A subordinated CIR intensity model with application to Wrong-Way risk CVA
AA subordinated CIR intensity model with application toWrong-Way risk CVA
Cheikh Mbaye Fr´ed´eric Vrins ∗ Louvain Finance Center (LFIN) & CORE † Universit´e catholique de Louvain, Belgium
Abstract
Credit Valuation Adjustment (CVA) pricing models need to be both flexible andtractable. The survival probability has to be known in closed form (for calibrationpurposes), the model should be able to fit any valid Credit Default Swap (CDS) curve,should lead to large volatilities (in line with CDS options) and finally should be ableto feature significant Wrong-Way Risk (WWR) impact. The Cox-Ingersoll-Ross model(CIR) combined with independent positive jumps and deterministic shift (JCIR++) isa very good candidate : the variance (and thus covariance with exposure, i.e. WWR)can be increased with the jumps, whereas the calibration constraint is achieved via theshift. In practice however, there is a strong limit on the model parameters that canbe chosen, and thus on the resulting WWR impact. This is because only non-negativeshifts are allowed for consistency reasons, whereas the upwards jumps of the JCIR++need to be compensated by a downward shift. To limit this problem, we consider thetwo-side jump model recently introduced by Mendoza-Arriaga & Linetsky, built bytime-changing CIR intensities. In a multivariate setup like CVA, time-changing the ∗ Voie du Roman Pays 34, B-1348 Louvain-la-Neuve, Belgium.E-mail: [email protected]. † Center for Operations Research and Econometrics. a r X i v : . [ q -f i n . M F ] J a n ntensity partly kills the potential correlation with the exposure process and destroysWWR impact. Moreover, it can introduce a forward looking effect that can lead toarbitrage opportunities. In this paper, we use the time-changed CIR process in away that the above issues are avoided. We show that the resulting process allows tointroduce a large WWR effect compared to the JCIR++ model. The computation costof the resulting Monte Carlo framework is reduced by using an adaptive control variateprocedure. Keywords: default intensity, time-changed diffusion, subordinator, credit value adjustment(CVA), wrong-way risk (WWR).
Since the 2008 crisis, regulators suggest financial institutions to pay specific attention toCounterparty Credit Risk (CCR) when valuing Over the Counter (OTC) deals. In thiscontext, CCR refers to the possibility that the counterparty of the transaction can defaultbefore the maturity of the contract. The CCR can be accounted for either by setting upa strong collateralisation agreement, or by charging a Credit Value Adjustment (CVA) toabsorb the corresponding expected losses. One of the main challenge when pricing suchadjustments is to account for the potential dependency of the exposure with counterparty’scredit quality, a phenomenon commonly referred to as wrong-way risk (WWR).In that respect, a popular framework fitting in the class of reduced-form models is toconsider the default intensity to be governed by the CIR++ [3] or the JCIR++ [5] process.In essence, the intensity is modeled as a CIR or a jump diffusion CIR (JCIR) dynamicsshifted in a deterministic way so as to fit a given CDS term structure. However, this modelsuffers from an important restriction: the resulting intensity process (including the shift)needs to be positive. Because for tractability reasons the jumps in the JCIR++ model areupwards only, increasing the jump activity (e.g. to increase the implied spread volatility)under the constraint of keeping the survival probability curve unchanged introduces a shift2unction that tends to be more and more negative. Because negative shifts should be ruledout for consistency reasons, this puts limits on the CIR or JCIR parameters that can bechosen. This will further limit the WWR impact in CVA applications. One way to limit theappearance of shift functions with negative values would be to allow for both upwards anddownwards jumps, without affecting the tractability of the model.In this paper, we consider the approach of Mendoza-Arriaga and Linetsky [11] to modelthe default intensity as a time-changed CIR in a CVA context. This poses several problemsthat need to be addressed. First, the time-change process will destroy the potential correla-tion between the intensity and the exposure increments. As a consequence, this would leadto a weak WWR impact. Second, the time-change approach may also introduce arbitrageopportunities via a forward looking effect. Eventually, even if some techniques exist to dealwith WWR in a semi-analytical way (see e.g. [6] and [14]), one generally has to rely onMonte Carlo simulations. Standard Monte Carlo methods are known to be computationallyintensive. In addition, because of the stochastic clock, the time-change model is very timeconsuming due to the fact that the simulation is done in a random grid.Our contribution in this paper is multiple. First, we propose a way to use the time-changed model of Mendoza-Arriaga and Linetsky in the context of CVA avoiding both thecorrelation destruction and the appearance of arbitrage opportunities. This is achievedby reconstructing the exposure process in a “synchronous” way with the intensity, whilepreserving the original exposure’s dynamics. Second, we show via numerical experimentsthat the corresponding model is indeed able to generate larger WWR CVA figures comparedto JCIR++ without facing the inconsistency issue resulting from a negative shift. Eventually,we propose a variance reduction technique based on the adaptive control variate to reducethe computational cost.The paper is organized as follows. In Section 2, we recall the theory of some reduced-form intensity models in the literature of credit risk such as diffusions intensity models andtheir extensions. In the third section, we introduce the subordinated model combined with3 reconstruction of the exposure process avoiding possible arbitrage opportunities resultingfrom the time-change. The fourth section reviews the basic concepts of CVA computationin the reduced-form setup for the diffusion models and the new subordinated model. InSection 5, we present the numerical experiments including the comparison of the diffusionmodels and the time-change model in term of WWR effects and the control variate technique.The last section contains some concluding remarks and perspectives.
Before defining the subordinated model, we recall the definitions of some existing models inthe credit risk modelling literature using the intensity approach. In this study, we considerthe well known square-root diffusion default intensity models and their extended shiftedversion SSRD [3] and SSRJD [5].
We consider a fixed time horizon
T > , F T , F , Q ) where F =( F t ) ≤ t ≤ T is the filtration generated by the vector W = ( W B , W V , W ⊥ ). In this setup, Q represents the risk-neutral probability measure and the components of W are risk drivers. Inparticular, W B governs the dynamics of the risk-free rate r , hence that of the bank accountnum´eraire: dB t = r t B t dt, B = 1 . Under Q , all prices of tradable assets divided by B are F -martingales between two cash-flowdates. The second Brownian motion W V drives the dynamics of the portfolio price process dV t = b ( V t ) dt + σ ( V t ) dW Vt , V > . (1)We assume the coefficients b, σ to be regular enough to guarantee that a unique strongsolution to this SDE exists. Finally, we model the default time τ of our counterparty as4 random time. It is defined as a first passage time of an increasing stochastic processΛ t := (cid:82) t λ s ds , ( λ s ) s ≥ ≥
0, above a unit-mean exponential random barrier E : τ := inf { t ≥ t ≥ E} . (2)In this setup, the default intensity λ is driven by a Brownian motion correlated to W V , W λ := ρW V + (cid:112) − ρ W ⊥ , W ⊥ ⊥ W V , ρ ∈ [ − , E is independentfrom F . In such a reduced form setup, the complete filtration G = ( G t ) ≤ t ≤ T is obtained byprogressively enlarging F with D = ( D t ) ≤ t ≤ T , the natural filtration of the default indicator D t = { τ ≤ t } : G t = F t ∨D t where D t := σ ( D u , ≤ u ≤ t ). Hence, τ is a D - and a G -stoppingtime, but not a F -stopping time. Generally speaking however, τ and V are related one toanother (via W V ). And from the Doob-Meyer decomposition of D , its G -intensity is givenby λ G t = (1 − D t ) λ t [10]. Under F , the intensity is simply λ . Since the random time τ isconstructed through a Cox process, H- Hypothesis which state that every F -local martingaleis also a G -local martingale holds between the filtrations F and G , F ⊂ G (from[2], and [10]Proposition 5 . . . . . . t survival probability to survive up to time T implied by the model is given by P ( t, T ) := Q ( τ > T |G t ) = { τ>t } E [ S T |F t ] S t = { τ>t } G t ( T ) G t ( t ) (3)where G t ( T ) := Q ( τ > T |F t ) is known as the risk-neutral survival probability in the filtration F and the survival process S t := G t ( t ) = Q ( τ > t |F t ) is the Az´ema supermartingale (see [8]for more details). Usually G ( T ) is parametrized as P M (0 , T ) = e − (cid:82) T h ( s ) ds , where h > E to denote expectationunder Q .To avoid arbitrage opportunities, one needs to calibrate the model curve to the market curve,i.e. make sure that P (0 , t ) = G ( t ) = P M (0 , t ) for all t > S t = e − Λ t .The Az´ema supermartingale associated to this type of models has a special Doob-Meyer5ecomposition: it is decreasing, meaning that the martingale part vanishes. Other types ofdefault models exist for which the martingale part is non-zero, see e.g. [9]. A convenient way to define the intensity process λ is to set λ t = k ( X t ) where k is a givenpositive function continuous on (0 , ∞ ) and X follows a Cox-Ingersoll-Ross (CIR) SDE dX t = κ ( β − X t ) dt + η (cid:112) X t dW λt , X = x > . (4)By doing so, the intensity process becomes (a function of) a mean-reverting square-rootprocess X with speed of mean reversion κ , long-term mean β and volatility η , usuallychosen to satisfy the Feller constraint 2 κβ > η .In order to describe the appearance of positive jumps in the default intensity process,we consider the jump-diffusion CIR model (JCIR) defined as dX t = κ ( β − X t ) dt + η (cid:112) X t dW λt + dJ t , X = x (5)where J t := N t (cid:88) i =1 Y i , t ≥ , (6) N t is a Poison process with intensity ω > Y , Y , . . . a sequence of identically dis-tributed exponential random variables with mean 1 /α , α >
0, independent of the PoissonProcess N t and W .A common choice is to consider k ( x ) = x , in which case the intensity is driven by CIR orJCIR dynamics respectively defined in equations (4) and (5). Adding non-negative jumpsindependent from W λ in the SDE (4) increases the volatility of the intensity process. Thesetwo choices belong to the class of Affine models: the time- t survival probability curve takesthe simple form P CIR ( t, T ) = { τ>t } A ( t, T ) e − B ( t,T ) X t (7)6nd P JCIR ( t, T ) = { τ>t } ¯ A ( t, T ) e − ¯ B ( t,T ) X t (8)for some deterministic functions A, B , ¯ A and ¯ B (see [4] for more details). Shifting the process X in a time-dependent way does not affect the above relationship as long as the shift isdeterministic but provides full flexibility in terms of calibration capabilities. Therefore, onetypically consider λ t = X t + ψ ( t ) where X is a CIR or JCIR process and ψ is chosen such thatthe model and market survival probability curves coincide at inception: P (0 , t ) = P M (0 , t ).The corresponding models are know as CIR++ and JCIR++, depending on whether X features jumps or not. The main advantage of adding the shift is that we can fit exactly anyterm structure of hazard rates and derive analytical formulas both for bonds and Europeanoptions. In particular, the CIR++ and JCIR++ models remain affine, we just need toreplace A by Ae − (cid:82) t ψ ( s ) ds in the CIR model and similarly for ¯ A in the JCIR model. Andthe shift ψ , at any time t , is given by (for both the CIR and JCIR model) ψ ( t ) = − ddt ln P M (0 , t ) P (0 , t ) . The weakness of this approach is that we can guarantee the positivity of intensities onlythrough restrictions on model parameters such that ψ ≥
0. Indeed, X can take valuesarbitrarily close to zero, so that the condition λ ≥ Q -a.s. is equivalent to saying ψ ≥ ψ has to correct the function P so as it sticks, thanks to the shift,to the target function P M (0 , . ). Given a set of model parameters for X , nothing prevents ψ to become negative, in general. But should it take negative values, the resulting modelfails to be a Cox-type, and the H- hypothesis does not hold anymore. This problem is ofhigh importance in practice. Indeed, the CIR model has low volatility to fit CDS curves,and increasing the volatility just breaks the Feller condition. It is possible to increase thevolatility without breaking the Feller constraint using JCIR. However, the affine form of theJCIR model requires the jumps to be independent from the diffusion part, so that positivityallows for upwards jumps only. This of course tends to increase the mean of λ , and hence7o decrease P JCIR (the shift can go quickly negative leading to negative intensities whichis inconsistent to the Cox model). Reciprocally, fitting a given target curve P M (0 , . ) withnon-negative shift only puts constraints on the jump sizes/rates, hence on the attainablevolatility.However, one cannot increase the activity of J without bounds. By doing so indeed, thecalibration constraint P (0 , t ) = P M (0 , t ) drives the implied shift function ψ downwards. As ψ cannot take negative values, there is a strong limit on the jump rates and/or sizes thatone can use while preserving the consistency of the model.In the next section, we propose an alternative model that is less subjected to suffer fromthe above problems. As explained earlier, the fact that JCIR jumps rapidly pushes ψ downwards results from thefact that the jumps are positive only. This would not be the case if the jumps could go inboth directions. Yet, it is not enough just to use symmetric jumps in JCIR++: this wouldbreak the positiveness of X if J is independent from W λ .One possibility consists in modeling λ as a time-changed version of a standard intensityprocess like X . On that respect, we define the time-changed CIR (TC-CIR) model bysubordinating the CIR process X t in (4) with a jump-process θ t := t + J (cid:48) t (9)where J (cid:48) is a compound Poison process independent of W defined as in (6) but with aPoisson process N (cid:48) t instead of N t . That is, we define a new process X θ by X θt = X θ t where θ is the stochastic clock defined above. If the stochastic clock features jumps, the resultingtime-changed process would still be positive, and would feature jumps in both directions. It is possible that θ is a subordinator with drift a > θ t = at + J (cid:48) t ) but we focus here on the case a = 1. λ avoiding the implied shift of thetime-change model to become negative too quickly as the jumps activity increases. As X θ is no longer affine, we need to apply the procedure developed by Mendoza-Arriaga andLinetsky [11] to get a closed formula for the survival probability. This approach is a time-changed CIR default intensity by mean of subordination in the sense of Bochner [11]. Basedon a Cox model, it is analytically tractable by means of eigenfunction expansions of relevantsemigroups, yielding closed-form pricing of defaultable zero coupon bonds. Consider the corresponding time-changed probability space (Ω , F θT , F θ , Q ) with F θt = F θ t and F θ = ( F θ t ) t ≥ . To introduce the time change defaultable market, we consider the defaulttime as defined in (2) in order to determine the corresponding intensity of the time-changemodel. Let’s define the corresponding indicator process of D by D θt := { τ ≤ θ t } , t ≥
0. Tointroduce the time-change filtration, we need first to define an inverse subordinator process( L t := inf { s ≥ θ s > t } , t ≥ L = ( L t ) t ≥ be its completed natural filtrationand H = ( H t ) t ≥ the enlarged filtration with H t = G t ∨ L t . We then define our time-changed filtration H θ = ( H θt ) t ≥ by H θt = H θ t . Hence, the time-changed bivariate process( X θt , D θt ) t ≥ is H θ -adapted and c`adl`ag and is an H θ -semimartingale (see [11] for details).In this setup, from the Doob-Meyer decomposition of D θ , our time-changed intensity is(see Theorem 3 . λ H θ t = (1 − D θt ) λ θt , λ θt = k θ ( X θt ) with k θ ( x ) = k ( x ) + (cid:90) (0 , ∞ ) (cid:16) − A (0 , s ) e − B (0 ,s ) x (cid:17) ν ( ds )where we set k ( x ) = x (as in the CIR intensity model), ν ( ds ) = ωαe − αs ds and A, B are thesame as in (7). Hence, if k θ is a function from R + to R + and X is an intensity process, λ θ defines a new intensity process and can be used to define a new default time using the Coxframework used above: τ θ := inf (cid:26) t ≥ (cid:90) t λ θs ds ≥ E (cid:27) { τ θ ≤ t } ≡ { τ ≤ θ t } Q - a . s . , with D θt = { τ θ ≤ t } . In this setup, the new time- t survival probability to survive up to time T is P θ ( t, T ) := Q (cid:0) τ θ > T |H θt (cid:1) = { τ θ >t } E [ S θT |F θt ] S θt = { τ θ >t } G θt ( T ) G θt ( t ) (10)where S θt = Q ( τ θ > t | F θt ) = e − (cid:82) t λ θs ds is the Az´ema supermartingale and G θt ( T ) = Q ( τ θ >T | F θt ) the risk-neutral survival probability in the time-change model.In a multivariate setup in general and in the specific case of CVA application in particular,the time-change approach presents a problem. Indeed, λ θ is an intensity, which typicallycan be correlated with other processes (e.g. V and B ). If we correlate these Brownianmotions, two problems arise. First, because of the time change, the correlation between theintensity λ θ and ( V, B ) is partially destroyed. Indeed, V t and B t depend on W V and W B on[0 , t ] whereas the intensity λ θ depends on W λ on [0 , θ t ] with θ t ≥ t . This methodology thusimpacts negatively the dependence between the processes λ and ( B, V ) as the intensity orthe size of jumps of θ increases. Another problem, probably even more important, is relatedto arbitrage opportunities. The knowledge of λ θ at t contains information on W λ up to θ t .If ρ (cid:54) = 0, this introduces a forward looking effect on V . It is therefore important to workwith a model in which all processes remain synchronized, but without changing the law of B and V as originally specified. To do this, it is enough to rebuild new Brownian motions( (cid:102) W V , (cid:102) W B ) so that the increments of ( B, V ) remain synchronized with those of λ θ . Lemma 3.1.1.
Let W be a Brownian motion and t i be the time of the i th jump of thePoisson process N (cid:48) t . Then the process (cid:102) W t := N (cid:48) t − (cid:88) i =0 (cid:90) t − i +1 ∧ tt i ∧ t dW θ s + ( W θ t − W θ tN (cid:48) t ) (11) is an F θ -Brownian motion and behaves exactly as W sampled on the time grid.Proof. (cid:102) W t is a continuous F θ -local martingale with (cid:102) W = 0 and for every t ≥ (cid:104) (cid:102) W (cid:105) t = t .By the L´evy’s characterization theorem, the process ( (cid:102) W t ) t ≥ is an F θ -Brownian motion.10he dynamics of V can thus be equivalently described in terms (cid:102) W V setting W = W V on F θ as: d (cid:101) V t = b ( (cid:101) V t ) dt + σ ( (cid:101) V t ) d (cid:102) W V , (cid:101) V = V . (12)Applying the same procedure of Lemma 3.1.1 on W B , the corresponding copy of W B on thetime changed grid is (cid:102) W B which will govern the new dynamics of the risk-free rate (cid:101) r leadingto that of the bank account num´eraire given by d (cid:101) B = (cid:101) r t (cid:101) Bdt, (cid:101) B = 1 . Remark.
It is worth stressing the fact that keeping W V as the driver of the exposure process V (instead of (cid:102) W V ) does not completely destroy the WWR effect resulting from the correlation ρ between W V , W λ . The reason is that even if the instantaneous correlation between theinfinitesimal increments of λ θ and V are mutually independent as from the first jump of θ ,the correlation between λ θt and V t is non-zero for any t > whenever ρ (cid:54) = 0 . This is becausethese processes depend on the integrals of increments of Brownian motions on some timeintervals. Between two jumps of the clock in particular, the Brownian increments drivingthe change in the intensity process λ θ can be independent from the Brownian incrementsdriving the exposure V on a same time period, but the increments of the first Brownianmotion on a given time interval can be dependent on the second Brownian motion on anotherinterval. This explains that two processes λ θ and V can be dependent on each other even ifthe instantaneous correlation of their increments vanishes because of the time-change. Bycontrast, the correlation between λ θt and V t is lower than that of λ θt and (cid:101) V t : by synchronizingthe changes of the Brownian motion driving V with the one driving λ θ , one maximizes theattainable correlation. For instance, let W, B be two Brownian motions with instantaneouscorrelation ρ , i.e. d (cid:104) W, B (cid:105) t = ρdt and define B δ as B δt = B t + δ where δ > . The correlationbetween the increments of W and B on the interval [ s, t ] is ρ whereas that between W and B δ is ρ ( t − ( s + δ )) + / ( t − s ) whose absolute value is no greater than | ρ | . Theorem 3.1.2.
Discounted payoffs driven by (cid:101) V and (cid:101) B are H θ -martingales under Q . roof. We know that the discounted payoffs driven by V and B are ( Q , F )-martingales,hence they are ( Q , G )-martingales due to the H- Hypothesis . By construction, (cid:101) V and (cid:101) B havethe same F θ dynamics as V and B under F (see Lemma 3.1.1). Hence, the discountedpayoffs driven by (cid:101) V and (cid:101) B are ( Q , F θ )-martingales. Because τ θ is modelled with a Coxprocess, immersion holds and they remains martingales when progressively enlarging F θ with τ θ . This shows that they are ( Q , G θ )-martingales, hence ( Q , H θ )-martingales (since L θ t = t ).As explained earlier, time-changing the intensity can introduce a forward-looking effectand thus arbitrage opportunities. Indeed, prices of non-dividend paying asset discounted atthe risk-free rate would not be a martingale under Q in the natural filtration generated by( W V , W λ θ ) when ρ (cid:54) = 0. This is formalized in the next lemma proven in the Appendix. Lemma 3.1.3.
Suppose that V is a martingale with respect to F W V , the natural filtrationof W V . It is also a martingale with respect to F W V ∨ F W λ . We note W λ θ the time-changedversion of W λ . For instance suppose that for some s ∈ [0 , t ] , we have s < θ s < t . Then, X may not be a F W V ∨ F W λθ -martingale. To define the shifted time-change CIR (TC-CIR++) model, we need to have a closed formof the survival probability of the TC-CIR model. For that, we only need to compute theLaplace transform of our time-change process θ in order to apply the idea devised in [11]. Laplace transform of a L´evy subordinator:
Our time-change jump-process θ is aL´evy subordinator and its Laplace transform can be found easily using the L´evy-Khintchineformula [1]. For any u ∈ R , the Laplace transform of θ reads E [ e − uθ t ] = E [ e − u ( t + J t ) ] = e − ut E [ e − uJ t ] . E [ e − uJ t ] = e − tϕ ( u ) where ϕ is the L´evy exponent given by ϕ ( u ) = (cid:90) R ‘ { } (1 − e − us ) ωαe − αs ds { s> } = (cid:90) (0 , ∞ ) ωα (1 − e − us ) e − αs ds. It comes that E [ e − uθ t ] = e − tφ ( u ) , φ ( u ) = u (cid:18) u + α + ωu + α (cid:19) . Knowing φ , the time-changed survival probability takes the closed form P θ ( t, T ) = { τ θ >t } ∞ (cid:88) n =1 e − φ ( λ n )( T − t ) f n (0) ϕ n ( X θt )where λ n , f n and ϕ n are given in [11].The TC-CIR++ model is obtained by defining the time-changed intensity process as λ θt = k θ ( X θt ) + ψ θ ( t ) and finding ψ θ such that P θ (0 , T ) = G θ ( T ) = P M (0 , T ). Hence ψ θ ( t ) = − ddt ln P M (0 , t ) P θ (0 , t ) . The reduced form approach relies on a change of filtrations. In this section, we derive theCVA formulas in both cases where the default intensity is given by the square-root diffusionsor the time-change model.CVA attempts to measure the expected loss due to missing the remaining payments ofthe OTC portfolio. Its mathematical expression is given in a risk-neutral pricing frameworkbased on a no-arbitrage setup. Let R be the recovery rate of the counterparty and theexposure processes V and (cid:101) V respectively given by (1) and (12). For the mathematical proofof the following CVA expressions, we refer to [10].13 .1 CVA formula in the diffusion model From the H-
Hypothesis , payoffs driven by V and B are G -martingale. In addition, since V /B is Q -integrable and F -predictable, assuming τ > R ,the time- t CVA expression readsCVA t = { τ>t } B t E (cid:20) (1 − R ) V + τ B τ { τ ≤ T } (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) = − { τ>t } B t S t E (cid:34) (1 − R ) (cid:90) Tt V + u B u dS u (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . (13)Discretizing the integral with a numerical scheme, the Monte-Carlo estimation of time-0CVA becomes (cid:91) CVA := − (1 − R ) 1 m m (cid:88) i =1 n (cid:88) k =1 V + , ( i ) t k B ( i ) t k ∆ S ( i ) t k , n = Tδ (14)where ∆ S ( i ) t k = S ( i ) t k − S ( i ) t k − . The right-hand side results from Monte Carlo approximation,by taking the sample mean of m time-integrals discretized in n intervals of length δ .In the specific case where τ is independent from the discounted exposure (i.e. ρ = 0),the independent CVA formula is given byCVA ⊥ = − (1 − R ) (cid:90) T E (cid:20) V + u B u (cid:21) dG ( u ) . (15)In other words, CVA only depends separately on the expected discounted exposure E (cid:104) V + u B u (cid:105) and the prevailing mrisk-neutral survival probability curve G ( . ). We refer to [6] for moredetails.Generally speaking however the later expression does not hold, and CVA depends onthe joint dynamics of the exposure and credit worthiness. We cannot get to such a simplerformula as in (15) and we have to take in account the dependency between credit andexposure. Wrong way risk (WWR) is the additional risk related to this dependency.14 .2 CVA formula in the time-change model As (cid:101) V / (cid:101) B is Q -integrable and F θ -predictable, assuming τ θ > (cid:101) V , (cid:101) B ) instead of ( V, B ) (having the same dynamics under F θ as ( V, B ) under F ), but using the intensity of τ θ with Az´ema supermartingale S θ , wehave CVA t = { τ θ >t } (cid:101) B t E (cid:34) (1 − R ) (cid:101) V + τ (cid:101) B τ { τ θ ≤ T } (cid:12)(cid:12)(cid:12)(cid:12) H θt (cid:35) = { τ θ >t } (cid:101) B t E (cid:34) (1 − R ) (cid:101) V + τ (cid:101) B τ { τ θ ≤ T } (cid:12)(cid:12)(cid:12)(cid:12) ( τ θ ≤ t ) ∨ F θt (cid:35) = − { τ θ >t } (cid:101) B t S θt E (cid:34) (1 − R ) (cid:90) Tt (cid:101) V + u (cid:101) B u dS θu (cid:12)(cid:12)(cid:12)(cid:12) F θt (cid:35) . (16)Therefore, the time-0 CVA in the time-changed model can be approximated using m pathsof Monte Carlo simulations as (cid:91) CVA := − (1 − R ) 1 m m (cid:88) i =1 n (cid:88) k =1 (cid:101) V + , ( i ) t k (cid:101) B t k ∆ S θ, ( i ) t k , n = Tδ . (17)In the specific case where τ θ is independent from the discounted exposure, we obtain the independent CVA formula in the time-changed modelCVA ⊥ = − (1 − R ) (cid:90) T E (cid:34) (cid:101) V + u (cid:101) B u (cid:35) dG θ ( u ) . (18)Observe that by construction of ψ and ψ θ , G ( t ) = G θ ( t ) = P M (0 , t ). Hence, CVA ⊥ agreesin either models under the calibration constraint. This meansCVA ⊥ = − (1 − R ) (cid:90) T E (cid:20) V + u B u (cid:21) dP M (0 , u ) = − (1 − R ) (cid:90) T E (cid:34) (cid:101) V + u (cid:101) B u (cid:35) dP M (0 , u ) . (19) In this section, we start by defining the simulation procedure of the bivariate process ( X θ , (cid:101) V ).The CVA is computed using standard Monte Carlo simulation and the performance of theshifted time-changed model in term of WWR is compared to the CIR++ and JCIR++15tochastic intensity models. For the sake of simplicity, the recovery rate R and the interestrate r are assumed to be constant and set to zero (i.e. R = 0 and B = (cid:101) B = 1) to put thefocus and the treatment of the credit-exposure dependency. X θ , (cid:101) V ) Let’s denote by T = { , δ, δ, . . . , T } the time- t grid and T θ = { , θ δ , θ δ , . . . , θ T } the gridat time θ t . To refine the grid T θ , we define grids T θi , i = 1 , . . . , n , as T θi := n i − (cid:91) j =1 (cid:26) θ t i − + j ∆ θ t i n i (cid:27) , n i = (cid:24) ∆ θ t i δ (cid:25) , T θfine := T θ (cid:91) (cid:40) n (cid:91) i =1 T θi (cid:41) The simulation grid T θfine contains T θ and is completed in such a way that (after sorting),the step between two consecutive points is no greater than the chosen time step δ to keepcontrol on the discretization error independently of the jump sizes of θ .To simulate X θ , we simulate the CIR process X in (4) on T θfine using Diop’s scheme [7]:¯ X ( i +1) δ = ¯ X iδ + κ ( β − ¯ X + iδ ) δ + η (cid:113) ¯ X + iδ ( W λ ( i +1) δ − W λiδ ) , ¯ X = X , i = 0 , . . . , n − .X θ is obtained by extracting in T θfine the corresponding values of X on the grid T θ .To obtain (cid:101) V , we simulate W V on T θfine , extract its corresponding values on T θ and userespectively (11) and (12). In this section, we compare the performances of the three models studied above in termsof WWR impact for a simple forward-type Gaussian exposure and Brownian swap bridgeexposure. We fix the CIR parameters ( κ, β, η, x ) as in [6] and search for the jump parameters( ω, α ) of JCIR (affecting directly the intensity) and of TC-CIR (affecting the stochasticclocks and only indirectly the jumps in the intensity) such that ψ ≥ .2.1 Brownian exposure We set the coefficients of the exposure dynamics in eq. (1) to b ( V t ) ≡ σ ( V t ) ≡ σ and V = 0 which leads to dV t = σdW Vt . This example illustrates a 3 years forward contract or total return swap, which does notpay dividends. In Figure 1 we plot the CVA in function of the correlation ρ for the treeconsidered models. We notice that due the the shift constraint, the JCIR model has slightlymore WWR impact than the CIR model (because of the jumps) while the TC-CIR modelhas a larger WWR impact. −1.0 −0.5 0.0 0.5 1.0 . . . . . . r C VA r = 0CIRJCIRTC−CIR (a) CIR (0 . , . , . , . , JCIR (0 . , . . , . −1.0 −0.5 0.0 0.5 1.0 . . . . . . r C VA r = 0CIRJCIRTC−CIR (b) CIR (0 . , . , . , . , JCIR (0 . , . . , . Figure 1: CVA figures, 3Y Gaussian exposure, σ = 8%. Hazard rate is h ( t ) = 5%. The drifted Brownian bridge is obtained by choosing in (1) b ( V t ) ≡ γ ( T − t ) − V t T − t , σ ( V t ) ≡ σ and V = 0, so that dV t = (cid:20) γ ( T − t ) − V t T − t (cid:21) dt + σdW Vt where γ stands for the future expected moneyness of an interest rate swap implied by theforward curve and σ controls the exposure volatility. We get similar results in term of WWR17mpact as in the previous example. −1.0 −0.5 0.0 0.5 1.0 . . . r C VA r = 0CIRJCIRTC−CIR (a) CIR (0 . , . , . , . , JCIR (0 . , . . , . −1.0 −0.5 0.0 0.5 1.0 . . . r C VA r = 0CIRJCIRTC−CIR (b) CIR (0 . , . , . , . , JCIR (0 . , . . , . Figure 2: CVA figures, 3Y Swap exposure, ( σ, γ ) = (8% , . h ( t ) = 5%. The standard Monte Carlo method applied to the TC-CIR++ model is time consuming.The reason is that the time-step needs to be kept relatively small (to limit the discretizationerrors) whereas the simulation horizon is governed by θ T which can be much larger than T . In order to reduce the computational cost, we propose to adopt a variance reductiontechnique called adaptive control variate . In this section, we briefly recall the idea of controlvariate, describe its adaptive implementation and then transpose it to our CVA application.Our purpose is to find an estimator of E [ Y ] from m i.i.d. observations of Y . The unbiasedsample-mean estimator of E [ Y ] is ˆ Y := m (cid:80) mk =1 Y k . The idea of control variate consists offinding an alternative unbiased estimator ˜ Y with lower variance compared to ˆ Y by using acontrol variate Z with known expectation. Consider a generic pair ( Y, Z ) of random variableswith i.i.d copies ( Y k , Z k ) , k ∈ { , , . . . , m } and define Y µk := Y k − µ Ξ k , Ξ k := Z k − E [ Z ] . (20)The sample-mean estimator of Y µk is an alternative unbiased estimator of E [ Y ], and is given18y ˆ Y µ = 1 m m (cid:88) k =1 Y µk = 1 m m (cid:88) k =1 ( Y k − µ Ξ k ) . Its variance is equal to that of ˆ Y for µ = 0 but is minimum for µ = µ ∗ where µ ∗ := Cov( Y, Ξ)Var(Ξ) = E [ Y Ξ] E [Ξ ] . Because Var( ˆ Y µ ∗ ) = (1 − Corr ( Y, Ξ))Var( ˆ Y ), this approach is interesting when choosingthe control variate Ξ highly correlated with Y .In practice however, the optimal constant µ ∗ needs to be itself estimated. The adaptive control variate uses a different value for µ for every index k ∈ { , , . . . , m } : V k := 1 k k (cid:88) i =1 Ξ i , C k := 1 k k (cid:88) i =1 Y i Ξ i and µ k := C k V k , where µ := 0 and µ k − is the best estimator of µ ∗ at step k (see [12] for more details).Eventually, the adaptive control variate estimator of E [ Y ] is given by˜ Y µ := 1 m m (cid:88) k =1 Y µ k − k = ˆ Y − m m (cid:88) k =1 µ k − Z k + E [ Z ] m m (cid:88) k =1 µ k − . In our CVA application, we are interested in estimating CVA which is nothing but E [ Y ] with Y = − (cid:82) T V + u dS u in the CIR and JCIR models (13) or Y = − (cid:82) T (cid:101) V + u dS θu in theTC-CIR model (16). We take as control variable Z = − (cid:82) T V + u dS ⊥ u or Z = − (cid:82) T (cid:101) V + u dS θ, ⊥ u ,respectively, where S ⊥ (resp. S θ, ⊥ ) is the survival process S (resp. S θ ) associated to theintensity λ (resp. λ θ ) simulated using W ⊥ instead of W λ . In light of the above development,this choice is appealing because Z is correlated with Y (via the exposure process as wellas the W ⊥ component of the survival process) whereas the expectation of Z is known inclosed form and corresponds to CVA ⊥ given in (19). In the adaptive procedure, the m pairs ( Y k , Z k ) are given by integrals ( Y, Z ) over the corresponding scenario. They are i.i.d.copies of Y and Z (up to discretization error resulting from the integral computation and Alternatively, if the trajectories of V and S are stored, it is enough to shuffle those of V (or S ), so asto combine, in the CVA computation, the i -th exposure’s sample path with the π ( i ) (cid:54) = i survival process’sample path. adaptive CVA estimator for the CIR and JCIR modelsreads (cid:93)
CVA = (cid:91) CVA − CVA ⊥ m m (cid:88) k =1 µ k − + 1 m m (cid:88) k =1 n (cid:88) j =1 µ k − V + , ( k ) t j ∆ S ( k ) t j (21)with (cid:91) CVA given in (14) and (17), respectively. The TC-CIR estimator takes a similar formprovided that one replaces ( V, S ) by ( (cid:101)
V , S θ ).In figure 3, we compare the confidence interval at level 95% of the CVA computationresulting from standard Monte Carlo (MC) and adaptive control variate (CV). We applythis technique for the three considered models (CIR, JCIR and TC-CIR) by using the sameset of parameters in figure 1 (a) with a Gaussian exposure. Similar results are detailed inthe Appendix when using the same Gaussian exposure profile but with the parameters offigure 1 (b). 20 e+00 2e+04 4e+04 6e+04 8e+04 1e+05 . . . . CIR, r = 0.8 m C VA MC CV (a) CIR (0 . , . , . , . . . . . CIR, r = 0.4 m C VA MC CV (b) CIR (0 . , . , . , . . . . . JCIR, r = 0.8 m C VA MC CV (c) CIR (0 . , . , . , . , JCIR (0 . , . . . . JCIR, r = 0.4 m C VA MC CV (d) CIR (0 . , . , . , . , JCIR (0 . , . . . . . TC−CIR, r = 0.8 m C VA MC CV (e) CIR (0 . , . , . , . , TC-CIR (0 . , . . . . . TC−CIR, r = 0.4 m C VA MC CV (f) CIR (0 . , . , . , . , TC-CIR (0 . , . Figure 3: Control variate CVA figures, 3Y Gaussian exposure, σ = 8%. h ( t ) = 5%.21e observe clearly that the variance reduction technique adopted allows to reduce signif-icantly the computational of the standard Monte Carlo as a solution to CVA computationin presence of WWR. Since Y and and the chosen control Z are more correlated as | ρ | decreases, we observe that the variance is reduced again when | ρ | is decreasing and theconvergence of the adaptive estimator is faster in this case. Among the reduced-form intensity models, affine models like CIR++ process received muchattention. The latter consists of a time-homogeneous mean-reverting square-root diffusionshifted in a deterministic way so as to fit a given probability term-structure. In order toincrease the attainable volatilities, one can add jumps to the CIR++ dynamics. If the jumpsare independent and positive, one obtains the so-called JCIR++ model, which remains affine.The problem however is that the model-implied survival probability curve decreases whenincreasing the activity of the jumps because they are one-sided. The calibration of the modelcurve to the market curve being achieved via the shift function, the latter decreases whenincreasing jumps’ activity. Consequently, in order to avoid facing “negative intensities”, oneis limited in the activity of the jumps that can be used. For instance, this specificity limitsthe attainable values for value-at-risk on CDS, CDS options or wrong-way risk CVA.An alternative intensity model that allows for two-side jumps is the time-changed idea ofMendoza-Arriaga & Linetsky. Because jumps can be both positive and negative, the model-implied survival probability curve is expected to decrease less rapidly when increasing thejumps’ activity compared to the JCIR++. Hence, the problem of facing negative shift func-tion (i.e. “negative intensities”) is expected to be less severe, so that larger “volatility‘seffect” could be generated. This motivates the use of the above model for CVA purposes.Yet, using the time-changed intensity approach in a multivariate framework requires spe-cific precautions. Without specific adjustments indeed, the time-change technique partly22estroys the potential correlation between intensity and exposure which impacts negativelythe attainable WWR effect. More importantly, it features forward-looking effects that cangenerate arbitrage opportunities. In this paper, we have shown how the time-changed modelcan be used in a consistent and efficient way by reconstructing the exposure dynamics ina “synchronous” way such that the above problems can be avoided. The computationalissue inherent to the time-changed technique is tackled by proposing a variance reductiontechnique based on adaptive control variate. Eventually, numerical simulations show thatunder calibration constraint to a given term structure, the time-change model can give largerWWR effects compared to the CIR++ and JCIR++ models.Analyzing the model’s ability to generate higher CDS spread’s volatility is another impor-tant question from both risk-management and pricing perspectives, which is left for futurework.
The research of Cheikh Mbaye is funded by the
National Bank of Belgium via an
FSR grant. We would like to thank M. Jeanblanc for discussion on the immersion propertywhen dealing with jumps and time-changed processes. We are also so grateful to G. Pag`esfor discussions about the adaptive control variate method. Eventually, we thank LauraBallotta for stimulating discussions about a previous version of this work.The opinions expressed in this paper are those of the authors and do not necessarilyreflect the views of the
National Bank of Belgium . In this section, we give the proof of Corollary 3.1.3 and the variance reduction figures usingthe parameter set of figure 1 (b). 23 roof.
Let V t = V s e − σ ( t − s )+ σ ( W Vt − W Vs ) . Clearly, because V is adapted to F W V , E (cid:104) V t | F W V s ∨ F W λ s (cid:105) = E (cid:104) V t | F W V s (cid:105) = V s e − σ ( t − s ) E (cid:104) e σ ( W Vt − W Vs ) (cid:105) = V s However, the increments of W V after θ s > s are independent both from F W V s and from F W λ θ s := F W λθ s . Hence, E (cid:20) V t | F W V s ∨ F W λθ s (cid:21) = V s e − σ ( t − s ) E (cid:104) e σ ( W Vt − W Vs ) |F W λ θ s (cid:105) = V s e − σ ( t − s ) e σ ( t − θ s ) E (cid:104) e σ ( W Vθs − W Vs ) |F W λ θ s (cid:105) = V s e − σ ( θ s − s ) E (cid:104) e σ ( W Vθs − W Vs ) |F W λ θ s (cid:105) Assuming in the sequel that ρ (cid:54) = 0, E (cid:104) e σρ [( W λθs − W λs ) − √ − ρ ( W ⊥ θs − W ⊥ s )] |F W λ θ s (cid:105) = e σρ ( W λθs − W λs ) E (cid:20) e − σ √ − ρ ρ ( W ⊥ θs − W ⊥ s ) |F W λ θ s (cid:21) Observe that W ⊥ θ s − W ⊥ s is not independent from F W λ θ s as this information set gives us thevalue of W λθ s − W λs : W λθ s − W λs = ρ (cid:0) W Vθ s − W Vs (cid:1) + (cid:112) − ρ (cid:0) W ⊥ θ s − W ⊥ s (cid:1) The computation of the above conditional expectation amounts to evaluate the momentgenerating function (MGF) ϕ (cid:18) − σ √ − ρ ρ (cid:19) associated to the Normal variable W ⊥ θ s − W ⊥ s forwhich one knows the value of its weighted sum with another independent variable. Moreexplicitly, we are looking for the MGF of X = (cid:112) − ρ √ θ s − sZ such that X + Y = W λθ s − W λs with Y = ρ √ θ s − sZ , Z , Z iid standard Normal. It can be shown (see eg [13])that, given X + ω Z = c , X = ω Z ∼ N (˜ µ, ˜ σ ) with ˜ σ = (cid:0) ω − + ω − (cid:1) − and ˜ µ = c ˜ σ /ω .Using the values of ω , ω and c = W λθ s − W λs , ϕ ( t ) = e (1 − ρ )( W λθs − W λs ) t + ρ (1 − ρ )( θ s − s ) t E (cid:20) V t | F W V s ∨ F W λθ s (cid:21) = V s e − σ ( θ s − s ) e σρ ( W λθs − W λs ) e (1 − ρ )( W λθs − W λs ) (cid:18) − σ √ − ρ ρ (cid:19) × e ρ (1 − ρ )( θ s − s ) (cid:18) − σ √ − ρ ρ √ (cid:19) = V s e σρ (cid:104) − (1 − ρ ) (cid:105) ( W λθs − W λs )+ σ [ (1 − ρ ) − ] ( θ s − s ) (cid:54) = V s . e+00 2e+04 4e+04 6e+04 8e+04 1e+05 . . . . CIR, r = 0.8 m C VA (a) CIR (0 . , . , . , . . . . . CIR, r = 0.4 m C VA MC CV (b) CIR (0 . , . , . , . . . . JCIR, r = 0.8 m C VA MC CV (c) CIR (0 . , . , . , . , JCIR (0 . , . . . . . JCIR, r = 0.4 m C VA MC CV (d) CIR (0 . , . , . , . , JCIR (0 . , . . . . . TC−CIR, r = 0.8 m C VA J C I R MC CV (e) CIR (0 . , . , . , . , TC-CIR (0 . , . . . . . TC−CIR, r = 0.4 m C VA MC CV (f) CIR (0 . , . , . , . , TC-CIR (0 . , . Figure 4: Control variate CVA figures, 3Y Gaussian exposure, σ = 8%. h ( t ) = 5%.26 eferences [1] D. Applebaum. L´evy processes and stochastic calculus, second edition. CambridgeUniversity Press , 2009.[2] T. Bielecki, M. Jeanblanc, and M. Rutkowski. Credit risk modeling.
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