be useful when reviewing the I regulatory CVA capital charge that incorporate dependent risk (wrong or right way). Appendix A Default Times with Stochastic Intensity and the Proof of the Dependent CVA Formula Itiswellknownthatadefaulttime, τ, definedonafilteredprobabilityspace, (Ω,F,{F } ,P), t t≥0 admits a stochastic intensity, λ, when the process, (cid:90) t∧τ 1{τ ≤ t}− λ du, u 0 isamartingale, (wheret∧τ ≡ min{t,τ}). Tomakethemartingalepropertyprecisethefiltration is to be specified. For the general case see Chapter 2 of Bremaud [1981]. In what follows, we do this for our setting. A consequence of the existence of an intensity is the identity: P(τ > t) = E (cid:104) e− (cid:82) 0 tλudu (cid:105) , which is used throughout this paper and in the proof Lemma 1. Doubly Stochastic Random Times Let τ be a default time on a filtered probability space (Ω,F,{F } ,P). Let {H } denote the filtration generated by the default indicator process t t≥0 t t≥0 1{τ ≤ t}. Suppose that the distribution of τ depends on additional information denoted by 13

{G } . Set F ≡ G ∨H where F is the smallest σ-algebra that contains G and H .13 The t t≥0 t t t t t t default time, τ, is called doubly stochastic when for all t > 0,14 P(τ ≤ t|G ) = P(τ ≤ t|G ), ∞ t and when conditional on G , (cid:82)t λ du is strictly increasing.15 t 0 u In our setting {G } is the filtration generated by the exposure process V. The first t t≥0 condition implies that given the past values, u ≤ t, of V, the future, s > t does not contain any extra information for predicting the probability that τ occurs before t.16 The credit exposure process, V, could have jumps due to the the expiration of trades prior to the maturity of the longest instrument in the portfolio. In this case, where V has points of discontinuity, τ may not be doubly stochastic. But, it can be shown that τ still admits a stochastic intensity λ, (see Definition D7 and Theorem D8 of Bremaud [1981]). Lemma 1. Consider a real-valued process V defined on the probability space (Ω,F,P). Let {G } , denote the filtration generated by V, i.e., G = σ{V ; 0 ≤ s ≤ t}, the smallest σ-field t t≥0 t s with respect to which V is measurable for every s ∈ [0,t], and let G ≡ G ⊂ F. Let D denote a s ∞ real-valued process that is adapted to {G } . Let τ denote a counterparty’s default time, which t t≥0 admits the stochastic intensity λ that is adapted to {G } . For t ≥ 0, t t≥0 P(τ > t|G) = e− (cid:82) 0 tλudu and P(τ > t) = E (cid:104) e− (cid:82) 0 tλudu (cid:105) . (16) Then, the following holds for any given T ≥ 0: E[D τ V τ 1{τ ≤ T}] = (cid:90) T E (cid:104) D t V t λ t e− (cid:82) 0 tλudu (cid:105) dt. 0 Proof. Conditional on G we can write, (cid:90) T E[D V 1{τ ≤ T}|G] = E[D V |G,τ = t]f (t)dt τ τ τ τ τ|G 0 (cid:90) T (cid:90) T = D t V t f τ|G (t)dt = D t V t λ t e− (cid:82) 0 tλududt, 0 0 where f is the conditional density of τ and is derived based on the left side of (16). Then, τ|G the Lemma follows by noting that: 13By definition, τ is an H -stopping time. Note that τ is also a F ≡G ∨H -stopping time for any {G } . t t t t t t≥0 14it represents the first event time of a conditional or doubly stochastic Poisson process. 15see, for instance, Chapter 9 of McNeil et al. McNiel et al. [2005]. 16Manyofthestochasticintensitymodelsinthecreditliteratureworkunderthisdoublystochasticframework, (see, for instance, Duffie and Singleton [2003]). 14

E[D V 1{τ ≤ T}] = E[E[D V 1{τ ≤ T}|G]], τ τ τ τ and E (cid:20)(cid:90) T D t V t λ t e− (cid:82) 0 tλududt (cid:21) = (cid:90) T E (cid:104) D t V t λ t e− (cid:82) 0 tλudu (cid:105) dt. 0 0 Remark 2 We would like to emphasize that the dependent CVA formula of Lemma 1 also applies to multi-factor settings. That is, when λ is defined based on more than one risk factor, the proof works by {G } denoting the filtration generated by all the risk factors. t t≥0 B Approximating Conditional Survival Probabilities from Zero- Coupon Bond Spreads Hereweuseastylizedsettingtoshowhowconditionalsurvivalprobabilitiescanbeapproximated fromzero-couponbondspreads. Letδ(t,T)denotetherisk-neutralpriceofamaturity-T defaultfree zero-coupon bond at time t > 0. It is well known that: δ(t,T) = E t [e− (cid:82) t Trudu], where r is the short rate process and E denotes the risk-neutral expectation conditional on t information available by time t, (see, for instance, Bjork [2009]). Let d(t,T) denote the risk neutralpriceofamaturity-T zero-recoverydefaultablezero-couponbondattimet > 0. Reduced form debt pricing for a default time τ with the risk-neutral default intensity process λ gives d(t,T) = E t [e− (cid:82) t T(ru+λu)du], as shown by Lando [1998]. Note that in a stylized setting where λ and r are independent, conditional survival probabilities are easily obtained from the defaultable and default free bond prices: P(τ > T|τ > t) = E t [e− (cid:82) t Tλudu] = d(t,T) . δ(t,T) More realistic corporate bond reduced form pricing models also allow the modeler to estimate conditionalsurvivalprobabilitiesfrommarketdata,(seeChapter6ofDuffieandSingleton[2003] and the references there). C Proof of the Result of Example 1 Assume zero short rate which gives D ≡ 1. First consider CVA : I 15

CVA = E[V 1{τ ≤ T}] = E[X]P(τ ∈ A )+E[nX]P(τ ∈ A ), I τ 1 2 where A = (t ,t ], i = 1,2. Note that i i−1 i P(τ ∈ A ) = E[e−bKi−1−ai−1]−E[e−bKi−ai], i where K = t = a = 0, K = X, and K = nX. Using the right side of the above in in the 0 0 0 1 2 CVA formula, we can write I (cid:16) (cid:17) (cid:16) (cid:17) CVA = E[X] 1−E[e−bX−a1] +E[nX] E[e−bX−a1]−E[e−bnX−a2] (17) I Now, consider CVA and recall the proof of Lemma 1: W CVA = E[V 1{τ ≤ T}] = E[E[V 1{τ ≤ T}|X]]. W τ τ Considertheconditionalexpectationontherightsideabove; furtherconditioningonthedefault time gives (cid:16) (cid:17) (cid:16) (cid:17) E[V 1{τ ≤ T}|X] = X 1−e−bX−a1 +nX e−bX−a1 −e−bnX−a2 , τ and so, (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) CVA = E X 1−e−bX−a1 +nX e−bX−a1 −e−bnX−a2 . (18) W Using Formula (17), Formula (18), and simple algebraic manipulations, we have CVA −CVA = e−a1(1−n)Cov(X,e−bX)+ne−a2Cov(X,e−bnX). (19) I W Simple calculations show 1 1 e−bn e−bn nCov(X,e−bnX) = − + − − . 2b b2n 2b b2n This term converges to a constant as n → ∞. Note that, using Chebyshev’s algebraic inequality, Cov(X,e−bX) < 0 when b > 0. So, CVA ≥ CVA for large values of n. I W 16

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