The Jarrow & Turnbull setting revisited
TTHE JARROW & TURNBULL SETTING REVISITED
THOMAS KRABICHLER AND JOSEF TEICHMANN
Abstract.
We consider a financial market with zero-coupon bonds that areexposed to credit and liquidity risk. We revisit the famous Jarrow & Turnbullsetting [13] in order to account for these two intricately intertwined risk types.We utilise the foreign exchange analogy that interprets defaultable zero-couponbonds as a conversion of non-defaultable foreign counterparts. The relevantexchange rate is only partially observable in the market filtration, which leadsus naturally to an application of the concept of platonic financial markets asintroduced in [6]. We provide an example of tractable term structure modelsthat are driven by a two-dimensional affine jump diffusion. Furthermore, wederive explicit valuation formulae for marketable products, e.g., for creditdefault swaps. Introduction
A zero-coupon bond is a financial contract that promises its holder the paymentof one monetary unit at maturity, with no intermediate payments. If adversecircumstances occur over the lifetime of the contract, the final redemption mayonly be a fraction of the promised payoff. The uncertainty about the recoverabilityof financial entitlements is referred to as default risk . Even though public awarenessis often not raised sufficiently, most financial contracts that one encounters in thereal world are subject to default risk. The distinction between defaultable andnon-defaultable zero-coupon bonds is mainly associated with aspects of the finalpayoff. Even if an issuer of defaultable zero-coupon bonds manages to meet theirfinal obligations in the end, losses could occur for some investors during the lifetimeof the contract all the same. This is particularly the case, if investors do not intendto keep the contract until maturity. Rumours about the credit quality of the issueror a respective down-grading by an external rating agency may affect the resalevalue adversely. Possible losses prior to maturity for the aforesaid reasons areusually referred to as migration risk . Default risk and migration risk are generallysubsumed under the broad term credit risk .Indisputably, there is an intricate connection between credit risk and aspectsof liquidity . These are diverse and relate to both the market and the commodityitself. More precisely, they are the asset liquidity , describing the immediacy and thetransaction cost with which the asset in scope can be converted into legal tender,and the institutional liquidity , standing for the considered issuer’s ability to meet itssettlement obligations; see also the IMF Working Paper WP/02/232 [18]. Creditand liquidity risk have received a lot of attention, especially since the subprimecrisis struck the financial markets in 2007/2008.It is the aim of this article to present a neat mathematical framework thatcaptures both phenomena. To this end, we utilise the foreign exchange (FX) analogy
Mathematics Subject Classification. a r X i v : . [ q -f i n . M F ] A p r THOMAS KRABICHLER AND JOSEF TEICHMANN for credit risk modelling, which was introduced by Jarrow & Turnbull in 1991; see[13]. They assumed inherently that, if a default has occurred, the recovery rateis known instantaneously. In the light of typically observing long and complicatedunwinding processes, we allow ourselves to slightly modify their setting. We proposethat all one can observe in the market filtration is the occurrence of a liquiditysqueeze , which results in a delay of due payments. The final recovery will only beknown after a while. If it happens to be strictly lower than one monetary unit, theliquidity squeeze has turned into a default event . This premise naturally motivatesthe existence of two filtrations. There is the filtration of full information on theone hand and that of genuinely observable market information on the other hand.We formalise this idea by applying a recently found fundamental theorem, see [6],while maintaining a tractable and arbitrage-free framework as demonstrated by ourexample.Roughly speaking, there are two common concepts to study credit risk, namely structural and reduced-form approaches. Either of them may be understood in theFX-analogy. Though, the design, the perspective and the assumptions are different.It is worth noticing that the idea of the FX-analogy fell into oblivion as quicklyas it appeared on the scene and several conceptual problems remained unsolved.Essentially, it was not debated how to deal with unobservable quantities such as,for instance, the FX rate. Moreover, clear notions for aspects of liquidity were notdefined. Bianchetti revived in 2009 the idea of the FX-analogy in order to explainthe occurrence of multiple yield curves ; see [1]. Jarrow & Turnbull studied theFX-analogy originally in a discrete time tree model as well as in a simple HJM-setup. They extended and refined it subsequently in [14], but moved on to a ratherintensity-based approach for rating purposes. Up until the previously mentionedfinancial crisis, the FX-analogy was cited only sporadically. Nguyen & Seifriedelaborate in [19] on the FX-analogy when they apply the potential approach from[20] in order to model multiple yield curves. The FX-analogy is also outlinedbriefly in the appendix of [5] or in Section 3.3 of [12], where the post-crisis LIBORmarket is modelled in both cases in terms of a multicurrency HJM-framework. TheFX-analogy can as well be applied for modelling inflation-linked risks; e.g., seeSection 15.1 in [3] or [15].The structure of the article is as follows. In Section 2, we recall the essentialelements of the FX-like venture. Subsequently, we specify the credit and liquidityrisk terminology in Section 3. Next, we move on to briefly discuss topics of theFX-like venture that matter from a practical viewpoint. Finally, we present thecentrepiece of this article in Section 5. In the last section, we derive explicitvaluation formulae for marketable products, e.g., for credit default swaps.2.
The Jarrow & Turnbull Setting
Let (Ω , G , P ) with G = ( G t ) t ≥ be a filtered probability space satisfying the usualconditions. We consider P as objective probability measure. By B = ( B t ) t ≥ wedescribe the accumulation of the domestic risk-free bank account with initial valueof one monetary unit. For any T ≥ , we denote by (cid:0) P ( t, T ) (cid:1) ≤ t ≤ T the càdlàgprice process of a non-defaultable zero-coupon bond with maturity T ≥ andpayoff P ( T, T ) = 1 . Furthermore, we denote by (cid:0) (cid:101) P ( t, T ) (cid:1) ≤ t ≤ T the càdlàg priceprocess of a defaultable zero-coupon bond with the same maturity and a randompayoff < (cid:101) P ( T, T ) ≤ . We assume that P ( T, T ) and (cid:101) P ( T, T ) are written in the HE JARROW & TURNBULL SETTING REVISITED 3 same currency. The distribution of the final recovery (cid:101) P ( T, T ) is strongly linked tothe riskiness of the issuer’s business model. The mappings ω (cid:55)−→ P (cid:0) t, T (cid:1) ( ω ) and ω (cid:55)−→ (cid:101) P (cid:0) t, T (cid:1) ( ω ) ought to be positive and G t -measurable for all ≤ t ≤ T < ∞ .We may introduce another term structure (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ via Q ( t, T ) := (cid:101) P ( t, T ) (cid:101) P ( t, t ) . Note that we have Q ( T, T ) = 1 and, hence, that this synthetic series is default-free.By setting S t := (cid:101) P ( t, t ) , we get(2.1) (cid:101) P ( t, T ) = S t Q ( t, T ) . Although this rewriting is very elementary, it opens an extremely nice modellingopportunity for defaultable zero-coupon bonds. We recognise that credit risk canbe analysed in an FX-like setting.
Paradigm 2.1 (Jarrow & Turnbull 1991) . The series P ( t, T ) and Q ( t, T ) areconsidered as non-defaultable zero-coupon bonds in different currencies. (cid:101) P ( t, T ) may be interpreted as conversion of foreign default-free counterparts. S t = (cid:101) P ( t, t ) is referred to as recovery rate or spot FX rate .The foreign market describes the unique default-free interest rate model inwhich yields are driven by (cid:8) (cid:101) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ and obligations are always met. Inmulti-currency settings, (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ is the main driver for so-called quantosecurities denominated in the domestic currency. Their basic feature is that theyare not exposed to any FX-risks whatsoever for the buyer, but certainly for theissuer. The involved payoffs are constituted as if the FX rate were kept constantafter conclusion of the deal.We shall deal with different informational structures here: it is natural to assumethat the recovery rate S t , or spot FX rate in our analogy, is not observable by thetrader’s filtration F at time t . Paradigm 2.2.
We denote the trader’s filtration by F ⊂ G and we assume thatthe bond prices of the domestic market P ( ., T ) are F -adapted for T ≥ , but (cid:101) P ( ., T ) and S are not necessarily.For such a two-filtration setting, the findings of [6] can be applied: in order toguarantee absence of arbitrage we therefore assume that existence of a measure Q ≈ P with respect to the F -adapted bank account numéraire B such that theoptionally projected discounted processes P ( t, T ) B t = E Q (cid:20) P ( t, T ) B t (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , E Q (cid:20) S t Q ( t, T ) B t (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = E Q (cid:20) (cid:101) P ( t, T ) B t (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) for ≤ t ≤ T and each T ≥ form Q -martingales.In particular, if F = G , it holds that(2.2) (cid:101) P ( t, T ) = S t Q ( t, T ) P ( t, T ) P ( t, T ) = E Q T (cid:2) S T (cid:12)(cid:12) F t (cid:3) P ( t, T ) for all ≤ t ≤ T < ∞ , where d Q T d Q (cid:12)(cid:12)(cid:12)(cid:12) F t := P ( t, T ) P (0 , T ) B t THOMAS KRABICHLER AND JOSEF TEICHMANN denotes the domestic T -forward measure. In multi-currency settings, the quantity(2.3) F ( t, T ) := (cid:101) P ( t, T ) P ( t, T ) = S t Q ( t, T ) P ( t, T ) is usually referred to as forward FX rate , i.e., as seen from time t , the agreementto exchange one foreign monetary unit for locked-in F ( t, T ) domestic monetaryunits at time T is at arm’s length and worth zero. Thus, F ( t, T ) is a natural basisfor currency forwards. It is the risk-neutral t -forecast of the recovery/FX rate attime T . In the following, we shall use the term forward recovery rate for F ( t, T ) equivalently.If the term structures T (cid:55)−→ P ( t, T ) and T (cid:55)−→ Q ( t, T ) are assumed to be C asin the classical HJM-framework, one may consider the continuously compoundedinstantaneous forward rates f dom ( t, T ) := − ∂∂T log P ( t, T ) , f for ( t, T ) := − ∂∂T log Q ( t, T ) . On the one hand, (2.1) and the product rule for logarithms yield − ∂∂T log (cid:101) P ( t, T ) = − ∂∂T log S t − ∂∂T log Q ( t, T ) = f for ( t, T ) . On the other hand, utilising (2.3) gives − ∂∂T log (cid:101) P ( t, T ) = − ∂∂T log P ( t, T ) − ∂∂T log F ( t, T )= f dom ( t, T ) − ∂∂T log F ( t, T ) . Thus, the negative logarithmic derivative of the forward recovery rate describes thespread of the foreign forward rates above their domestic counterparts. High spreadsindicate a precarious period of financial distress. In this sense, − ∂∂T log F ( t, T ) maybe attributed to the intensity of the credit risk. Remark 2.3 (Construction of Credit Risk Models) . The FX-like approach can beimplemented easily by considering a genuine multi-currency setting and restrictingoneself to a certain class of FX rates, which take values only in the target zone (0 , .Equation (2.2) is very helpful in this regard. Exemplarily, inspired by Dirichletproblems related to the Beta distribution on (0 , (e.g., see Table 1 in [2]) andFX rate models in a target zone (e.g., see [17]), one might model the recovery rate S = ( S t ) t ≥ under the pricing measure Q as a Jacobi process. Alternatively, onemight consider a recovery rate S t = e −(cid:104) ξ,X t (cid:105) for a d -dimensional non-negative affineMarkov process X = ( X t ) t ≥ and a parameter ξ ∈ R d + . Both assumptions lead tofairly tractable models. This prevails also in the presence of jumps; see [16].3. Credit and Liquidity Risk Terminology
For the Jarrow & Turnbull setting, we introduce the following terminology.
Definition 3.1 (Defaultable Term Structures) . A zero-coupon bond with priceprocess (cid:0) (cid:101) P ( t, T ) (cid:1) ≤ t ≤ T for a fixed maturity T > is called defaultable , if it holds P (cid:2) S T < (cid:3) > ; otherwise it is called default-free or non-defaultable . A whole termstructure of zero-coupon bonds (cid:8) (cid:101) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ is called defaultable , if for any HE JARROW & TURNBULL SETTING REVISITED 5 bound T ∗ > there exists another maturity T (cid:48) > T ∗ for which (cid:0) (cid:101) P ( t, T (cid:48) ) (cid:1) ≤ t ≤ T (cid:48) isdefaultable.In order not to end up in a standard framework, we will tacitly assume forthe remainder of this article that (cid:8) (cid:101) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ is defaultable. Generally, thenotion riskiness for a zero-coupon bond or a whole term structure does not relateto the uncertainty about the underlying payoffs, but to the stochastic nature of itsprice evolution. Still, we use the terms risk-free for non-defaultable and risky fordefaultable interchangeably. Loans are usually only marketable for virtuous issuerswho are duly concerned about their obligations. Definition 3.2 (Proper Term Structures) . A defaultable zero-coupon bond withprice process (cid:0) (cid:101) P ( t, T ) (cid:1) ≤ t ≤ T for a fixed maturity T > is called proper , if itholds P (cid:2) S T = 1 (cid:3) > ; otherwise it is called improper . A whole term structureof zero-coupon bonds (cid:8) (cid:101) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ is called proper , if any zero-coupon bond (cid:0) (cid:101) P ( t, T ) (cid:1) ≤ t ≤ T for T > is proper.Any downturn of the recovery rate relates to a shortage of liquid funds. Definition 3.3 (Liquidity Squeeze) . Let the term structure (cid:8) (cid:101) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ berisky. A liquidity squeeze at time t is the probabilistic event { S t < } .Nonetheless, a liquidity squeeze may be without consequences as long as nophysical payments become due. The instances in which payments have to be settledare described by means of a sequence of stopping times. The first time a liquiditysqueeze coincides with a payment date, a default event is deemed to occur. Definition 3.4 (Payment Schedule and Default Event) . Let (cid:8) (cid:101) P ( t, T ) (cid:9) ≤ t ≤ T < ∞ berisky. Denote by ( τ n ) n ∈ N a payment schedule , that is a sequence of finite F -stoppingtimes without accumulation points. The default time τ is defined as the possiblyinfinite stopping time τ := inf (cid:8) τ n (cid:12)(cid:12) S τ n < , n ∈ N (cid:9) . A default event at time t isthe probabilistic event { τ = t } ∩ { S t < } .The default time τ really is a stopping time, since we have { τ ≤ t } = (cid:91) n ∈ N (cid:16) { τ n ≤ t } ∩ { S τ n ∧ t < } (cid:17) . Typically, a default event triggers either a restructuring or a liquidation of theissuer. As the case may be, the issuer of the defaultable bonds succeeds in revivingthe business model in the aftermath of a default event. Because S τ might notbe readily observable in a sub-filtration describing genuinely accessible marketdata, the event { S τ = 1 } is not unlikely; see also the discussion in the nextsection. Consequently, one can refine the modelling approach by allowing formultiple defaults. Notably, the occurrence of { S τ < } does not prevent therecovery rate from returning to the level . Remark 3.5 (Bankruptcy) . It is tempting to postulate that the occurrence of { S t < } always causes bankruptcy of the referenced entity. However, the FX-likesetting can also be associated with an arbitrary investment portfolio in defaultablecorporate bonds. In that case, the resulting recovery rate is a superposition of manydefaultable bonds and respective recoveries. Thus, full recovery may almost neverbe given and { S t < } may occur without further consequences for the portfolio’sexistence. (cid:3) THOMAS KRABICHLER AND JOSEF TEICHMANN Practical Matters
Realistically or symptomatic of general market models, one realisation of thestochastic processes (cid:0) P ( t, T i ) (cid:1) ≤ t ≤ T i for i = 0 , , , . . . n and (cid:0) (cid:101) P ( t, (cid:101) T j ) (cid:1) ≤ t ≤ (cid:101) T j for j = 0 , , , . . . (cid:101) n can be inferred from market quotes for two generally differinggrids of rather remote maturities ≤ T < T < T < . . . < T n together with ≤ (cid:101) T < (cid:101) T < (cid:101) T < . . . < (cid:101) T (cid:101) n . Hence, neither S t = (cid:101) P ( t, t ) nor lim T → t + (cid:101) P ( t, T ) arealways accessible, given the latter expression makes sense at all. In contrast, theshort end P ( t, t ) = 1 of the non-defaultable term structure is known at any time t ≥ . Thus, the extrapolation exercise is much easier for the non-defaultable termstructure than it is for its defaultable counterpart; yet the non-defaultable shortrate r t = − ∂∂T (cid:12)(cid:12)(cid:12)(cid:12) T = t log P ( t, T ) , given this notion makes sense at all, is normally also unknown. The third series ofzero-coupon bonds (cid:8) Q ( t, T ) (cid:9) ≤ t ≤ T < ∞ is synthetic and its introduction is subjectto a redundancy. As S t and Q ( t, T ) /P ( t, T ) are only observable sporadically andpredominantly only on aggregated level F ( t, T ) , the market (more precisely, acertain sub-filtration of G describing the genuinely observable events) captures onlyvery little information of the FX-like setting. The recovery rate is only observableon a discrete payment schedule; see Definition 3.4 above. Therefore, further modelassumptions have to be imposed. This is in strong contrast to multi-currencysettings in which the spot FX rate is usually known. However, this is not ofutmost relevance in some applications. Typically, one assumes full recovery anywayand is rather interested in the likelihood of future downturns. Given full recovery { S t = 1 } , marketable instruments and credit derivatives can help to infer spreadfactors Q ( t, T ) /P ( t, T ) and to calibrate the models against the current marketconditions.Mathematically, the situation can be modelled as described in Paradigm 2.2.One equips the general FX-like setting with two filtrations. The trader’s filtration F comprises only genuinely observable market information and a larger filtration G also carries hidden information; the recovery rate process is G -adapted but notnecessarily F -adapted. Both filtrations are augmented in order to satisfy the usualconditions. Let ( τ n ) n ∈ N denote a payment schedule, τ the [0 , ∞ ] -valued defaulttime and τ (cid:48) ≥ τ be an F -stopping time that fixes the final recovery for generalmarket participants. Besides the price evolution of selected bonds up to time t , thecore part of the σ -algebra F t is generated by the sets (cid:8) { τ n ≤ u } (cid:12)(cid:12) u ≤ t (cid:9) , whichsimply are the payment dates, (cid:110) { τ n ≤ u } ∩ (cid:8) S τ n ∈ A (cid:9) (cid:12)(cid:12)(cid:12) n ∈ N , u ≤ t, A ∈ (cid:8) (0 , , { } (cid:9)(cid:111) , which distinguish payment dates qualitatively on whether any liquidity squeeze hasoccurred previously or not, and, for the Borel σ -algebra B , (cid:110) { τ (cid:48) ≤ u } ∩ (cid:8) S τ ∈ A (cid:9) (cid:12)(cid:12)(cid:12) u ≤ t, A ∈ B (cid:0) (0 , (cid:1)(cid:111) , which is the knowledge about the final post-default recovery. The filtering problemconsists then in calculating E P (cid:2) S t (cid:12)(cid:12) F t (cid:3) for t ≥ . It does not necessarily hold HE JARROW & TURNBULL SETTING REVISITED 7 E P (cid:2) S t (cid:12)(cid:12) F t (cid:3) ≡ on { τ > t } , since low prices for defaultable bonds may announce anupcoming default. We are making this notion more precise in the next section.5. Partial Observability of the Recovery Rate
It is an idiosyncrasy of the general FX-like setting that, given a default hashappened, the involved recovery rate is known instantaneously. From the practicalviewpoint, this feature may legitimately be questioned. All one usually knows isthat some sort of default event has happened. The final recovery will typically onlybe determined after a long and complicated unwinding process. The settlementof a default event always comes with a negotiation process. It is the aim of thissection to generalise the FX-analogy in this direction. To this end, we trade offphenomenological richness against full analytical or at least numerical tractability.Let us consider a two-dimensional [0 , ∞ ) -valued conservative regular affine jumpdiffusion ( X, Y ) with dX t = σ X (cid:112) X t dW Xt + dJ Xt , dY t = σ Y (cid:112) Y t dW Yt + dJ Yt and some initial condition ( X , Y ) = (0 , y ) with y ≥ . ( W X , W Y ) is a two-dimensional Brownian motion on a filtered probability space (Ω , F , Q ) with F =( F t ) t ≥ satisfying the usual conditions. ( J X , J Y ) is a right-continuous pure jumpprocess, whose jump heights have a fixed distribution ν on (0 , ∞ ) and arrive withintensity m + µ X X t – + µ Y Y t – for some parameters m > and µ X , µ Y ≥ . TheBrownian motions, the jumps heights and the arrival of the jumps are assumed tobe independent under the risk-neutral measure Q .For simplicity, we assume a trivial non-defaultable term structure B ≡ . Thepartially observable recovery rate is derived from the auxiliary process e − X t andtypically starts with a constant trajectory at the level one. Y features the intensityof liquidity squeezes. Once the recovery has jumped below one, the recovery ratepursuits an unsteady course. The further X jumps away from zero, the more likelybecome positive jumps for Y . This itself triggers yet another surge for a potentialdepreciation of the recovery rate. To this extent, downturns of the recovery rateare self-exciting . Nonetheless, X may also return to zero, since it is after all aFeller diffusion, and full recovery prevails for a certain period. See Figure 1 for anillustration of the above setting. In more involved settings, Y could stand for therisk-free short rate process. In this sense, defaults would be more likely with anincreasing interest rate burden; see Remark 5.2 below.We assume here that the trader’s filtration F is generated by X, Y . This meansif a liquidity squeeze occurs and contingencies of a zero-coupon bond maturing attime T cannot be paid off, this is usually known. However, the actual recovery isgenerally not observable: we assume that (cid:101) P ( T, T ) pays off { X T =0 } + e − X T + h { X T > } at the time instances T or T + h respectively, where h > is a positive parameter.Whence, this corresponds to G t = F t + h in the language of the two-filtration setting.By utilising the affine Markov structure, we aim at deriving an explicit valuationformula for a defaultable zero-coupon bond (cid:101) P ( t, T ) , i.e., we calculate the optionalprojections on F . It is remarkable that this is possible in this setting. THOMAS KRABICHLER AND JOSEF TEICHMANN
Partially Observable CIR Recovery Rate with Jumps term in years i n % o f pa r v a l ue Figure 1.
The sample path illustrates the setting of this section.If liquidity squeezes prevail for too long, they turn into defaultevents. The final recovery will only be known after a while.Furthermore, only the segments marked red are observable forgeneral market participants. Either full institutional liquidity isgiven or there are payment delays. In the latter case, the actuallevel of the recovery rate is completely unknown.Consistent with the above premises, the generalised Riccati equations for u, v ∈ R and ≤ t ≤ T are given by ∂ t φ ( t, iu, iv ) = mκ (cid:0) ψ X ( t, iu, iv ) , ψ Y ( t, iu, iv ) (cid:1) ,∂ t ψ X ( t, iu, iv ) = 12 σ X ψ X ( t, iu, iv ) + µ X κ (cid:0) ψ X ( t, iu, iv ) , ψ Y ( t, iu, iv ) (cid:1) ,∂ t ψ Y ( t, iu, iv ) = 12 σ Y ψ Y ( t, iu, iv ) + µ Y κ (cid:0) ψ X ( t, iu, iv ) , ψ Y ( t, iu, iv ) (cid:1) with the initial conditions φ (0 , iu, iv ) = 0 , ψ X (0 , iu, iv ) = iu and ψ Y (0 , iu, iv ) = iv ,where κ ( iu, iv ) := (cid:90) (0 , ∞ ) (cid:0) e iux + ivy − (cid:1) dν ( x, y ); see Theorem 2.7 in [8] for a general treatment. Accordingly, it holds for all u, v ∈ R and ≤ t ≤ TE Q (cid:104) e iuX T + ivY T (cid:12)(cid:12)(cid:12) F t (cid:105) = e φ ( T − t,iu,iv )+ X t ψ X ( T − t,iu,iv )+ Y t ψ Y ( T − t,iu,iv ) , where F t = σ (cid:0) X u , Y u ; u ≤ t (cid:1) . This can be exploited for the implied forward recoveryrates, since F ( t, T ) = E Q (cid:104) { X T =0 } + e − X T + h { X T > } (cid:12)(cid:12)(cid:12) F t (cid:105) = E Q (cid:104) { X T =0 } + E Q (cid:2) e − X T + h (cid:12)(cid:12) F T (cid:3) { X T > } (cid:12)(cid:12)(cid:12) F t (cid:105) . (5.1) HE JARROW & TURNBULL SETTING REVISITED 9
Due to the Riemann-Lebesgue lemma, the mass of the atom { X T = 0 } can becalculated in terms of Q (cid:2) X T = 0 (cid:3) = lim u →∞ E Q (cid:2) e iuX T (cid:3) = lim u →∞ e φ ( T,iu, y ψ Y ( T,iu, . The projection of the second summand in (5.1) can be rewritten as E Q (cid:104) E Q (cid:2) e − X T + h (cid:12)(cid:12) F T (cid:3) { X T > } (cid:12)(cid:12)(cid:12) F t (cid:105) = E Q (cid:104) e φ ( h, − , X T ψ X ( h, − , Y T ψ Y ( h, − , (cid:0) − { X T =0 } (cid:1)(cid:12)(cid:12)(cid:12) F t (cid:105) = E Q (cid:104) e φ ( h, − , X T ψ X ( h, − , Y T ψ Y ( h, − , (cid:12)(cid:12)(cid:12) F t (cid:105) − E Q (cid:104) e φ ( h, − , Y T ψ Y ( h, − , { X T =0 } (cid:12)(cid:12)(cid:12) F t (cid:105) . The minuend is yet another evaluation of the Fourier transform. The subtrahendcan also be calculated explicitly by utilising dominated convergence and the relation E Q (cid:104) e vY T { X T =0 } (cid:12)(cid:12)(cid:12) F t (cid:105) = lim u →−∞ E Q (cid:104) e uX T + vY T (cid:12)(cid:12)(cid:12) F t (cid:105) . Hence, Lévy’s inversion theorem is not required. Generally, already humble modelassumptions lead to Riccati equations, which are intractable analytically. Therefore,numerical approximation procedures are indispensable anyway; however, the affinestructure reduces the complexity of the approximation schemes considerably. Thecomplexity remains low if one incorporates drift terms for X and Y in order toincrease market consistency. Let us illustrate the above framework by a couple ofsimple examples. Example 5.1 (Deterministic Jump Intensity, σ X (cid:54) = 0 ) . As a starting point, wedisable the stochastic intensity Y and consider the limiting case m > , µ X = 0 and µ Y = 0 . Regarding the jump sizes, let us choose for λ X > the product measure dν ( x, y ) = λ X e − λ X x dx ⊗ δ { } ( dy ) , where the Dirac measure for A ∈ B ( R ) is defined as δ { } ( A ) = (cid:40) , if ∈ A, , otherwise . Whenever a jump occurs, it is entirely in the x -direction. Each jump size itselfis exponentially distributed with the parameter λ X . Since κ ( iu, iv ) = iuλ X − iu , thismodel choice translates into the system of Riccati equations ˙ φ = mψ X λ X − ψ X , ˙ ψ X = 12 σ X ψ X with the solution φ ( t, iu, iv ) = 2 mλ X σ X log (cid:32) λ X iu − λ X (cid:0) iu − σ X t (cid:1) − (cid:33) ,ψ X ( t, iu, iv ) = 1 iu − σ X t . It obviously holds φ ( t, iu, iv ) u →∞ −→ mλ X σ X log (cid:18) λ X σ X t + 1 (cid:19) . This results in the initial forward recovery rate F (0 , T ) = (cid:0) − e φ ( h, − , (cid:1) Q (cid:2) X T = 0 (cid:3) + e φ ( h, − , E Q (cid:104) e X T ψ X ( h, − , (cid:105) , where Q (cid:2) X T = 0 (cid:3) = (cid:18) λ X σ X T + 1 (cid:19) − mλXσX and E Q (cid:104) e X T ψ X ( h, − , (cid:105) = e φ (cid:0) T,ψ X ( h, − , , (cid:1) . Thus, analytical pricing formulas are available. For time instances t > , theforward recovery rates T (cid:55)−→ F ( t, T ) depending on the state variable X t can becalculated analogously by utilising the Markov property. (cid:3) Remark 5.2 (Time Value of Money) . If we modify Example 5.1 in the sense that Y does not only describe the jump intensity, but also the risk-free short rate, thenthe model remains fully tractable. All one has to do is introduce an additionalstructural component Z = ( Z t ) t ≥ with Z t := z + (cid:90) t Y u du. In this case, ( X, Y, Z ) denotes a three-dimensional affine jump diffusion. Thecalculations get a bit more cumbersome, but the derivation of a closed-form valuationformula is possible all the same; e.g., see Example 3.20 in [16]. (cid:3) Remark 5.3 (Generalisation) . The presented recipe in order to calculate (5.1) ispurposeful for the next upcoming critical maturity. For a discrete payment schedule < T < T , the payoff due at time T depends on whether a default event occurredat time T or not. Consistently, one might consider for ≤ t ≤ T < T + h ≤ T the generalised functional E Q (cid:20) { X T h =0 } (cid:16) { X T =0 } + e − X T h { X T > } (cid:17) + e − X T h { X T > } { X T h > } (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . For ≤ t ≤ T < T < T + h , the adaptation works analogously. Explicit pricingformulae become more complex but they are still available by cascading the aboverecipe. This may be useful in the market-consistent valuation of fully collateralisedover-the-counter deals with daily margining. If one fails to post collateral, one hasa time limit of a few days h , as agreed upon in the indenture, to supply the dueamount. If the deficiency still prevails thereafter, the deal is closed. (cid:3) Example 5.4 ( µ X = 0 , σ X = 0 ) . We consider the case m > , µ X = 0 and µ Y > as well as σ X = 0 . Correspondingly, the stochastic intensity is indifferentwith respect to the level of X . Once a liquidity squeeze has occurred, full recoveryis not possible any more. Regarding the jump sizes, we choose the same productmeasure as in the previous example. In this case, the Riccati equations reduce to ˙ φ = m ψ X λ X − ψ X , HE JARROW & TURNBULL SETTING REVISITED 11 ˙ ψ X = 0 , ˙ ψ Y = 12 σ Y ψ Y + µ Y ψ X λ X − ψ X with the solution φ ( t, iu, iv ) = m iuλ X − iu t,ψ X ( t, iu, iv ) = iu,ψ Y ( t, iu, iv ) = (cid:115) µ Y iuσ Y ( λ X − iu ) tan (cid:32) C + t (cid:115) iuµ Y σ Y λ X − iu ) (cid:33) , where the integration constant, certainly depending on iu and iv , C = arctan (cid:32) iv (cid:115) σ Y ( λ X − iu )2 µ Y iu (cid:33) is chosen such that the initial condition is met. It clearly holds φ ( t, iu, iv ) u →∞ −→ − mt .Moreover, since tan iu = i tanh u for all u ∈ R , one simply derives ψ Y ( t, iu, u →∞ −→ − (cid:114) µ Y σ Y tanh (cid:32) t (cid:114) µ Y σ Y (cid:33) . All in all, we get for F (0 , T ) the expression Q (cid:2) X T = 0 (cid:3) + e φ ( h, − , φ (cid:0) T, − ,ψ Y ( h, − , (cid:1) + y ψ Y (cid:0) T, − ,ψ Y ( h, − , (cid:1) (5.2) − e φ ( h, − , − mT + (cid:114) − µYσY tan (cid:18) arctan (cid:16) ψ Y ( h, − , (cid:114) − σY µY (cid:17) + T √ − µ Y σ Y (cid:19) , where Q (cid:2) X T = 0 (cid:3) = e − mT − y (cid:114) µYσY tanh (cid:0) T √ µ Y σ Y (cid:1) . (5.3)Consequently, the state of the initial term structure is captured by the six modelparameters h , λ X , m , µ Y , σ Y and y . Full analytical tractability is given as in theprevious example. (cid:3) Remark 5.5 (Deterministic Jump Sizes) . It is tempting to consider the aboveexamples in the case of deterministic jump sizes. For instance, one might want toconsider a product measure(5.4) dν ( x, y ) = δ { J X } ( dx ) ⊗ δ { J Y } ( dy ) for parameters J X > and J Y ≥ . Note that if J X was set to zero, the recoveryrate could not depreciate. This choice translates into S having the discrete support (cid:8) e − kJ X (cid:12)(cid:12) k ∈ N (cid:9) . Exemplarily, (5.2) and (5.3) would still prevail, if one replacedthe product measure in Example 5.4 by (5.4) for J Y = 0 , ceteris paribus. The onlydifference is that the parameterisation φ ( t, iu, iv ) = m (cid:0) e iuJ X − (cid:1) t changes from λ X to J X . (cid:3) Remark 5.6 ( µ X = 0 , σ X (cid:54) = 0 ) . If we let σ X (cid:54) = 0 in Example 5.4, then thederivation of the forward recovery rates is more involved. The Riccati equationsread ˙ φ = m ψ X λ X − ψ X , ˙ ψ X = 12 σ X ψ X , ˙ ψ Y = 12 σ Y ψ Y + µ Y ψ X λ X − ψ X . The solutions ψ X ( t, iu, iv ) = 2 iu − σ X t , φ ( t, iu, iv ) = 2 mλ X σ X log (cid:18) λ X − iu λ X − iu − iuλ X σ X t (cid:19) are readily at hand. If the auxiliary function x ( t ) = x ( t, iu, iv ) denotes a solutionto the second order linear differential equation ¨ x + iuµ Y σ Y λ X − iu − iuλ X σ X t x = 0 , ˙ x (0) x (0) = − ivσ Y , then we can characterise ψ Y in terms of ψ Y ( t, iu, iv ) = − x ( t ) σ Y x ( t ) . However, since the matrix A ( t ) = A ( t, iu, iv ) = (cid:32) − iuµ Y σ Y λ X − iu − iuλ X σ X t (cid:33) is non-commutative for different time parameters, there is no straightforward closed-form solution for x ( t ) as matrix exponential. (cid:3) Remark 5.7 (Link to Classical Credit Risk Models) . The approach of this sectionis a neat modification of classical credit risk models. Furthermore, it incorporatesnaturally the phenomenon of liquidity. On the one hand, one may perceive therecovery rate process as being arisen from a not directly observable balance sheetvariable. This is the structural component of the model. On the other hand, themodel features in the background an intensity process for jumps that trigger bothliquidity squeezes and defaults. The existence of two filtrations is idiosyncratic fordoubly-stochastic settings. Similarly, defaults are not predictable. A generalisationin this regard can be found in [4] and [11]. (cid:3)
Remark 5.8 (Real-World vs. Risk-Neutral Default Probabilities) . Sometimes, asfor instance in the current mortgage business, one encounters that the defaultableterm structure is rather pronounced, whereas the real-world default probabilitiesseem to be marginal. We can incorporate this case of a high market price of risk intoour setting by considering another measure P ≈ Q . Under this real-world measure,the jump size distribution of the recovery rate remains unaffected. However, thejump intensity might be arbitrarily low; e.g., see Section 4 in [9]. Q may be seenas the result of pricing claims under P together with a highly risk averse utilityfunction. (cid:3) HE JARROW & TURNBULL SETTING REVISITED 13 Valuation Formulae for Marketable Products
Mathematical Setup.
For the non-defaultable term structure T (cid:55)−→ P (0 , T ) , weconsider the extended Cox-Ingersoll-Ross (CIR++) short rate model r t = x t + ϕ ( t ) , dx t = ( b x − β x x t ) dt + σ x √ x t dW xt for some deterministic function t (cid:55)−→ ϕ ( t ) , positive parameters b x , β x , σ x with b x ≥ σ x and some initial condition r ∈ R ; see Section 3.9 in [3]. Then it holds P (0 , T ) = E Q (cid:104) e − (cid:82) T r u du (cid:105) = e − A (0 ,T ) − B (0 ,T ) r , (6.1)where A (0 , T ) = − b x σ x log (cid:26) λ x e ( λ x + β x ) T/ ( λ x + β x ) (cid:0) e λ x T − (cid:1) + 2 λ x (cid:27) − ϕ (0) B (0 , T ) + (cid:90) T ϕ ( u ) du,B (0 , T ) = 2 (cid:0) e λ x T − (cid:1) ( λ x + β x ) (cid:0) e λ x T − (cid:1) + 2 λ x ,λ x = (cid:113) β x + 2 σ x . Regarding the defaultable term structure T (cid:55)−→ (cid:101) P (0 , T ) , we consider the affinejump diffusion dX t = σ X √ X t dW Xt + dJ Xt with the initial condition X = x . ( W x , W X ) is a two-dimensional Brownian motion and J X is a right-continuous purejump process, whose jump heights have the fixed distribution dν ( x ) = λ X e − λ X x dx and arrive with intensity m X + µ X X t – . The recovery rate is modelled as S t := e − X t .By construction, it holds E Q (cid:2) e iuX T (cid:3) = e φ X ( T,iu )+ x ψ X ( T,iu ) for all T ≥ , where φ X and ψ X satisfy the generalised Riccati equations ˙ φ X ( t, iu ) = m X ψ X ( t, iu ) λ X − ψ X ( t, iu ) , ˙ ψ X ( t, iu ) = 12 σ X ψ X ( t, iu ) + µ X ψ X ( t, iu ) λ X − ψ X ( t, iu ) . Provided that µ X = 0 , these equations can be solved explicitly; φ X ( t, iu ) = 2 m X λ X σ X log (cid:32) λ X iu − λ X (cid:0) iu − σ X t (cid:1) − (cid:33) ,ψ X ( t, iu ) = 1 iu − σ X t . In general, the expected recovery is given by F (0 , T ) = (cid:101) P (0 , T ) P (0 , T ) = E Q (cid:2) S T (cid:3) = E Q (cid:2) e − X T (cid:3) = e φ X ( T, − x ψ X ( T, − . (6.2)Provided that µ X = 0 , it holds F (0 , T ) = (cid:18) λ X + 1 λ X (1 + σ X T ) + 1 (cid:19) mXλXσX e − x
01+ 12 σX T . Marketable Bonds.
For some parameter n and maturity T , let T < T In a CDS in-line with the concept outlined inSection 2.7 of [45], counterparties exchange a stream of coupon payments for asingle default protection payment in the event of a default by a reference entity.The coupon payments have to be paid by the one counterparty either until maturityor, if a contractually agreed upon default event occurs earlier, only up to butincluding that incident. The other counterparty is obliged to pay a contingentdefault compensation in the case of the predefined event. Otherwise, if nothing thelike happens until maturity, no payments become due for the other counterparty.Correspondingly, the CDS spread CDS n (0 , T ) is defined implicitly as solution to CDS n (0 , T ) n (cid:88) i =1 ( T i − T i − ) E Q (cid:20) e − (cid:82) Ti r u du { τ ≥ T i } (cid:21) ! = n (cid:88) i =1 E Q (cid:20) e − (cid:82) Ti r u du (1 − S T i ) { τ = T i } (cid:21) , where τ := min (cid:8) T , T , . . . , T n (cid:12)(cid:12) S T i < (cid:9) . Under the suitable assumptions that wemade, this equation simplifies to CDS n (0 , T ) n (cid:88) i =1 ( T i − T i − ) P (0 , T i ) E Q (cid:2) { τ ≥ T i } (cid:3) ! = n (cid:88) i =1 P (0 , T i ) (cid:16) E Q (cid:2) { τ = T i } (cid:3) − E Q (cid:2) S T i { τ = T i } (cid:3)(cid:17) . Furthermore, as derived in the next two sections, CDS spreads can be calculatedexplicitly. HE JARROW & TURNBULL SETTING REVISITED 15 Probability of Default (PD). We proceed consecutively. Since T is the firstpossible default time at all, we have PD(0 , T ) := E Q (cid:2) { τ ≤ T } (cid:3) = E Q (cid:2) { τ = T } (cid:3) =1 − E Q (cid:2) { τ>T } (cid:3) and E Q (cid:2) { τ ≥ T } (cid:3) = 1 . As seen from time T , no default occurs attime T with probability E Q (cid:2) { τ>T } (cid:3) = E Q (cid:2) { X T =0 } (cid:3) = lim u →−∞ E Q (cid:2) e uX T (cid:3) = lim u →−∞ e φ X ( T ,u )+ x ψ X ( T ,u ) . Provided that µ X = 0 , we get E Q (cid:2) { τ>T } (cid:3) = 1 − PD(0 , T ) = (cid:18) λ X σ X T + 1 (cid:19) − mXλXσX e − x σX T . For i ≥ , it holds E Q (cid:2) { τ = T i } (cid:3) = PD(0 , T i ) − PD(0 , T i − ) and E Q (cid:2) { τ ≥ T i } (cid:3) =1 − PD(0 , T i − ) . Similarly, we get E Q (cid:2) { τ>T } (cid:3) = E Q (cid:104) { X T =0 } E Q (cid:2) { X T =0 } (cid:12)(cid:12) F T (cid:3)(cid:105) = E Q (cid:20) { X T =0 } lim u →−∞ e φ X ( T − T ,u )+ X T ψ X ( T − T ,u ) (cid:21) . Provided that µ X = 0 , induction yields the formula E Q (cid:2) { τ>T i } (cid:3) = 1 − PD(0 , T i ) = e − x σX T i (cid:89) j =1 (cid:18) λ X σ X ( T j − T j − ) + 1 (cid:19) − mXλXσX . Loss Given Default (LGD). Again, we proceed consecutively. As seen fromtime T , recovery given default at time T is given by E Q (cid:2) S T { τ = T } (cid:3) = E Q (cid:2) S T (cid:3) − E Q (cid:2) S T { S T =1 } (cid:3) = F (0 , T ) − E Q (cid:2) { S T =1 } (cid:3) = F (0 , T ) − (cid:0) − PD(0 , T ) (cid:1) . Analogously, E Q (cid:2) S T { τ = T } (cid:3) = E Q (cid:104) { S T =1 } E Q (cid:2) S T { S T < } (cid:12)(cid:12) F T (cid:3)(cid:105) = E Q (cid:20) { S T =1 } (cid:16) F ( T , T ) − E Q (cid:2) { S T =1 } (cid:12)(cid:12) F T (cid:3)(cid:17)(cid:21) = e φ X ( T − T , − E Q (cid:2) { τ>T } (cid:3) − E Q (cid:2) { τ>T } (cid:3) . Induction yields for i ≥ the formula E Q (cid:2) S T i { τ = T i } (cid:3) = e φ X ( T i − T i − , − E Q (cid:2) { τ>T i − } (cid:3) − E Q (cid:2) { τ>T i } (cid:3) = e φ X ( T i − T i − , − (cid:0) − PD(0 , T i − ) (cid:1) − (cid:0) − PD(0 , T i ) (cid:1) . (6.5)The formula (6.5) is also valid for i = 1 if it holds x = 0 (i.e., S = 1 ) and if weadhere to the convention PD(0 , T ) := 0 . CDS Spreads. All in all, provided that x = 0 , we end up with the explicitformula(6.6) CDS n (0 , T ) = (cid:80) ni =1 P (0 , T i ) (cid:0) − e φ X ( T i − T i − , − (cid:1)(cid:0) − PD(0 , T i − ) (cid:1)(cid:80) ni =1 ( T i − T i − ) P (0 , T i ) (cid:0) − PD(0 , T i − ) (cid:1) . In this case, the factor e φ X ( T i − T i − , − coincides with F (0 , T i − T i − ) ; see (6.2). Ifwe had x > , the expression e φ X ( T i − T i − , − in the numerator of (6.6) for i = 1 (and only for i = 1 ) would need to be replaced by F (0 , T ) . In the special case n = 1 , the formula reduces to CDS (0 , T ) = 1 − F (0 , T ) T . If x = 0 and the partition is chosen equidistant, then it holds(6.7) CDS n (0 , T ) = 1 − F (0 , T /n ) T /n . The Calibration Task. We parameterised the two term structures T (cid:55)−→ P (0 , T ) and T (cid:55)−→ (cid:101) P (0 , T ) in terms of a deterministic function ϕ and eight model parameters r , b x , β x , σ x , λ X , m X , σ X as well as x ; the recovery rate is not assumed tobe self-exciting (i.e., µ X = 0 ) and starts at S = 1 . Given that we also knowthe term structure T (cid:55)−→ L (0 , T ) , the mapping from these eight parameters ontomarket quotes V n,c (0 , T ) , (cid:101) V n,c (0 , T ) and CDS n (0 , T ) is straightforward. The otherway round, however, is non-trivial. The bootstrapping of P (0 , T ) , (cid:101) P (0 , T ) and L (0 , T ) from marketable products is often cumbersome. We propose the followingnon-parametric calibration procedure for the initial yield curves on an equidistantpartition of [0 , T ] :(i) Bootstrap the non-defaultable term structure T (cid:55)−→ P (0 , T ) from liquidgovernment bonds utilising equation (6.3). This can be achieved in thesense of least squares by solving the corresponding normal equation.(ii) Derive the defaultable term structure T (cid:55)−→ (cid:101) P (0 , T ) from CDS spreadsutilising the relation (cid:101) P (0 , T ) = F (0 , T ) P (0 , T ) and equation (6.7).(iii) Bootstrap the illiquidity premium T (cid:55)−→ L (0 , T ) from issued corporatebonds according to (6.4) in order to explain the residual spread of corporatebonds above their governmental counterparts. To this end, one can proceedanalogously as in (i).This is a simple yet powerful algorithm to take a snapshot of the current marketsituation. For dynamic approaches, liquid derivatives should be incorporated inorder to account for the volatility surface of the term structure. Remark 6.1 (Machine Learning) . If we parameterise ϕ in terms of ϕ ( t ) = f ϕ ( t )+ f ϕ ( t ) + f ϕ ( t ) , where f k ∈ R and t (cid:55)−→ ϕ k ( t ) is a suitably chosen set of basisfunctions/principal components for k = 1 , , , then the inverse of the mappingfrom the parameter set (cid:8) r , b x , β x , σ x , λ X , m X , σ X , f , f , f (cid:9) onto the backed-outgrid of the quotes (cid:8) P (0 , T i ) , (cid:101) P (0 , T i ) , L (0 , T i ) (cid:12)(cid:12) T i = iT /n for i = 0 , , , . . . , n (cid:9) canbe learnt by a sufficiently complex neural network. The target function in theminimisation can certainly be extended to financial derivatives with optionalities. HE JARROW & TURNBULL SETTING REVISITED 17 References 1. Bianchetti, M. (2009). Two Curves, One Price: Pricing & Hedging InterestRate Derivatives Using Different Yield Curves for Discounting and Forwarding. http://ssrn.com/abstract=1334356 .2. Bibby, B. M., Skovgaard, I. M. and Sorensen, M. (2005). Diffusion-type models with givenmarginal distribution and autocorrelation function. Bernoulli . Vol. 11, No. 2, pp. 191–220.3. Brigo, D. and Mercurio, F. (Corrected 3rd Printing 2007). Interest Rate Models – Theory andPractice. Springer Verlag Berlin Heidelberg .4. Chen, L. and Filipovic, D. (2005). A simple model for credit migration and spread curves. Finance and Stochastics . Vol. 9, No. 2, pp. 211–231.5. Cuchiero, C., Fontana, C. and Gnoatto, A. (2016). A general HJM framework for multipleyield curve modeling. Finance and Stochastics . Vol. 20, No. 2, pp. 267–320.6. Cuchiero, C., Klein. I. and Teichmann, J. (2017). A fundamental theorem of asset pricingfor continuous time large financial markets in a two filtration setting. to appear in Theory ofProbability and its Applications, arXiv:1705.02087 .7. Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem ofasset pricing. Mathematische Annalen . Vol. 300, No. 1, pp. 463–520.8. Duffie, D., Filipovic, D. and Schachermayer, W. (2003). Affine processes and applications infinance. Annals of Applied Probability . Vol. 13, No. 3, pp. 984–1053.9. Duffie, D. (2004). Credit risk modeling with affine processes. Springer Verlag BerlinHeidelberg .10. Filipović, D. and Trolle, A. B. (2013). The Term Structure of Interbank Risk. Journal ofFinancial Economics . Vol. 109, No. 3, pp. 707–733.11. Gehmlich, F. and Schmidt, T. (2015). Dynamic defaultable term structure modelling beyondthe intensity paradigm. Mathematical Finance . Vol. 28, No. 1, pp. 211–239.12. Grbac, Z. and Runggaldier, W. (2015). Interest Rate Modeling: Post-Crisis Challenges andApproaches. Springer Verlag Berlin Heidelberg .13. Jarrow, R. A. and Turnbull, S. M. (1995). Pricing Derivatives on Financial Securities Subjectto Credit Risk. The Journal of Finance . Vol. 50, No. 1, pp. 53–85.14. Jarrow, R. A., Lando, D. and Turnbull, S. M. (1997). A Markov Model for the Term Structureof Credit Risk Spreads. The Review of Financial Studies . Vol. 10, No. 2, pp. 481–523.15. Jarrow, R. A. and Yildirim, Y. (2003). Pricing Treasury Inflation Protected Securities andRelated Derivatives using an HJM Model. Journal of Financial and Quantitative Analysis .Vol. 38, No. 2, pp. 409–430.16. Krabichler, T. (2017). Term Structure Modelling Beyond Classical Paradigms – An FX-likeApproach. Dissertation .17. Larsen, K. S. and Sorensen, M. (2007). Diffusion Models for Exchange Rates in a Target Zone. Mathematical Finance . Vol. 17, No. 2, pp. 285–306.18. Lybek, T. and Sarr, A. (2002). Measuring Liquidity in Financial Markets. IMF Working PaperWP/02/232 .19. Nguyen, T. A. and Seifried, F. T. (2015). The Multi-Curve Potential Model. InternationalJournal of Theoretical and Applied Finance . Vol. 18, No. 7, pp. 1–32.20. Rogers, L. C. G. (1997). The potential approach to the term structure of interest rates andforeign exchange rates. Mathematical Finance . Vol. 7, No. 2, pp. 157–164. E-mail address : [email protected] E-mail address ::