Affine term structure models : a time-changed approach with perfect fit to market curves
AAffine term-structure models : a time-changedapproach with perfect fit to market curves
Cheikh Mbaye & Fr´ed´eric Vrins ∗ Louvain Finance Center (LFIN), UC Louvain, Belgium
Abstract
We address the so-called calibration problem which consists of fitting in a tractable waya given model to a specified term structure like, e.g., yield, prepayment or default probabil-ity curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possiblycoupled with compound Poisson jumps, JCIR, a.k.a. SRJD), are tractable processes buthave limited flexibility; they fail to replicate actual market curves. The deterministic shiftextension of the latter, Hull-White or JCIR++ (a.k.a. SSRJD) is a simple but yet effi-cient solution that is widely used by both academics and practitioners. However, the shiftapproach may not be appropriate when positivity is required, a common constraint whendealing with credit spreads or default intensities. In this paper, we tackle this problem byadopting a time change approach, leading to the TC-JCIR model. On the top of providingan elegant solution to the calibration problem under positivity constraint, our model featuresadditional interesting properties in terms of variance. It is compared to the shift extensionon various credit risk applications such as credit default swap, credit default swaption andcredit valuation adjustment under wrong-way risk. The TC-JCIR model is able to gener-ate much larger implied volatilities and covariance effects than JCIR++ under positivityconstraint, and therefore offers an appealing alternative to the shift extension in such cases.
Keywords: model calibration, credit risk, stochastic intensity, jump-diffusions, term-structuremodels, time-change techniques ∗ Voie du Roman Pays 34, B-1348 Louvain-la-Neuve, Belgium. E-mail: [email protected]. Theresearch of Cheikh Mbaye is funded by the National Bank of Belgium and an FSR grant. The opinions expressedin this paper are those of the authors and do not necessarily reflect the views of the National Bank of Belgium.The work of F. Vrins was supported by the Fonds de la Recherche Scientifique F.S.R.-FNRS, Grant J.0037.18. a r X i v : . [ q -f i n . M F ] J a n Introduction
Model calibration is a standard problem in many areas of finance Brigo and Mercurio (2006);Joshi (2003); Veronesi (2010). It consists of tuning a model such that it “best fits” marketquotes at a given time. As an example, financial markets provide a set of prices associatedwith liquid instruments, that openly trade on the market. Alongside with risk management(hedging), the main purpose of a model here is to act as an “interpolation/extrapolation” tool,i.e., to obtain the value of products at a given time t for which the market does not discloseprices in a transparent way. This could happen because either the product to be priced is“exotic” (i.e., is too “special”, it does not quote openly on a platform, only on a bilateral basis)or because its cashflow schedule is not in line with the products that currently trade openly at t (a situation that commonly happens since products that were “standard” at inception, mayhave time-to-expiry or moneyness levels that are no longer “standard” afterwards).Mathematically, model calibration is nothing but an optimization problem. Starting froma set of prices quoted on the market (called “market prices”) for a set of specific financialproducts (called “calibration instruments”), model calibration consists of computing the modelparameters such that the prices generated by the model (called “model prices”) best fit tothe market prices, according to some error function. Model calibration is crucial in finance;it is strongly related to arbitrage opportunities. In practice, only models that are able toreproduce the market prices of “simple instruments” (either in a perfect way, or at least up tothe bid-ask spread) are trustworthy enough when it comes to pricing other instruments. Forinstance, one can price exotic derivatives (like barrier options) using stochastic volatility modellike Heston in a semi-analytical way Carr and Madan (1999); Heston (1993). The parametersof the Heston model will be obtained by “calibration” to a volatility surface, i.e., to a set ofliquid (“plain vanilla”) options, like European calls and puts of various strikes and maturities.The justification behind this is that in a no-arbitrage, complete market setup, the price of anoption can be obtained by computing the cost of setting up a self-financing hedging strategy.This cost depends on the prevailing prices of the hedging instruments. If the model fails tocorrectly price the latter, there is no chance it can correctly price the option.In this work, we focus on financial calibration problems arising in other asset classes: interest-rates and credit Brigo and Mercurio (2006); Duffie and Singleton (2003). When specifying aninterest rate model to price a derivative on, say, the Libor 3M index, one needs to make surethat the model generates a discount curve that is in line with that extracted from market quotes2f simpler Libor 3M-indexed products. In this case, the set of calibration instruments couldbe forward rate agreements (FRA), interest rate swaps (IRS), as well as vanilla cap/floors orswaptions. Similarly, adjusting the value of derivatives for counterparty risk (a problem knownas credit valuation adjustment, or CVA) generally involves a stochastic model to represent thedefault of the counterparty with whom the trade is executed. The default probability of thecounterparty can be extracted from a set of calibration instruments, which prices are drivenby the default likelihood of the counterparty, e.g., corporate bonds of credit default swaps(CDS). In this context, the default model must be “calibrated” in such a way that the defaultprobability curve generated by the stochastic model agrees with that implied from the pricesof the corresponding instruments (see, e.g., Gregory (2010) and Stein and Pong (2011) for ageneral overview of CVA and Brigo et al. (2014) for a discussion of bilateral CVA in presenceof collateralization agreements).The calibration constraint rises practical issues. Indeed, the models that are actually usedin the industry must have a tractability that is compatible with real-time pricing but, as ex-plained above, must be flexible enough to match the information conveyed by the calibrationinstruments. Affine term structure models (ATSM) have been extensively used in fixed incomemodeling because of their analytical tractability. See, e.g., Duffie et al. (2003) and Duffie andKan (1996) for an excellent review and a mathematical analysis of this class of processes. Inpractice, homogeneous affine jump diffusion (HAJD) models are extremely popular. The Va-sicek (Ornstein-Uhlenbeck) model Vasicek (1977) is a short-rate model being widely used inboth industry and academia. It is a time-homogeneous affine diffusion model that postulatesGaussian dynamics. If negative rates need to be ruled out, positive dynamics like the CIR(Cox-Ingersoll-Ross, also known as square-root diffusion, SRD) Cox et al. (1985) can be pre-ferred, possibly with independent compounded Poisson jumps (JCIR or SRJD). However, itis in general impossible with either models (even in a multi-factor setup) to achieve a perfectfit: the flexibility of HAJD is limited, they are in general unable to generate a given discountcurve. The same problem arises when dealing with credit derivatives: it is generally impossibleto make sure that the default intensity process, modeled with HAJD dynamics, will generate adefault probability curve that is in line with the corresponding curve, exogenously given by themarket via the calibration instruments.Several routes can be followed to deal with this issue. The first one consists of disregardingthis lack of flexibility. Nevertheless, working with a model that fails to yield a perfect fit to3he market is often unacceptable in practice. Indeed, as explained above, the models are usedto value derivatives positions, and a mismatch with the market can introduce a tremendousbias in the valuation of the book of companies or financial institutions. Another possibility isto significantly increase the complexity of the models. This is often to be avoided in practice,for computational, identification or over-fitting issues. A trade-off consists of extending the“simple models” in such a way that they can fit to the market. In fact, several authors showthat a great flexibility can be obtained by shifting HAJD models in a deterministic way. Dybvig(1997) for instance show that the term structure of interest rates can be reproduced by addinga deterministic shift to Ho and Lee (1986) or Vasicek processes. Later, Brigo and Merccurio(2001) extended this idea to a broader class of models, thereby providing a simple but veryclever solution to the calibration problem. Instead of considering a HAJD, one could simplyadjust it with a deterministic function ϕ : the resulting process will have the required flexibility.When shifted this way, the Vasicek, CIR and JCIR models respectively correspond to the Hull-White, the CIR++ or the JCIR++ (a.k.a. SSRJD) models Brigo and Mercurio (2006). Thistrick is actually very powerful: it solves the calibration problem at no cost, since the model’sdynamics remain affine. Moreover, the shift function is known analytically, as a function of theparameters of the underlying HAJD and the market curve to be fitted by the model.Yet, this approach suffers from an important limitation. Because of the shift, there is noreason that the range of the shifted process agrees with that of the underlying HAJD process.For instance, shifting a positive process with a deterministic function may result in a processthat could take on negative values. It all depends on the mismatch between the informationconveyed by the calibration instruments on the one hand, and the parameters of the underlyingHAJD on the other hand. In general, there is no reason to believe that the implied shiftfunction will preserve the range of the HAJD model. This is problematic in many cases, and incredit risk modeling in particular: negative default intensities, for instance, make no sense. Tocircumvent this issue, one could think of adding a non-negativity constraint on the shift in thecalibration step. But, as we will show, this drastically restricts the parameters of the underlyingHAJD, hence the randomness embedded in the model. This explains why this solution is oftennot considered by practitioners: the shift approach (without positivity constraint) remains thestandard approach, even if positivity is required, theoretically speaking. It seems that in absenceof a valid alternative, one actually prefers to rely on a model providing a perfect fit, even thoughthe latter suffers from theoretical inconsistencies.4n this paper we introduce an alternative to the deterministic shift. Using an equally simple– but intrinsically different – technique, we adjust a HAJD so as to allow for a perfect fit to agiven market curve, without affecting the model’s tractability, but also without introducing theaforementioned inconsistencies. More specifically, instead of shifting a HAJD, we time-changeit. Time change techniques were first studied in 1965 Dambis (1965); Dubins and Schwartz(1965). The first application to finance dates back from the early 2000. Geman et al. usedL´evy processes and interpreted the new time scale as the business time, in contrast with thecalendar time Geman et al. (2001). This was then applied to stochastic volatility models Carret al. (2003). Thanks to subordinated L´evy models, the authors introduced the leverage effect,as well as a long-term skew. Many other financial applications of time change techniquescan be found in the review Swishchuk (2016). More recently, Mendoza-Arriaga and Linetskyused stochastic time change processes to introduce two-side jumps in positive processes. Theanalytical tractability of the resulting model is preserved to some extend. This model has beenrecently applied to counterparty credit risk Mbaye and Vrins (2018). In this work, we exploitthe time change idea in yet another way, to solve a completely different problem. Our purposeis to time-change HAJDs so as to obtain models with the desired calibration flexibility, withoutaffecting tractability and preserving the range of the original process. The intuition is that byslowing down or speeding up the time of the latent HAJD, at the appropriate rate, one wouldobtain a model that could fit most discount curves, and actually every default probabilitycurve. Moreover, the time change function is easily found using simple numerical methods(namely, inversion of easy functions or ordinary differential equation). Eventually, our time-changed HAJD is proven to feature larger implied volatilities compared to the correspondingvalid (i.e., non-negative) shifted HAJD. To illustrate the power of our approach, we provide twoapplications taken from credit: pricing of CDS options and computation of derivatives pricingaccounting for counterparty risk under exposure-credit dependence (wrong-way risk, WWR). Ineither cases, all the considered default models perfectly fit the risk-neutral default probabilitycurve extracted from market quotes associated to the CDS of the reference entity. The obtainedresults illustrate the nice feature of large implied volatilities : they are able to generate largeroption prices compared to the shift approach calibrated on a same probability curve undernon-negativity constraint.Eventually, observe that although we focus on examples featuring reduced-form modelswhen pricing of credit-sensitive instruments, our approach is of potential use for other models,in many areas of finance and insurance. Ongoing work suggests that it can be applied to other5efault models, including the firm-value (structural) models Merton (1974), but also to linear-rational (polynomial) models Filipovic et al. (2017). Other models could be considered as well,like Jeanblanc and Vrins (2018) or Cr´epey et al. (2012).In terms of applications, the proposedmethod can be used in life insurance, to calibrate mortality rate to mortality tables. Our time-changed process could also be used to model prepayment rates in mortgage-backed securities(MBS). These products naturally exhibit a negative convexity due to a the negative relationshipbetween interest and prepayment rates: householders tend to refinance their loans when interestrates drop. This calls for a stochastic prepayment (i.e., positive) rate, that will be negativelycorrelated with interest rates, and which parameters could be calibrated so as to agree withthe averaged values given in the PSA measure, the indicator attached to MBS securities thatcharacterizes the prepayment speed in MBS Veronesi (2010). Eventually, the proposed methodcould be applied in many other applications, including the modeling of performance degradationof devices or materials through time, which average outstanding performances evolution aregiven according to some quality standards.The paper is organized as follows. In Section 2 the calibration problem is introduced andtwo specific cases (cashflow discounting and probability curves) are discussed. We then recallin Section 3 how a shifted version of time-homogeneous affine jump diffusions can fit everydiscount curve. We pay specific attention to the case where the resulting process needs to meet apositivity constraint. We then introduce in Section 4 our alternative model, specifically devotedto this case, focusing on the most common HAJD, namely the Vasicek and JCIR (generalizingthe CIR) models. Eventually, we compare in Section 5 our model’s performance to that of theshift approach on three different pricing problems taken from credit risk: CDS curve calibration,pricing of CDS options and pricing of credit valuation adjustment under wrong-way risk. Consider a given time- s market curve P markets ( t ), t ≥ s . The calibration problem consistsof finding, for a given model, the (set of) parameter(s) Ξ = Ξ (cid:63) such that the corresponding model curve P models ( t ) = P models ( t ; Ξ) “best fits” the market curve, according to some criterion.Mathematically speaking, this is an optimization problem that consists of finding a set ofparameters that minimizes an error function between model and market values,Ξ (cid:63) := arg min Ξ (cid:107) P models ( · ; Ξ) − P markets ( · ) (cid:107) , (1)6here (cid:107) f ( · ) − g ( · ) (cid:107) represents a divergence measure between two functions f, g . In practice,one often computes the mean-square error (MSE) between f and g on a set of maturities T := { T , . . . , T n } : (cid:107) f ( · ) − g ( · ) (cid:107) := 1 n n (cid:88) i =1 (cid:0) f ( T i ) − g ( T i ) (cid:1) . (2)A model with parameter Ξ is said to perfectly fit the market up to horizon T whenever P models ( t ; Ξ) = P markets ( t ) for all s ≤ t ≤ T or using a shorthand notation, P models ≡ P markets . A model can be either static or dynamic. For instance, the Nelsen-Siegel model Nelson andSiegel (1987) postulates a parametric form for the yield curve, but is a static model: the resultingcurve does not correspond to the yield curve generated by the dynamics of a stochastic model.We focus on continuous-time dynamic models in the sequel.We consider a frictionless market free of arbitrage opportunities in which trading takesplace continuously over the time interval [0 , T ], where T is a fixed time horizon. Uncertaintyin the market is modelled through a filtered probability space (Ω , G , G , Q ). In this setup, G = ( G t , t ∈ [0 , T ]) represents the information flow and corresponds to the filtration generatedby the stochastic market variables (risk factors, prices, interest rates, default intensities, defaultevent, etc), G := G T , and Q denotes the risk-neutral probability measure referred to as the pricing measure . In the sequel, we shall focus on a specific class of P model and P market functions:we assume they are discount curves , a set of functions that we now define. Definition 1 (Discount curve) . A time- s discount curve is any differentiable function of theform P s : [ s, ∞ ) → R +0 , t (cid:55)→ P s ( t ) satisfying P s ( s ) = 1 . In the specific s = 0 case, a time-0 discount curve P ( t ) is simply called a discount curve andis noted P ( t ), assuming implicitly that t ≥
0. Any time- s discount curve admits an exponential-integral from: Lemma 1.
Every time- s discount curve P s admits a representation in terms of time- s instan-taneous forward rate curve f s : P s ( t ) = e − (cid:82) ts f s ( u ) du , t ≥ s. f moreover P s ( t ) is strictly decreasing on ( s, ∞ ) , then f s ( t ) is strictly positive for all t > s .Proof. Since P s ( t ) > t ≥ s and P s ( t ) is differentiable with respect to t on ( s, ∞ ), then onecan define the time- s instantaneous forward rate function as f s ( t ) := − P s ( t ) ddt P s ( t ) = − ddt ln P s ( t )for all t > s ; the value f s ( s ) is not identified but can be defined by, e.g., the limit as t ↓ s .Moreover, if P s is strictly decreasing on ( s, ∞ ) then f s ( t ) > t > s .Discount curves are of paramount importance in finance. As suggested by the name, P s ( t )allows one to compute the time- s value of a cashflow paid at time t ≥ s , both in a credit risk-free and credit risky setup. As a special case of the second framework, they encompass survivalprobability curves, defined as one minus cumulative distribution functions. This is elaboratedin the next two subsections. In this application, P s ( t ) stands for the time- s price of a risk-free zero-coupon bond (ZCB) withmaturity t and face value 1, denominated in a given currency. In particular, P markets ( t ) and P models ( t ; Ξ) respectively give the market and the model prices of that instrument.Indeed, in a no-arbitrage setup, the price of a financial instrument paying a single cashflow(payoff) at a given maturity is given by the risk-neutral conditional expectation of the payoff,discounted at the risk-free rate from the payment date (maturity) back to the valuation date.Adopting a short-rate model, P models ( t ) corresponds to the Q -expectation of the stochasticdiscount factor D s ( t ) := e − (cid:82) ts r u du , the negative exponential of the risk-free short-rate process r ,integrated from the valuation time s up to the payment time t , conditional upon the informationprevailing at the pricing time Brigo and Mercurio (2006): P models ( t ) = E [ 1 D s ( t ) | G s ] = E (cid:104) e − (cid:82) ts r u du (cid:12)(cid:12)(cid:12) G s (cid:105) =: P rs ( t ) . In this context, we aim at finding a model x to depict the risk-free short rate dynamics r that would be tractable enough, and provide a perfect fit to any yield curve, the curve thatgives the set of prices of ZCBs with increasing maturities. Adopting the same framework as before, the time- s price of a zero-coupon bond paying oneunit of currency at time t ≥ s contingent on the fact that the issuer doesn’t default prior to he payment date is given by a similar expression as before. It suffices to replace the risk-freepayoff 1 by the risky one, namely { τ>t } , where the random variable τ represents the defaulttime of the issuer and A is the indicator function defined as 1 if A is true and zero otherwise.Mathematically, P models ( t ) = E (cid:2) { τ>t } D s ( t ) (cid:12)(cid:12) G s (cid:3) =: ¯ P rs ( t ) . In such a context, ¯ P rs corresponds to a risky discounting, where the term risk is referring tothe possibility for the issuer not to meet her financial obligations.To proceed, we need to model the default event. To that end, we consider a reduced-form (a.k.a. intensity ) default model. We refer the reader to Duffie and Singleton (1999)and Lando (2004) for an extensive exposition of this class of models. In this framework, thedefault time τ := τ ( λ ) is defined as the passage time of the process Λ := (Λ t , t ∈ [0 , T ]) definedas Λ t := (cid:82) t λ s ds above a unit-mean exponential random variable E independent from everyother processes. The process λ is an intensity , i.e., it is positive, so that Λ is almost surelyincreasing. In this model, the default event { τ ≤ t } is modeled as { Λ t ≥ E} and the survivalprobability is given by Q ( τ > t ) = Q (Λ t < E ) = Q (cid:0) U < e − Λ t (cid:1) = E (cid:2) e − Λ t (cid:3) , where U := e −E is a random variable uniformly distributed on [0 , P rs ( t ) can be proven to be a time- s discount curve in many cases. To show this,we first define a sub-filtration F such that all processes are F -adapted except those featuring τ (i.e., those featuring E or U , which are independent from F T ). We then define a second filtration H = ( H t , t ∈ [0 , T ]), the filtration generated by the default process H t = σ ( { τ
Key lemma . This fundamental theoremallows one to write the G s -conditional expectation of X { τ>t } as the F s -conditional expectationof Xe − (cid:82) ts λ u du , rescaled by { τ>s } , for every integrable and F t -measurable random variable X .It is originally due to Dellacherie and Meyer Dellacherie and Meyer (1980), although its use9n financial applications have been put forward by Bielecki, Jeanblanc and Rutkowski Bieleckiand Rutkowski (2002) (see, e.g., Bielecki et al. (2011) for numerous examples in credit riskand Brigo and Vrins (2018) for a specific application in counterparty credit risk). Applying theKey lemma to the risky ZCB formula above yields, with X ← e − (cid:82) ts r u du ,¯ P rs ( t ) = { τ>s } E (cid:104) e − (cid:82) ts λ u du D s ( t ) (cid:12)(cid:12)(cid:12) F s (cid:105) = { τ>s } E (cid:104) e − (cid:82) ts ( λ u + r u ) du (cid:12)(cid:12)(cid:12) F s (cid:105) =: { τ>s } P λ + rs ( t ) . Eventually, in the special case where r ≡
0, so that ¯ P rs ( t ) collapses to¯ P rs ( t ) = E (cid:2) { τ>t } |G s (cid:3) = Q ( τ > t |G s ) = { τ>s } P λs ( t ) . Hence, on the event { τ > s } , ¯ P rs ( t ) = P λs ( t ) agrees with the survival probability functionassociated with τ , conditional upon G s .In this specific context, we are interested in a model x to depict the dynamics of the inten-sity process λ that would be tractable enough, and provide a perfect fit to any valid survivalprobability curve extracted from the prices of defaultable instruments like corporate bonds orcredit default swaps (CDS). Remark 1.
Notice that in contrast to rates, that can – and some of them currently do – takenegative value, non-negativity is a formal requirement when x represents an intensity process λ . A default model featuring “negative intensities” is theoretically flawed, and is problematic.Indeed, modelling the event { τ > t } as { Λ t < E} yields a survival indicator process { τ>t } thatmight jump both up and down, i.e., the reference entity could be “brought back to life”. Onecould of course think of replacing the default event using a first passage time, thereby revisitingthe default time definition as τ := inf { t ≥ t ≥ E} . However, one looses the analyticaltractability for the survival probability since in this case, Q ( τ > t ) does no longer agree with Q (Λ t < E ) = E (cid:104) e − (cid:82) t λ u du (cid:105) = P λ ( t ) . The x ≥ constraint is also a natural requirement whenit represents a credit spread. Equation (1) suggests that the calibration problem consists of finding the parameters of a givenmodel to minimize the discrepancies between market and model curves up to a time horizon T . However, it is clear that the choice of the model class will also have a substantial impact.Indeed, depending on the model chosen, the minimum of the error function could be large, smallor even zero, in which case the perfect fit is obtained.10nspired by the financial problems mentioned in Section 2.1, we consider the following prob-lems directly related to the perfect fit constraint (up to a given time horizon T , that is implicitin the sequel). The first one does not impose any constraint on the process x to consider. Problem 1.
Find a tractable process x satisfying P xs ( t ) := E (cid:104) e − (cid:82) ts x u du (cid:12)(cid:12)(cid:12) F s (cid:105) = P s ( t ) for t ∈ [0 , T ] and every given discount curve P s . Depending on the application at hand, one may need to impose additional constraints on x .As suggested by the risky discounting example, non-negativity is a crucial one. This leads usto consider a second (constrained) problem. Problem 2.
Find a tractable positive process x (i.e., such that Q ( x t ≥
0) = 1 and Q ( x t > > for all t ∈ [0 , T ] ) satisfying P xs ( t ) := E (cid:104) e − (cid:82) ts x u du (cid:12)(cid:12)(cid:12) F s (cid:105) = P s ( t ) for t ∈ [0 , T ] and every strictly decreasing discount curve P s . In either problems, tractability refers to the fact that model calibration (1) – that featuresan optimization over the parameter space – is not too cumbersome, computationally. Solvingthis optimization problem typically requires many iterations, hence numerous evaluations ofthe objective function. This suggests that a highly desirable feature of the model is to admit aclosed form expression for P model or, at least, that the latter can be computed without havingto rely on time-consuming numerical methods like, e.g., Monte Carlo simulations. In order to solve these two problems, we consider what is probably the most tractable familyof models, namely affine processes and, more specifically time-homogeneous affine processes .Indeed, for a one-factor affine model y := ( y t , t ∈ [0 , T ]), many expressions are available ana-lytically, as well as for its integrated version Y := ( Y t , t ∈ [0 , T ]), Y t := (cid:82) t y u du . In particular, P ys ( t ) := E (cid:104) e − (cid:82) ts y u du (cid:12)(cid:12)(cid:12) F s (cid:105) is merely the conditional moment generating function of Y t − Y s , t ≥ s . 11 .1 Affine processes and affine jump-diffusions As recalled in the introduction, ATSM models are widely used in finance because they offeran appealing modeling framework : they are scarce, and empirical evidences suggest that theydepict relatively well the market dynamics. Affine models are characterized as follows Filipovic(2005).
Definition 2 (Affine process) . An affine process is any process y satisfying P ys ( t ) := E (cid:104) e − (cid:82) ts y u du (cid:12)(cid:12)(cid:12) F s (cid:105) = e A ys ( t ;Ξ) − B ys ( t ;Ξ) y s =: P ys ( t ; Ξ) (3) where Ξ is the (set of ) parameter(s) governing y and A ys , B ys are differentiable functions satis-fying A ys ( s ; Ξ) = B ys ( s ; Ξ) = 0 . Provided that the
A, B functions are known, the analytical form (3) facilitates in a tremen-dous way calibration procedures such as (1) when the considered P model function takes theform of the conditional expectation in (3), as illustrated on the risk-free and risky discountingapplications. This explains why such models are so popular in term-structure modeling.For such processes, the function P ys is thus well-defined for every s , is positive, and satisfies P ys ( s ) = 1. It is therefore a time- s discount curve in the sense of Definition 1 since it is obviouslydifferentiable on ( s, ∞ ). For instance, P rs and P λs in the above two examples are time- s discountcurves whenever r and λ are affine processes, respectively.It is known (see, e.g., Brigo and Mercurio (2006) and Duffie and Kan (1996)) that everydiffusion with affine drift and diffusion coefficients, regular enough so that a solution exists, is anaffine process. Similarly, every jump-diffusion with such types of drift and variance coefficientsand independent compounded Poisson jumps (i.e., exponentially-distributed jumps arrivingaccording to a Poisson process) is also affine. Definition 3 (Affine jump-diffusions, AJD) . A stochastic process y is called an affine jump-diffusion if its dynamics take the form dy t = ( a ( t ) + b ( t ) y t ) dt + (cid:112) c ( t ) + d ( t ) y t dW t + dJ t (4) with W an F -Brownian motion and J an F -adapted compound Poisson process independent from W , defined according to J t := (cid:80) N t j =1 ζ i where N is a Poisson process with instantaneous jumprate ω ( t ) ≥ and ζ i ’s are i.i.d. exponentially distributed random variables with mean α ≥ . Inthe special case where the parameters ( a, b, c, d, α, ω ) are constant, y is said time-homogeneous,or simply homogeneous, or HAJD.
12s explained above, affine models are specifically relevant in our context when
A, B areknown in closed form. This is the case for HAJD. Three important homogeneous cases arethe Ornstein-Uhlenbeck, the square-root diffusion and the square-root jump-diffusion. Thefirst one, widely known as the Vasicek (VAS) model, corresponds to the special case where( a ( t ) , b ( t ) , c ( t ) , d ( t ) , α, ω ( t )) = ( κβ, − κ, η , , ,
0) and y ∈ R . The second model is the Cox-Ingersoll-Ross with (CIR), and is associated to ( a ( t ) , b ( t ) , c ( t ) , d ( t ) , α, ω ( t )) = ( κβ, − κ, , δ , , y , β >
0. Eventually, the JCIR is an extension of the CIR, associated with parameters( a ( t ) , b ( t ) , c ( t ) , d ( t ) , α, ω ( t )) = ( κβ, − κ, , δ , α, ω ). The speed of mean-reversion κ is assumedto be positive in all models. When the initial value y is part of the parameters, we note theparameter set Ξ . In contrast to VAS which is a Gaussian model, the CIR and JCIR modelsare non-negative. We recall (and derive) some properties of these processes in the Appendix(Section 7.1) for further references.Observe that the sum of two affine processes x, y is, generally speaking, not an affine process.Hence, it is not clear whether the risky discounting curve P λ + rs is a time- s discount curve, evenin the simple case where both r, λ are affine processes. Some special cases are discussed in theAppendix (Section 7.2). In the sequel, we consider a specific pricing time, say s = 0 withoutloss of generality, and drop the observation time subscript for conciseness.HAJD models like VAS, CIR and JCIR seem appropriate to solve problems 1 and 2. Unfor-tunately, they do not allow for a perfect fit to a given discount curve P , except in very specialcases. Indeed, it is not possible in general, for such type of processes x , to find Ξ (or Ξ ) suchthat P model := P x ( · ; Ξ) ≡ P , even up to a finite horizon T . The starting point is to notice that the limited capacities of homogeneous models result fromtheir rigid parametric form. Therefore, an interesting route is to consider a family of models x defined as time-dependent transform of a base HAJD model y in such a way that the model’stractability is not affected. In this section, we recall the general deterministic shift extensionapproach. The latter has been introduced in the seminal paper Brigo and Merccurio (2001)in order, precisely, to address calibration issues such as Problem 1. In this model, x := x ϕ isdefined as a HAJD ( y ) that is shifted in a time-dependent way using a deterministic function ϕ : x ϕt := y t + ϕ ( t ) . (5)13nterestingly, P model ( t ) := P x ϕ ( t ; Ξ) where x ϕ remains affine (although no longer homoge-neous) and is hence analytically tractable in terms of calibration since : P x ϕ ( t ; Ξ) = e A xϕ ( t ;Ξ) − B xϕ ( t ;Ξ) x with A x ϕ ( t ; Ξ) = A y ( t ; Ξ) − (cid:90) t ϕ ( u ) du + B y ( t ; Ξ) ϕ (0) ,B x ϕ ( t ; Ξ) = B y ( t ; Ξ) . Clearly, the dynamics of x ϕ are easily obtained from that of y . Indeed, assuming dy t = µ ( t, y t ) dt + σ ( t, y t ) dW t + dJ t , (6)the dynamics of x ϕ read, when ϕ is differentiable, as dx ϕt = dy t + ϕ (cid:48) ( t ) dt = ( µ ( t, x ϕt − ϕ ( t )) + ϕ (cid:48) ( t )) dt + σ ( t, x ϕt − ϕ ( t )) dW t + dJ t , x ϕ = y + ϕ (0) . It can be shown that in the particular case where y is a HAJD, then x ϕ remains an AJD, eventhough no longer homogeneous, unless ϕ ( t ) is constant. For instance, if the dynamics of y obey(4), then x ϕ is governed by the same type of dynamics since dx ϕt = ( a ϕ ( t ) + b ( t ) x ϕt ) dt + (cid:113) c ϕ ( t ) + d ( t ) x ϕt dW t + dJ t . (7)where a ϕ ( t ) := a ( t ) + ϕ (cid:48) ( t ) − b ( t ) ϕ ( t ) and c ϕ ( t ) := c ( t ) − d ( t ) ϕ ( t ). As already noticed in Brigoand Merccurio (2001), whatever the base model y , the parameter Ξ and the discount curve P market , there always exists a shift function ϕ ( t ) = ϕ (cid:63) ( t ; Ξ) that provides a perfect fit betweenthe x ϕ -model and the market. This is summarized in the next lemma. Remark 2.
The shift approach may look suspicious: adding a deterministic function to astochastic process is arguably a somewhat artificial way to fix the model’s limitations in termsof calibration. However, as clear from (7) , shifting the model in a deterministic way actuallyamounts to consider an inhomogeneous model. For instance, the Vasicek model ( a ( t ) , b ( t ) , c ( t ) , d ( t )) =(0 , − κ, η, shifted with ϕ ( t ) ← (cid:82) t β ( s ) e − κ ( t − s ) ds yields a HAJD with ( a ( t ) , b ( t ) , c ( t ) , d ( t )) =( κβ ( t ) , − κ, η, , which is known as the Hull-White (HW) model Hull and White (1990). More-over, the later is itself a particular case of the Heath-Jarrow-Morton (HJM) model Heathet al. (1992) which consists of modeling the entire instantaneous forward curve f s ( t ) with df s ( t ) = µ ( s, t ) dt + ηe − κ ( t − s ) dW s where the drift µ ( s, t ) is given by no-arbitrage, and pro-vided that the initial discount curve and the long-term mean obey the relationship β ( t ) =14 dt f ( t ) + κf ( t ) + η κ (1 − e − κt ) . Therefore, any instantaneous forward curve f market (hence dis-count curve P market ) can be fitted with either models provided that one takes f ( t ) ← f market ( t ) as initial curve (HJM), the corresponding long-term mean β ( t ) (HW), or the associated shift ϕ ( t ) (shifted Vasicek). These models became very popular among practitioners, essentially be-cause of their ability to replicate market curves, i.e., to solve Problem 1. Lemma 2.
The x -model defined according to (5) where y is a HAJD solves Problem 1 providedthat ϕ ( t ) ← ϕ (cid:63) ( t ; Ξ) := ddt ln P y ( t ; Ξ) P market ( t ) = f market ( t ) − f y ( t ; Ξ) . (8) where f market and f y are the instantaneous forward rate functions associated with P market and P y , respectively.Proof. Indeed, because y is a HAJD, P y is a discount curve and from Lemma 1, it admitsa representation in terms of forward rates f y . By assumption, same holds true for P market .Eventually, P x ϕ ( t ; Ξ) = E (cid:104) e − (cid:82) t x u du (cid:105) = e − (cid:82) t ϕ (cid:63) ( u ;Ξ) du E (cid:104) e − (cid:82) t y u du (cid:105) = e − (cid:82) t f market ( u ) du = P market ( t ) . The model is tractable since f y ( t ; Ξ) = − ddt ln P y ( t ; Ξ) can be computed in closed form.It is worth noting that, for a given model y , the perfect fit can be attained for everyparameters Ξ. This suggests that the calibration problem (1) is ill-posed. Indeed, the choice ofΞ is completely arbitrary since the error between P x ϕ and P market can be set to zero for anyΞ, provided that one chooses ϕ ( t ) ← ϕ (cid:63) ( t ; Ξ). In particular, one could take the null process for y and ϕ ( t ) = f market ( t ). This trivial choice rends x ϕ deterministic, which is most likely not thedesired result. A common practice to circumvent this indeterminacy is thus either (i) to extendthe set of calibration instruments, incorporating products that are sensitive to volatility (likeinterest-rate or credit options in the above asset classes), or (ii) to require the y -model to fitthe market “as best as possible” (to get Ξ (cid:63) ) and then take ϕ ( t ; Ξ (cid:63) ) as shift function: P model ( t ) := P x ϕ ( t ; Ξ (cid:63) ) where Ξ (cid:63) := arg min Ξ (cid:107) P y ( · ; Ξ) − P market ( · ) (cid:107) , ϕ ( t ) ← ϕ (cid:63) ( t ) := ϕ (cid:63) ( t ; Ξ (cid:63) ) . (9)This approach is particularly relevant when no or little “volatility-sensitive” instrumentsare quoted on the market. The role of the shift is thus merely to compensate the remainingdiscrepancies between the market curve P market and the one generated by the “best” parametric15odel y , P y ( · ; Ξ (cid:63) ). Adding a shift to the VAS, CIR or JCIR models yield the Hull-White,CIR++ or JCIR++, respectively Brigo and Mercurio (2006). Remark 3.
On the top of the appealing affine structure, the shifted model is highly tractablebecause many statistical properties of the process are available in closed form. Indeed, as recalledin Section 7.1, the k -th moment m y ( k, t ) := E [ y kt ] and the moment generating function (MGF) ψ y ( u, t ) := E [ e uy t ] of a time-homogeneous affine model y are known analytically, as well as thoseof their time-integrals Y t := (cid:82) t y u du , m Y ( k, t ) and ψ Y ( u, t ) . Due to the simple shift structure,the corresponding expressions for x ϕ , the shifted model, are readily available. For instance, the k -th moment of x ϕt and X ϕt are given by Newton’s binomial formula applied to ( y t + ϕ ( t )) k andthe MGFs simply collapse to ψ x ϕ ( u, t ) = e uϕ ( t ) ψ y ( u, t ) and ψ X ϕ ( u, t ) = e u (cid:82) t ϕ ( s ) ds ψ Y ( u, t ) . As discussed above, the deterministic shift extension nicely solves Problem 1. In order to solveProblem 2 however, one first considers a non-negative base process y . Yet, there is no reasonthat the shifted process x ϕ would remain non-negative. For instance, taking CIR dynamics for y , x ϕ is non-negative on [ s, t ] if and only if min u ∈ [ s,t ] ϕ ( u ) ≥
0. From (8), the shift functiondepends both on the y model (and its parameters Ξ) and on the market curve. Remark 4.
Observe that the optimization problem (9) is contradictory with non-negative shiftfunctions. Indeed, by construction of Ξ (cid:63) , P y ( · ; Ξ (cid:63) ) passes through P market . Consequently, theshift ϕ ( t ) ← ϕ (cid:63) ( t ; Ξ (cid:63) ) will lead to a perfect fit, but will correct for both negative and positiveerrors. In other words, ϕ will change of sign. Therefore, this strategy does not provide a validsolution to Problem 2. This will be illustrated on a real example in Section 5.1. In order to satisfy the non-negativity constraint mentioned in Problem 1, one needs to forcethe non-negativity constraint on the shift at the optimal parameters. The shift function underpositivity constraint is referred to with the notation ϕ (cid:63), + ( t ) to stress the difference with theunconstrained counterpart, ϕ (cid:63) ( t ). Lemma 3.
Let y be a HAJD that is non-negative on [0 , T ] with parameters Ξ (cid:63) given by Ξ (cid:63), + := arg min Ξ (cid:107) P y ( · ; Ξ) − P market ( · ) (cid:107) subject to f y ( t ; Ξ) ≤ f market ( t ) , ∀ ≤ t ≤ T . (10)
Then, the x ϕ -model (5) with ϕ ( t ) ← ϕ (cid:63), + ( t ) := ϕ (cid:63) ( t ; Ξ (cid:63), + ) solves Problem 2. Notice that the Hull-White model is a Vasicek model where the long-term mean parameter is replaced by adeterministic function of time. roof. The condition on the instantaneous forward rates ensures that shift function ϕ (cid:63), + will benon-negative on [0 , T ]; this is obvious from (8). Hence, since y is assumed to be non-negative,so is the shifted process x ϕ . Moreover, taking ϕ ( s ) ← ϕ (cid:63) ( s ; Ξ) yields a perfect fit for every Ξ,by construction, including Ξ = Ξ (cid:63), + .Notice that there always exists a set of parameters Ξ such that the constraint is met. Indeed,all parameters Ξ associated to the deterministic case y ≡ f y ( · ; Ξ) ≡
0. Clearly, theconstraint is met since f market ( t ) is strictly positive given that P market is strictly decreasing, byassumption. The shift is simply given by the market forward rate ϕ ( t ) ← ϕ (cid:63), + ( t ) = f market ( t ).However, the trivial process parameter is likely not to be satosfactory.In order to deal with Problem 2, we need to consider a non-negative base model y . Given thatwe focus on HAJDs, we consider the CIR and JCIR models. To make the distinction betweenthe two shifted models, we call S-(J)CIR the (J)CIR process shifted with ϕ ( t ) ← ϕ (cid:63) ( t ) = ϕ (cid:63) ( t ; Ξ (cid:63) ) (i.e., without positivity constraint, and parameter Ξ (cid:63) given by (9)) and PS-(J)CIRthe (J)CIR process shifted with ϕ ( t ) ← ϕ (cid:63), + ( t ) = ϕ (cid:63) ( t ; Ξ (cid:63), + ) (i.e., under positivity constraint,and parameter Ξ (cid:63) given by (10)). Although the PS-(J)CIR allows both for a perfect fit and thenon-negativity constraint, one may argue that it is not as tractable as the (J)CIR. Indeed, theoptimization problem (10) is more difficult than (9) due to the constraint on the instantaneousforwards, even if some sufficient conditions on the parameters can be found. Second, andprobably more importantly, this constraint is binding, in the sense that it often deeply impactsthe optimal parameter Ξ (cid:63), + . Even if it is unlikely that the optimal solution corresponds to thedeterministic case, it often yields dynamics associated to rates that feature “little randomness”.This will be illustrated in Section 5, first by comparing the variance of the integrated S-CIR andPS-CIR processes, as well as the impact when dealing with financial applications. These twopoints are discussed in (Brigo and Mercurio, 2006, sec. 3.9.3, p.107-109). To circumvent thisissue in an interest rate framework, the authors suggest to relax the strict positivity constraint.By working in a setup where positivity is expected but not guaranteed, they obtain a processthat yields much more realistic results in terms of implied volatility levels. This is perfectly finein such a context as positivity of rates might be desirable (in some cases), but zero is by no meansa strict lower bound (neither theoretically nor practically). Yet, this is more problematic whenit comes to model such things as default intensities, because this kind of applications requiresboth strict positivity and, typically, large volatility. Increasing the variance of the CIR++17rocess without breaking Feller’s constraint can be achieved by incorporating compoundedPoisson jumps (JCIR++) but, unfortunately, increasing the jump activity while maintainingthe calibration to a given market curve f market is difficult under the positivity constraint.Indeed, the minimum of the implied shift function is driven down when increasing the jumpactivity because the difference f JCIR ( t ) − f CIR ( t ) is non-negative and increases with ω, α for α, ω > ϕ for the JCIR++ than for the corresponding CIR++. For this reason, there isa need for an alternative to CIR++ and JCIR++ that would combine (i) tractability, (ii) theprefect fit feature, (iii) the large implied volatility and (iv) positivity. In order to circumvent the drawbacks of the deterministic shift extension with regards to Prob-lem 2, we propose a different approach. In the same spirit as the shift, we aim at finding a model x by adjusting a time-homogeneous affine model y , that would benefit from a set of desirableproperties. The x -model is obtained by time-changing a HAJD y using a specific (but deterministic) clock Θ that may differ from the calendar clock. A clock is a time change function that can differfrom identity, but having specific properties.
Definition 4 (Clock) . A clock is an application
Θ : R + → R + , t (cid:55)→ Θ( t ) that is a grounded, increasing and differentiable. In other words, a clock is any function Θ ofthe form Θ( t ) := (cid:90) t θ ( u ) du where θ ( u ) > , ∀ u ≥ . Clearly, Θ( t ) = t is the calendar clock, and any function of the form Θ( t ) = kt , k >
0, isagain a clock, corresponding to a constant rescaling of the calendar time. Increasing the volatility of the CIR++ process by increasing the diffusion paramter δ just breaks the Feller’scondition (2 κβ ≥ δ ) and leads to an intensity process that almost surely equals to zero at a given time interval. x = x θ , obtained from the following transform ofthe base process y : x θt := θ ( t ) y Θ( t ) . (11)The dynamics of x θ are given by Ito’s product rule. Defining the process y θ := (cid:0) y Θ( t ) , t ∈ [0 , T ] (cid:1) ,one gets dx θt = y θt dθ ( t ) + θ ( t ) dy θt , (12)where the dynamics of y θ are given in the below lemma. Lemma 4.
Let Θ be a clock and consider a base model y with dynamics (6) . Then, the dynamicsof y θ take the form dy θt = µ (cid:16) Θ( t ) , y θt (cid:17) θ ( t ) dt + σ (cid:16) Θ( t ) , y θt (cid:17) (cid:112) θ ( t ) dB t + dJ θt , y θ = y (13) where B is an F θ -Brownian motion, F θ := ( F Θ( t ) , t ∈ [0 , T ]) and J θ an inhomogeneous com-pounded Poisson process with jump size mean α and time- t intensity ωθ ( t ) .Proof. By definition, we have y θt := y Θ( t ) = y + (cid:90) Θ( t )0 µ ( u, y u ) du + (cid:90) Θ( t )0 σ ( u, y u ) dW u + (cid:90) Θ( t )0 dJ u . Hence, (cid:90) Θ( t )0 µ ( u, y u ) du = (cid:90) t µ (cid:0) Θ( u ) , y Θ( u ) (cid:1) θ ( u ) du = (cid:90) t µ (cid:16) Θ( u ) , y θu (cid:17) θ ( u ) du , and (cid:90) Θ( t )0 σ ( u, y u ) dW u = (cid:90) t σ (cid:0) Θ( u ) , y Θ( u ) (cid:1) dW Θ( u ) = (cid:90) t σ (cid:16) Θ( u ) , y θu (cid:17) (cid:112) θ ( u ) dB u . Indeed, Θ is a clock, hence θ > W θ := ( W Θ( t ) , t ∈ [0 , T ]) is a local martingalewith quadratic variation (cid:104) W θ , W θ (cid:105) t = Θ( t ). From Jeanblanc et al. (2009), the process B :=( B t , t ∈ [0 , T ]) defined as B t := (cid:90) t (cid:112) θ ( u ) dW Θ( u ) (14)is then a Brownian motion. Differentiating y θt leads to (13). With regards to the compoundedPoisson process, notice that dJ t = ζ N t dN t and dJ θt = dJ Θ( t ) = ζ N Θ ( t ) dN Θ( t ) . The process N θ defined as N θt := N Θ( t ) is a Poisson process with instantaneous intensity ωθ ( t ). Hence, thedynamics of J θ are given by J θ = 0 and ζ N θt dN θt , so that J θ is a compounded Poisson processwith jump size mean α and time- t instantaneous rate of jumps arrival, ωθ ( t ).19his model looks appealing for several reasons. First, just as the shift extension, it is adeterministic adjustment of a base model and is hence expected to be tractable when the latteris, say, a HAJD. Second, because x θt is a positive rescaling of the process y sampled at timeΘ( t ), the range of x θ is linked to that of y . In particular, if the range of y is R , as for the Vasicekmodel, then so is the range of x θ . However, if y is non-negative as in the (J)CIR case, then sois x θ . Hence, this solves the drawback of the shift approach related to Problem 2. Eventually,the time-dependent feature of the clock rate θ is expected to provide additional flexibility in thecalibration properties of x θ with respect to that of the homogeneous model y . Two questionsremain open in this respect. First, we need to clarify the circumstances under which the modelprovides a perfect fit. Second, in the case where the perfect fit can be achieved, we need toprovide an efficient procedure to compute the resulting “optimal clock”, Θ (cid:63) . The price to pay isthat, in contrast with the shift extension, the time-changed model is not fully flexible. Indeed,starting with a given model y , the x θ model can only generate specific shapes for discountcurves. We are thus more dependent on the initial choice of the base model y . Fortunately, itturns out that a perfect fit is achievable for a wide set of market curves, including all decreasingdiscount curves, considered in Problem 2. This is clearly the most important case since (i) itcorresponds to the case where the shift approach fails to provide a convincing solution and (ii) itis probably the most common case in practice, since it encompasses the class of discount curveswith non-negative rates (or, more generally, with non-negative instantaneous forward rates), aswell as the set of all continuous survival probability curves. Moreover, even if the mathematicalexpression of the clock Θ (cid:63) is not available in closed form, its numerical computation turns outto be easy. This leads us to the first fundamental result of the paper. Theorem 1.
Let P market be a discount curve and y a model such that P y is a discount curve.Define the x θ -model as in (11) . Then, P x θ ≡ P market provided that Θ ← Θ (cid:63) where Θ (cid:63) satisfiesthe first-order ODE θ (cid:63) ( t ) := ddt Θ (cid:63) ( t ) = f market ( t ) f y (Θ (cid:63) ( t )) , (15) with f market , f y the corresponding instantaneous forward curves. Moreover, if P market and P y are strictly decreasing, the solution to (15) exists, is a clock, and is given by Θ (cid:63) ( t ) := Q y (cid:16) P market ( t ) (cid:17) , (16) where Q y is the inverse of the base-model discount curve, P y . When no confusion is possible, the explicit reference to the model parameters Ξ is avoided to ease thenotations. roof. See Section 7.3.Observe that the optimal clock Θ (cid:63) actually depends from the y -model parameters Ξ. Justlike for the shift, we actually have Θ (cid:63) ( t ) = Θ (cid:63) ( t ; Ξ). Although other frameworks are possible,we set Ξ = Ξ (cid:63) as in (9). Similar to the function ϕ in the shift approach, the purpose of theclock Θ is then to absorb the remaining errors between P y ( · ; Ξ (cid:63) ) and P market . As in the shift extension, a time-changed model x θ enjoys a similar tractability level to thatof the base model y . Indeed the k -th moment is m x θ ( k, t ) = θ ( t ) k m y ( k, Θ( t )) and momentgenerating function is ψ x θ ( u, t ) = e uθ ( t ) ψ y ( u, Θ( t )), whereas those of X θt coincide with thoseof Y Θ( t ) . Hence, a tractable model x θ can be obtained by considering HAJD processes as basemodel y . We illustrate our method by analyzing two calibration problems that can be solvedby considering the Vasicek and the JCIR processes.It is clear from Lemma 4 that in the particular case where y is a HAJD, then x θ is a scaledversion of an inhomogeneous affine jump diffusion (AJD), unless θ ( t ) is a positive constant, inwhich case it remains a HAJD. To see this, suppose that the dynamics of y obey (4). FromLemma 4, y θ is governed by dy θt = (cid:16) a (Θ( t )) + b (Θ( t )) y θt (cid:17) θ ( t ) dt + (cid:113)(cid:0) c (Θ( t )) + d (Θ( t )) y θt (cid:1) θ ( t ) dB t + dJ θt . (17)Interestingly, y θ is still an AJD. In the sequel, we focus on the special case where the basemodel y is a HAJD, i.e., takes the form (4) with constant parameters ( a ( t ) , b ( t ) , c ( t ) , d ( t ) , α, ω ( t )) =( κβ, − κ, η , δ , α, ω ). To simplify the notation, we specify the model parameters using the vectorΞ = ( κ, β, η, δ, α, ω ). Our time change approach can be easily used to solve Problem 1 in the most common case wherethe forward curve f market is arbitrary (monotonic, humped, etc) provided that it is positive.As there is no constraint on the range of the process x θ , let us postulate Vasicek dynamics forthe base process with parameters Ξ = ( κ, β, η, , ,
0) : dy t = κ ( β − y t ) dt + ηdW t , y ∈ R . The forward curve associated to this model is given by f y ( t ) = f VAS ( t ) := f VAS0 ( t ) in (24): f VAS ( t ) = (1 − e − κt ) κ β − η / κ + η κ e − κt (1 − e − κt ) + y e − κt . (18)21t can thus be used to select an appropriate Vasicek model. The next corollary providesguidelines to generate decreasing discount curve, associated with the most common case ofpositive instantaneous forwards. Corollary 1.
Let P market be a strictly decreasing market curve. Then, for every Vasicek modelwith parameters satisfying y ≥ and κ β > η , there exists a clock Θ (cid:63) such that P x θ ≡ P market .Proof. Because y is a Vasicek process, P y is a discount curve. Moreover, the conditions y ≥ κ β > η guarantee that the forward curve (18) is strictly positive, hence P y is strictlydecreasing. From Theorem 1, the clock Θ (cid:63) exists and is given by (15) with f y given in (18).Notice that the dynamics of the time-changed Vasicek model x θt are given by (12) with dy θt = κ ( β − y θt ) θ ( t ) dt + η (cid:112) θ ( t ) dB t , y θ = y , showing that y θ remains a Gaussian process. Fitting perfectly a strictly decreasing discountcurve (without further constraints on the process) is a special case of Problem 1, that can alsobe solved using the shift approach (5) by taking x ← x ϕ where y is a Vasicek with arbitraryparameters Ξ and ϕ ( t ) ← ϕ (cid:63) ( t ; Ξ) = f market ( t ) − f VAS ( t ). The main interest of the time-changedapproach is actually when considering Problem 2. The following result is the second main contribution of the paper. It shows that the time changeapproach x ← x θ provides a solution to Problem 2. Corollary 2.
Let y be an almost-surely positive HAJD with parameters Ξ . Then, the model x θ defined in (11) with Θ ← Θ (cid:63) ( t ; Ξ) solves Problem 2.Proof. Because y is a HAJD, P y is a discount curve and is tractable analytically. Moreover, thelatter is strictly decreasing since y is almost-surely positive. We conclude the proof by relyingon Theorem 1.Let us now consider the JCIR model, i.e., the HAJD with Ξ = ( κ, β, , δ, α, ω ). The CIR isrecovered as a special case by choosing ( α, ω ) such that αω = 0.Then, dy t = κ ( β − y t ) dt + δ √ y t dW t + dJ t , y > , (19)22here κ, β, δ are strictly positive constants and ω, α are non-negative. The optimal clock Θ (cid:63) leading to the perfect fit to a given strictly decreasing curve P market is given by (15) where theforward curve associated to this model is given by f y ( t ) = f JCIR ( t ) := f JCIR0 ( t ) in (28) : f JCIR ( t ) = 2 κβ ( e tγ − γ + ( κ + γ )( e tγ −
1) + y γ e tγ [2 γ + ( κ + γ )( e tγ − + 2 ωα ( e tγ − γ + ( κ + γ + 2 α )( e tγ − , (20)where γ := √ κ + 2 δ . The dynamics of the time-changed process x θt = θ ( t ) y θt are given by (12)with dy θt = κ ( β − y θt ) θ ( t ) dt + δ (cid:113) θ ( t ) y θt dB t + dJ θt , y θ = y . where B is an F θ -Brownian motion and J θ is an inhomogeneous compound Poisson process withjump size mean α and time- t instantaneous rate of arrival ωθ ( t ).The time change technique applied to a JCIR (TC-JCIR) therefore solves Problem 2. Inparticular, in contrast to the S-JCIR (that focuses on parameters such that ϕ is positive), thepositivity constraint on x θ is automatically satisfied for every (strictly decreasing) market curveand every Ξ (such that y is not trivially equal to 0). However, we have shown that it is possible toensure positivity by considering the PS-JCIR, x ϕ, + . Working with Ξ (cid:63), + instead of Ξ (cid:63) can makethe job, but at the expenses of having a process x ϕ, + that is, to a large extend, deterministic(i.e., x ϕ, + t varies in a small neighborhood around f market ( t )). Consequently, TC-JCIR modelare expected to feature a higher volatility compared to the corresponding PS-JCIR, at least upto some time horizon. This is summarized in the next theorem, which is the third main resultof the paper. Theorem 2.
Let P market be a strictly decreasing discount curve and y be a JCIR++ processwith parameter Ξ such that the perfect fit JCIR++ model x ϕ (cid:63) t is positive. Then, the ODE (15) with f y ( t ) = f JCIR ( t ; Ξ) given by (20) admits a solution that satisfies Θ (cid:63) ( t ) = Θ (cid:63) ( t ; Ξ) ≥ t . Moreover, the variance of the corresponding perfect fit TC-JCIR model x θ (cid:63) t satisfies:1) V (cid:2) X θ (cid:63) t (cid:3) ≥ V (cid:104) X ϕ (cid:63) t (cid:105) , ∀ t ≥ ,2) V (cid:2) x θ (cid:63) t (cid:3) ≥ V (cid:104) x ϕ (cid:63) t (cid:105) if one of the following holds:i) y = β + ωακ ,ii) f market constant and y ≤ β + ωακ ,iii) y > β + ωακ and t < Θ (cid:63) − ( t ) ,iv) ( κβ + ωα ) /γ < y < β + ωακ and t > Θ (cid:63) − ( t )23 here t := 1 κ ln (cid:18) y + 2 ωα /γ y − β − ωα/κ (cid:19) and t := 1 γ ln ( γ − κ )( κβ + y γ + ωα ) − ωα ( κ + γ )( y γ − κβ − ωα ) − ωα . Proof.
See Section 7.4.To sum up, the TC-JCIR model (including the TC-CIR) provides an elegant solution toProblem 2: the process x θ (cid:63) is non-negative (in contrast with the S-JCIR x ϕ (cid:63) ), is almost astractable as the simple JCIR diffusion (in contrast with the PS-JCIR x ϕ (cid:63), + ), provides a perfectfit to every strictly decreasing discount curve (as both JCIR++ models) and features, to someextend, a larger variance (compared to the PS-JCIR x ϕ (cid:63), + ). In particular, it is observed,empirically, that the variance of the integral of the TC-JCIR remains similar to that of theunconstrained (i.e., flawed, but high-volatility) S-JCIR model x ϕ (cid:63) . Therefore, when a positivityconstraint is required, the TC-JCIR avoids the drawbacks of the JCIR++ models. The onlyprice to pay is that the clock is not available in closed form, but requires a (simple) numericalinversion. The properties of the model, namely the perfect fit and high-variance features, areillustrated in the next section on various applications taken from credit risk modeling. We consider a reduced-form default model as in Section 2.1.2 by using a CIR base model y (i.e.,(19) with J ≡ λ is modelled either as a CIR++ ( λ ← λ ϕt := y t + ϕ ( t ))or using the TC-CIR ( λ ← λ θt := θ ( t ) y Θ( t ) ). Observe that depending on the pair ( P market , Ξ),the CIR++ process can feature negative values. This will be the case when taking Ξ ← Ξ (cid:63) given using the MSE approach (9), unless there is an explicit constraint as in (10), leading totake Ξ ← Ξ (cid:63), + . Bear in mind that when λ represents an intensity process, the S-CIR model( λ ϕ ) is actually flawed as there is a non-zero probability to observe negative intensities, and P λ ϕ ( t ) cannot be interpreted as a survival probability associated to a Cox model. Yet, we givethe results of the model as a benchmark since, as explained in the introduction, it is a verystandard approach.We compare the CIR++ (S-CIR and PS-CIR) to the TC-CIR on several aspects related toa real case example where the reference entity is Ford Inc. We also discuss the TC-JCIR casewhen relevant. We first analyze the perfect fit feature of both types of models, as well as thenon-negativity property of λ . We then compare the variance of the integrated processes Λ. Wethen analyse their behaviors in two different applications, namely the pricing of various credit24efault swaptions (a.k.a. CDS options, or CDSO) with Ford as reference entity, or on the creditvaluation adjustment (CVA) of prototypical FRA and IRS exposures where Ford is the tradecounterparty.It it well-admitted that “pure credit instruments” like CDS or CDSO are quite insensitive tothe stochasticity of the interest rates in realistic conditions. This has been discussed explicitlyfor the CIR base model in Brigo and Alfonsi (2005) and Brigo and Cousot (2006). Hence, weconsider a deterministic short rate process, which is stressed by the notation r u = r ( u ). In thiscase, one simply gets P rs ( t ) = D s ( t ) = e − (cid:82) ts r ( u ) du . In the sequel, we first illustrate the perfect fit feature of S-CIR, PS-CIR and TC-CIR whenthe default model is calibrated on the survival probability curve of Ford Inc. We then use themodel to price CDSO and compute CVA figures.
We consider the CDS term-structure of Ford Inc, and show that considering a set of parameterΞ, there exist ϕ and Θ that yield a perfect fit. In the sequel, we drop the star superscript onthe shift and clock functions. Hence, Ξ (cid:63) corresponds to the CIR parameter optimized withoutconstraint to a given P market curve, and ϕ and Θ refer to the corresponding optimal shift andclock functions. The corresponding parameters found under a non-negativity constraint arenoted Ξ (cid:63) + , ϕ + and Θ + , respectively.A credit default swap (CDS) is a financial instrument used by two parties – called theprotection buyer and the protection seller – to transfer to the protection seller the financialloss that the protection buyer would suffer if a particular default event happened to a thirdparty called the reference entity . Typically, we set τ as the default time of the latter. In adefault swap contracted at time t , started at time T a with maturity T b , the protection buyerpays a coupon (of spread) k at a set of payment dates T a , . . . , T b as long as the reference entitydoes not default. The protection seller agrees to make a single payment LGD to the protectionbuyer if the default occurs between T a and T b . When applicable, the protection buyer makes afinal payment corresponding to the spread accrued since the last payment date before default.For more details about the mechanics of this product, we refer to Brigo and Alfonsi (2005) andBrigo and El-Bachir (2010). Given that the interest rates have little impact on the figure and that our main objective is to discuss theimpact of the default model, we considered zero risk-free rate in the numerical applications below. For more details about the actual market conventions, we refer the interested reader to Markit (2004) and t par spread s t ( T i ) of a CDS contract of maturity T i is defined as thecontract spread k that sets the value of the CDS contract to 0 at time t . The par spreads havebeen taken from Bloomberg on November 12, 2018 and are shown on the table below.Maturity (years) 1 3 5 7 10Spread (bps) 18.3 136.6 191.9 267.6 280.6Table 1: CDS spread term structure of Ford Inc. on November 12, 2018. Source: Bloomberg.In this context, the market curve P market to be fitted is the risk-neutral survival probabilitycurve, defined as G ( t ) := Q ( τ > t ) associated with the default time τ of a given referenceentity (here, Ford Inc.). It can be extracted from CDS quotes by inverting the no-arbitragepricing formulae of the corresponding financial instruments. In practice, one only has a couple ofcalibration equations, say n , given by the number of market quotes (here, n = 5). It is thereforenot possible to estimate the full (i.e., infinite-dimensional) market curve G without furtherassumptions. It is common market practice to consider the CDS model from the InternationalSwap and Derivative Association (ISDA) – a.k.a the JP Morgan model – Markit (2004), thatprovides a slightly simplified version of the actual no-arbitrage pricing formula applying toCDSs. In this approach, the curve G is parametrized via a positive hazard rate function h ,playing a similar role as the instantaneous forward rate f market , G ( t ) := e − (cid:82) t h ( s ) ds , where h is itself parametrized by n constants h , h , . . . , h n bootstrapped from the spreads s , s , . . . , s n associated with the maturities T , T , . . . , T n . Let us focus on the horizon T = T n .It is market practice to assume that h is piecewise constant between the maturities, i.e., topostulate the parametric form: h ( t ) = n (cid:88) i =1 { T i − ≤ t
0) isobtained from eq. (9) and y = h . The shift function ϕ ( t ) ← ϕ (cid:63) ( t, Ξ (cid:63) ) is shown in panel (b).Panel (c) gives the clock Θ( t ) ← Θ (cid:63) ( t ; Ξ (cid:63) ). Eventually, panel (d) yields the survival probabilitycurves given by the market ( G ( t ), green), or associated to Q ( τ ( λ ) > t ) for various intensitymodels λ : the best base model λ ← y (leading to Q ( τ ( y ) > t ) = P y ( t, Ξ (cid:63) ), dashed blue), λ ← λ ϕ (S-CIR) and λ ← λ θ (TC-CIR model). By construction of ϕ and Θ, the last two curvescoincide (magenta) and agree with G ( t ). 29he parameters used in the numerical examples in the rest of the paper are given in Table2. Ξ κ β δ y Ξ (cid:63) h Ξ (cid:63), + h Ξ (cid:63) (cid:63), +0 . . − . . − . . − . . − Table 2: Calibration parameters using Ford piecewise constant hazard rate. Parameters Ξ (cid:63) andΞ (cid:63), + correspond to the parameters of the CIR model y with and without positivity constraint,where y is set exogenously to the first level of the piecewise hazard rate function, h = 0 . (cid:63) and Ξ (cid:63), +0 , correspond to the similar cases but where y is a parameterthat enters the optimization procedure. In all cases, we have taken α = ω = 0.Notice that in both Figure 1 and 2, the shift function ϕ can take negative values. Thismeans that the shift approach, S-CIR, yields negative default intensities λ ϕ and, calibratedthat way, is flawed. In particular, we cannot interpret λ ϕ as a default intensity associated toa Cox process. This contrasts with the TC-CIR approach since λ θ is a positive process if sois y . To fix this issue in a CIR++ framework, one needs to rely on PS-CIR. We note thecorresponding processes y + and λ ϕ, + . As illustrated on Figure 3 with our Ford example, thisprocedure is very restrictive: it leads to a curve P y that is decreasing at a very low rate. Inparticular, the shape of P λ ϕ, + essentially results from the shift, not from the base model y . Thisis problematic: it basically amounts to say that h ≈ ϕ , i.e., that the PS-CIR process λ ϕ, + isessentially deterministic. This will put strong limitations on the resulting default model, andwill be further discussed in the remaining subsections.30 . . . . . t G ( t ) l l l l l l (a) Survival probability ( P y + , G = P λ ϕ, + ) . . . . . t G ( t ) l l l l l l (b) Survival probability ( P y + , G = P λ ϕ, + ) . . . . . t j ( t ) (c) Shift function ( ϕ = ϕ (cid:63), + ) . . . . t j ( t ) (d) Shift function ( ϕ = ϕ (cid:63), + ) Figure 3: Fitting Ford Inc. CDS term-structure using λ ϕ (cid:63) , + (PS-CIR). Panels (a) and (c) cor-respond to y +0 = h whereas panels (b) and (d) correspond to the case where y +0 is one of theoptimized parameters. The survival probability curve G ( t ) is parametrized with a piecewise con-stant hazard rate function h ( t ) extracted from market prices taken from Bloomberg on November12 2018. The parameters Ξ (cid:63), + are computed under the constraint ϕ ( t ) ← ϕ (cid:63) ( t ; Ξ (cid:63), + ) ≥ y + is a CIR with parameters Ξ = Ξ (cid:63), + (left) and Ξ = Ξ (cid:63), +0 (right).31 .2 Variance analysis Interesting observations can be made regarding the variance of the various integrated processes.As shown in the next two sections, they will have important consequences when consideringfinancial applications, where Λ plays a central role in governing volatility and covariance effects.First, observe that the integrated CIR process with optimal parameter Ξ (cid:63) is expected tofeature a larger variance compared to the integrated CIR with parameter Ξ (cid:63), + . Because of theshift constraint, the discount curve P y in the latter case rends to be much flatter than in theformer case i.e., one expects to have, in general P y ( t ; Ξ (cid:63), + ) ≥ P y ( t ; Ξ (cid:63) ) . This can be observed from panels (a) and (b) of Figure 3. When working with Ξ (cid:63), + , a substantialpart of the shape of P market = G comes from the deterministic shift. This amounts to limitthe randomness of the process. Not surprisingly, this will impact the variance of the integratedprocess Y . Indeed, because the discount curve of the CIR process with parameter Ξ (cid:63), + generallydominates that of the CIR process with parameter Ξ (cid:63) , one intuitively expects the variance ofthe CIR with parameter Ξ (cid:63) to be larger than that of the CIR with parameter Ξ (cid:63), + , due tothe zero lower bound. In other words, even if it seems difficult to provide a formal proof, oneexpects intuitively the following to hold, in general: v Λ ϕ ( t ) = v Y ( t ; Ξ (cid:63) ) ≥ v Y ( t ; Ξ (cid:63), + ) = v Λ ϕ, + ( t ) . This is indeed the case on Figure 4: v Λ ϕ (dotted blue) dominates v Λ ϕ, + (solid blue).Second, observe that for a given base process y , the variance of the integrated TC-CIR isalways larger than that of the integrated PS-CIR. Indeed, when working under the positivityconstraint (i.e., when y is driven by Ξ (cid:63), + ), we necessarily have Θ + ( t ) := Θ( t ; Ξ (cid:63), + ) ≥ t , inagreement with Theorem 2. Because for any parameter, the variance of Y is an increasingfunction of time (Lemma 6 in the Appendix, Section 7.1.2) we have, for Ξ ← Ξ (cid:63), + in particular, v Λ θ, + ( t ) = v Y (Θ + ( t ); Ξ (cid:63), + ) ≥ v Y ( t ; Ξ (cid:63), + ) = v Λ ϕ, + ( t ) . Third, we observe from Figure 4 that, in this example at least, the variance of the TC-CIRusing Ξ (cid:63) is comparable to the variance of the S-CIR: v Λ θ ( t ) ≈ v Y ( t ; Ξ (cid:63) ) = v Λ ϕ ( t ) . The fact that the variance of the S-CIR is expected to be close to that of the correspondingTC-CIR model can be understood intuitively as follows. As explained above the parameter32 ← Ξ (cid:63) computed using (9) leads the HAJD y to best fits the market curve, and the clock isused to absorb the remaining discrepancies. Therefore, one expects the clock not to deviatemuch from the actual time, i.e θ ( t ) ≈ y θ are those of y scaled by θ ( t ), and x θt = θ ( t ) y θt ≈ y θt , at least when the fitbetween P market and the base HAJD model P y is not too poor; see (17).To sum up, we observe that when dealing with CIR++ under a positivity constraint, onehas to choose between a valid (but low-volatility) PS-CIR process λ ϕ, + , or a flawed (by high-volatility) S-CIR one λ ϕ . By contrast, the TC-CIR model λ θ is always valid (Corollary 2),always feature a variance that is larger than the PS-CIR counterpart (Theorem 2), and itsvariance is actually comparable to the large levels generated by the S-CIR. The TC-CIR thusproves to be a solid challenger to CIR++ models. In particular, its features are specificallyinteresting when dealing with actual credit risk applications, as we now point out based on twocase studies. . . . . . t V a r (a) y = h . . . . . . t V a r (b) y optimized Figure 4: Variances of the integrated versions of λ ϕ, + (PS-CIR Ξ = Ξ (cid:63), + (left) and Ξ = Ξ (cid:63), +0 (right), solid blue), λ ϕ (S-CIR Ξ = Ξ (cid:63) (left) and Ξ = Ξ (cid:63) (right), dashed blue), and λ θ (TC-CIRwith Ξ = Ξ (cid:63) (left) and Ξ = Ξ (cid:63) (right), magenta). We deal with the pricing of a CDS option (CDSO). Because CDSO is an option on CDS, westart by recalling the no-arbitrage pricing equation of a CDS. We note t the valuation time and33ssume τ > t as it is pointless to price a CDS (or a CDSO) post-default. From the perspective ofthe protection buyer, the time- t value of a 1 dollar notional CDS CDS t ( a, b, k ) starting at time T a with maturity T b , t ≤ T a < T b , a spread k and (known) loss given default LGD = (1 − R ) isgiven by the difference of the conditional risk-neutral expectation of the protection and premiumcashflows : CDS t ( a, b, k ) = E (cid:2) (1 − R ) { T a ≤ τ ≤ T b } P rt ( τ ) |G t (cid:3) − k E (cid:34) b (cid:88) i = a +1 (cid:18) { τ ≥ T i } α i P rt ( T i ) + { T i − ≤ τ 25 (quarterly payment dates). In a reduced-form setup, when the default is triggered by thefirst jump of a Cox process with intensity λ , this expression can be developped explicitely thanksto the Key lemma: CDS t ( a, b, k ) = { τ>t } (cid:18) − (1 − R ) (cid:90) T b T a P rt ( u ) ∂ u P λt ( u ) du − k C t ( a, b ) (cid:19) , (21)where C t ( a, b ) is the risky duration, i.e., the time- t value of the CDS premia paid during thelife of the contract when the spread is 1: C t ( a, b ) := b (cid:88) i = a +1 α i P rt ( T i ) P λt ( T i ) − (cid:90) T i T i − u − T i − T i − T i − α i P rt ( u ) ∂ u P λt ( u ) du . The spread which, at time t , sets the forward start CDS at 0, called par spread, is given by: { τ>t } s t ( a, b ) := { τ>t } − (1 − R ) (cid:82) T b T a P rt ( u ) ∂ u P λt ( u ) duC t ( a, b ) . (22)The no-arbitrage price of a call option on such a contrat at time t = 0 becomes P SO ( a, b, k ) = E (cid:2) ( CDS T a ( a, b, k )) + P r ( T a ) (cid:3) = P r ( T a ) E e − Λ Ta (cid:32) (1 − R ) − b (cid:88) i = a +1 (cid:90) T i T i − g i ( u ) P r + λT a ( u ) du (cid:33) + , where g i ( u ) := (1 − R )( r ( u ) + δ T b ( u )) + k α i r ( u ) T i − T i − (1 − ( u − T i − )), with δ s ( u ) the Dirac deltafunction centered at s .Replacing the base intensity model ( λ ) by its shifted ( λ ϕ , λ ϕ, + ) or time-changed ( λ θ ) ver-sions leads to model prices noted P SO ϕ ( a, b, k ) , P SO ϕ, + ( a, b, k ) and P SO θ ( a, b, k ), respectively.Interestingly, these models are equally tractable as they feature similar expressions that can be34ritten in terms of the base process λ or its time integral, Λ. For instance, dropping the Ξ forshort, Λ ϕT a = Λ T a + (cid:90) T a ϕ ( u ) du , Λ θT a = Λ Θ( T a ) , and P λ ϕ T a ( u ) = P λT a ( u ) e (cid:82) uTa ϕ ( s ) ds = e A Ta ( u ) − B TA ( u ) λ Ta + (cid:82) uTa ϕ ( s ) ds ,P λ θ T a ( u ) = P λ Θ( T a ) ( u ) = e A Θ( Ta ) ( u ) − B Θ( Ta ) ( u ) λ Θ( Ta ) . Recall that these expressions have a closed form when λ is a (J)CIR process.Such kind of options has little liquidity. Models are then often compared in terms of theircapabilities to generate large “implied volatilities”. Indeed, empirical evidences show that thisis a typical feature of CDS option quotes, when disclosed. Therefore, we compare the modelsin terms of their “Black volatilities”: the volatility that one needs to plug in a “Black-Scholes”type of model to reproduce the model prices. Black model for P SO is recalled in the Appendix,Section 7.5. The Black volatility associated to a model price P SO model ( a, b, k ) is thus thevolatility ¯ σ satisfying P SO model ( a, b, k ) = P SO Black ( a, b, k, ¯ σ ). Recall that in all cases, theintensity process λ is calibrated to the market, i.e., P λ ( t ) = G ( t ). In other words, choosing,e.g., a CIR process for the base intensity process λ combined with the correct shift with ( ϕ + ) orwithout ( ϕ ) positivity constraint, or eventually using the correct clock rate θ , all three modelsyield the same survival probability curve ( P λ ϕ ( t ) = P λ ϕ, + ( t ) = P λ θ ( t ) = G ( t )). Hence, all thesemodels agree on the par spread: s ( a, b ) = (1 − R ) (cid:82) T b T a P r ( u ) h ( u ) G ( u ) duC ( a, b ) . We compare the S-CIR, the PS-CIR and the TC-CIR. The base HAJD process y in TC-CIRis taken to be the same as that of the S-CIR. One can see from Table 3 that the S-CIR featureslarge implied volatilities. Recall however that it allows for negative intensities, hence is notappropriate. The PS-CIR model are not capable of generating large volatility levels, in linewith the previous discussion. The TC-CIR fits in between: it rules out negative intensities,while maintaining substantial volatility levels.One might be concerned by the fact that the implied volatilities of the TC-CIR remainrelatively small. This can be addressed in two ways. First, one can play with the parameterΞ. However, the Feller constraint is often required to hold, which sets limits on the process’volatility. Another approach consists of considering a JCIR model as HAJD. Indeed, JCIR35s often considered when large volatilities are required. However, as explained in Section 3.3,increasing the volatility by boosting the jump activity while maintaining the calibration to agiven market curve G reinforces the positivity issue. Fortunately, we do not have this problemin the TC-JCIR. One can drastically increase the jump activity without impacting the positivityof the TC-JCIR. As a consequence, the TC-JCIR seems very much appropriate when one needsa positive but yet high-volatility process. This is illustrated on Table 4 using the same jumpparameters as those given in Brigo and Mercurio (2006). We keep the same parameter Ξ asbefore for the diffusion part, and play with the jump rate ( ω ) and jump size ( α ) in the compoundPoisson process J . In every case, the clock Θ is chosen such that the model perfectly fits Ford’ssurvival probability curve. Interestingly, the (positive) TC-JCIR model can feature much largerimplied volatility levels than the PS-CIR. The results for the S-JCIR are also larger than thePS-CIR, but they are not shown because the negative intensity problem is magnified. T a T b CIR++ TC-CIR λ ϕ λ ϕ, + λ θ T a T b CIR++ TC-CIR λ ϕ λ ϕ, + λ θ Table 3: Black volatilities for at-the-money ( k = s ( a, b )) CDS options implied by CIR++models (S-CIR and PS-CIR) and the TC-CIR model with y = h (left) and y optimized(right) using Monte Carlo simulation (500K paths with time step 0.01). In all the consideredcases, the CIR++ model without shift is not valid since inf { λ ϕt , t ∈ [0 , T a ] } < 0. Among thetwo valid intensity models ( λ ϕ, + t and λ θt ), the latter exhibits a much higher implied volatility.36 a T b TC-JCIR ( ω, α )(0 , 0) (0 . , . 1) (0 . , . Table 4: Black volatilities for at-the-money ( k = s ( a, b )) CDS options implied by the TC-JCIRmodel (jump arrival rate ω and jump size α ) using Monte Carlo simulation (10 paths withtime step 0.01) and paramter set Ξ = Ξ (cid:63) but for various jump parameters ( α, ω ). A major concern of the post-crisis regulation is the modeling of the capital requirement of firmstacking into account some credit adjustment to the valuation under credit risk. Counterpartycredit risk is defined as the risk that the counterparty of an over-the-counter (OTC) deal willdefault before the maturity of the contract. The latter can be seen as an option given to thecounterparty, and can be priced in a risk-neutral setup by adjusting the OTC derivative, leadingto CVA. The latter is nothing but the expected losses due to the missed payments associatedto the OTC portfolio. In a risk-neutral specification and assuming τ > 0, the current ( t = 0)value of the CVA is expressed as:CVA = E (cid:2) (1 − R ) V + τ { τ ≤ T } (cid:3) where V stands for the discounted exposure (i.e., the exposure process rescaled by the stochasticdiscount factor D ). A straightforward application of the Key lemma (under some technicalconditions that are valid here) yieldsCVA = E (cid:20) (1 − R ) (cid:90) T V + u λ u e − Λ u du (cid:21) . (23)37he CVA of the shifted and the time-changed models, CVA ϕ and CVA θ , correspond toabove expression, replacing ( λ, Λ) by ( λ ϕ , Λ ϕ ) and ( λ θ , Λ θ ), respectively. The purpose of thissection is to illustrate the order of magnitude of CVA figures that can be obtained with eithermodels. In particular, we do not aim at representing a specific exposure. Instead, we simplifythe analysis by considering two prototypical dynamics: dV t = νdW Vt ,dV t = (cid:18) γ ( T − t ) − V t T − t (cid:19) dt + νdW Vt . where W V is an F -Brownian motion. The first SDE is that of a martingale, and can depict theevolution of the discounted price of a forward contract prior to its cashflow date. The secondSDE corresponds to a Brownian bridge with drift, and mimics the dynamics of the discountedprice of an asset paying continuous dividends. These two models have been previously used inVrins (2017) and Brigo and Vrins (2018) to describe, in a schematic way, exposures of FRA andIRS. Calibration to actual exposures give indicative value for the parameters.In general, there is no reason to assume that the Brownian motion driving the defaultintensity ( W ) would be independent of the Brownian motion driving the exposure ( W V ): itdepends on the problem at hand. Usually, we consider the general case of wrong-way risk (WWR) effect, obtained by introducing a correlation between the Brownian drivers. For theCIR++ we assume dW t dW Vt = ρdt , whereas for the TC-CIR, we apply the synchronisationprocedure devised in Mbaye and Vrins (2018) in order to preserve the correlation after time-changing the intensity process. In the special case where the default time of the counterpartyis independent from the discounted exposure (i.e., ρ = 0, that is no wrong-way risk) one caneasily deduce from (23) the independent CVA formulaCVA ⊥ = − (1 − R ) (cid:90) T E (cid:2) V + u (cid:3) d E (cid:2) e − Λ u (cid:3) = (1 − R ) (cid:90) T f λ ( u ) E (cid:2) V + u (cid:3) P λ ( u ) du . Recall that whatever the chosen model, it is assumed to be calibrated to the survival proba-bility curve G , extracted from CDS prices. This leads to P λ ( u ) = G ( t ), and to the optimal shiftand clock functions, namely ϕ or ϕ + in the S-CIR and PS-CIR cases, and Θ for the TC-CIR.In this case, CVA ⊥ does not depend on the default model:CVA ⊥ = − (1 − R ) (cid:90) T E (cid:2) V + u (cid:3) dG ( u ) = (1 − R ) (cid:90) T h ( u ) E (cid:2) V + u (cid:3) G ( u ) du . However, the independent case ρ = 0 is unrealistic, and may lead to severe over or underes-timations of CVA Kim and Leung (2016); Brigo and Vrins (2018); Breton and Marzouk (2018).38nder WWR, CVA becomes model-dependent. Figure 5 shows the evolution of CVA with re-spect to ρ for three different models: λ ϕ (CIR++ without constraint, solid blue), λ ϕ, + (CIR++with constraint, dashed blue) and λ θ (TC-CIR, dashed magenta), all calibrated to Ford’s sur-vival probability curve G as before. Under no-WWR, the CVA is equal to the independent CVA(cyan): it is flat, model-free and can be computed using a simple integration. Under WWR,the CVAs are computed using Monte Carlo simulations (100K paths, time step of 0.01) andadaptive control variate . The TC-CIR and S-CIR models exhibit the largest WWR effectsand seem therefore appropriate to deal with high WWR applications. Recall that only TC-CIRis valid here as S-CIR gives room to negative intensities. The PS-CIR however is almost flat,equal to the independent CVA. This can be understood from the fact that WWR is essentiallya covariance effect between V and e − Λ . Hence, the models featuring large variance for Λ exhibitlarger WWR effects at any (non-zero) fixed correlation level ρ . Eventually, TC-CIR provides anappealing trade-off: on the one hand, as the PS-CIR, it rules out the negative intensity probleminherent to the S-CIR model. But on the other hand, it preserves, to some extend, the varianceof the S-CIR model, and therefore exhibits a much larger variance compared to PS-CIR. see Mbaye and Vrins (2018) for the implementation of the adaptive control variate applied on CVA compu-tation. . . . . . . . r C VA (a) dV t = νdW Vt , y = h −1.0 −0.5 0.0 0.5 1.0 . . . . . . . . r C VA (b) dV t = νdW Vt , y optimized −1.0 −0.5 0.0 0.5 1.0 . . . . . r C VA (c) dV t = (cid:16) γ ( T − t ) − V t T − t (cid:17) dt + νdW Vt , y = h −1.0 −0.5 0.0 0.5 1.0 . . . . . r C VA (d) dV t = (cid:16) γ ( T − t ) − V t T − t (cid:17) dt + νdW Vt , y optimized Figure 5: Impact of the exposure-credit correlation ρ on CVA levels for prototypical 5Y Forward(top) and Swap exposure (bottom) with y = h (left) or y optimized (right), ν = 8% and γ = 0 . λ ϕ, + (PS-CIR Ξ = Ξ (cid:63), + (left)and Ξ = Ξ (cid:63), +0 (right), dotted blue), λ ϕ (S-CIR Ξ = Ξ (cid:63) (left) and Ξ = Ξ (cid:63) (right), solid blue), and λ θ (TC-CIR Ξ = Ξ (cid:63) (left) and Ξ = Ξ (cid:63) (right), dotted magenta). The case without wrong-wayrisk corresponds to the flat (cyan) line. 40 Conclusion The calibration problem consists of finding the parameters of a model x so as to perfectly fita given market curve. The perfect fit is an important feature in a pricing context, that isconnected to no-arbitrage opportunities and corrects valuation of trading positions. This callsfor two important features: the model x must be (i) flexible enough (to be able to generatevarious shapes) and (ii) tractable enough (to facilitate the parameters’ optimization procedure).Time-homogeneous affine models like Vasicek, CIR or JCIR are very good candidates in thisrespect, and are widely used in interest rates and credit risk modeling. However, as such, theyonly feature a couple of constants and hence lack calibration flexibility. The deterministic shiftextension offers an appealing solution. It consists of starting with a tractable base model y ,that is shifted in a deterministic way with a function ϕ . The resulting process x t = y t + ϕ ( t )becomes fully flexible. Indeed, any discount or survival probability curve can be generated bysuch a model. Moreover, it has a tractability level that is very similar to that of y because ϕ is deterministic. Eventually, for every market curve, the shift ϕ (cid:63) that leads to the perfect fit isknown in closed form, as a function of the y parameters and the market curve. However, thismethod is less appealing when the model x needs to fulfill some range constraints. Among those,non-negativity is of primary importance when modeling interest rates (depending on the type ofeconomy at hand), mortality rate, prepayment rate or default intensities. In the deterministicshift approach indeed, starting with a non-negative base process y is not enough to guaranteethat so will be x , without additional constraint on ϕ . Furthermore, this constraint becomesmore and more severe when increasing the process volatility, due to the zero lower bound.It seems obvious to rule out models allowing for “negative volatilities”. However, surpris-ingly, the same does not seem to apply when it comes to “negative intensities”. Yet, bothare equally flawed. We believe the reason is twofold: first, negative intensities do not directlygenerate numerical problems (in contrast with volatilities that often appear in square-roots), sothat the issue is less “obvious”, second, there is a lack of a sound alternative. The positivityconstraint can be dealt with by including a non-negativity constraint on ϕ . However, this againraises two problems. First the parameter optimization problem becomes more difficult and sec-ond, the resulting process x then features a much lower variance than without the constraint,which contradicts empirical evidences. Therefore, one often prefers to disregard the “negativeintensities” issue, giving the priority to stochasticity and perfect fit.In this paper, we develop such an alternative. It simply consists of time-changing a positive41omogeneous affine jump-diffusion. The model remains tractable, positive, the optimal clock isfound by simple inversion and features larger implied volatility compared to the shift approach.Moreover, the perfect fit is achievable for a broad class of discount curves, including all decreas-ing discount curves. The features of the model have been illustrated on topical examples takenfrom credit risk, but other applications could be considered as well. This method thus proves tobe a competitive challenger to the shift approach, at least under the (very common) positivityconstraint, and when large volatility levels are needed. Let y be a F -adapted jump-diffusion introduced in Definition 3 and Y t := (cid:82) t y u du its integratedversion. We denote v x ( t ) = v x ( t ; Ξ) := V [ x t ] the variance of a stochastic process x parametrizedby Ξ at time t . Without explicit mention, all the results below that are given without proofscan be found in, e.g., Brigo and Mercurio (2006). New results are given in lemmas for furtherreference. The Vasicek model corresponds to the special HAJD case ( a ( t ) , b ( t ) , c ( t ) , d ( t ) , α, ω ( t )) = ( κβ, − κ, η , , , A y , B y functions in (3) are given by: A ys ( t ; Ξ) = (cid:18) β − η κ (cid:19) ( B ys ( t ; Ξ) + s − t ) − η κ B ys ( t ; Ξ) ,B ys ( t ; Ξ) = 1 κ (cid:16) − e − κ ( t − s ) (cid:17) . The forward curve associated to this model is proven to be f VAS s ( t ) := (1 − e − κ ( t − s ) ) κ β − η / κ + η κ e − κ ( t − s ) (1 − e − κ ( t − s ) ) + y t e − κ ( t − s ) . (24)Moreover, both y and Y are Normally distributed at all times, with E [ y t ] = y − κt + βκB y ( t ; Ξ) , E [ Y t ] = y B y ( t ; Ξ) + β ( t − B y ( t ; Ξ)) ,v y ( t ) = η κ (cid:0) − e − κt (cid:1) ,v Y ( t ) = η κ (cid:20) t + 1 − e − κt κ − B y ( t ; Ξ) (cid:21) . emma 5. Let y be a Vasicek process and Y its time integral. The functions v y ( t ) and v Y ( t ) are increasing with respect to t .Proof. It is obvious for v y ( t ), and for v Y ( t ), a few manipulations lead to ddt v Y ( t ) = (cid:16) ηκ (1 − e − κt ) (cid:17) ≥ . The CIR model corresponds to the special HAJD case ( a ( t ) , b ( t ) , c ( t ) , d ( t ) , α, ω ( t )) = ( κβ, − κ, , δ , , A y , B y functions in eq. (3) are given by: A ys ( t ; Ξ) = 2 κβδ ln 2 γ exp { ( κ + γ )( t − s ) / } γ + ( κ + γ )(exp { ( t − s ) γ } − ,B ys ( t ; Ξ) = 2(exp { ( t − s ) γ } − γ + ( κ + γ )(exp { ( t − s ) γ } − . where γ := √ κ + 2 δ . The forward curve associated to this model is given by Brigo andMercurio (2006) f CIR s ( t ) := 2 κβ ( e ( t − s ) γ − γ + ( κ + γ )( e ( t − s ) γ − 1) + y t γ e ( t − s ) γ [2 γ + ( κ + γ )( e ( t − s ) γ − . (25)Important characteristics of the CIR processes can be computed explicitly, (see, e.g., Dufresne(2001)). For instance, y is distributed as a non-central chi-squared. The two first order momentsof y and Y are respectively given by E [ y t ] = y e − κt + β (cid:0) − e − κt (cid:1) , E [ y t ] = y e − κt + (cid:18) δ κ + β (cid:19) (cid:104) y (cid:0) e − κt − e − κt (cid:1) + β (cid:0) − e − κt (cid:1) (cid:105) , E [ Y t ] = tβ + ( y − β ) κ (1 − e − κt ) , E [ Y t ] = (cid:18) y − βκ (cid:19) + δ (cid:18) y κ − β κ (cid:19) + tβ (cid:18) y κ − βκ + δ κ (cid:19) + t β + e − κt (cid:34) − (cid:18) y − βκ (cid:19) + 2 βδ κ + t (cid:18) − y βκ + 2 β κ − y δ κ + 2 βδ κ (cid:19)(cid:35) + e − κt (cid:34)(cid:18) y − βκ (cid:19) − y δ κ + βδ κ (cid:35) . In contrast with the Vasicek model, the variance of the CIR is not always increasingmonotonously with time; it depends on the parameters. However, the variance of the inte-grated CIR is increasing. These properties are proven in the next lemma, and will be centralin the proof of Theorem 2. 43 emma 6. Let y be a CIR process and Y its time integral. Then, v y ( t ) = δ κ (cid:20) y ( e − κt − e − κt ) + β κ (1 − e − κt ) (cid:21) , (26) v Y ( t ) = δ κ (cid:2)(cid:0) β − κt ( y − β ) − ( y − β/ e − κt (cid:1) e − κt + tκβ + ( y − β/ (cid:3) . (27) The function v y ( t ) is increasing if β ≥ y . Otherwise, it is first increasing up to a time t (cid:63) , andthen decreasing on ( t (cid:63) , ∞ ) . By contrast, v Y ( t ) is always increasing.Proof. The computation of the variances is trivial from the first two moments recalled above.The derivative of the variance of the CIR is given by ddt v y ( t ) = y δ ( − e − κt + 2 e − κt ) + βδ e − κt (1 − e − κt )= δ e − κt ( β − y ) − δ e − κt ( β − y ) . This expression has a root on the positive half-line at t (cid:63) = 1 κ ln (cid:18) y y − β (cid:19) only if y > β . Otherwise, v y ( t ) is always increasing in t . The derivative of the variance of theintegrated CIR with respect of time can be written, after some manipulations, as ddt v Y ( t ) = δ κ (cid:2) y e − κt ( e − κt − (1 − κt )) + β (1 − (2 κt + e − κt ) e − κt ) (cid:3) . Because e − x ≥ − x for all x ≥ y , κ are positive constants, the first term is positive forall t ≥ 0. On the other hand, β > 0, and it is enough to check that 1 − g ( κt ) ≥ t ≥ g ( x ) := (2 x + e − x ) e − x . Clearly, g (0) = 1 and g (cid:48) ( x ) = 2 e − x (1 − x − e − x ) ≤ x ≥ − g ( κt ) ≥ t ≥ The characteristics of the JCIR can be obtained by adjusting those of the corresponding CIR,i.e., with same initial value and diffusion parameters. We note z the former and y the latter,and similarly for their integrated versions ( Z and Y , respectively). Hence, if the parameter setfor the CIR ( y, Y ) is Ξ = ( κ, β, δ, , , y ), the parameter set of the corresponding JCIR ( z, Z )is Ξ = ( κ, β, δ, α, ω, z ) with z = y and α, ω ≥ 0. The functions associated to the discountcurve are given by A zs ( t ; Ξ) = A ys ( t ; Ξ) + αωδ / − κα − α ln 2 γe γ + κ +2 α ( t − s ) γ + ( κ + γ + 2 α )( e ( t − s ) γ − ,B zs ( t ; Ξ) = B ys ( t ; Ξ) . f JCIR s ( t ) := f CIR s ( t ) + 2 ωα ( e ( t − s ) γ − γ + ( κ + γ + 2 α )( e ( t − s ) γ − , (28)where f CIR s ( t ) is given in (25). For every valid parameters, f JCIR s ( t ) ≥ f CIR s ( t ) for all t ≥ s .Regarding the moments, we have the following result. Lemma 7. Let y (resp. Y ) be a CIR (resp. integrated CIR) and z (resp. Z ) be a JCIR (resp.integrated JCIR) with same initial value, same diffusion parameters but with jumps governedby ( ω, α ) . Then, E [ z t ] = E [ y t ] + ωακ (1 − e − κt ) , E [ Z t ] = E [ Y t ] + ωακ (cid:0) κt − (1 − e − κt ) (cid:1) .v z ( t ) = v y ( t ) + ωα (cid:34) δ (cid:18) − e − κt κ (cid:19) + α − e − κt κ (cid:35) ,v Z ( t ) = v Y ( t ) + αωκ (cid:20) − e − κt κ (cid:0) ξ (3 − e − κt ) − δ (cid:1) + 2 δ te − κt + t (cid:0) ακ + δ (cid:1)(cid:21) , where ξ := δ / − ακ . The function v z ( t ) is increasing with respect to t unless y > β + ωα/κ ,in which case it is first increasing up to a time t , and then decreasing on ( t , ∞ ) . Moreover, v z ( t ) ≥ v y ( t ) , v Z ( t ) ≥ v Y ( t ) and v Z ( t ) is always increasing.Proof. Applying Ito’s lemma we can solve the JCIR SDE (19) by z t = z e − κt + β (1 − e − κt ) + δ (cid:90) t e − κ ( t − s ) √ z s dW s + (cid:90) t e − κ ( t − s ) dJ s , (29)and find the SDE governing the integrated JCIR process Z t = tβ + ( z − β ) κ (1 − e − κt ) + δ (cid:90) t (cid:90) s e − κ ( s − u ) √ z u dW u ds + (cid:90) t (cid:90) s e − κ ( s − u ) dJ u ds . (30)From (29), we can write E [ z t ] = z e − κt + β (1 − e − κt ) + E (cid:20)(cid:90) t e − κ ( t − s ) dJ s (cid:21) = E [ y t ] + ωα (cid:90) t e − κ ( t − s ) ds = E [ y t ] + ωακ (1 − e − κt ) . v z ( t ) = E [( z t − E [ z t ]) ]= E (cid:34)(cid:18) δ (cid:90) t e − κ ( t − s ) √ z s dW s + (cid:90) t e − κ ( t − s ) dJ s − ωα (cid:90) t e − κ ( t − s ) ds (cid:19) (cid:35) = E (cid:34)(cid:18) δ (cid:90) t e − κ ( t − s ) √ z s dW s (cid:19) (cid:35) + E (cid:34)(cid:18)(cid:90) t e − κ ( t − s ) ( dJ s − ωαds ) (cid:19) (cid:35) = v y ( t ) + δ ωακ (cid:90) t e − κ ( t − s ) (1 − e − κs ) ds + 2 ωα (cid:90) t e − κ ( t − s ) ds = v y ( t ) + ωα (cid:34) δ (cid:18) − e − κt κ (cid:19) + α − e − κt κ (cid:35) . Using a similar procedure applied to (30) combined with Fubini’s theorem, one can derive theexpectation and variance of the integrated JCIR : E [ Z t ] = tβ + ( z − β ) κ (1 − e − κt ) + E (cid:20)(cid:90) t (cid:90) s e − κ ( s − u ) dJ u ds (cid:21) = E [ Y t ] + E (cid:20)(cid:90) t (cid:90) ts e − κu due κs dJ s (cid:21) = E [ Y t ] + ωακ (cid:90) t (1 − e − κ ( t − s ) ) ds = E [ Y t ] + ωακ (cid:0) κt − (1 − e − κt ) (cid:1) and v Z ( t ) = E [( Z t − E [ Z t ]) ]= E (cid:34)(cid:18) δ (cid:90) t (cid:90) s e − κ ( s − u ) √ z u dW u ds + (cid:90) t (cid:90) s e − κ ( s − u ) dJ u ds − E (cid:20)(cid:90) t (cid:90) s e − κ ( s − u ) dJ u ds (cid:21)(cid:19) (cid:35) = E (cid:34)(cid:18) δκ (cid:90) t (1 − e − κ ( t − s ) ) √ z s dW s + 1 κ (cid:90) t (1 − e − κ ( t − s ) ) dJ s − ωακ (cid:90) t (1 − e − κ ( t − s ) ) ds (cid:19) (cid:35) = E (cid:34)(cid:18) δκ (cid:90) t (1 − e − κ ( t − s ) ) √ z s dW s (cid:19) (cid:35) + E (cid:34)(cid:18) κ (cid:90) t (1 − e − κ ( t − s ) )( dJ s − ωαds ) (cid:19) (cid:35) = v Y ( t ) + δ ωακ (cid:90) t (1 − e − κ ( t − s ) ) (1 − e − κs ) ds + 2 ωα κ (cid:90) t (1 − e − κ ( t − s ) ) ds = v Y ( t ) + αωκ (cid:20) − e − κt κ (cid:0) ( δ / − ακ )(3 − e − κt ) − δ (cid:1) + 2 δ te − κt + t (cid:0) ακ + δ (cid:1)(cid:21) . Notice that the above results can be obtained using another procedure, namely by deriving onceor twice the characteristic function Ψ t ( u, v ) = E [ e uz t + vZ t ] of ( z t , Z t ) which can be recovered fromeq. (A.1) in Duffie and Gˆarleanu (2001). This procedure is however much heavier. Be aware that there are typos in this formula. The correct expression can be found in eq. (B.9) in the draftversion of Duffie and Gˆarleanu’s paper, that is available for download on the authors’ webpage. ddt v z ( t ) = y δ ( − e − κt + 2 e − κt ) + βδ e − κt (1 − e − κt ) + δ ωακ e − κt (1 − e − κt ) + 2 ωα e − κt = δ e − κt ( β − y + ωα/κ ) − δ e − κt ( β − y + ωα/κ − ωα /δ ) . This expression has a root on the positive half-line at t := 1 κ ln (cid:18) y + 2 ωα /δ y − β − ωα/κ (cid:19) (31)only if y > β + ωα/κ . Otherwise, v y ( t ) is always increasing in t .It seems intuitive that the variance of the JCIR cannot be smaller than that of the CIR, andsimilarly for the integrated versions. Because of the mean reverting effect however, this needsto be confirmed. It is obvious that v z ( t ) − v y ( t ) ≥ 0. The term associated to v Z ( t ) − v Y ( t )starts at zero (since obviously v Z (0) = v Y (0) = 0). This difference is increasing: ddt ( v Z ( t ) − v Y ( t )) = δ ωακ (cid:0) − (2 κt + e − κt ) e − κt (cid:1) + 2 ωα κ (cid:0) − e − κt (cid:1) ≥ . Indeed, the second term is obviously positive and the first term takes the form δ ωακ (1 − g ( κt ))where the function g ( x ) = (2 x + e − x ) e − x is shown to be bounded by 1 for x ≥ v Z ( t ) ≥ v Y ( t ). Because both v Y ( t ) (from Lemma 6) and v Z ( t ) − v Y ( t )are increasing; v Z ( t ) is itself increasing. P x + y is a discount curve Observe first that P x + y is a discount curve whenever P x , P y are in the case where x, y areindependent since then P x + ys = P xs P ys , and the product of two time- s discount curves is itselfa time- s discount curve. The next lemma provides sufficient conditions on y for P y to be adiscount curve in the general case. Lemma 8. Let T be a fixed time horizon. Then, P y is a discount curve whenever y is positiveand sup t ∈ [0 ,T ] y t is integrable.Proof. We start with the lemma giving sufficient conditions to swap the expectation and deriva-tive operators which can be found in, e.g., Pag`es (2018). Lemma 9. Let I be a nontrivial interval of R , B ( I ) the Borel set of I and Ψ : I × Ω → R , ( x, ω ) (cid:55)→ Ψ( x, ω ) be a B ( I ) ⊗ G -measurable function. If the function Ψ satisfies:(i) For every x ∈ I , the random variable Ψ( x, ω ) ∈ L , ii) Ψ x ( x, ω ) := ∂ Ψ( x,ω ) ∂x exists for all x ∈ I a.s.,(iii) There exists Z ∈ L such that for every x ∈ I , | Ψ x ( x, ω ) | ≤ Z ( ω ) a.s. . Then the function ψ ( x ) := E [Ψ( x, ω )] is defined and differentiable at every x ∈ I with derivative dψ ( x ) dx = E [Ψ x ( x, ω )] . We now proceed with the proof of Lemma 8.Let us fix t ≤ T . Hence, (cid:12)(cid:12)(cid:12)(cid:12) ddt e − (cid:82) t y u du (cid:12)(cid:12)(cid:12)(cid:12) = | y t e − (cid:82) t y u du | ≤ y t ≤ sup t ∈ [0 ,T ] y t for all t ∈ [0 , T ]. Noting that sup t ∈ [0 ,T ] y t is integrable, one can use Lemma 9 with Ψ( t, w ) ← e − (cid:82) t y u ( w ) du and Z ( ω ) ← S yT := sup t ∈ [0 ,T ] y t , justifying the swap between the derivative andexpectation operators: ddt P y ( t ) = ddt E (cid:104) e − (cid:82) t y u du (cid:105) = E (cid:20) ddt e − (cid:82) t y u du (cid:21) = − E (cid:104) y t e − (cid:82) t y u du (cid:105) , where the right-hand side is bounded by the expectation of Z , which is integrable. This con-cludes the proof.In order for thE assumption about the integrability of the running supremum of y to beuseful in practice, it needs to be “checkable’. Hence, we need to give simpler sufficient conditions(e.g., based on the coefficients of the SDE of y ) that would guarantee that S yT := sup t ∈ [0 ,T ] | y t | satisfies E [ S yT ] < ∞ . Lemma 10. Let W be a Brownian motion, J a compound Poisson process with constant jumpintensity ω and the jump sizes are exponentially distributed with mean α , and y solving dy t = µ ( t, y t ) dt + σ ( t, y t ) dW t + dJ t where y is positive, E [ (cid:82) T | µ ( t, y t ) | dt ] < ∞ and E [ (cid:82) T σ ( t, y t ) dt ] < ∞ . Then, E [ | S yT | ] = E [ S yT ] < ∞ where S yT := sup t ∈ [0 ,T ] | y t | . Proof. The solution of the SDE is y t = y + (cid:90) t µ ( s, y s ) ds (cid:124) (cid:123)(cid:122) (cid:125) A t + (cid:90) t σ ( s, y s ) dW s (cid:124) (cid:123)(cid:122) (cid:125) M t + J t ⇒ | y t | ≤ | A t | + | M t | + J t , S yT ≤ sup t ∈ [0 ,T ] | A t | (cid:124) (cid:123)(cid:122) (cid:125) S AT + sup t ∈ [0 ,T ] | M t | (cid:124) (cid:123)(cid:122) (cid:125) S MT + sup t ∈ [0 ,T ] J t (cid:124) (cid:123)(cid:122) (cid:125) S JT . We show in the sequel that S AT , S MT and S JT are integrable. This would conclude the proof sinceit would lead to E [ | S yT | ] = E [ S yT ] ≤ E [ S AT ] + E [ S MT ] + E [ S JT ] < ∞ . Suppose that E [ (cid:82) T | µ ( s, y s ) | ds ] < ∞ . Then, S AT = sup t ∈ [0 ,T ] | y + (cid:90) t µ ( s, y s ) ds | ≤ | y | + sup t ∈ [0 ,T ] (cid:90) t | µ ( s, y s ) | ds ≤ | y | + (cid:90) T | µ ( s, y s ) | ds . showing that E [ | S AT | ] = E [ S AT ] ≤ | y | + E [ (cid:82) T | µ ( s, y s ) | ds ] < ∞ .On the other hand, M is a martingale, so that | M | is a submartingale: E [ | M t | |F s ] ≥ | E [ M t |F s ] | = | M s | . We can then apply Doob’s inequality, E [ S MT ] = E [ sup t ∈ [0 ,T ] | M t | ] ≤ ee − (cid:0) E [ | | M T | log + | M T | | ] (cid:1) . Using − e ≤ x log x ≤ x for x ≥ | x log + x | = x log + x ≤ | x log x | ≤ max( e − , x ): E [ | M T | log + | M T | ] ≤ E [max( e − , M T )]= E [ e − { M T ≤ e − } + M T { M T >e − } ] ≤ e − + E [ M T ] . Hence, E [ S MT ] ≤ ee − (cid:0) e − + E [ M T ] (cid:1) . Using Ito isometry, E [ M T ] = E (cid:104)(cid:82) T σ ( t, y t ) dt (cid:105) which is bounded, by assumption.Similarly, one can prove that E [ S JT ] is finite by applying the Doob’s inequality to the mar-tingale ( J t − ωαt ) , t ≤ T . Indeed, J t = | J t − ωαt + ωαt | ≤ | J t − ωαt | + ωαt, ∀ t ≤ T , which implies that E [ S JT ] ≤ E [ sup t ∈ [0 ,T ] | J t − ωαt | ] + ωαT ≤ ee − (cid:0) e − + E [( J T − ωαT ) ] (cid:1) + ωαT = ee − (cid:0) e − + 2 ωα T (cid:1) + ωαT < ∞ . P x + y is a discount curve when x, y are HAJD, possibly driven bycorrelated Brownian motions. Indeed, they satisfy the assumptions of Lemma 10. Observe first that for every θ and every t , one gets (cid:90) t x θu du = (cid:90) t θ ( u ) y θ ( u ) du = (cid:90) Θ( t )0 y u du . Hence, the expectation of their negative exponentials agree as well: P x θ ( t ) = P y (Θ( t )) . The specific clock rate θ (cid:63) given by the calibration equation thus satisfies, for all t , P market ( t ) = P x θ(cid:63) ( t ) = P y (Θ (cid:63) ( t )) . (32)Turning this equality in terms of instantaneous forward rates yields (cid:90) t f market ( u ) du = (cid:90) Θ (cid:63) ( t )0 f y ( u ) du . Eq. (15) is just the differential form of the latter.It is not clear, in general, to determine when this ODE admits a solution. However, asimple case is when P y is a strictly decreasing discount curve. In this case indeed, P y admitsan inverse on the positive half line, noted Q y . Apply Q y to (32) yields Θ (cid:63) = Q y ( P market ( t )).Furthermore, the inverse of a decreasing function is decreasing, and the combination of twodecreasing functions is itself increasing. Hence, if P market is decreasing, Θ (cid:63) ( t ) is continuous andstrictly increasing. Moreover, Θ (cid:63) (0) = Q y (cid:0) P market (0) (cid:1) = Q y (1) = 0. Hence, Θ (cid:63) exists, and isa clock. It is known from Corollary 2 that for any (non-trivial) (J)CIR process y with parameter Ξ,there exits a clock Θ (cid:63) ( t ) = Θ (cid:63) ( t ; Ξ) that yields a perfect fit between the curves P x generated ythe TC-JCIR x θ (cid:63) t := θ (cid:63) ( t ) y Θ (cid:63) ( t ) . Same holds true for the JCIR++, x ϕ (cid:63) t . This means: P x ϕ(cid:63) ( t ; Ξ) = P market ( t ) = P x θ(cid:63) ( t ; Ξ) , 50r equivalently, e − (cid:82) t ϕ (cid:63) ( u ) du P y ( t ; Ξ) = P market ( t ) = P y (Θ (cid:63) ( t ); Ξ) . Because y is a JCIR, it can be arbitrarilly close to 0 at any time t , hence the calibration con-straint amounts to force ϕ (cid:63) ( t ) ≥ f JCIR ( t ) ≤ f market ( t )) ∀ t ≥ 0. This impliesthat P y (Θ (cid:63) ( t ); Ξ) ≤ P y ( t ; Ξ). Because P y ( . ; Ξ) is a decreasing function, the last inequality isequivalent to Θ (cid:63) ( t ) ≥ t .To prove 1), we start from the increasingness of v Y ( t ) (Lemma 7). Hence, v Y (Θ (cid:63) ( t )) ≥ v Y ( t )since Θ (cid:63) ( t ) ≥ t .From (28), we have, after some computations, ddt f JCIR ( t ) = 4 γ e tγ (cid:2) κβ ( γ − κ + ( κ + γ ) e tγ ) + y γ ( γ − κ − ( κ + γ ) e tγ ) + ωα (2 γ + ( κ + γ )( e tγ − (cid:3) [2 γ + ( κ + γ )( e tγ − = 4 γ e tγ (cid:2) ( γ − κ )( κβ + y γ + ωα ) − ωα (cid:3) + 4 γ e tγ (cid:2) ( γ + κ )( κβ − y γ + ωα ) + 2 ωα (cid:3) [2 γ + ( κ + γ )( e tγ − . From this expression, one can check that f JCIR is strictly increasing if y < β + ωα/κ and y γ ≤ κβ + ωα . It is strictly decreasing if y ≥ β + ωα/κ . Otherwise, i.e., if y < β + ωα/κ and y γ > κβ + ωα , the derivative has a root at t := 1 γ ln ( γ − κ )( κβ + y γ + ωα ) − ωα ( κ + γ )( y γ − κβ − ωα ) − ωα . i.e., f JCIR is first increasing, then decreasing.The constraint ϕ (cid:63) ( t ) ≥ t , simply means that f market ( t ) ≥ f JCIR ( t ) and so θ (cid:63) ( t ) ≥ f JCIR ( t ) f JCIR (Θ (cid:63) ( t )) . Observe that the condition y = β + ωα/κ corresponds to the case where v y ( t ) isincreasing and f JCIR ( t ) is decreasing, hence ( i ) holds.If f market is constant, we have that f market ( t ) ≥ f JCIR ( t ) which implies that f market ( t ) ≥ f JCIR (Θ (cid:63) ( t )). Clearly, if f market ( t ) is constant or f JCIR is decreasing, then θ (cid:63) ( t ) ≥ 1. Andif v y ( t ) is increasing, v y (Θ (cid:63) ( t )) ≥ v y ( t ) since Θ (cid:63) ( t ) ≥ t . From the fact that V [ x θ (cid:63) t ] := θ (cid:63) ( t ) v y (Θ (cid:63) ( t )) and the variation of v y ( t ) (Lemma 7), ( ii ), ( iii ) and ( iv ) follow.In particular, taking ωα = 0, we recover the CIR case which corresponds to the TC-CIR model. In this context, the Black-Scholes model works as follows. We start by noting that the forwardstart CDS can be written in terms of the difference between the fair and the agreed premiumcashflows. Indeed, the former corresponds to the protection leg. Inserting (22) in (21) yields CDS t ( a, b, k ) = { τ>t } ( s t ( a, b ) − k ) C t ( a, b ) . P SO ( a, b, k ) = E (cid:2) ( s T a ( a, b ) − k ) + C T a ( a, b ) D ( T a ) (cid:3) = C ( a, b ) E ( a,b ) (cid:2) ( s T a ( a, b ) − k ) + (cid:3) (33)where E ( a,b ) stands for the expectation under the equivalent measure Q ( a,b ) , associated withthe num´eraire C ( a, b ). Interestingly, it is clear from (22) that the par spread s ( a, b ) is a Q ( a,b ) -martingale on [0 , T a ]. Hence, the Black-Scholes model for CDSO naturally postulates Q ( a,b ) -martingale dynamics for the par spread ds t ( a, b ) = ¯ σs t ( a, b ) dW st , t ≤ T a where W s is a Q ( a,b ) -Brownian motion. Eventually, the expectation in (33) is given by thestandard Black-Scholes formula by setting r ← 0. Hence, the Black-Scholes price of the PSO isgiven by P SO Black ( a, b, k, ¯ σ ) = C ( a, b ) [ s ( a, b )Φ( d ) − k Φ( d )]where d = ln s ( a,b ) k + ¯ σ T a ¯ σ √ T a , d = d − ¯ σ (cid:112) T a and Φ is the distribution function of a standard Normal random variable. References T. Bielecki and M. Rutkowski. Credit risk : modeling, valuation and hedging . Springer finance.Springer, 2002.T. Bielecki, M. Jeanblanc, and M. Rutkowski. Credit risk modeling. Technical report, Centerfor the Study of Finance and Insurance, Osaka University, Osaka (Japan), 2011.M. Breton and O. Marzouk. Evaluation of counterparty risk for derivatives with early-exercisefeatures. Journal of Economic Dynamics and Control , 88:1–20, 2018.D. Brigo and A. Alfonsi. Credit default swaps calibration and option pricing with the SSRDstochastic intensity and interest rate model. Finance and Stochastics , 9:29–42, 2005.D. Brigo and L. Cousot. A comparison between the SSRD model and the market model for cdsoptions pricing. International Journal of Theoretical and Applied Finance , 9(3), 2006.D. Brigo and N. El-Bachir. An exact formula for default swaptions pricing in the SSRJDstochastic intensity model. Mathematical Finance , 20(3):365–382, 2010.52. Brigo and F. Merccurio. A deterministic-shift extension to analytically-tractable and time-homogeneous short-rate models. Finance and Stochastics , 5:369–388, 2001.D. Brigo and F. Mercurio. Interest Rate Models - Theory and Practice . Springer, 2006.D. Brigo and F. Vrins. Disentangling Wrong-Way Risk: Pricing CVA via change of measuresand drift adjustment. European Journal of Operational Research , 269:254–264, 2018.D. Brigo, A. Capponi, and A. Pallavicini. Arbitrage-free bilateral counterparty risk valuationunder collateralization and application to credit default swaps. Mathematical Finance , 24(1):125–146, 2014.P. Carr and D. Madan. Option valuation using the fast fourier transform. Journal of Compu-tational Finance , 2(4):61–73, 1999.P. Carr, H. Geman, D. Madan, and M. Yor. Stochastic volatility for L´evy processes. Mathe-matical Finance , 13:345–382, 2003.JC. Cox, J.E. Ingersoll, and S.A. Ross. A theory of the term structure of interest rates. Econo-metrica , 53:385–407, 1985.S. Cr´epey, M. Jeanblanc, and D. Wu. Informationally dynamized gaussian copula. Technicalreport, 2012. URL .K.E. Dambis. On the decomposition of continuous submartingales. Theory of Probability andits Applications , 10:40914100, 1965.B. Dellacherie and P.-A. Meyer. Probabilit´es et Potentiel - Th´eorie des martingales . Hermann,1980.L. Dubins and G. Schwartz. On continuous martingales. Proceedings of the National Academyof Sciences , 53:913916, 1965.D. Duffie and N. Gˆarleanu. Risk and valuation of collateralized debt obligations. FinancialAnalyst Journal , 57:41–59, 2001.D. Duffie and R. Kan. A yield-factor model of interest rates. Mathematical Finance , 6:379406,1996.D. Duffie and K. Singleton. Modeling term structures of defaultable bonds. Review of FinancialStudies , 12:687–720, 1999. 53. Duffie and K. Singleton. Credit risk: pricing, measurement, and management. PrincetonUniversity Press , 2003.D. Duffie, D. Filipovic, and W. Schachermayer. Affine processes and applications in finance. Annals of Applied Probability , 13(3):984–1056, 2003.D. Dufresne. The integrated square-root process. Technical report, Research Paper 90, Centerfor Actuarial Studies, Department of Economics, University of Melbourne, 2001.P.H. Dybvig. Bond and bond option pricing based on the current term structure. In M. Demp-ster and S. Pliska, editors, Mathematics of Derivatives Securities (pp. 271-293) . CambridgeUniversity Press, 1997.D. Filipovic. Time-inhomogeneous affines processes. Stochastic processes and their application ,pages 639–659, 2005.D. Filipovic, M. Larsson, and A.B. Trolle. Linear-rational term structure models. Journal ofFinance , 72(2):655–704, 2017.H. Geman, D. Madan, and M. Yor. Time changes for L´evy processes. Mathematical Finance ,11:79–96, 2001.J. Gregory. Counterparty Credit Risk . Wiley Finance, 2010.D. Heath, R. Jarrow, and A. Morton. Bond pricing and the term structure of interest rates. Econometrica , 60:77–105, 1992.S. Heston. A closed-form solution for options with stochastic volatility with applications tobond and currency options. Review of Financial Studies , 6(2):327343, 1993.T. Ho and S.-B. Lee. Term structure movements and pricing interest rate contingent claims. Journal of Finance , 41(5):10111029, 1986.J. Hull and A. White. Pricing interest rate derivative securities. Review of Financial Studies ,3:573–592, 1990.M. Jeanblanc and F. Vrins. Conic martingales from stochastic integrals. Mathematical Finance ,28(2):516–535, 2018.M. Jeanblanc, M. Yor, and M. Chesney. Mathematical Methods for Financial Markets . SpringerVerlag, Berlin, 2009. 54. Joshi. The Concepts and Practice of Mathematical Finance . Cambridge University Press,2003.J. Kim and T. Leung. Pricing derivatives with counterparty risk and collateralization: A fixedpoint approach. European Journal of Operational Research , 249(2):525–539, 2016.D. Lando. Credit Risk Modeling: Theory and Applications International Journal of Theoretical and Applied Finance , 21(7):1850045, 2018.R. Merton. On the pricing of corporate debt: The risk structure of interest rates. Journal ofFinance , 29:449470, 1974.C. Nelson and A. Siegel. Parsimonious modelling of yield curves. Journal of Business , 60, 1987.G. Pag`es. Numerical Probability: An Introduction with Applications to Finance . Springer,Cham, 2018.H. Stein and K. Pong. Counterparty valuation adjustments. In Brigo Bielecki and Patras,editors, Credit Risk Frontiers: Subprime crisis, Pricing and Hedging, CVA, MBS, Ratingsand Liquidity , chapter 15. Wiley/Bloomberg Press, 2011.A. Swishchuk. Change of Time Methods in Quantitative Finance . SpringerBriefs in Mathemat-ics. Springer, 2016.O. Vasicek. An equilibrium characterization of the term structure. Journal of Financial Eco-nomics , 5:177–188, 1977.P. Veronesi. Fixed Income Analysis: Valuation, Risk, and Risk Management . Wiley, 2010.F. Vrins. Wrong-way risk CVA models with analytical EPE profiles under Gaussian exposuredynamics.