A Consistent Stochastic Model of the Term Structure of Interest Rates for Multiple Tenors
AA Consistent Stochastic Model of the Term Structureof Interest Rates for Multiple Tenors * Mesias Alfeus † , Martino Grasselli ‡ and Erik Schl ¨ogl §September 19, 2018 Abstract
Explicitly taking into account the risk incurred when borrowing at a shortertenor versus lending at a longer tenor (“roll-over risk”), we construct a stochasticmodel framework for the term structure of interest rates in which a frequency basis(i.e. a spread applied to one leg of a swap to exchange one floating interest rate foranother of a different tenor in the same currency) arises endogenously. This roll-over risk consists of two components, a credit risk component due to the possibilityof being downgraded and thus facing a higher credit spread when attempting to rollover short–term borrowing, and a component reflecting the (systemic) possibilityof being unable to roll over short–term borrowing at the reference rate (e.g., LI-BOR) due to an absence of liquidity in the market. The modelling framework is of“reduced form” in the sense that (similar to the credit risk literature) the source ofcredit risk is not modelled (nor is the source of liquidity risk). However, the frame-work has more structure than the literature seeking to simply model a differentterm structure of interest rates for each tenor frequency, since relationships betweenrates for all tenor frequencies are established based on the modelled roll-over risk.We proceed to consider a specific case within this framework, where the dynamicsof interest rate and roll-over risk are driven by a multifactor Cox/Ingersoll/Ross–type process, show how such model can be calibrated to market data, and used forrelative pricing of interest rate derivatives, including bespoke tenor frequencies notliquidly traded in the market.
Keywords: tenor swap, basis, frequency basis, liquidity risk, swap market
JEL Classfication: C6, C63, G1, G13 * We would like to thank Alex Backwell, Bruno Bouchard, Alan Brace, Jos´e da Fonseca, Marc Hen-rard, Andrea Macrina, Michael Nealon and David Skovmand for helpful discussions on earlier versionsof this paper. The usual disclaimer applies. † University of Technology Sydney, Australiae-mail:
[email protected] ‡ Dipartimento di Matematica, Universit`a degli Studi di Padova (Italy) and L´eonard de Vinci PˆoleUniversitaire, Research Center, Finance Group, 92 916 Paris La D´efense Cedex (France).e-mail: [email protected] § University of Technology Sydney, Australiae-mail:
[email protected] a r X i v : . [ q -f i n . P R ] S e p Introduction
The phenomenon of the frequency basis (i.e. a spread applied to one leg of a swap toexchange one floating interest rate for another of a different tenor in the same currency)contradicts textbook no–arbitrage conditions and has become an important feature ofinterest rate markets since the beginning of the Global Financial Crisis (GFC) in 2008.As a consequence, stochastic interest rate term structure models for financial risk man-agement and the pricing of derivative financial instruments in practice now reflect theexistence of multiple term structures, i.e. possibly as many as there are tenor frequen-cies. While this pragmatic approach can be made mathematically consistent (see Gras-selli and Miglietta (2016) and Grbac and Runggaldier (2015) for a recent treatise, aswell as the literature cited therein), it does not seek to explain this proliferation of termstructures, nor does it allow the extraction of information potentially relevant to riskmanagement from the basis spreads observed in the market.In the pre-GFC understanding of interest rate swaps (see explained in, e.g., Hull(2008)), the presence of a basis spread in a floating–for–floating interest rate swap wouldpoint to the existence of an arbitrage opportunity, unless this spread is too small torecover transaction costs. As documented by Chang and Schl ¨ogl (2015), post–GFCthe basis spread cannot be explained by transaction costs alone, and therefore theremust be a new perception by the market of risks involved in the execution of textbook“arbitrage” strategies. Since such textbook strategies to profit from the presence ofbasis spreads would involve lending at the longer tenor and borrowing at the shortertenor, the prime candidate for this is “roll–over risk.” This is the risk that in the future,once committed to the “arbitrage” strategy, one might not be able to refinance (“rollover”) the borrowing at the prevailing market rate (i.e., the reference rate for the shortertenor of the basis swap). This “roll–over risk,” invalidating the “arbitrage” strategy,can be seen as a combination of “downgrade risk” (i.e., the risk faced by the potentialarbitrageur that the credit spread demanded by its creditors will increase relative to themarket average) and “funding liquidity risk” (i.e., the risk of a situation where fundingin the market can only be accessed at an additional premium).We propose to model this roll-over risk explicitly, which endogenously leads to thepresence of basis spreads between interest rate term structures for different tenors. Thisis in essence the “reduced–form” or “spread–based” approach to multicurve modelling,similar to the approach taken in the credit risk literature, where the risk of loss due todefault gives rise to credit spreads. The model allows us to extract the forward–looking“market’s view” of roll-over risk from the observed basis spreads, to which the model iscalibrated. Preliminary explorations using a simple model of deterministic basis spreadsin Chang and Schl¨ogl (2015) indicate an improving stability of the calibration, suggest-ing that the basis swap market has matured since the turmoil of the GFC and pointingtoward the practicability of constructing and implementing a full stochastic model.The bulk of the literature on modelling basis spreads is in a sense even more “reduced–form” than what we propose here, in the sense that basis spreads are recognised to exist,and are modelled to be either deterministic or stochastic in a mathematically consistentfashion, but there are no structural links between term structures of interest rates fordifferent tenors (in a sense, the analogue of this approach applied to credit risk would2e to model stochastic credit spreads directly, without any link to probabilities of de-fault and losses in the event of default). This strand of the literature can be tracedback to Boenkost and Schmidt (2004), who used this approach to construct a modelfor cross currency swap valuation in the presence of a basis spread. This was subse-quently adapted by Kijima, Tanaka and Wong (2009) to modelling a single–currencybasis spread. Henrard (2010) took an axiomatic approach to the problem, modellinga deterministic multiplicative spread between term structures associated with differenttenors. Initially, these models were not reconciled with the requirement of the absenceof arbitrage. Subsequent work, however, such as Fujii, Shimada and Takahashi (2009),gave explicit consideration to this requirement. This pragmatic way of modelling inter-est rates in the presence of spreads between term structures of interest rates for differenttenors has been pursued further in a number of papers, including Mercurio (2009,2010)in a LIBOR Market Model setting, Kenyon (2010) in a short–rate modelling frame-work, stochastic additive basis spreads in Mercurio and Xie (2012), and Henrard (2013)for stochastic multiplicative basis spreads. Moreni and Pallavicini (2014) construct amodel of two curves, riskfree instantaneous forward rates and forward LIBORs, whichis Markovian in a common set of state variables. Macrina and Mahomed (2018) con-struct a pricing kernel framework for multicurve models (be it discount count curvesin different currencies, or real vs. nominal interest rates, or for different tenors), butagain this approach does not attempt to model the structural links between differentterm structures.Early work incorporating some of the potential causes of basis spreads into modelsof the single–currency “multicurve” environment post–GFC includes Morini (2009) andBianchetti (2010), who focus on counterparty credit risk. The model of Cr´epey (2015)links funding cost and counterparty credit risk in a credit valuation adjustment (CVA)framework, but does not explicitly consider spreads between different tenor frequenciesarising from roll–over risk.Recently, there has been an emerging view that “roll–over risk” is what prevents pre–crisis textbook arbitrage strategies to exploit the basis spreads between tenor frequen-cies, and that modelling this risk can provide the link between overnight index swaps(OIS), the XIBOR (e.g. LIBOR, EURIBOR, etc.) style money market, the vanilla swapmarket, and the basis swap market. An important contribution in this vein is Filipovi´cand Trolle (2013), who estimate the dynamics of interbank risk from time series datafrom these markets. They define “interbank risk” as “the risk of direct or indirect lossresulting from lending in the interbank money market.” Decomposing the term structureof interbank risk into what they identify as default and non-default (liquidity) compo-nents, they study the associated risk premia. Filipovi´c and Trolle interpret the “default”component in terms of the risk of a deterioration of creditworthiness of a LIBOR ref-erence panel bank resulting in it dropping out of the LIBOR panel, in which case thisbank immediately would no longer be able to roll over debt at the overnight referencerate, while the rate on any LIBOR borrowing would remain fixed until the end of the ac-crual period (i.e., typically for several months). In their analysis, this differential impactof downgrade risk on rolling debt explains part of the LIBOR/OIS spread; the residual is This is also known as the “renewal effect,” see Collin–Dufresne and Solnik (2001) and Grinblatt(2001). It is important to note that both components man-ifest themselves in the risk of additional cost when rolling over debt, i.e., “downgraderisk” and “funding liquidity risk” combining to form a total “roll–over risk.” Based on similar considerations, Cr´epey and Douady (2013) model the spread be-tween LIBOR and OIS as a combination of credit and liquidity risk premia, where inparticular they focus on providing some model structure for the latter. They constructa stylised equilibrium model of credit risk and funding liquidity risk to explain the LI-BOR/OIS spread, arguing (unlike Filipovi´c and Trolle) that the overnight rate underly-ing OIS (e.g., the Fed Funds or EONIA rate) is riskfree (we will return to this point inour model setup below).An alternative approach at the more fundamental end of the modelling spectrumis the recent work by Gallitschke, M¨uller and Seifried (2014), who propose a modelfor interbank cash transactions and the relevant credit and liquidity risk factors, whichendogenously generates multiple term structures for different tenors. In particular, theyexplicitly model a mechanism by which XIBOR is determined by submissions of themember banks of a panel, which adds substantial complexity to the model.Our aim is to construct a consistent stochastic model encompassing OIS, XIBOR,vanilla and basis swaps in a single currency. Our “reduced–form” approach explicitlymodels both the credit and the funding component of roll–over risk to link multipleyield curves. In that, it is more parsimonious than the “pragmatic” way of modellingextant in the literature (reviewed above), where stochastic dynamics for basis spreadsare specified directly without recourse to the underlying roll–over risk. In particular,this allows the relative pricing of bespoke tenors in a model calibrated to basis spreadsbetween tenor frequencies for which liquid market data is available. It does not requirethe introduction of a new stochastic factor (or deterministic spread) for each new tenorfrequency. However, the approach is “reduced–form” in the sense that it abstracts fromstructural causes of downgrade risk and funding liquidity risk — in this sense, our ap-proach is closest in spirit to the “reduced–form” models of credit risk, doing for basisspreads what those models have done for credit spreads. The framework which we pro-pose below departs from that of Chang and Schl¨ogl (2015) in that, rather than focusingexclusively on basis swaps, we treat OIS, XIBOR, vanilla and basis swaps in a unifiedframework, as well as calibrating credit risk to credit default swaps, and model both the Past empirical studies, in particular of the GFC, also indicate that credit risk alone is insufficient toexplain the LIBOR/OIS spread; see e.g. Eisenschmidt and Tapking (2009). “Funding liquidity risk” has also been considered explicitly in a separate strand of the literature. Forexample, Acharya and Skeie (2011) model liquidity hoarding by participants in the interbank market. Intheir model, there is a positive feedback effect between roll–over risk and liquidity hoarding (via termpremia on interbank lending rates), which in the extreme case can lead to a freeze of interbank lending.Brunnermeier and Pedersen (2009) model a similar adverse feedback effect between market liquidity andfunding liquidity. Since the GFC a cross currency basis exceeding pre-crisis textbook arbitrage bounds has alsoemerged, see for example Chang and Schl¨ogl (2012). The approach presented here could be extendedto multiple currencies, but it is our view that across currencies there may be other factors than variousforms of roll–over risk giving rise to a basis spread. For example, Andersen et al. (2017) demonstratehow funding value adjustments (FVAs) can prevent potential arbitrageurs from enforcing covered interestparity (CIP) across currencies. If the CIP arbitrage channel is blocked, then this would allow for CIPviolations driven by, say, different supply/demand equilibria in FX spot versus FX forward markets.
We model a frictionless market free of arbitrage opportunities in which trading takesplace continuously over the time interval [0 , T ] , where T is an arbitrary positive fi-nal date. Uncertainty in the market is modeled through a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , Q ) , supporting all the price processes we are about to introduce. Here Q denotes the risk neutral measure, so that we are modeling the market directly underthe pricing measure. Also, E Q t is a shorthand for E Q [ . |F t ] .Denote by r c the continuously compounded short rate abstraction of the interbankovernight rate (e.g., Fed funds rate or EONIA). This is equal to the riskless (default–free) continuously compounded short rate r plus a credit spread. In the simplest case,one could adopt a “fractional recovery in default,” a.k.a. “recovery of market value,”model and denote by q the (assumed constant) loss fraction in default. Then r c ( s ) = r ( s ) + Λ( s ) q (2.1)where Λ( s ) q is the average (market aggregated) credit spread across the panel and Λ( s ) is the corresponding default intensity. Note that although a significant part of the “mul-ticurve” interest rate modelling literature mentioned in the introduction heuristicallyadvances the argument that (mainly due to its short maturity) the interbank overnightrate is essentially free of default risk, in intensity–based models of default this is incor-rect even in the instantaneous limit. In our modelling, we do not require this argument.Instead, it suffices that r c is the appropriate rate at which to discount payoffs of any fully See Duffie and Singleton (1999). Here, with “risk neutral measure” we mean an equivalent martingale measure associated with dis-counting by the instantaneous, default–free interest rate r (i.e., the numeraire is the associated continu-ously compounded savings account). In the presence of risks (such as credit risk) with respect to whichthe market may be incomplete, this measure is not necessarily unique, but we follow the bulk of thecredit derivatives literature in assuming that the model, calibrated to the extent possible to available liq-uid market instruments, gives us the “correct” dynamics under the assumed pricing measure. In otherwords, model prices are arbitrage–free with respect to the prices observed in the market, but this doesnot necessarily imply that any departure from these model prices would result in an exploitable arbitrageopportunity. ction Time t T Borrow overnight, 1 − e (cid:82) Tt r c ( s ) ds rolling from time t to T .Enter OIS 0 e (cid:82) Tt r c ( s ) ds − (1 + ( T − t ) OIS ( t, T )) Lend at LIBOR -1 T − t ) L ( t, T ) Net outcome ( T − t )( L ( t, T ) − OIS ( t, T )) Table 1: Strategy to exploit the LIBOR/OIS spread assuming the absence of roll–overrisk.collateralised derivative transaction, because the standard ISDA Credit Support Annex(CSA) stipulates that posted collateral accrues interest at the interbank overnight rate. Roll–over risk is modelled via the introduction of a π ( s ) , denoting the spread over r c ( s ) which an arbitrary but fixed entity must pay when borrowing overnight. π ( s ) hastwo components, π ( s ) = φ ( s ) + λ ( s ) q (2.2)where φ ( s ) is pure funding liquidity risk (both idiosyncratic and systemic) and λ ( s ) q is the idiosyncratic credit spread over r c (initially, e.g. at time 0, λ (0) = 0 by virtueof the fact that at time of calibration to basis spread data, we are considering marketaggregated averages). The default intensity of any given (but representative) XIBORpanel member is Λ( s ) + λ ( s ) . Thus λ ( s ) q represents the “credit” (a.k.a. “renewal”) riskcomponent of roll–over risk, i.e. the risk that a particular borrower will be unable to rollover overnight (or instantaneously, in our mathematical abstraction) debt at r c , insteadhaving to pay an additional spread λ ( s ) q because their credit quality is lower than thatof the panel contributors determining r c . Let us consider first the simplest case, i.e. the LIBOR/OIS spread over a single accrualperiod. In the absence of roll–over risk, one could construct a strategy to take advan-tage of this spread (similar strategies can be constructed for multiple accrual periods, orto take advantage of the frequency (tenor) basis): Borrow at the overnight rate, rollingover the borrowing daily. Enter into an OIS receiving floating and paying fixed, thuseliminating the exposure to interest rate risk. Lend at LIBOR. Although this not strictly For a detailed discussion of this latter point, see Piterbarg (2010). In Section 3, φ ( s ) will be modelled as a diffusion as a “first–cut” concrete specification of our model.Modelling liquidity freezes properly may require permitting φ ( s ) to jump — the framework laid out inthe present section would allow for such an extension. Note again that for mathematical convenience we are equating “daily” with the continuous–timeinfinitesimal limit — if this simplification is considered to have material impact, it could be lifted at thecost of some additional mathematical tedium.
6n arbitrage strategy, if we assume that one is guaranteed to be able to roll over theborrowing at r c (i.e., in the absence of roll–over risk), and credit spreads apply symmet-rically to the borrower and the lender, then this strategy results in a profit equal to theLIBOR/OIS spread, as summarised in Table 1. We interpret this profit as a compensa-tion for roll–over risk, in a manner which we will now proceed to make specific.Borrowing overnight from t to T (setting δ = T − t ), rolling principal and interestforward until maturity, an arbitrary but fixed entity pays at time T : − e (cid:82) Tt r c ( s ) ds e (cid:82) Tt π ( s ) ds (2.3)Assuming symmetric treatment of credit risk when borrowing and lending, lending toan arbitrary but fixed entity from t to T will incur credit risk with intensity Λ( s ) + λ ( s ) .Discounting with r ( s ) + (Λ( s ) + λ ( s )) q = r c ( s ) + λ ( s ) q the present value of (2.3) is − E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) (2.4)Enter OIS to receive the overnight rate and pay the fixed rate OIS ( t, T ) , discountingwith r c (due to collateralisation of OIS), the present value of the payments is E Q t (cid:104) − e − (cid:82) Tt r c ( s ) ds − e − (cid:82) Tt r c ( s ) ds OIS ( t, T ) δ (cid:105) (2.5)Spot LIBOR observed at time t for the accrual period [ t, T ] is denoted by L ( t, T ) . Lend-ing at LIBOR L ( t, T ) , we receive (at time T ) the credit risky payment δL ( t, T ) Discounting as above with r c ( s ) + λ ( s ) q , the present value of this is E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (1 + δL ( t, T )) (cid:105) (2.6)We must have (because the initial investment in the strategy is zero) that the sum of thethree terms ((2.4), (2.5) and (2.6)) is zero, i.e. E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) = E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (1 + δL ( t, T )) − e − (cid:82) Tt r c ( s ) ds (1 + δ OIS ( t, T )) (cid:105) (2.7) This strategy is not strictly riskless, because of the potential impact of default risk. However, thisdefault risk should be reflected in any interbank borrowing credit spread, be it overnight or for a longerperiod. It is the roll–over risk which introduces a new distinction between shorter and longer tenorborrowing, as we demonstrate below. We need to assume symmetric treatment of roll–over risk, both the “credit” and the “funding liq-uidity” component, in order to maintain additivity of basis spreads: Swapping a one–month tenor into athree–month tenor, and then swapping the three–month tenor into a 12–month tenor, is financially equiv-alent to swapping the one–month tenor into the 12–month tenor, and thus (ignoring transaction costs) the1m/12m basis spread must equal the the sum of the 1m/3m and 3m/12m basis spreads. λ ( s ) ≡ , then the LIBOR/OIS spread issolely due to funding liquidity risk: E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) = E Q t (cid:104) e − (cid:82) Tt r c ( s ) ds δ ( L ( t, T ) − OIS ( t, T )) (cid:105) (2.8)Define the discount factor implied by the overnight rate as D OIS ( t, T ) = E Q t (cid:104) e − (cid:82) Tt r c ( s ) ds (cid:105) (2.9)Since the mark–to–market value of the OIS at inception is zero, we haveOIS ( t, T ) = 1 − D OIS ( t, T ) δD OIS ( t, T ) (2.10) ⇔ D OIS ( t, T ) = 11 + δ OIS ( t, T ) (2.11)Thus the dynamics of r c should be consistent with (2.11) (the term structure of the D OIS ( t, T ) ), and the dynamics of φ ( s ) and λ ( s ) should be consistent with (2.7).The following remarks are worth noting:• (2.11) implies E Q t (cid:104) e − (cid:82) Tt r c ( s ) ds (1 + δ OIS ( t, T )) (cid:105) = 1 Therefore (2.7) implies that we cannot have E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (1 + δL ( t, T )) (cid:105) = 1 (2.12)unless φ ( s ) = 0 , i.e. unless the LIBOR/OIS spread is solely due to renewal risk.• It may seem counterintuitive that (2.12) doesn’t hold, but this is due to the fact thatthe discounting in (2.12) only takes into account credit risk (including “renewalrisk”), i.e. it does not take into account the premium a borrower of LIBOR (asopposed to rolling overnight borrowing) would pay for avoiding the roll–over riskinherent in φ ( s ) . OIS may pay more frequently than once at T for an accrual period [ t, T ] (especiallywhen the period covered by the OIS exceeds one year). In this case the strategy of theprevious section needs to be modified as follows.Borrowing overnight from t = T to T n (normalising T j − T j − = δ, ∀ < j ≤ n ),rolling principal until maturity and interest forward until each T j , an arbitrary but fixedentity pays at time T j ( ∀ < j < n ) : − e (cid:82) TjTj − r c ( s ) ds e (cid:82) TjTj − π ( s ) ds (2.13)8nd at time T n : − e (cid:82) TnTn − r c ( s ) ds e (cid:82) TnTn − π ( s ) ds (2.14)Discounting with r c ( s ) + qλ ( s ) , the present value of this is n − (cid:88) j =1 (cid:18) E Q t (cid:104) e − (cid:82) Tjt ( r c ( s )+ qλ ( s )) ds (cid:105) − E Q t (cid:20) e − (cid:82) Tj − t ( r c ( s )+ qλ ( s )) ds e (cid:82) TiTj − φ ( s ) ds (cid:21)(cid:19) − E Q t (cid:104) e − (cid:82) Tn − t ( r c ( s )+ qλ ( s )) ds e (cid:82) TnTn − φ ( s ) ds (cid:105) (2.15)Enter OIS to receive the overnight rate and pay the fixed rate OIS ( t, T n ) at each T j ( ∀ < j ≤ n ) , discounting with r c , the present value of the payments is n (cid:88) j =1 ( D OIS ( t, T j − ) − (1 + δ OIS ( t, T n )) D OIS ( t, T j )) (2.16)If lending to time T n at L ( t, T n ) is possible (i.e., a LIBOR L ( t, T n ) is quoted in themarket), and supposing that LIBOR is quoted with annual compounding, with T n − t = m , the present value of interest and repayment of principal is E Q t (cid:104) e − (cid:82) Tnt ( r c ( s )+ qλ ( s )) ds (1 + L ( t, T n )) m (cid:105) (2.17)We must have (because the initial investment in the strategy is zero) that the sum of thethree terms ((2.15), (2.16) and (2.17)) is zero. Since the mark–to–market value of theOIS at inception is zero, we can drop (2.16) and write this directly as n (cid:88) j =1 E Q t (cid:20) e − (cid:82) Tj − t ( r c ( s )+ qλ ( s )) ds e (cid:82) TjTj − φ ( s ) ds (cid:21) = E Q t (cid:104) e − (cid:82) Tnt ( r c ( s )+ qλ ( s )) ds (1 + L ( t, T n )) m (cid:105) + n − (cid:88) j =1 E Q t (cid:104) e − (cid:82) Tjt ( r c ( s )+ qλ ( s )) ds (cid:105) (2.18)Analogously to the single–period case, since the mark–to–market value of the OIS atinception is zero, we have OIS ( t, T n ) = 1 − D OIS ( t, T n ) δ (cid:80) nj =1 D OIS ( t, T j ) (2.19) Substituting (2.11) into (2.7), we obtain the dynamics of the L ( t, T ) as conditional ex-pectations over the dynamics of φ , r c and λ : L ( t, T ) = 1 δ E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (cid:105) − (2.20)9ewriting this as a discount factor D L ( t, T ) = (1 + δL ( t, T )) − = E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (cid:105) E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) (2.21)we see that if we assume independence of the dynamics of r c ( s ) + λ ( s ) q from φ ( s ) , the“instantaneous spread” (admittedly a theoretical abstraction) over r c inside the expecta-tion becomes simply π ( s ) = φ ( s ) + λ ( s ) q : D L ( t, T ) = E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ φ ( s )+ λ ( s ) q ) ds (cid:105) (2.22) The application of roll–over risk must be symmetric, i.e. applied to roll–over of bothborrowing and lending (with opposite sign for borrowing vs. lending), because other-wise contradictions are inescapable. Suppose we are swapping LIBOR with tenor structure T ( L ) , from T ( L )0 = T = t to T ( L ) n L into a fixed rate s (1) ( t, T (1) n ) , paid based on a tenor structure T (1) , with T ( L ) n L = T (1) n ,using an interest rate swap. For a (fully collateralised) swap transaction, it must holdthat n L (cid:88) j =1 E Q t (cid:34) e − (cid:82) T ( L ) jt r c ( s ) ds ( T ( L ) j − T ( L ) j − ) L ( T ( L ) j − , T ( L ) j ) (cid:35) = n (cid:88) j =1 D OIS ( t, T (1) j )( T (1) j − T (1) j − ) s (1) ( t, T (1) n ) (2.23)Typically, we have market data on (vanilla) fixed–for–floating swaps of different matu-rities, for a single tenor frequency. Combining this with market data on basis swaps, weget a matrix of market calibration conditions (2.23), i.e. one condition for each (matu-rity,tenor) combination. Note that the basis spread is typically added to the shorter tenorof a basis swap, so (2.23) needs to be modified accordingly. For example, if T ( L ) cor-responds to a three–month frequency, T (1) corresponds to a six–month frequency, and T (2) corresponds to a one–month frequency of the same maturity ( T ( L ) n L = T (2) n ), then Such contradictions would arise in particular because asymmetry in the treatment of roll-over riskwould prevent additivity of basis spreads: Combing a position of paying one–month LIBOR versus re-ceiving three–month LIBOR with a position of paying three–month LIBOR versus receiving six–monthLIBOR is equivalent to paying one–month LIBOR versus receiving six–month LIBOR; therefore (up totransaction costs), the one–month versus six–month basis spread should equal the sum of the one–monthversus three–month and three–month versus six–month spreads. For example, a vanilla USD swap would exchange three–month LIBOR (paid every three months)against a stream of fixed payments paid every six months. b ( t, T ( L ) n L , T (2) n ) , we obtain n (cid:88) j =1 E Q t (cid:34) e − (cid:82) T (2) jt r c ( s ) ds ( T (2) j − T (2) j − ) L ( T (2) j − , T (2) j ) (cid:35) = n (cid:88) j =1 D OIS ( t, T (1) j )( T (1) j − T (1) j − ) s (1) ( t, T (1) n ) − n (cid:88) j =1 D OIS ( t, T (2) j )( T (2) j − T (2) j − ) b ( t, T ( L ) n L , T (2) n ) (2.24)Conversely, because the convention of basis swaps is that the basis swap spread is addedto the shorter tenor, if T ( L ) corresponds to a three–month frequency, T (1) correspondsto a six–month frequency, and T (3) corresponds to a twelve–month frequency of thesame maturity ( T ( L ) n L = T (3) n ), then combining (2.23) with a basis swap with spread b ( t, T ( L ) n L , T (3) n ) , we obtain n (cid:88) j =1 E Q t (cid:34) e − (cid:82) T (3) jt r c ( s ) ds ( T (3) j − T (3) j − ) L ( T (3) j − , T (3) j ) (cid:35) = n (cid:88) j =1 D OIS ( t, T (1) j )( T (1) j − T (1) j − ) s (1) ( t, T (1) n )+ n (cid:88) j =1 D OIS ( t, T (3) j )( T (3) j − T (3) j − ) b ( t, T ( L ) n L , T (3) n ) (2.25) We assume that the model is driven by a time-homogeneous affine Markov process X taking values in a non-empty convex subset E of R d ( d ≥ ), endowed with theinner product (cid:104)· , ·(cid:105) . We assume that X = ( X t ) t ∈ [0 , ∞ ) admits the transition semigroup ( P t ) t ∈ [0 , ∞ ) acting on B ( E ) b (the space of bounded Borel functions on E ).Let us now define the model variables r c ( t ) , λ ( t ) and φ ( t ) as follows: r c ( t ) = a ( t ) + (cid:104) a, X ( t ) (cid:105) (2.26) λ ( t ) = b ( t ) + (cid:104) b, X ( t ) (cid:105) (2.27) φ ( t ) = c ( t ) + (cid:104) c, X ( t ) (cid:105) (2.28)where a, b, c are arbitrary constant projection vectors in E and a , b , c are scalar de-terministic functions. Note that the process ( r c , λ, φ ) does not enjoy, in general, theMarkov property and when it does, it is a priori time-inhomogeneous.In order to preserve analytical tractability, in this paper we will take the Markovprocess X in the class of affine processes, introduced by Duffie and Kan (1996) and thenclassified by Duffie et al. (2003) in the canonical state space domain E = R m + × R n .Affine processes have been recently recovered thanks to the interesting extension to thestate space of positive semidefinite matrices (see Bru (1991), Gourieroux and Sufana11RS OIS 1m/3m 3m/6mT bid ask bid ask bid ask bid ask0.5 0.50825 0.50825 0.13 0.17 9.6 9.6 19.32 21.321 0.311 0.331 0.125 0.165 8.29 9 16.25 18.252 0.37 0.395 0.125 0.165 7.8 9.8 13.56 15.563 0.5 0.5 0.12 0.16 7.21 9.21 11.7 13.74 0.657 0.667 0.12 0.16 7.7 7.7 10.64 12.645 0.83 0.87 0.109 0.16 7.3 7.3 9.74 11.746 1.0862 1.0978 0.13 0.17 6.8 6.8 10.2 10.28 1.5001 1.5149 0.226 0.276 6 6 9.7 9.79 1.6496 1.6684 0.368 0.418 5.7 5.7 9.7 9.710 1.836 1.837 0.563 0.613 5.3 5.3 8.67 10.67Table 2: Market data quotes on 01/01/2013. Source: Bloomberg(2003), Da Fonseca et al. (2007), Grasselli and Tebaldi (2008) and Cuchiero et al.(2011)). In Appendix A we recall the definition of the affine processes as well as their char-acterisation in terms of the solution of Riccati ODEs. Then, in Appendix B, we developthe computation of the relevant expectations involved in the pricing of swaps as wellas non-linear instruments like caps. We emphasise that our framework is extremelyanalytically tractable.
In this section, we proceed to calibrate the version of the model based on multifactorCox/Ingersoll/Ross (CIR) dynamics to market data for overnight index swaps, vanillaand basis swaps, step by step adding more instruments to the calibration. Thus, theobjective is to obtain a single calibrated, consistent roll–over risk model for interest rateterm structures of all tenors. More explicit separation of roll–over risk into its credit andfunding liquidity components is left for Section 4 below, which includes credit defaultswaps in the set of calibration instruments. This version of the model is obtained by Affine processes belong to the more general family of polynomial processes recently investigated bye.g. Filipovic and Larsson (2016), see also references therein. Our approach can be easily extended to thislarger class with minor changes and a slightly different technique in the computation of the expectations.The same holds true as well as for other processes, like for example the L´evy driven models, see e.g.Eberlein and Raible (1999). Actually, the choice of the stochastic model is just instrumental as far as thecomputation of the relevant expectations involved in the sequel can be efficiently performed, in view ofour calibration exercise. r c ( t ) = a ( t ) + d (cid:88) i =1 a i y i ( t ) (3.1) λ ( t ) = b ( t ) + d (cid:88) i =1 b i y i ( t ) (3.2) φ ( t ) = c ( t ) + d (cid:88) i =1 c i y i ( t ) (3.3)where the y i follow the Cox–Ingersoll–Ross (CIR) dynamics under the pricing measure,i.e. dy i ( t ) = κ i ( θ i − y i ( t )) dt + σ i (cid:112) y i ( t ) dW i ( t ) , (3.4)where dW i ( t ) ( i = 1 , · · · , d ) are independent Wiener processes. In order to keep themodel analytically tractable, in keeping with (2.26)–(2.28) we do not allow for time–dependent coefficients at this stage. Since each of the factors follow independentCIR–type dynamics, the sufficient condition for each factor to remain positive is κ i θ i ≥ σ i , ∀ i , as discussed for the one–factor case in Cox et al. (1985).We begin our calibration with OIS, then include money market instruments such asvanilla swaps and basis swaps. Calibration to instruments with optionality (caps/floorsand swaptions) is in principle possible, but not included in the scope of the presentpaper. At each step calibration consists in minimising the sum of squared deviationsfrom calibration conditions such as (2.9) and (2.23). For example, following Schl¨ogl and Schl¨ogl (2000), the coefficients could be made piecewise con-stant to facilitate calibration to at–the–money option price data, while retaining most analytical tractabil-ity.
Our goal is to jointly calibrate the model to three different tenor frequencies, i.e., to 1-month, 3-month and 6-month tenors, as well as to the OIS–based discount factors. Table2 to 7 show market data quoted on the 01/01/2013, 08/09/2014, 18/06/2015, 20/04/2016,22/03/2017 and 31/10/2017, respectively. In the USD market, the benchmark interest rate swap (IRS) pays 3-month LIBORfloating vs. 6-month fixed, i.e. δ = 0 . and ∆ = 0 . . To get IRS indexed to LIBORof other maturities, we can use combinations of basis swaps with the benchmark IRS.Calibration conditions for 1-month, 3-month and 6-month tenors are given in Equations With maturities T = { . , , , , , , , , , } , as there are no quotes for maturity of 7 years inthe basis swap market. IRS and OIS are given in % while basis swaps are quoted in basis points, i.e.,one–hundredths of one percent. a for both OIS bid and ask and then interpolate the bond pricesusing the OIS equation to get the corresponding bid and ask values for intermediatematurities. Model calibration is a process of finding model parameters such that model prices matchmarket prices as closely as possible, i.e., in equation (2.23), we want the left hand side tobe equal to the right hand side. In essence, we are implying model parameters from themarket data. The most common approach is to minimise the deviation between modelprices and market prices, i.e., in the “relative least–squares” sense of arg min θ ∈ Θ N (cid:88) i =1 (cid:18) P model ( θ ) − P market P market (cid:19) , (3.5)where N is the number of calibration instruments. θ represents a choice of parametersfrom Θ , the admissible set of model parameter values. P model and P market are the model andmarket prices, respectively. In our case, P model is the left hand side of (2.23) and P market is the right hand side of the same equation. Market instruments are usually quoted asbid and ask prices, therefore we consider market prices to be accurate only to withinthe bid/ask spread. Thus, we solely require the model prices to fall within bid–askdomain rather than attempting to fit the mid price exactly. The problem in (3.5) can bereformulated as arg min θ ∈ Θ N (cid:88) i =1 (cid:18) max (cid:26) P model ( θ ) − P askmarket P askmarket , (cid:27) + max (cid:26) P bidmarket − P model ( θ ) P bidmarket , (cid:27)(cid:19) . (3.6)Gradient-based techniques typically are inadequate to handle high–dimensional prob-lems of this type. Instead, we utilise two methods for global optimisation, DifferentialEvolution (DE) (see Storn and Price (1995, 1997)) and Adaptive Simulated Annealing(ASA) (see Ingber (1996, 1993)). Recall that the model is made up of the following quantities: r c which is the short rateabstraction of the interbank overnight rate, φ the pure funding liquidity risk, and λ whichrepresents the credit or renewal risk portion of roll-over risk.In order to keep the dimension of the optimisation problem manageable, we split thecalibration into 3 stages: 16 Forward rate curve
BidAsk
Figure 1: OIS–implied forward rates on 1 January 2013 (based on data from Bloomberg)• Stage 1: Fit the model to OIS data. The OIS model parameters consist of the set θ = { a i , y i (0) , σ i , κ i , θ i , ∀ i = 1 , · · · , d, and a ( t ) , t ∈ [ T k − , T k ] , k = 1 , , · · · , } . where d is the number of factors (one or three in the cases considered here).• Stage 2: Using Stage 1 calibrated parameters, calibrate the model to basis swaps.From the results in Section B.1, note that basis swaps depend on c ( t ) and b ( t ) only via c ( t ) + qb ( t ) , i.e. the two cannot be separated based on basis swap dataalone — this will be done when calibration to credit default swaps is implementedin Section 4. Define d ( t ) = c ( t ) + qb ( t ) and in this stage assume constant d .Thus, the parameters which need to be determined in this step are θ = { b i , c i , q, ∀ i = 1 , · · · , d, and d } . • Stage 3: Now, keeping all other parameters fixed, we improve the fit to vanillaand basis swaps by allowing d ( t ) to be a piecewise constant function of time,i.e. d ( t ) = d ( k )0 for t ∈ [ T k − , T k ) . Since the shortest tenor is one month, the [ T k − , T k ) are chosen to be one–month time intervals. The calibration problemis then underdetermined, so we aim for the d ( k )0 to vary as little as possible byimposing an additional penalty in the calibration objective function based on thesum of squared deviations between consecutive values, i.e. (cid:88) k ( d ( k +1)0 − d ( k )0 ) . This type of smoothness criterion on time–varying parameters is well establishedin the literature on the calibration of interest rate term structure models, with itsuse in this context dating back to the seminal paper by Pedersen (1998).17
Maturity -3-2-1012 10 -3 model - market bid model - market ask (a) 1 January 2013 Maturity -3-2.5-2-1.5-1-0.500.511.5 10 -3 model - market bid model - market ask (b) 08 September 2014 Maturity -3-2-10123 10 -3 model - market bid model - market ask (c) 18 June 2015 Maturity -2.5-2-1.5-1-0.500.5 10 -3 model - market bid model - market ask (d) 20 April 2016 Maturity -1-0.500.511.522.533.54 10 -3 model - market bid model - market ask (e) 22 March 2017 Maturity -3-2-1012 10 -3 model - market bid model - market ask (f) 31 October 2017 Figure 2: One–factor model fit to OIS discount factors (based on data from Bloomberg)
We use overnight index swap (OIS) data to extract the OIS discount factors D OIS ( t, T ) as per equations (2.11) and (2.19). Note that this is model–independent.18loomberg provides OIS data out to a maturity of 30 years. However, there seemsto be a systematic difference in the treatment of OIS beyond 10 years compared tomaturities of ten years or less. This becomes particularly evident when one looks at theforward rates implied by the OIS discount factors: As can be seen in Figure 1, there is amarked spike in the forward rate at the 10–year mark. Therefore, for the time being wewill focus on maturities out to ten years only in our model calibration.The OIS discount factors depend only on the dynamics of r c , thus calibrating to OISdiscount factors only involves the parameters in equation (2.26). As is standard practicein CIR–type term structure models, we can ensure that the model fits the observed(initial) term structure by the time dependence in the (deterministic) function a ( t ) . Inthe one–factor case ( d = 1 ), choosing a = 1 , we first fit the initial y (0) and constantparameters a , κ , θ and σ in such a way as to match the observed OIS discount factorson a given day as closely as possible, and then choose time–dependent (piecewiseconstant) a ( t ) to achieve a perfect fit, as illustrated in Figure 2. In this figure, thedifference between the discount factor based on OIS bid and the model discount factoris always negative, and the difference between the discount factor based on OIS ask andthe model discount factor is always positive. This means that we have fitted the modelto the market in the sense that the model price always lies between the market bid andask.The fitted parameters are shown in Tables 8 to 14. The calibration condition for each vanilla swap is given by equation (2.23), and for eachbasis swap we obtain a condition as in (2.24) or (2.25). Specifically, in USD we havevanilla swaps exchanging a floating leg indexed to three–month LIBOR paid quarterlyagainst a fixed leg paid semi-annually. We combine this with basis swaps for threemonths vs. six months and one months vs. three months, giving us three calibrationconditions for each maturity (again, we restrict ourselves to maturities out to ten yearsfor which data is available from Bloomberg, i.e. maturities of six months, 1, 2, 3, 4, 5,6, 8, 9 and 10 years).Figure 4 shows the fits of a one–factor model to vanilla and basis swap data obtainedfor 1 January 2013, 8 September 2014, 18 June 2015, 20 April 2016, 22 March 2017 and31 October 2017, respectively. Table 15 gives the corresponding model parameters. As the fit of the one–factor model to the market is no longer perfect (though close)once vanilla and basis swap data are taken into account, we expand the number of factorsto three ( d = 3 ), modifying the staged procedure to further facilitate the non-linearoptimisation involved in the calibration: We first fit a one–factor model to OIS data as See e.g. Brigo and Mercurio (2006), Section 4.3.4. If one were to include market instruments with option features in the calibration, the volatility pa-rameters would be calibrated primarily to those instruments, rather than the shape of the term structure ofOIS discount factors. Given that there is insufficient information contained on vanilla and basis swaps to calibrate the loss–in–default fraction q ( q is essentially just a scaling parameter), we fix q = 0 . , i.e. a default recovery rateof 40%. y (0) κ θ σ a y (0) κ θ σ a y (0) κ θ σ a y (0) κ θ σ a y (0) κ θ σ a y (0) κ θ σ a a (1)0 a (3)0 (a) 01 January 2013 T a (1)0 a (3)0 (b) 08 September 2014 T a (1)0 a (3)0 (c) 18 June 2015 T a (1)0 a (3)0 (d) 20 April 2016 T a (1)0 a (3)0 (e) 22 march 2017 T a (1)0 a (3)0 (f) 31 October 2017 Table 14: Time-dependent parameter calibrations21
Maturity -4-3-2-101234 10 -3 model - market bid model - market ask (a) 1 January 2013 Maturity -1-0.500.511.522.533.54 10 -3 model - market bid model - market ask (b) 08 September 2014 Maturity -1-0.500.511.522.533.54 10 -3 model - market bid model - market ask (c) 18 June 2015 Maturity -2-1.5-1-0.500.51 10 -3 model - market bid model - market ask (d) 20 April 2016 Maturity -2-1.5-1-0.500.511.52 10 -3 model - market bid model - market ask (e) 22 March 2017 Maturity -2-1.5-1-0.500.511.522.53 10 -3 model - market bid model - market ask (f) 31 October 2017 Figure 3: Three–factor model fit to OIS discount factors (based on data fromBloomberg) 22
Maturity -2-1.5-1-0.500.511.522.5 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (a) 1 January 2013
Maturity -1-0.8-0.6-0.4-0.200.20.40.60.81 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (b) 08 September 2014
Maturity -3-2.5-2-1.5-1-0.500.511.52 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (c) 18 June 2015
Maturity -1.5-1-0.500.511.52 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (d) 20 April 2016
Maturity -1.5-1-0.500.51 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (e) 22 March 2017
Maturity -1-0.500.51 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (f) 31 October 2017
Figure 4: One–factor model fit to Basis Swaps (based on data from Bloomberg)23 (0) a b c q σ κ θ (a) 01 January 2013 y (0) a b c q σ κ θ (b) 08 September 2014 y (0) a b c q σ κ θ (c) 18 June 2015 y (0) a b c q σ κ θ (d) 20 April 2016 y (0) a b c q σ κ θ (e) 22 March 2017 y (0) a b c q σ κ θ (f) 31 October 2017 Table 15: Basis model parametersdescribed in Section 3.4, and then keep those parameters fixed in the three-factor model,setting a = a = 0 to maintain the OIS calibration, and then fit the initial y (0) , y (0) and constant parameters d , b , b , b , c , c , c , κ , κ , θ , θ and σ , σ in such a wayas to match the swap calibration conditions on a given day as closely as possible.Lastly, in order to improve the fit further, we allow for time–dependent (piecewiseconstant) d . Figure 5 shows the resulting fits for 1 January 2013, 8 September 2014, 18June 2015, 20 April 2016, 22 March 2017 and 31 October 2017, respectively, and Table15 gives the corresponding model parameters.24 Maturity -2-1.5-1-0.500.511.522.5 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (a) 1 January 2013
Maturity -6-4-202468 10 -4
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (b) 08 September 2014
Maturity -3-2-1012 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (c) 18 June 2015
Maturity -4
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (d) 20 April 2016
Maturity -8-6-4-202468 10 -4
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (e) 22 March 2017
Maturity -1-0.500.511.5 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (f) 31 October 2017
Figure 5: Tenors fit to market data (based on data from Bloomberg)From these results, we note that on a given day, the model can be simultaneouslycalibrated to all available OIS and vanilla/basis swap data, resulting in a fit between thebid and ask prices for all maturities, except for small discrepancies, mainly for maturitiesup to a year. 25
Separating Funding Liquidity Risk and Credit RiskUsing CDS
This section aims to disentangle the components φ ( s ) and λ ( s ) by including credit de-fault swaps (CDS) in the calibration. The coupon leg of the CDS consists of a streamsof payments while there is no default and the protection leg of the CDS only pays whendefault happens before maturity of the CDS. Since the reference rates for the vanillaand basis swaps are LIBORs, in our CDS calibration we focus on banks on the LIBORpanel. Because the payoff of a CDS is contingent on default of its reference entity, de-fault must now be modelled explicitly, i.e. we also have to disentangle the default–freeinterest rate r and the default intensity, rather than directly modelling the LHS of 2.1, r c . To do this, one needs information on the systemic default risk component Λ of r c as specified in (2.1). However, identifying this from market information is not reliablypossible, as Filipovi´c and Trolle (2013) also have found. They fix Λ at 5 basis points(bp), noting that “reasonable variations” in the value of Λ do not change their results, apoint which also applies in our context, and thus we follow this modelling choice. Let τ j be the default time of the bank j in the panel. Define a full filtration F jt = F τ j t ∨ H t , where F τ j t = σ ( { τ j > u } : u ≤ t ) is a subfiltration generated by τ j and H t is a subfiltration that contains default–freeinformation, also known as “market filtration”. Then the default intensity ˆ λ j ( s ) =Λ( s )+ λ j ( s ) of bank j is H t -adapted, and λ j gives the departure of ˆ λ j of bank j from thepanel’s systemic Λ . We then take the λ required by the roll–over risk pricing conditionsderived in Section 2 to be the average over the λ j in the panel and consequently thecoefficients of the dynamics of λ are the averages (due to linearity in coefficients) of thecorresponding coefficients of the λ j .Analogously to (2.26)–(2.28), define r c as r c ( t ) = α ( t ) + (cid:104) α, X ( t ) (cid:105) (4.1)where as before the coefficients α ( t ) , α are calibrated to OIS discount factors via con-dition 2.9. Following Filipovi´c and Trolle (2013), Λ( t ) is fixed identically equal to 5basis points, and we obtain the default–free rate r via r ( t ) = r c ( t ) − Λ q (4.2) Typical candidates as proxies for a risk–free rate, i.e. government bonds or repo rates, often result interm structures which intersect the term structure implied by OIS, indicating that the spread between OISand these rates is not a reliable proxy for Λ . We note that LIBOR is determined by taking the trimmed average of the corresponding simple–compounded rates quoted by the panel banks. Thus, for the sake of tractability, with this assumption weare making two approximations: Firstly, taking the relationship between λ j and the simple–compoundedrate quoted by the j -th bank to be approximately linear, and secondly assuming that the trimmed outlierslie approximately symmetrically above and below the panel average (in which case the trimmed averageis approximately equal to the untrimmed average). j –th bank follows ˆ λ j ( t ) = Λ + λ j ( t ) = ˆ b j ( t ) + (cid:104) ˆ b j , X ( t ) (cid:105) , (4.3)and the aim is to calibrate the coefficients of these dynamics to the corresponding creditdefault swaps. As noted above, λ ( t ) is calculated as the average of λ j ( t ) . We can thenwrite r c ( t ) + λ ( t ) q = d ( t ) + (cid:104) d, X ( t ) (cid:105) , Fixing q = 0 . as before and noting d ( t ) = a ( t ) + q ˆ b ( t ) − q Λ (4.4) d i = a i + qb i , (4.5)we are left with the freedom to determine the coefficients c ( t ) , c of the dynamics (2.28)of the “funding liquidity” component φ of roll–over risk. We obtain c ( t ) , c by calibrat-ing to vanilla and basis swaps via conditions (2.23) to (2.25). The time t -value of a zero–recovery zero coupon bond with notional issued by the j –th bank within the LIBOR panel is given by B j ( t, T ) = E Q (cid:104) e − (cid:82) Tt r ( s ) ds I τ j >T | F t (cid:105) = E Q (cid:104) e − (cid:82) Tt r ( s )+ˆ λ j ( s ) ds | H t (cid:105) = E Q (cid:104) e − (cid:82) Tt r c ( s ) − ( q Λ − ˆ λ j ( s )) ds | H t (cid:105) = D OIS ( t, T ) E Q T (cid:104) e − (cid:82) Tt (ˆ λ j ( s ) − q Λ) ds | H t (cid:105) . (4.6)In Appendix C we develop the previous expression in terms of our affine specification.Let us now consider the pricing of a Credit Default Swap , i.e. a contractual agree-ment between the protection buyer and the protection seller, typically designed to pro-vide protection from a credit event associated with a risky bond issued by the referenceentity . CDS represent the most liquid credit derivative contracts by far.• The protection buyer pays rate C (the CDS spread) at times T , · · · , T N = T .• The protection seller agrees to make a single protection payment called LossGiven Default (LGD) (i.e. L = 1 − R , where R is assumed to be a fixed cashrecovery) in case the default event happens between T and T .Let ( T k ) k =1 , ··· ,N be the CDS spread payment dates and τ the default time. The CDSdiscounted payoff from the perspective of a protection buyer, with unit notional andprotection payment − R , is at time t ≤ T We drop the subscript corresponding to the reference entity bank j in order simplify the notation,when there is no ambiguity. ( t, T ) = E Q (cid:104) e − (cid:82) τt r c ( s ) ds (1 − R ) I { τ ≤ T } | F t (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) (1) (4.7) − E Q (cid:34) N (cid:88) k =1 e − (cid:82) Tkt r c ( s ) ds ( T k − T k − ) C I { τ>T k } | F t (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) (2) (4.8) − E Q (cid:34) N (cid:88) k =1 e − (cid:82) τt r c ( s ) ds ( τ − T k − ) C I { τ ∈ [ T k − ,T k ] } | F t (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) (3) , where:(1) Protection leg payment, price of any proceeds from default before T ;(2) No-default spread payments;(3) Payments of accrued spread interest at default.We proceed to evaluate each of these components. E Q (cid:104) e − (cid:82) Tt r c ( s ) ds I { τ>T } | F t (cid:105) = D OIS ( t, T ) E Q T (cid:2) I { τ>T } | F t (cid:3) (4.9) = D OIS ( t, T ) E Q T (cid:104) e − (cid:82) Tt ˆ λ j ( s ) ds | F t (cid:105) := D OIS ( t, T ) Z ( t, T ) . Here the dynamics of ˆ λ j ( s ) have to be adjusted in accordance with T -forward measure.It is easy to see that dZ ( t, u ) du = − E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds ˆ λ j ( u ) | F t (cid:105) , (4.10)where the change of order of differentiation and expectation is justified by the dominatedconvergence theorem. 28onsider a mesh { t p } on the interval [ T k − , T k ] , E Q (cid:34) N (cid:88) k =1 e − (cid:82) τt r c ( s ) ds C ( τ − T k − ) I { τ ∈ [ T k − ,T k ] } | F t (cid:35) (4.11) = E Q (cid:34) N (cid:88) k =1 e − (cid:82) τt r c ( s ) ds C ( τ − T k − ) I { τ ∈ [ T k − ,T k ] } (cid:88) p I { τ ∈ [ t p ,t p +1 ] } | F t (cid:35) = N (cid:88) k =1 C (cid:88) p D OIS ( t, t p ) E Q T (cid:2) ( t p − T k − ) I { τ ∈ [ t p ,t p +1 ] } | F t (cid:3) = N (cid:88) k =1 C (cid:88) p D OIS ( t, t p )( t p − T k − ) E Q T (cid:2) I { τ>t p } − I { τ>t p +1 } | F t (cid:3) = N (cid:88) k =1 C (cid:88) p D OIS ( t, t p )( t p − T k − ) [ Z ( t, t p ) − Z ( t, t p +1 )] ≈ C N (cid:88) k =1 (cid:90) T k T k − D OIS ( t, u )( u − T k − ) dZ ( t, u )= −C N (cid:88) k =1 (cid:90) T k T k − D OIS ( t, u )( u − T k − ) E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds ˆ λ j ( u ) | F t (cid:105) du. Finally, the protection leg can be decomposed as follows: E Q (cid:104) e − (cid:82) τt r c ( s ) ds (1 − R ) I { τ
BACORP Bank of America CorporationMUFJ-BTMUFJ The Bank of Tokyo–Mitsubishi UFJ, Ltd.BACR-Bank BARCLAYS BANK PLCC Citigroup Inc.ACAFP CREDIT AGRICOLE SACSGAG Credit Suisse Group AGDB DEUTSCHE BANK AKTIENGESELLSCHAFTHSBC HSBC HOLDINGS plcJPM JPMorgan Chase & Co.LBGP LLOYDS BANKING GROUP PLCCOOERAB Cooeperatieve Rabobank U.A.RY Royal Bank of CanadaSOCGEN SOCIETE GENERALESUMIBK-Bank Sumitomo Mitsui Banking CorporationNORBK The Norinchukin BankRBOS-RBOSplc The Royal Bank of Scotland public limited companyUBS UBS AGTable 16: LIBOR panel banksthe panel can now be written asCDS ( t, T ) = − (1 − R ) (cid:90) Tt D OIS ( t, u ) E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds ˆ λ j ( u ) | F t (cid:105) du (4.13) − C N (cid:88) k =1 ( T k − T k − ) D OIS ( t, T k ) Z ( t, T k ) − C N (cid:88) k =1 (cid:90) T k T k − D OIS ( t, u )( u − T k − ) E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds ˆ λ j ( u ) | F t (cid:105) du. The market–quoted CDS spread s t at time t is the value of C such that CDS ( t, T ) = 0 .Thus the CDS calibration problem is to determine the coefficients of the dynamics of ˆ λ j such that the RHS of (4.13) equals zero when the market–quoted CDS spread forthe j –th bank is substituted for C , for all maturities T for which market–quoted CDSspreads for the j –th bank are available. The calibration procedure outlined above thus proceeds in three steps:1. Calibrate the coefficients α ( t ) , α of the dynamics of r c to the term structure ofdiscount factors implied by OIS. This is identical to the first step of the calibrationdescribed in Section 3.5, so we do not present the results of this step here.30. Calibrate the coefficients ˆ b j ( t ) , ˆ b j of the dynamics of the idiosyncratic defaultintensity ˆ λ j to the term structure of CDS spreads of bank j . Taking the averageover all j of each coefficient in ˆ b j ( t ) , ˆ b j and subtracting Λ = 5 bp from ˆ b ( t ) givesthe coefficients b ( t ) , b of the dynamics of λ .3. Calibrate the coefficients c ( t ) , c of the dynamics of the “funding liquidity” com-ponent φ of roll–over risk simultaneously to all available vanilla and basis swaps.For the reasons stated in Section 3.4, we restrict ourselves to maturities up to ten years.We report the calibration results for six exemplary dates, 1 January 2013, 8 September2014, 18 June 2015, 20 April 2016, 22 March 2017 and 31 October 2017. Results forother days in our data set are qualitatively similar. For IRS, OIS and basis swaps here weuse the same instruments as in Section 3. The USD LIBOR panel is made up of the 17banks listed in Table 16. Where CDS quotes (sourced from Markit) were not availableon a given day for a given panel bank (e.g. for Lloyds Banking Group, Rabobank andRoyal Bank of Canada), we have dropped these from our calibration. For the remainingpanel banks, we used all available maturities in the CDS calibration, i.e. 0.5, 1, 2, 3, 4,5, 7 and 10 years.The calibrated coefficients b for each day, in the one–factor and in the three–factormodel, are given in Tables 17–22. Both the one–factor and three–factor model fit theMarkit quotes to well within one basis point, as Tables 23 to 28 show.Moving to Step 3 in the calibration procedure, as in Section 3.5 we plot the bid andask values for each tenor relative to the value produced by the calibrated model. Theresults in this step differ from those reported in Section 3.5, because now part of theroll–over risk (the part which a borrower faces due to the possibility of having to pay ahigher credit spread than the LIBOR panel average in the future) is already determinedby the credit spread dynamics calibrated in Step 2. Nevertheless, the one–factor modelstill fits the market reasonably well, as Figure 6 shows — and this could be improvedfurther by a staged fit of a three–factor model in the same manner as in Section 3.Thus we see that the model, even in its one–factor version, can be calibrated si-multaneously to market data on a given day for OIS, CDS, interest rate and basis swaps.This is achieved in the usual fashion of fitting interest rate term structure models to themarket, using “term structures” of time–dependent coefficients α ( t ) , b ( t ) and c ( t ) . Inthis sense, our approach extends the interest rate term structure modelling frameworkto roll–over risk in a relatively straightforward and consistent fashion. However, whencalibrating simultaneously to all terms (1–month, 3–month and 6–month), the calibrated c ( t ) are not very smooth, see Figure 7, which may be due to market frictions not cap-tured by the model, or an indication that the market across tenors has yet to maturefully. This approach was pioneered in interest–rate term structure modelling by Hull and White (1990),and developed in the fashion used here by Brigo and Mercurio (1998) (it is called “the CIR++ Model” inBrigo and Mercurio (2006)). In a simpler model of roll–over risk, Chang and Schl¨ogl (2015) show how the ability of that modelto fit the market for basis swaps has improved in the years since the tenor basis became an economicallysignificant phenomenon as a result of the financial crisis of 2007/8. a) One-factor CIR Average ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) (b) Three-factor CIR Average ˆ b ˆ b ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) Table 17: Model calibration to CDS data for USD LIBOR panel banks on 01/01/2013 (c) One-factor CIR
Average ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) (d) Three-factor CIR Average ˆ b ˆ b ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) Table 18: Model calibration to CDS data for USD LIBOR panel banks on 08/09/2014 (a) One-factor CIR
Average ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) (b) Three-factor CIR Average ˆ b ˆ b ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) Table 19: Model calibration to CDS data for USD LIBOR panel banks on 18/06/201532 a) One-factor CIR
Average ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) (b) Three-factor CIR Average ˆ b ˆ b ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) Table 20: Model calibration to CDS data for USD LIBOR panel banks on 20/04/2016 (c) One-factor CIR
Average ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) (d) Three-factor CIR Average ˆ b ˆ b ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) Table 21: Model calibration to CDS data for USD LIBOR panel banks on 22/03/2017 (a) One-factor CIR
Average ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) (b) Three-factor CIR Average ˆ b ˆ b ˆ b ˆ b (1) ˆ b (2) ˆ b (3) ˆ b (4) ˆ b (5) ˆ b (6) ˆ b (7) ˆ b (8) Table 22: Model calibration to CDS data for USD LIBOR panel banks on 31/10/201733
IBOR Panel Maturity 0.5 1 2 3 4 5 7 10
BACORP
Market 0.04948% 0.06922% 0.13891% 0.22322% 0.27845% 0.32756% 0.37376% 0.41302%1-factor 0.04945% 0.06920% 0.13898% 0.22321% 0.27829% 0.32760% 0.37385% 0.41284%3-factor 0.04948% 0.06922% 0.13888% 0.22327% 0.27840% 0.32730% 0.37385% 0.41312%
MUFJ-BTMUFJ
Market 0.06353% 0.07170% 0.10938% 0.14172% 0.18472% 0.22588% 0.27046% 0.30236%1-factor 0.06353% 0.07170% 0.10938% 0.14172% 0.18472% 0.22588% 0.27046% 0.30236%3-factor 0.06365% 0.07156% 0.10944% 0.14181% 0.18485% 0.22450% 0.27106% 0.30287%
BACR-Bank
Market 0.04903% 0.05996% 0.12029% 0.18846% 0.25487% 0.32914% 0.38240% 0.41731%1-factor 0.04903% 0.05996% 0.12029% 0.18846% 0.25487% 0.32914% 0.38240% 0.41731%3-factor 0.04891% 0.05991% 0.12028% 0.18887% 0.25496% 0.32860% 0.38249% 0.41923% C Market 0.05000% 0.07509% 0.13787% 0.21259% 0.27657% 0.32116% 0.36793% 0.41046%1-factor 0.05000% 0.07509% 0.13787% 0.21259% 0.27657% 0.32116% 0.36793% 0.41046%3-factor 0.05013% 0.07542% 0.13760% 0.21265% 0.27718% 0.32107% 0.36872% 0.41217%
ACAFP
Market 0.03341% 0.05218% 0.13774% 0.22943% 0.31452% 0.39119% 0.44453% 0.48030%1-factor 0.03341% 0.05218% 0.13774% 0.22943% 0.31452% 0.39119% 0.44453% 0.48030%3-factor 0.03339% 0.05219% 0.13739% 0.22914% 0.31422% 0.39080% 0.44504% 0.48011%
CSGAG
Market 0.02276% 0.02462% 0.05648% 0.11484% 0.17199% 0.22845% 0.27654% 0.31271%1-factor 0.02276% 0.02462% 0.05648% 0.11484% 0.17199% 0.22845% 0.27654% 0.31271%3-factor 0.02276% 0.02461% 0.05647% 0.11484% 0.17195% 0.22844% 0.27656% 0.31272% DB Market 0.02084% 0.02521% 0.06536% 0.12177% 0.18313% 0.24353% 0.29449% 0.32557%1-factor 0.02084% 0.02521% 0.06536% 0.12177% 0.18313% 0.24353% 0.29449% 0.32557%3-factor 0.02084% 0.02521% 0.06487% 0.12163% 0.18316% 0.24370% 0.29418% 0.32584%
HSBC
Market 0.05374% 0.06313% 0.09683% 0.12872% 0.16578% 0.20440% 0.23195% 0.25383%1-factor 0.05374% 0.06313% 0.09683% 0.12872% 0.16578% 0.20440% 0.23195% 0.25383%3-factor 0.05366% 0.06291% 0.09676% 0.12889% 0.16564% 0.20424% 0.23311% 0.25359%
JPM
Market 0.04090% 0.04977% 0.09009% 0.14186% 0.18265% 0.22204% 0.25815% 0.29288%1-factor 0.04090% 0.04977% 0.09009% 0.14186% 0.18265% 0.22204% 0.25815% 0.29288%3-factor 0.04092% 0.04950% 0.08994% 0.14124% 0.18152% 0.22219% 0.25875% 0.29332%
SOCGEN
Market 0.05892% 0.08162% 0.17974% 0.27959% 0.35927% 0.44248% 0.49035% 0.52445%1-factor 0.05892% 0.08162% 0.17974% 0.27959% 0.35927% 0.44248% 0.49035% 0.52445%3-factor 0.05876% 0.08163% 0.17928% 0.27965% 0.35956% 0.44160% 0.49045% 0.52414%
SUMIBK-Bank
Market 0.07395% 0.09135% 0.12479% 0.16631% 0.20062% 0.23445% 0.28110% 0.31914%1-factor 0.07395% 0.09135% 0.12479% 0.16631% 0.20062% 0.23445% 0.28110% 0.31914%3-factor 0.07400% 0.09127% 0.12486% 0.16639% 0.20065% 0.23440% 0.28129% 0.31900%
NORBK
Market 0.04795% 0.06673% 0.09658% 0.12932% 0.16512% 0.19870% 0.23620% 0.26255%1-factor 0.04795% 0.06673% 0.09658% 0.12932% 0.16512% 0.19870% 0.23620% 0.26255%3-factor 0.04796% 0.06708% 0.09608% 0.12917% 0.16521% 0.19866% 0.23557% 0.26241%
RBOS-RBOSplc
Market 0.02482% 0.04344% 0.12412% 0.22640% 0.31477% 0.39681% 0.45233% 0.49291%1-factor 0.02482% 0.04344% 0.12412% 0.22640% 0.31492% 0.39673% 0.45211% 0.49292%3-factor 0.02486% 0.04350% 0.12385% 0.22701% 0.31479% 0.39628% 0.45189% 0.49229%
UBS
Market 0.02381% 0.02630% 0.06334% 0.11498% 0.17802% 0.23502% 0.28481% 0.31691%1-factor 0.02380% 0.02635% 0.06331% 0.11494% 0.17801% 0.23498% 0.28452% 0.31711%3-factor 0.02371% 0.02638% 0.06321% 0.11479% 0.17905% 0.23549% 0.28596% 0.31753%
Table 23: Model calibration to USD LIBOR panel banks on 01/01/2013
Since seeking to profit from the frequency basis (be it the LIBOR/OIS spread or basisswaps involving different LIBOR tenors) entails lending at a longer tenor against rollingover borrowing at a shorter tenor, modelling the risk of such a strategy, i.e. “roll–overrisk,” is a natural approach to obtain a consistent model of all possible tenors in a rela-tively parsimonious fashion. Such a model — as we have shown — can be calibratedto a large number of relevant market instruments simultaneously. Here, we have delib-erately included two distinct calibration exercises: In the first, the model is calibratedto OIS and all available USD interest rate swaps (vanilla IRS and basis swaps) — thisdemonstrates that the model can represent the interest rate markets for different tenorfrequencies as effectively as the more common, ad hoc models in the literature, which34
IBOR Panel Maturity 0.5 1 2 3 4 5 7 10
BACORP
Market 0.000739 0.000895 0.001260 0.001607 0.002029 0.002402 0.002998 0.0035921-factor 0.000739 0.000895 0.001259 0.001607 0.002028 0.002402 0.002997 0.0035923-factor 0.000739 0.000895 0.001266 0.001603 0.002026 0.002402 0.002992 0.003588
MUFJ-BTMUFJ
Market 0.000360 0.000464 0.000680 0.000945 0.001286 0.001630 0.002037 0.0022691-factor 0.000360 0.000464 0.000681 0.000944 0.001286 0.001631 0.002036 0.0022703-factor 0.000361 0.000465 0.000681 0.000943 0.001290 0.001631 0.002027 0.002270
BACR-Bank
Market 0.000443 0.000576 0.000897 0.001264 0.001663 0.002062 0.002642 0.0029991-factor 0.000443 0.000576 0.000897 0.001265 0.001663 0.002062 0.002642 0.0029993-factor 0.000443 0.000575 0.000898 0.001265 0.001667 0.002061 0.002634 0.003002 C Market 0.000666 0.000905 0.001279 0.001646 0.002090 0.002436 0.003051 0.0035261-factor 0.000666 0.000905 0.001279 0.001646 0.002089 0.002437 0.003050 0.0035263-factor 0.000666 0.000905 0.001278 0.001649 0.002091 0.002432 0.003051 0.003527
ACAFP
Market 0.000317 0.000402 0.000639 0.000874 0.001138 0.001410 0.001896 0.0021951-factor 0.000317 0.000402 0.000638 0.000874 0.001139 0.001410 0.001896 0.0021943-factor 0.000317 0.000400 0.000639 0.000873 0.001127 0.001413 0.001902 0.002192
CSGAG
Market 0.000302 0.000399 0.000608 0.000831 0.001046 0.001258 0.001631 0.0020101-factor 0.000302 0.000399 0.000608 0.000831 0.001045 0.001259 0.001632 0.0020113-factor 0.000301 0.000397 0.000611 0.000832 0.001045 0.001255 0.001632 0.002012 DB Market 0.000374 0.000579 0.000944 0.001287 0.001714 0.002082 0.002667 0.0030481-factor 0.000374 0.000578 0.000944 0.001288 0.001714 0.002083 0.002668 0.0030483-factor 0.000374 0.000577 0.000942 0.001287 0.001715 0.002082 0.002670 0.003047
HSBC
Market 0.000498 0.000599 0.000892 0.001190 0.001534 0.001847 0.002381 0.0027471-factor 0.000498 0.000599 0.000893 0.001190 0.001533 0.001849 0.002381 0.0027463-factor 0.000499 0.000599 0.000893 0.001197 0.001528 0.001847 0.002384 0.002757
JPM
Market 0.000480 0.000806 0.001095 0.001336 0.001655 0.001988 0.002537 0.0030601-factor 0.000480 0.000806 0.001095 0.001336 0.001656 0.001988 0.002538 0.0030593-factor 0.000480 0.000807 0.001095 0.001337 0.001657 0.001995 0.002542 0.003058
SOCGEN
Market 0.000287 0.000458 0.000894 0.001327 0.001798 0.002215 0.002864 0.0032961-factor 0.000288 0.000458 0.000892 0.001327 0.001800 0.002214 0.002864 0.0032973-factor 0.000288 0.000457 0.000891 0.001322 0.001795 0.002215 0.002870 0.003301
SUMIBK-Bank
Market 0.000381 0.000472 0.000694 0.000943 0.001279 0.001624 0.001992 0.0022721-factor 0.000382 0.000472 0.000694 0.000943 0.001279 0.001624 0.001992 0.0022713-factor 0.000383 0.000468 0.000696 0.000945 0.001279 0.001623 0.001997 0.002271
NORBK
Market 0.000348 0.000442 0.000635 0.000855 0.001160 0.001453 0.001787 0.0020281-factor 0.000387 0.000488 0.000729 0.000971 0.001304 0.001576 0.001889 0.0021493-factor 0.000349 0.000442 0.000638 0.000854 0.001160 0.001450 0.001791 0.002030
RBOS-RBOSplc
Market 0.000819 0.001055 0.001550 0.002062 0.002557 0.003061 0.003851 0.0042451-factor 0.000819 0.001056 0.001550 0.002062 0.002557 0.003058 0.003851 0.0042443-factor 0.000819 0.001056 0.001548 0.002065 0.002559 0.003061 0.003847 0.004246
UBS
Market 0.000248 0.000329 0.000590 0.000847 0.001148 0.001425 0.001959 0.0023411-factor 0.000248 0.000329 0.000590 0.000847 0.001148 0.001425 0.001959 0.0023413-factor 0.000248 0.000329 0.000591 0.000840 0.001140 0.001425 0.001970 0.002336
Table 24: Model calibration to USD LIBOR panel banks on 08/09/2014typically model spreads between term structures for different tenors directly, withoutrecourse to any underlying theoretical justification for the existence of such spreads. Inthe second calibration exercise, the component of roll–over risk associated with the riskof a credit downgrade of the entity attempting to roll over their borrowing is explicitlyseparated from the component interpreted as funding liquidity risk, by including creditdefault swap data in the calibration. Again, it is demonstrated that the model can fitthe market represented by this expanded set of calibration instruments (OIS, IRS, basisswaps and CDS).Thus, in the sense of the bulk of the derivatives pricing literature, the present paper35
IBOR Panel Maturity 0.5 1 2 3 4 5 7 10
BACORP
Market 0.04813% 0.06260% 0.08497% 0.11369% 0.14062% 0.17582% 0.23483% 0.27894%1-factor 0.04814% 0.06257% 0.08499% 0.11366% 0.14060% 0.17582% 0.23474% 0.27887%3-factor 0.04823% 0.06287% 0.08451% 0.11443% 0.14098% 0.17649% 0.23487% 0.27936%
MUFJ-BTMUFJ
Market 0.03513% 0.04003% 0.05647% 0.07192% 0.09724% 0.12071% 0.15429% 0.17695%1-factor 0.03513% 0.04003% 0.05646% 0.07193% 0.09724% 0.12070% 0.15429% 0.17695%3-factor 0.03507% 0.04006% 0.05633% 0.07184% 0.09746% 0.12045% 0.15454% 0.17679%
BACR-Bank
Market 0.10031% 0.11363% 0.13322% 0.15291% 0.17759% 0.20189% 0.24070% 0.26574%1-factor 0.10026% 0.11365% 0.13322% 0.15294% 0.17761% 0.20185% 0.24076% 0.26586%3-factor 0.10028% 0.11363% 0.13322% 0.15280% 0.17759% 0.20169% 0.24179% 0.26568% C Market 0.05332% 0.07192% 0.09909% 0.13358% 0.16440% 0.20218% 0.25854% 0.30506%1-factor 0.05331% 0.07196% 0.09908% 0.13356% 0.16446% 0.20212% 0.25846% 0.30504%3-factor 0.05328% 0.07186% 0.09916% 0.13363% 0.16437% 0.20237% 0.25850% 0.30516%
ACAFP
Market 0.10370% 0.11174% 0.13671% 0.16195% 0.18846% 0.21685% 0.25641% 0.28629%1-factor 0.10371% 0.11174% 0.13670% 0.16195% 0.18847% 0.21685% 0.25642% 0.28624%3-factor 0.10373% 0.11179% 0.13666% 0.16178% 0.18855% 0.21689% 0.25657% 0.28638%
CSGAG
Market 0.11119% 0.12543% 0.14612% 0.16801% 0.18889% 0.21226% 0.24583% 0.27880%1-factor 0.11120% 0.12545% 0.14612% 0.16796% 0.18889% 0.21226% 0.24586% 0.27877%3-factor 0.11124% 0.12547% 0.14611% 0.16797% 0.18892% 0.21218% 0.24597% 0.27880% DB Market 0.12448% 0.14204% 0.16512% 0.18943% 0.21609% 0.24378% 0.28444% 0.31339%1-factor 0.12448% 0.14208% 0.16510% 0.18945% 0.21609% 0.24379% 0.28444% 0.31338%3-factor 0.12443% 0.14182% 0.16506% 0.18932% 0.21594% 0.24392% 0.28462% 0.31361%
HSBC
Market 0.08689% 0.09758% 0.11594% 0.13616% 0.16099% 0.18521% 0.22451% 0.26311%1-factor 0.08688% 0.09758% 0.11594% 0.13618% 0.16100% 0.18518% 0.22441% 0.26311%3-factor 0.08688% 0.09751% 0.11638% 0.13624% 0.16107% 0.18524% 0.22416% 0.26374%
JPM
Market 0.04570% 0.06190% 0.08312% 0.10283% 0.13030% 0.16602% 0.21851% 0.26588%1-factor 0.04570% 0.06190% 0.08312% 0.10283% 0.13029% 0.16603% 0.21852% 0.26588%3-factor 0.04573% 0.06202% 0.08344% 0.10310% 0.13020% 0.16599% 0.21850% 0.26598% Ry Market 0.04640% 0.06612% 0.08624% 0.09515% 0.12247% 0.14505% 0.18275% 0.21404%1-factor 0.04641% 0.06606% 0.08620% 0.09515% 0.12242% 0.14505% 0.18275% 0.21406%3-factor 0.04634% 0.06599% 0.08656% 0.09521% 0.12270% 0.14518% 0.18237% 0.21442%
SOCGEN
Market 0.10419% 0.11758% 0.14766% 0.17816% 0.20979% 0.24109% 0.28484% 0.31386%1-factor 0.10420% 0.11763% 0.14759% 0.17813% 0.20983% 0.24110% 0.28476% 0.31378%3-factor 0.10419% 0.11756% 0.14766% 0.17820% 0.20968% 0.24114% 0.28464% 0.31383%
SUMIBK-Bank
Market 0.02772% 0.03362% 0.04944% 0.06761% 0.09643% 0.12386% 0.16048% 0.18282%1-factor 0.02773% 0.03363% 0.04945% 0.06759% 0.09643% 0.12387% 0.16050% 0.18277%3-factor 0.02773% 0.03362% 0.04943% 0.06759% 0.09653% 0.12393% 0.16049% 0.18278%
NORBK
Market 0.02549% 0.03455% 0.04727% 0.06671% 0.08839% 0.12041% 0.15002% 0.18693%1-factor 0.02550% 0.03455% 0.04725% 0.06670% 0.08822% 0.12042% 0.15018% 0.18690%3-factor 0.02556% 0.03452% 0.04710% 0.06679% 0.08820% 0.12041% 0.15025% 0.18717%
RBOS-RBOSplc
Market 0.10865% 0.12172% 0.14396% 0.16540% 0.19014% 0.21605% 0.25592% 0.28547%1-factor 0.10864% 0.12171% 0.14397% 0.16534% 0.19015% 0.21607% 0.25593% 0.28551%3-factor 0.10873% 0.12158% 0.14402% 0.16525% 0.19015% 0.21601% 0.25571% 0.28557%
UBS
Market 0.08423% 0.09542% 0.11551% 0.13517% 0.15375% 0.17305% 0.20454% 0.23801%1-factor 0.08429% 0.09543% 0.11552% 0.13519% 0.15376% 0.17316% 0.20446% 0.23784%3-factor 0.08432% 0.09548% 0.11519% 0.13567% 0.15359% 0.17299% 0.20448% 0.23780%
Table 25: Model calibration to USD LIBOR panel banks on 18/06/2015takes the practitioners’ approach of static calibration to (cross–sectional) market data ona given day, for a model to be used for pricing less liquid derivatives in an arbitrage–free36
IBOR Panel Maturity 0.5 1 2 3 4 5 7 10
BACORP
Market 0.06923% 0.09045% 0.11964% 0.14343% 0.17157% 0.21044% 0.26665% 0.30669%1-factor 0.06922% 0.09047% 0.11973% 0.14347% 0.17164% 0.21039% 0.26676% 0.30681%3-factor 0.06928% 0.09030% 0.11980% 0.14317% 0.17159% 0.21089% 0.26678% 0.30672%
MUFJ-BTMUFJ
Market 0.04612% 0.05479% 0.08152% 0.10809% 0.14762% 0.18391% 0.21351% 0.23720%1-factor 0.04613% 0.05479% 0.08153% 0.10809% 0.14762% 0.18388% 0.21350% 0.23720%3-factor 0.04613% 0.05479% 0.08136% 0.10812% 0.14766% 0.18394% 0.21363% 0.23741%
BACR-Bank
Market 0.14546% 0.16644% 0.19214% 0.21354% 0.24068% 0.26680% 0.31066% 0.33777%1-factor 0.14552% 0.16637% 0.19215% 0.21353% 0.24066% 0.26673% 0.31066% 0.33777%3-factor 0.14551% 0.16623% 0.19208% 0.21348% 0.24092% 0.26681% 0.31075% 0.33791% C Market 0.06438% 0.09030% 0.12128% 0.14588% 0.17473% 0.21120% 0.26621% 0.30696%1-factor 0.06444% 0.09036% 0.12129% 0.14592% 0.17464% 0.21114% 0.26639% 0.30695%3-factor 0.06438% 0.09023% 0.12118% 0.14597% 0.17485% 0.21130% 0.26644% 0.30672%
ACAFP
Market 0.05232% 0.06163% 0.08805% 0.11431% 0.14472% 0.17637% 0.22025% 0.24761%1-factor 0.05232% 0.06163% 0.08806% 0.11431% 0.14471% 0.17639% 0.22025% 0.24754%3-factor 0.05234% 0.06175% 0.08791% 0.11412% 0.14429% 0.17647% 0.22031% 0.24795%
CSGAG
Market 0.20990% 0.22439% 0.24518% 0.26489% 0.29034% 0.31339% 0.35276% 0.38414%1-factor 0.20990% 0.22433% 0.24521% 0.26492% 0.29037% 0.31341% 0.35278% 0.38414%3-factor 0.21043% 0.22343% 0.24633% 0.26485% 0.29072% 0.31435% 0.35247% 0.38384% DB Market 0.24014% 0.28283% 0.31705% 0.34429% 0.36361% 0.38440% 0.41072% 0.42937%1-factor 0.24022% 0.28297% 0.31675% 0.34430% 0.36353% 0.38447% 0.41066% 0.42953%3-factor 0.24011% 0.28295% 0.31703% 0.34462% 0.36344% 0.38429% 0.41068% 0.42953%
HSBC
Market 0.12261% 0.13272% 0.15764% 0.18317% 0.20849% 0.23459% 0.27675% 0.30796%1-factor 0.12256% 0.13270% 0.15755% 0.18311% 0.20855% 0.23470% 0.27691% 0.30798%3-factor 0.12273% 0.13260% 0.15773% 0.18311% 0.20854% 0.23438% 0.27714% 0.30798%
JPM
Market 0.05800% 0.07182% 0.09415% 0.10954% 0.12903% 0.16619% 0.21402% 0.26331%1-factor 0.05800% 0.07181% 0.09415% 0.10956% 0.12903% 0.16618% 0.21401% 0.26334%3-factor 0.05792% 0.07184% 0.09409% 0.10977% 0.12921% 0.16588% 0.21419% 0.26323% Ry Market 0.05722% 0.06608% 0.08607% 0.10522% 0.12680% 0.14766% 0.18556% 0.21962%1-factor 0.04641% 0.06606% 0.08620% 0.09515% 0.12242% 0.14505% 0.18275% 0.21406%3-factor 0.04634% 0.06599% 0.08656% 0.09521% 0.12270% 0.14518% 0.18237% 0.21442%
COOERAB
Market 0.10539% 0.13421% 0.16112% 0.18228% 0.22540% 0.25207% 0.28264% 0.30758%1-factor 0.10530% 0.13421% 0.16112% 0.18226% 0.22536% 0.25224% 0.28263% 0.30748%3-factor 0.10533% 0.13432% 0.16107% 0.18253% 0.22448% 0.25197% 0.28260% 0.30778%
SOCGEN
Market 0.05087% 0.06263% 0.09305% 0.12067% 0.15037% 0.18207% 0.22809% 0.25887%1-factor 0.05086% 0.06263% 0.09309% 0.12064% 0.15034% 0.18203% 0.22815% 0.25886%3-factor 0.05089% 0.06264% 0.09320% 0.12057% 0.15039% 0.18198% 0.22829% 0.25893%
SUMIBK-Bank
Market 0.05089% 0.05631% 0.08055% 0.11043% 0.15166% 0.19291% 0.22353% 0.25009%1-factor 0.05086% 0.05633% 0.08056% 0.11046% 0.15165% 0.19294% 0.22352% 0.25009%3-factor 0.05066% 0.05641% 0.08039% 0.11046% 0.15147% 0.19241% 0.22434% 0.25052%
NORBK
Market 0.03872% 0.04881% 0.07293% 0.09713% 0.13045% 0.15755% 0.18881% 0.21476%1-factor 0.03872% 0.04884% 0.07293% 0.09715% 0.13039% 0.15757% 0.18891% 0.21492%3-factor 0.03866% 0.04856% 0.07263% 0.09704% 0.13057% 0.15732% 0.19038% 0.21470%
RBOS-RBOSplc
Market 0.16161% 0.18295% 0.20569% 0.22655% 0.25696% 0.28531% 0.33445% 0.34308%1-factor 0.16160% 0.18295% 0.20570% 0.22655% 0.25696% 0.28534% 0.33445% 0.34307%3-factor 0.16157% 0.18303% 0.20578% 0.22654% 0.25690% 0.28526% 0.33447% 0.34300%
UBS
Market 0.06047% 0.07226% 0.09013% 0.10946% 0.13127% 0.15226% 0.18996% 0.22196%1-factor 0.06043% 0.07222% 0.09005% 0.10947% 0.13139% 0.15227% 0.19013% 0.22204%3-factor 0.06038% 0.07226% 0.09004% 0.10952% 0.13134% 0.15223% 0.18984% 0.22190%
Table 26: Model calibration to USD LIBOR panel banks on 20/04/201637
IBOR Panel Maturity 0.5 1 2 3 4 5 7 10
BACORP
Market 0.07252% 0.11828% 0.14502% 0.17841% 0.20794% 0.25230% 0.30848% 0.36520%1-factor 0.07252% 0.11827% 0.14505% 0.17840% 0.20790% 0.25232% 0.30847% 0.36533%3-factor 0.07253% 0.11807% 0.14483% 0.17842% 0.20798% 0.25241% 0.30817% 0.36527%
MUFJ-BTMUFJ
Market 0.05547% 0.06740% 0.08868% 0.10900% 0.14743% 0.18566% 0.21703% 0.23960%1-factor 0.05548% 0.06744% 0.08876% 0.10883% 0.14740% 0.18556% 0.21729% 0.23964%3-factor 0.05551% 0.06738% 0.08871% 0.10911% 0.14725% 0.18562% 0.21704% 0.23945%
BACR-Bank
Market 0.06269% 0.08022% 0.13055% 0.18031% 0.22667% 0.26994% 0.32310% 0.35594%1-factor 0.06268% 0.08021% 0.13062% 0.18028% 0.22676% 0.26979% 0.32327% 0.35606%3-factor 0.06272% 0.08031% 0.13047% 0.18052% 0.22670% 0.27021% 0.32296% 0.35596% C Market 0.07404% 0.10033% 0.13763% 0.17862% 0.22290% 0.26959% 0.33201% 0.39175%1-factor 0.07400% 0.10030% 0.13752% 0.17860% 0.22300% 0.26962% 0.33191% 0.39149%3-factor 0.07401% 0.10059% 0.13723% 0.17897% 0.22305% 0.26950% 0.33196% 0.39148%
ACAFP
Market 0.16709% 0.18774% 0.25196% 0.32503% 0.38096% 0.44389% 0.51201% 0.55655%1-factor 0.16710% 0.18777% 0.25195% 0.32512% 0.38097% 0.44387% 0.51199% 0.55660%3-factor 0.16701% 0.18770% 0.25189% 0.32510% 0.38049% 0.44415% 0.51191% 0.55640%
CSGAG
Market 0.13103% 0.15823% 0.20877% 0.25801% 0.30053% 0.34177% 0.40034% 0.43667%1-factor 0.13100% 0.15824% 0.20882% 0.25801% 0.30052% 0.34180% 0.40033% 0.43666%3-factor 0.13138% 0.15844% 0.20812% 0.25822% 0.30030% 0.34178% 0.39971% 0.43678% DB Market 0.18094% 0.20880% 0.31204% 0.42938% 0.54998% 0.64002% 0.70736% 0.74787%1-factor 0.18100% 0.20880% 0.31205% 0.42947% 0.54995% 0.63988% 0.70739% 0.74784%3-factor 0.18100% 0.20884% 0.31215% 0.42926% 0.55027% 0.63976% 0.70702% 0.74744%
HSBC
Market 0.11167% 0.12932% 0.19583% 0.25457% 0.31632% 0.37958% 0.45059% 0.50503%1-factor 0.11167% 0.12932% 0.19583% 0.25457% 0.31631% 0.37959% 0.45060% 0.50503%3-factor 0.11172% 0.12928% 0.19581% 0.25455% 0.31622% 0.37972% 0.45070% 0.50510%
JPM
Market 0.06314% 0.09585% 0.12173% 0.14363% 0.17416% 0.20975% 0.27020% 0.33598%1-factor 0.06315% 0.09586% 0.12168% 0.14363% 0.17414% 0.20979% 0.27019% 0.33599%3-factor 0.06315% 0.09615% 0.12195% 0.14350% 0.17442% 0.20962% 0.27055% 0.33605%
COOERAB
Market 0.12140% 0.14572% 0.20139% 0.25495% 0.30128% 0.34992% 0.40392% 0.43743%1-factor 0.12143% 0.14571% 0.20140% 0.25498% 0.30123% 0.34997% 0.40393% 0.43743%3-factor 0.12144% 0.14576% 0.20138% 0.25478% 0.30130% 0.34982% 0.40400% 0.43739% RY Market 0.05537% 0.07156% 0.07753% 0.10519% 0.15142% 0.18154% 0.24223% 0.28360%1-factor 0.05537% 0.07155% 0.07750% 0.10519% 0.15142% 0.18158% 0.24222% 0.28358%3-factor 0.05535% 0.07169% 0.07747% 0.10495% 0.15157% 0.18165% 0.24171% 0.28342%
SOCGEN
Market 0.20716% 0.24002% 0.32344% 0.39096% 0.46318% 0.53272% 0.59547% 0.63315%1-factor 0.20715% 0.24002% 0.32345% 0.39098% 0.46318% 0.53275% 0.59548% 0.63315%3-factor 0.20715% 0.23995% 0.32341% 0.39109% 0.46337% 0.53286% 0.59522% 0.63325%
SUMIBK-Bank
Market 0.05614% 0.07030% 0.09399% 0.11720% 0.15934% 0.20025% 0.23365% 0.25986%1-factor 0.05614% 0.07026% 0.09397% 0.11721% 0.15956% 0.20033% 0.23345% 0.26000%3-factor 0.05614% 0.07034% 0.09400% 0.11713% 0.15942% 0.20026% 0.23364% 0.25986%
NORBK
Market 0.03408% 0.04891% 0.07764% 0.10794% 0.14182% 0.17389% 0.20219% 0.23017%1-factor 0.03407% 0.04891% 0.07764% 0.10790% 0.14181% 0.17390% 0.20220% 0.23017%3-factor 0.03413% 0.04919% 0.07751% 0.10796% 0.14196% 0.17415% 0.20221% 0.22967%
RBOS-RBOSplc
Market 0.17319% 0.20029% 0.26063% 0.32684% 0.39763% 0.45950% 0.51386% 0.54946%1-factor 0.17311% 0.20027% 0.26069% 0.32696% 0.39764% 0.45955% 0.51391% 0.54947%3-factor 0.17326% 0.20028% 0.26056% 0.32689% 0.39785% 0.45968% 0.51400% 0.54950%
UBS
Market 0.13648% 0.15646% 0.20061% 0.26107% 0.30586% 0.35330% 0.40325% 0.44214%1-factor 0.13649% 0.15645% 0.20060% 0.26112% 0.30588% 0.35330% 0.40325% 0.44211%3-factor 0.13653% 0.15645% 0.20062% 0.26108% 0.30584% 0.35334% 0.40330% 0.44226%
Table 27: Model calibration to USD LIBOR panel banks on 22/03/201738
Maturity -2-1.5-1-0.500.511.5 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (a) 1 January 2013
Maturity -10-8-6-4-20246 10 -4
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (b) 08 September 2014
Maturity -3-2.5-2-1.5-1-0.500.511.52 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (c) 18 June 2015
Maturity -1.5-1-0.500.51 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (d) 20 April 2016
Maturity -1-0.500.511.5 10 -3
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (e) 22 March 2017
Maturity -6-4-2024681012 10 -4
1 month model - 1 month market bid1 month model - 1 month market ask3 month model - 3 month market bid3 month model - 3 month market ask6 month model - 6 month market bid6 month model - 6 month market ask (f) 31 October 2017
Figure 6: One–factor model fit to Basis Swaps on 1 January 2013, 18 June 2015 and 20April 2016 (based on data from Bloomberg)manner relative to a set of calibration instruments. The consistent model guarantees the39 T -1.5-1-0.500.511.5 c Figure 7: Time-dependent calibrated parameter c absence of arbitrage, though as in other applications where markets are incomplete (credit derivatives are a prominent example), we remain agnostic as to the extent towhich arbitrage–free prices are unique. Unlike approaches which do not model the risksdriving tenor spreads, the model presented here can be used to provide basis spreadsfor bespoke tenors not quoted in the market. This could also be employed to “fill inthe gaps” in markets where key term structures (such as OIS) are missing, potentiallyimproving on less structured, regression–based approaches hitherto available (such asJakarasi, Labuschagne and Mahomed (2015)).The relationship between tenor basis spreads (in particular the LIBOR/OIS spread)and roll–over risk, as expressed in the model presented in this paper and calibrated tomarket data, also serves as a cautionary note in the present debate on the replacement ofLIBOR–type benchmarks with benchmarks based on overnight lending, such as the Se-cured Overnight Financing Rate (SOFR). As Henrard (2018) notes in a critical overviewof the current discussions on this issue, most proposals suggest no more than adjustingovernight benchmarks for the existing LIBOR/OIS spread by adding a fixed spread toe.g. SOFR. This would imply taking the view that roll–over risk (as priced by the mar-ket) is known and constant for all time, and even a cursory analysis (such as calibratingthe model presented here to market data on a few exemplary dates) shows that this isnot the case. In the sense of Radner (1972). T c Figure 8: Time-dependent calibrated parameter c on a 3–month tenor A Affine process specification
The Markov process X is affine if it is stochastically continuous and its Fourier-Laplacetransform has exponential-affine dependence on the initial state, that is there exist somedeterministic functions Φ u : R + → C and Ψ u : R + → C d such that the semigroup P acts as follows: (cid:90) E e (cid:104) u,w (cid:105) P t ( x, dw ) = E Q (cid:2) e (cid:104) u,X ( t ) (cid:105) (cid:3) (A.1) = e Φ u ( t )+ (cid:104) Ψ u ( t ) ,x (cid:105) (A.2)for all t ≥ , X (0) = x ∈ E and u ∈ D T , where D T ⊆ C d is such that E Q (cid:2) e (cid:104) u,X ( t ) (cid:105) (cid:3) < ∞ for all u ∈ D T and t ≤ T . It can be shown (see e.g. Cuchiero et al. (2011)) that theprocess X is a semimartingale with characteristics A t = (cid:90) t α ( X ( s − )) ds,B t = (cid:90) t β ( X ( s − )) ds,ν ( ω, dt, dξ ) = K ( X ( t − , ω ) , dξ ) dt, α ( x ) , β ( x ) , K ( x, dξ ) affine functions: α ( x ) = α + x α + ... + x d α d ,β ( x ) = β + x β + ... + x d β d ,K ( x, dξ ) = µ ( dξ ) + x µ ( dξ ) + ... + x d µ d ( dξ ) , where α ( x ) (the diffusion coefficient) is a positive semidefinite d × d matrix, β ( x ) is the R d -vector of the drift, and K ( x, dξ ) is a Radon measure on R d associated to the affinejump part and it is such that (cid:90) R d (cid:0) (cid:107) ξ (cid:107) ∧ (cid:1) K ( x, dξ ) < ∞ and K ( x, { } ) = 0 .The deterministic functions Φ u ( t ) , Ψ u ( t ) solve the generalized Riccati equations∂∂t Φ u ( t ) = 12 (cid:104) Ψ u ( t ) , α Ψ u ( t ) (cid:105) + (cid:104) β , Ψ u ( t ) (cid:105) + (cid:90) R d \{ } (cid:0) e −(cid:104) ξ, Ψ u ( t ) (cid:105) − − (cid:104) h ( ξ ) , Ψ u ( t ) (cid:105) (cid:1) µ ( dξ ) , Φ u (0) = 0 , and for all i = 1 , ..., d : ∂∂t Ψ iu ( t ) = 12 (cid:104) Ψ u ( t ) , α i Ψ u ( t ) (cid:105) + (cid:104) β i , Ψ u ( t ) (cid:105) + (cid:90) R d \{ } (cid:0) e −(cid:104) ξ, Ψ u ( t ) (cid:105) − − (cid:104) h ( ξ ) , Ψ u ( t ) (cid:105) (cid:1) µ i ( dξ ) , Ψ u (0) = u, where h ( ξ ) = {(cid:107) ξ (cid:107)≤ } ξ is a truncation function.It is also useful to consider the process ( X, Y γ ) := ( X, (cid:82) · (cid:104) γ, X ( u ) (cid:105) du ) which is anaffine process with state space E × R starting from ( x, . Lemma A.1.
Let ˜ P γ be the semigroup of the process ( X, Y γ ) . Then we have (cid:90) E × R e (cid:104) u,w (cid:105) + vz ˜ P γt (( x, y ) , ( dw, dz )) = e Φ ( u,v ) ( t,γ )+ (cid:104) Ψ ( u,v ) ( t,γ ) ,x (cid:105) + vy (A.3) for every ( u, v ) ∈ V T ⊆ C d +1 such that the LHS of Eq. (A.3) is finite. Here the functions Φ ( u,v ) ( · , γ ) and Ψ ( u,v ) ( · , γ ) satisfy the following system of generalized Riccati ODEs ∂∂t Φ ( u,v ) ( t, γ ) = 12 (cid:104) Ψ ( u,v ) ( t, γ ) , α Ψ ( u,v ) ( t, γ ) (cid:105) + (cid:104) β , Ψ ( u,v ) ( t, γ ) (cid:105) + (cid:90) R d \{ } (cid:0) e −(cid:104) ξ, Ψ ( u,v ) ( t,γ ) (cid:105) − − (cid:104) h ( ξ ) , Ψ ( u,v ) ( t, γ ) (cid:105) (cid:1) µ ( dξ ) , (A.4) Φ ( u,v ) (0 , γ ) = 0 , nd for i = 1 , ..., d∂∂t Ψ i ( u,v ) ( t, γ ) = vγ i + 12 (cid:104) Ψ ( u,v ) ( t, γ ) , α i Ψ ( u,v ) ( t, γ ) (cid:105) + (cid:104) β i , Ψ ( u,v ) ( t, γ ) (cid:105) + (cid:90) R d \{ } (cid:0) e −(cid:104) ξ, Ψ ( u,v ) ( t,γ ) (cid:105) − − (cid:104) h ( ξ ) , Ψ ( u,v ) ( t, γ ) (cid:105) (cid:1) µ i ( dξ ) , (A.5) Ψ ( u,v ) (0 , γ ) = u. In our calibration example we consider the important case where X follows a d − dimensional vector of independent Cox–Ingersoll–Ross (CIR), for which the RiccatiODE admit a closed form expression. This dynamics is of particular interest because,as discussed for the one–factor case in Cox et al. (1985), a simple condition ensures thepositivity of the modelled object. In order to keep the model analytically tractable, wedo not allow for time–dependent coefficients at this stage . What is more, in the CIRcase there exists a complete explicit characterization of the maximal domains D T , V T for the transforms. This turns out to be very useful in the calibration procedure whenwe shall provide some bounds for the parameters. B Computation of the relevant expectations
Let us first consider the expectations involved in formula (2.7), i.e. E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) , E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (cid:105) and D OIS ( t, T ) = E Q t (cid:104) e − (cid:82) Tt r c ( s ) ds (cid:105) . Assume that the deterministic functions a ( t ) , b ( t ) , c ( t ) are integrable and a, b, c ∈ E are constant. Thanks to the time-homogeneity of the affine process X , it follows thatall these conditional expectations will depend on time through the difference ( T − t ) .Using the notation of the Appendix A, it is easy to show that E Q t (cid:104) e (cid:82) Tt φ ( s ) ds (cid:105) = e (cid:82) Tt c ( s ) ds E Q t (cid:104) e (cid:82) Tt (cid:104) c,X ( s ) (cid:105) ds (cid:105) = e (cid:82) Tt c ( s ) ds +Φ (0 , ( T − t,c )+ (cid:104) Ψ (0 , ( T − t,c ) ,X ( t ) (cid:105) , (B.1)where the deterministic functions Φ (0 , , Ψ (0 , satisfy the generalized Riccati ODE inLemma A.1.Analogously, E Q t (cid:104) e − (cid:82) Tt ( r c ( s )+ λ ( s ) q ) ds (cid:105) = e − (cid:82) Tt ( a ( s )+ b ( s ) q ) ds E Q t (cid:104) e − (cid:82) Tt (cid:104) a + bq,X ( s ) (cid:105) ds (cid:105) = e − (cid:82) Tt ( a ( s )+ b ( s ) q ) ds +Φ (0 , ( T − t, − ( a + bq ))+ (cid:104) Ψ (0 , ( T − t, − ( a + bq )) ,X ( t ) (cid:105) (B.2) For example, following Schl¨ogl and Schl¨ogl (2000), the coefficients could be made piecewise con-stant to facilitate calibration to at–the–money option price data, while retaining most analytical tractabil-ity. D OIS ( t, T ) = E Q t (cid:104) e − (cid:82) Tt r c ( s ) ds (cid:105) = e − (cid:82) Tt a ( s ) ds E Q t (cid:104) e − (cid:82) Tt (cid:104) a,X ( s ) (cid:105) ds (cid:105) = e − (cid:82) Tt a ( s ) ds +Φ (0 , ( T − t, − a )+ (cid:104) Ψ (0 , ( T − t, − a ) ,X ( t ) (cid:105) . (B.3)The proof follows immediately from the definition of r c , λ, φ and the Lemma A.1.Let us now turn out attention to formula (2.18), which involves the following expec-tation: E Q t (cid:20) e − (cid:82) Tj − t ( r c ( s )+ λ ( s ) q ) ds e (cid:82) TjTj − φ ( s ) ds (cid:21) = E Q t (cid:20) e − (cid:82) Tj − t ( r c ( s )+ λ ( s ) q ) ds E Q T j − (cid:20) e (cid:82) TjTj − φ ( s ) ds (cid:21)(cid:21) = E Q t (cid:20) e − (cid:82) Tj − t ( r c ( s )+ λ ( s ) q ) ds e (cid:82) TjTj − c ( s ) ds +Φ (0 , ( T j − T j − ,c )+ (cid:104) Ψ (0 , ( T j − T j − ,c ) ,X ( T j − ) (cid:105) (cid:21) = e − (cid:82) Tj − t ( a ( s )+ b ( s ) q ) ds + (cid:82) TjTj − c ( s ) ds +Φ (0 , ( T j − T j − ,c ) . E Q t (cid:104) e (cid:82) Tj − t (cid:104)− ( a + bq ) ,X ( s ) (cid:105) ds + (cid:104) Ψ (0 , ( T j − T j − ,c ) ,X ( T j − ) (cid:105) (cid:105) = e − (cid:82) Tj − t ( a ( s )+ b ( s ) q ) ds + (cid:82) TjTj − c ( s ) ds +Φ (0 , ( T j − T j − ,c ) .e Φ (Ψ(0 , Tj − Tj − ,c ) , ( T j − − t, − ( a + bq ))+ (cid:104) Ψ (Ψ(0 , Tj − Tj − ,c ) , ( T j − − t, − ( a + bq )) ,X ( t ) (cid:105) (B.4)In conclusion, all the expectations in formulae (2.7) and (2.18) can be explicitlycomputed. B.1 The LIBOR and the pricing of swaps
In this subsection we compute the expectations appearing in swap transaction formula(2.23), which is crucial to the calibration procedure. The expression of D OIS has alreadybeen computed in (B.3), so we focus now on the first expectation in (2.23) involving theforward LIBOR.From (2.20) it follows immediately δL ( T j − , T j ) = − e (cid:82) TjTj − ( c ( s )+ a ( s )+ qb ( s )) ds +Φ (0 , ( T j − T j − ,c ) − Φ (0 , ( T j − T j − , − ( a + qb )) .e (cid:104) Ψ (0 , ( T j − T j − ,c ) − Ψ (0 , ( T j − T j − , − ( a + qb )) ,X ( T j − ) (cid:105) . We have E Q t (cid:104) e − (cid:82) Tjt r c ( s ) ds L ( T j − , T j ) (cid:105) = E Q t (cid:104) e − (cid:82) Tj − t r c ( s ) ds L ( T j − , T j ) D OIS ( T j − , T j ) (cid:105) = E Q t (cid:20) e − (cid:82) Tj − t ( a ( s )+ (cid:104) a,X ( s ) (cid:105) ) ds − (cid:82) TjTj − a ( s ) ds +Φ (0 , ( T j − T j − , − a )+ (cid:104) Ψ (0 , ( T j − T j − , − a ) ,X ( T j − ) (cid:105) L ( T j − , T j ) (cid:21) = e − (cid:82) Tjt a ( s ) ds +Φ (0 , ( T j − T j − , − a ) E Q t (cid:104) e − (cid:82) Tj − t (cid:104) a,X ( s ) (cid:105) ds + (cid:104) Ψ (0 , ( T j − T j − , − a ) ,X ( T j − ) (cid:105) L ( T j − , T j ) (cid:105) . E Q t (cid:104) e − (cid:82) Tjt r c ( s ) ds L ( T j − , T j ) (cid:105) = − δ e − (cid:82) Tjt a ( s ) ds +Φ (0 , ( T j − T j − , − a ) E Q t (cid:104) e − (cid:82) Tj − t (cid:104) a,X ( s ) (cid:105) ds + (cid:104) Ψ (0 , ( T j − T j − , − a ) ,X ( T j − ) (cid:105) (cid:105) + 1 δ e − (cid:82) Tjt a ( s ) ds + (cid:82) TjTj − ( c ( s )+ a ( s )+ qb ( s )) ds +Φ (0 , ( T j − T j − , − a )+Φ (0 , ( T j − T j − ,c ) − Φ (0 , ( T j − T j − , − ( a + qb )) . E Q t (cid:104) e − (cid:82) Tj − t (cid:104) a,X ( s ) (cid:105) ds + (cid:105) Ψ (0 , ( T j − T j − , − a )+Ψ (0 , ( T j − T j − ,c ) − Ψ (0 , ( T j − T j − , − ( a + qb )) ,X ( T j − ) (cid:105) (cid:105) = − δ e − (cid:82) Tjt a ( s ) ds +Φ (0 , ( T j − T j − , − a )+Φ (Ψ(0 , Tj − Tj − , − a ) , ( T j − − t, − a )+ (cid:104) Ψ (Ψ(0 , Tj − Tj − , − a ) , ( T j − − t, − a ) ,X ( t ) (cid:105) + 1 δ e − (cid:82) Tjt a ( s ) ds + (cid:82) TjTj − ( c ( s )+ a ( s )+ qb ( s )) ds +Φ (0 , ( T j − T j − , − a )+Φ (0 , ( T j − T j − ,c ) − Φ (0 , ( T j − T j − , − ( a + qb )) .e Φ (Ψ(0 , Tj − Tj − , − a )+Ψ(0 , Tj − Tj − ,c ) − Ψ(0 , Tj − Tj − , − ( a + qb )) , ( T j − − t, − a ) .e (cid:104) Ψ (Ψ(0 , Tj − Tj − , − a )+Ψ(0 , Tj − Tj − ,c ) − Ψ(0 , Tj − Tj − , − ( a + qb )) , ( T j − − t, − a ) ,X ( t ) (cid:105) . Therefore, also the expectations in (2.23) can be computed explicitly and the formulacan be obtained in closed form.
B.2 The pricing of caps
Let us first consider a caplet on the LIBOR with maturity T j , whose payoff is given by Kδ ( L ( T j − , T j ) − R ) + , where K denotes the notional and R is the strike price of the caplet.The price at time t ≤ T j − is given by Caplet t = E Q t (cid:104) e − (cid:82) Tjt r c ( s ) ds Kδ ( L ( T j − , T j ) − R ) + (cid:105) = D OIS ( t, T j ) E Q Tj t (cid:2) Kδ ( L ( T j − , T j ) − R ) + (cid:3) = D OIS ( t, T j ) Kδ E Q Tj t δ E Q T j − (cid:20) e (cid:82) TjTj − φ ( s ) ds (cid:21) E Q T j − (cid:20) e − (cid:82) TjTj − ( r c ( s )+ λ ( s ) q ) ds (cid:21) − (1 + δR ) + = D OIS ( t, T j ) K (1 + δR ) E Q Tj t (cid:104)(cid:0) e Z ( T j − ) − (cid:1) + (cid:105) , (B.5)where e Z ( T j − ) = 11 + δR E Q T j − (cid:20) e (cid:82) TjTj − φ ( s ) ds (cid:21) E Q T j − (cid:20) e − (cid:82) TjTj − ( r c ( s )+ λ ( s ) q ) ds (cid:21) . (B.6)(B.7)45sing (B.1) and (B.2), we can now write e Z ( T j − ) = 11 + δR e (cid:82) TjTj − c ( s ) ds +Φ (0 , ( T j − T j − ,c )+ (cid:104) Ψ (0 , ( T j − T j − ,c ) ,X ( T j − ) (cid:105) e − (cid:82) TjTj − ( a ( s )+ b ( s ) q ) ds +Φ (0 , ( T j − T j − , − ( a + bq ))+ (cid:104) Ψ (0 , ( T j − T j − , − ( a + bq )) ,X ( T j − ) (cid:105) = e f ( T j − )+ (cid:104) g ( T j − ) ,X ( T j − ) (cid:105) , (B.8)where f ( T j − ) = − ln(1 + δR ) + (cid:90) T j T j − ( c ( s ) + a ( s ) + b ( s ) q ) ds + Φ (0 , ( T j − T j − , c ) − Φ (0 , ( T j − T j − , − ( a + bq )) , (B.9) g ( T j − ) =Ψ (0 , ( T j − T j − , c ) − Ψ (0 , ( T j − T j − , − ( a + bq )) . (B.10)In order to compute Equation (B.5), we apply Fourier transform techniques well-explored by e.g. Carr and Madan (1999). For this purpose, we derive the expres-sion for the characteristic function or the moment generating function of Z under the Q T j − forward probability measure. Proposition B.1.
The conditional characteristic function under the Q T j − forward prob-ability measure of the random variable Z defined in (B.6) is given by ϕ T j Z ( u ) = E Q Tj t (cid:2) e iuZ ( T j − ) (cid:3) = 1 D OIS ( t, T j ) e iuf ( T j − ) − (cid:82) Tjt a ( s ) ds +Φ (0 , ( T j − T j − , − a ) .e Φ ( iug ( Tj − , Tj − Tj − , − a ) , ( T j − − t, − a )+ (cid:104) Ψ ( iug ( Tj − , Tj − Tj − , − a ) , ( T j − − t, − a ) ,X ( t ) (cid:105) . The transform is well defined for u ∈ C such that Φ ( iug ( T j − )+Ψ (0 , ( T j − T j − , − a ) , ( T j − − t, − a ) and Φ ( iug ( T j − )+Ψ (0 , ( T j − T j − , − a ) , ( T j − − t, − a ) are bounded.Proof. From (B.3) it follows that ϕ T j Z ( u ) = E Q Tj t (cid:2) e iuZ ( T j − ) (cid:3) = e iuf ( T j − ) E Q Tj t (cid:2) e iu (cid:104) g ( T j − ) ,X ( T j − ) (cid:105) (cid:3) = e iuf ( T j − ) D OIS ( t, T j ) E Q t (cid:104) e − (cid:82) Tjt r c ( s ) ds + iu (cid:104) g ( T j − ) ,X ( T j − ) (cid:105) (cid:105) = e iuf ( T j − ) D OIS ( t, T j ) E Q t (cid:20) e − (cid:82) Tj − t r c ( s ) ds + iu (cid:104) g ( T j − ) ,X ( T j − ) (cid:105) E Q T j − (cid:20) e − (cid:82) TjTj − r c ( s ) ds (cid:21)(cid:21) = e iuf ( T j − ) D OIS ( t, T j ) E Q t (cid:20) e − (cid:82) Tj − t r c ( s ) ds + iu (cid:104) g ( T j − ) ,X ( T j − ) (cid:105) E Q T j − (cid:20) e − (cid:82) TjTj − a ( s ) ds − (cid:82) TjTj − (cid:104) a,X ( s ) (cid:105) ds (cid:21)(cid:21) = e iuf ( T j − ) − (cid:82) Tjt a ( s ) ds +Φ (0 , ( T j − T j − , − a ) D OIS ( t, T j ) . E Q t (cid:104) e − (cid:82) Tj − t (cid:104) a,X ( s ) (cid:105) ds + (cid:104) iug ( T j − )+Ψ (0 , ( T j − T j − , − a ) ,X ( T j − ) (cid:105) (cid:105) . Now we apply once again Lemma A.1 and we get the result.46ith the explicit expression of the characteristic function of the process Z we canapply the Carr and Madan (1999) methodology in order to get the price of a capleton LIBOR. For more complex product like swaptions analytical formulas are no moreavailable. However, one can easily apply efficient approximations in the spirit of Cal-dana et al. (2016). C Expressions for the Pricing of a CDS
First of all, let us consider the time t -value of a zero-recovery loan with notional to a j bank within the LIBOR panel given by Formula (4.6): B j ( t, T ) = D OIS ( t, T ) E Q T (cid:104) e − (cid:82) Tt (ˆ λ j ( s ) − q Λ( s )) ds | H t (cid:105) = D OIS ( t, T ) e − (cid:82) Tt (ˆ b j ( s ) − qβ ( s )) ds +Φ (0 , ( T − t, ˆ b j − qβ )+ (cid:104) Ψ (0 , ( T − t, ˆ b j − qβ ) ,X ( t ) (cid:105) , (C.1)where the functions Φ (0 , , Ψ (0 , solve the Riccati system (A.4), (A.5).Now let us find an expression for E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds ˆ λ j ( u ) (cid:105) .This can be done as follows: E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds ˆ λ j ( u ) (cid:105) = − ∂∂u E Q T (cid:104) e − (cid:82) ut ˆ λ j ( s ) ds (cid:105) = − ∂∂u E Q T (cid:104) e − (cid:82) ut ˆ b j ( s ) ds − (cid:82) ut (cid:104) ˆ b j ,X ( s ) (cid:105) ds (cid:105) = − ∂∂u (cid:110) e − (cid:82) ut ˆ b j ( s ) ds +Φ (0 , ( u − t, ˆ b j )+ (cid:104) Ψ (0 , ( u − t, ˆ b j ) ,X ( t ) (cid:105) (cid:111) = − (cid:16) − ˆ b j ( u ) + ∂ u Φ (0 , ( u − t, ˆ b j ) + (cid:104) ∂ u Ψ (0 , ( u − t, ˆ b j ) , X ( t ) (cid:105) (cid:17) .e − (cid:82) ut ˆ b j ( s ) ds +Φ (0 , ( u − t, ˆ b j )+ (cid:104) Ψ (0 , ( u − t, ˆ b j ) ,X ( t ) (cid:105) , where the derivatives of the functions Φ (0 , , Ψ (0 , can be easily computed from theRiccati system (A.4), (A.5). References
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IBOR Panel Maturity 0.5 1 2 3 4 5 7 10
BACORP
Market 0.04765% 0.07869% 0.10314% 0.12627% 0.15811% 0.19605% 0.25590% 0.31003%1-factor 0.04765% 0.07868% 0.10315% 0.12630% 0.15813% 0.19609% 0.25582% 0.31001%3-factor 0.04769% 0.07869% 0.10324% 0.12620% 0.15799% 0.19595% 0.25577% 0.31000%
MUFJ-BTMUFJ
Market 0.04112% 0.05484% 0.07999% 0.10463% 0.13901% 0.17205% 0.20889% 0.23103%1-factor 0.04114% 0.05484% 0.07991% 0.10465% 0.13904% 0.17205% 0.20874% 0.23104%3-factor 0.05551% 0.06738% 0.08871% 0.10911% 0.14725% 0.18562% 0.21704% 0.23945%
BACR-Bank
Market 0.06283% 0.07470% 0.10530% 0.13799% 0.17674% 0.20894% 0.24773% 0.27961%1-factor 0.06282% 0.07474% 0.10533% 0.13798% 0.17674% 0.20897% 0.24770% 0.27971%3-factor 0.06272% 0.08031% 0.13047% 0.18052% 0.22670% 0.27021% 0.32296% 0.35596% C Market 0.05522% 0.08068% 0.10889% 0.14458% 0.17606% 0.21353% 0.28439% 0.33442%1-factor 0.05524% 0.08068% 0.10891% 0.14457% 0.17602% 0.21352% 0.28443% 0.33436%3-factor 0.05521% 0.08071% 0.10912% 0.14441% 0.17536% 0.21343% 0.28317% 0.33564%
ACAFP
Market 0.04400% 0.05628% 0.09576% 0.13468% 0.17967% 0.22376% 0.29454% 0.34602%1-factor 0.04400% 0.05626% 0.09580% 0.13461% 0.17970% 0.22379% 0.29464% 0.34621%3-factor 0.04401% 0.05625% 0.09537% 0.13449% 0.17980% 0.22304% 0.29384% 0.34697%
CSGAG
Market 0.04169% 0.04965% 0.07940% 0.10701% 0.13990% 0.17717% 0.23487% 0.27539%1-factor 0.04170% 0.04964% 0.07940% 0.10700% 0.13991% 0.17723% 0.23486% 0.27539%3-factor 0.04172% 0.04962% 0.07949% 0.10710% 0.14020% 0.17683% 0.23508% 0.27529% DB Market 0.07469% 0.09145% 0.16837% 0.24921% 0.31997% 0.39129% 0.49154% 0.56485%1-factor 0.07469% 0.09145% 0.16842% 0.24920% 0.31997% 0.39133% 0.49157% 0.56499%3-factor 0.07476% 0.09132% 0.16828% 0.24916% 0.32015% 0.39153% 0.49200% 0.56504%
HSBC
Market 0.05794% 0.05791% 0.09311% 0.14675% 0.18578% 0.23833% 0.29412% 0.34312%1-factor 0.05794% 0.05790% 0.09306% 0.14672% 0.18567% 0.23851% 0.29431% 0.34314%3-factor 0.05813% 0.05785% 0.09321% 0.14707% 0.18517% 0.23790% 0.29413% 0.34258%
JPM
Market 0.04781% 0.07509% 0.09724% 0.11621% 0.14779% 0.17934% 0.23899% 0.29312%1-factor 0.04779% 0.07510% 0.09725% 0.11620% 0.14780% 0.17938% 0.23899% 0.29313%3-factor 0.04779% 0.07514% 0.09736% 0.11631% 0.14783% 0.17842% 0.23941% 0.29317%
LBGP
Market 0.14239% 0.11944% 0.16880% 0.22706% 0.28200% 0.33578% 0.41119% 0.45180%1-factor 0.14241% 0.11946% 0.16879% 0.22705% 0.28193% 0.33577% 0.41122% 0.45187%3-factor 0.14236% 0.11955% 0.16888% 0.22714% 0.28172% 0.33582% 0.41142% 0.45265%
COOERAB
Market 0.03703% 0.04606% 0.07299% 0.10144% 0.13410% 0.16635% 0.22954% 0.26655%1-factor 0.03699% 0.04606% 0.07299% 0.10146% 0.13418% 0.16619% 0.22962% 0.26655%3-factor 0.03699% 0.04614% 0.07290% 0.10136% 0.13469% 0.16620% 0.22941% 0.26645% RY Market 0.03909% 0.04316% 0.04805% 0.05553% 0.09077% 0.11238% 0.16119% 0.19930%1-factor 0.03909% 0.04313% 0.04804% 0.05552% 0.09077% 0.11245% 0.16118% 0.19940%3-factor 0.03924% 0.04309% 0.04830% 0.05523% 0.09077% 0.11242% 0.16143% 0.19889%
SOCGEN
Market 0.06425% 0.07306% 0.12102% 0.16241% 0.20321% 0.24923% 0.32346% 0.38950%1-factor 0.06427% 0.07309% 0.12100% 0.16236% 0.20305% 0.24905% 0.32357% 0.38948%3-factor 0.06425% 0.07305% 0.12109% 0.16242% 0.20312% 0.24911% 0.32354% 0.38949%
SUMIBK-Bank
Market 0.02385% 0.02955% 0.04556% 0.06148% 0.08704% 0.11248% 0.13883% 0.15339%1-factor 0.02387% 0.02954% 0.04554% 0.06145% 0.08708% 0.11246% 0.13850% 0.15339%3-factor 0.02384% 0.02956% 0.04548% 0.06150% 0.08701% 0.11241% 0.13898% 0.15344%
NORBK
Market 0.03292% 0.04668% 0.07367% 0.10251% 0.13718% 0.16995% 0.20108% 0.23021%1-factor 0.03294% 0.04661% 0.07368% 0.10256% 0.13720% 0.17025% 0.20103% 0.23022%3-factor 0.03289% 0.04664% 0.07364% 0.10256% 0.13724% 0.16999% 0.20138% 0.23017%
RBOS-RBOSplc
Market 0.05447% 0.05456% 0.08235% 0.11239% 0.15152% 0.18195% 0.20423% 0.22949%1-factor 0.05446% 0.05455% 0.08243% 0.11246% 0.15152% 0.18193% 0.20462% 0.22950%3-factor 0.05465% 0.05474% 0.08219% 0.11210% 0.15123% 0.18164% 0.20444% 0.22944%
UBS