Impact Of Multiple-Curve Dynamics In Credit Valuation .
Impact of Multiple-Curve Dynamicsin Credit Valuation AdjustmentsGiacomo Bormetti, Damiano Brigo, Marco Francischelloand Andrea PallaviciniAbstract We present a detailed analysis of interest rate derivatives valuation undercredit risk and collateral modeling. We show how the credit and collateral extendedvaluation framework in Pallavicini et al. (2011) can be helpful in defining the keymarket rates underlying the multiple interest rate curves that characterize currentinterest rate markets. We introduce the collateralized valuation measures and formulate a consistent realistic dynamics for the rates emerging from our analysis. Wepoint out limitations of multiple curve models with deterministic basis consideringvaluation of particularly sensitive products such as basis swaps.Keywords Multiple curvesHJM model· Evaluation adjustments · Basis swaps · Collateral ·1 IntroductionAfter the onset of the crisis in 2007, all market instruments are quoted by takinginto account, more or less implicitly, credit- and collateral-related adjustments. Asa consequence, when approaching modeling problems one has to carefully checkstandard theoretical assumptions which often ignore credit and liquidity issues. Onehas to go back to market processes and fundamental instruments by limiting oneselfto use models based on products and quantities that are available on the market.G. Bormetti (B)University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italye-mail: giacomo.bormetti@unibo.itD. Brigo · M. FrancischelloImperial College London, London SW7 2AZ, UKe-mail: damiano.brigo@imperial.ac.ukM. Francischelloe-mail: m.francischello14@imperial.ac.ukA. PallaviciniImperial College London and Banca IMI, Largo Mattioli, 3, 20121 Milan, Italye-mail: a.pallavicini@imperial.ac.uk The Author(s) 2016K. Glau et al. (eds.), Innovations in Derivatives Markets, Springer Proceedingsin Mathematics & Statistics 165, DOI 10.1007/978-3-319-33446-2 12251
252G. Bormetti et al.Referring to market observables and processes is the only means we have to validateour theoretical assumptions, so as to drop them if in contrast with observations. Thisgeneral recipe is what is guiding us in this paper, where we try to adapt interest ratemodels for valuation to the current landscape.A detailed analysis of the updated valuation problem one faces when includingcredit risk and collateral modeling (and further funding costs) has been presentedelsewhere in this volume, see for example [6, 7]. We refer to those papers andreferences therein for a detailed discussion. Here we focus our updated valuationframework to consider the following key points: (i) focus on interest rate derivatives;(ii) understand how the updated valuation framework can be helpful in defining thekey market rates underlying the multiple interest rate curves that characterize currentinterest rate markets; (iii) define collateralized valuation measures; (iv) formulate aconsistent realistic dynamics for the rates emerging from the above analysis; (v) showhow the framework can be applied to valuation of particularly sensitive productssuch as basis swaps under credit risk and collateral posting;(vi) point out limitationsin some current market practices such as explaining the multiple curves throughdeterministic fudge factors or shifts where the option embedded in the credit valuationadjustment (CVA) calculation would be priced without any volatility. For an extendedversion of this paper we remand to [3]. This paper is an extended and refined versionof ideas originally appeared in [24].2 Valuation Equation with Credit and CollateralClassical interest-rate models were formulated to satisfy no-arbitrage relationshipsby construction, which allowed one to price and hedge forward-rate agreements interms of risk-free zero-coupon bonds. Starting from summer 2007, with the spreadingof the credit crunch, market quotes of forward rates and zero-coupon bonds beganto violate usual no-arbitrage relationships. The main driver of such behavior was theliquidity crisis reducing the credit lines along with the fear of an imminent systemicbreak-down. As a result the impact of counterparty risk on market prices could notbe considered negligible any more.This is the first of many examples of relationships that broke down with the crisis. Assumptions and approximations stemming from valuation theory should bereplaced by strategies implemented with market instruments. For instance, inclusion of CVA for interest-rate instruments, such as those analyzed in [8], breaks therelationship between risk-free zero-coupon bonds and LIBOR forward rates. Also,funding in domestic currency on different time horizons must include counterpartyrisk adjustments and liquidity issues, see [15], breaking again this relationship. Wethus have, against the earlier standard theory,L(T0 , T1 ) 1T1 T0 1Pt (T0 )1 1 , Ft (T0 , T1 ) 1 ,PT0 (T1 )T1 T0 Pt (T1 )(1)
Impact of Multiple-Curve Dynamics 253where Pt (T ) is a zero-coupon bond price at time t for maturity T , L is the LIBOR rateand F is the related LIBOR forward rate. A direct consequence is the impossibilityto describe all LIBOR rates in terms of a unique zero-coupon yield curve. Indeed,since 2009 and even earlier, we had evidence that the money market for the Euroarea was moving to a multi-curve setting. See [1, 19, 20, 27].2.1 Valuation FrameworkIn order to value a financial product (for example a derivative contract), we have todiscount all the cash flows occurring after the trading position is entered. We followthe approach of [25, 26] and we specialize it to the case of interest-rate derivatives,where collateralization usually happens on a daily basis, and where gap risk is notlarge. Hence we prefer to present such results when cash flows are modeled ashappening in a continuous time-grid, since this simplifies notation and calculations.We refer to the two names involved in the financial contract and subject to defaultrisk as investor (also called name “I”) and counterparty (also called name “C”). Wedenote by τI , and τC , respectively, the default times of the investor and counterparty.We fix the portfolio time horizon T 0, and fix the risk-neutral valuation model(Ω, G , Q), with a filtration (Gt )t [0,T ] such that τC , τI are (Gt )t [0,T ] -stopping times.We denote by Et [ · ] the conditional expectation under Q given Gt , and by Eτi [ · ]the conditional expectation under Q given the stopped filtration Gτi . We exclude thepossibility of simultaneous defaults, and define the first default event between thetwo parties as the stopping time τ : τC τI .We will also consider the market sub-filtration (Ft )t 0 that one obtains implicitlyby assuming a separable structure for the complete market filtration (Gt )t 0 . Gt is thengenerated by the pure default-free market filtration Ft and by the filtration generatedby all the relevant default times monitored up to t (see for example [2]).We introduce a risk-free rate r associated with the risk-neutral measure. We therefore need to define the related stochastic discount factor D(t, u, r) that in general willdenote the risk-neutral default-free discount factor, given by the ratioD(t, u, r) Bt /Bu , dBt rt Bt dt,where B is the bank account numeraire, driven by the risk-free instantaneous interestrate rt and associated to the risk-neutral measure Q. This rate rt is assumed to be(Ft )t [0,T ] adapted and is the key variable in all pre-crisis term structure modeling.We now want to price a collateralized derivative contract, and in particular weassume that collateral re-hypothecation is allowed, as done in practice (see [4] for adiscussion on re-hypothecation). We thus write directly the adjustment payout termsas carry costs cash flows, each accruing at the relevant rate, namely the price Vt of aderivative contract, inclusive of collateralized credit and debit risk, margining costs,can be derived by following [25, 26], and is given by:
254G. Bormetti et al. TVt E D(t, u; r) 1{u τ } dπu 1{τ du} θu (ru cu )Cu du Gt (2)twhere πu is the coupon process of the product, without credit or debit risk and withoutcollateral cash flows; Cu is the collateral process, and we use the convention that Cu 0 while I is thecollateral receiver and Cu 0 when I is the collateral poster. (ru cu )Cu are thecollateral margining costs and the collateral rate is defined as ct : ct 1{Ct 0} ct 1{Ct 0} with c defined in the CSA contract. In general we may assume theprocesses c , c to be adapted to the default-free filtration Ft . θu θu (C, ε) is the on-default cash flow process that depends on the collateralprocess Cu and the close-out value εu .1 It is primarily this term that originates thecredit and debit valuation adjustments (CVA/DVA) terms, that may also embedcollateral and gap risk due to the jump at default of the value of the considereddeal (e.g. in a credit derivative), see for example [5].Notice that the above valuation equation (2) is not suited for explicit numericalevaluations, since the right-hand side is still depending on the derivative price via theindicators within the collateral rates and possibly via the close-out term, leading torecursive/nonlinear features. We could resort to numerical solutions, as in [11], but,since our goal is valuing interest-rate derivatives, we prefer to further specialize thevaluation equation for such deals.2.2 The Master Equation Under Change of FiltrationIn this first work we develop our analysis without considering a dependence betweenthe default times if not through their spreads, or more precisely by assuming thatthe default times are F -conditionally independent. Moreover, we assume that thecollateral account and the close-out processes are F -adapted. Thus, we can simplifythe valuation equation given by (2) by switching to the default-free market filtration.By following the filtration switching formula in [2], we introduce for any Gt -adaptedprocess Xt a unique Ft -adapted process Xt , defined such that 1{τ t} Xt 1{τ t} Xt .Hence, we can write the pre-default price process as given by 1{τ t} Vt Vt wherethe right-hand side is given in Eq. (2) and where Ṽt is Ft -adapted. Before changingfiltration, we have to specify the form of the close-out payoff:θτ ετ (τ, T ) 1{τC τI } LGDC (ετ (τ, T ) Cτ ) 1{τI τC } LGDI (ετ (τ, T ) Cτ ) 1 Thecloseout value is the residual value of the contract at default time and the CSA specifies theway it should be computed.
Impact of Multiple-Curve Dynamics 255where LGD 1 is the loss given default, (x) indicates the positive part of x and(x) ( x) . For an extended discussion of the term θτ we refer to [3]. Moreover,to derive an explicit valuation formula we assume that gap risk is not present, namelyṼτ Ṽτ , and we consider a particular form for collateral and close-out prices,namely we model the close-out value as Tεs (t, T ) E D(t, u, r)dπu Gs., Ct αt εt (t, T )twith 0 αt 1 and where αt is Ft -adapted. This means that the close-out is therisk-free mark to market at first default time and the collateral is a fraction αt of theclose-out value. An alternative approximation that does not impose a proportionalitybetween the account value processes can be found in [9]. We obtain, by switching tothe default-free market filtration F the following.2Proposition 1 (Master equation under F -conditionally independent default times,no gap risk and Ft measurable payout πt ) Under the above assumption, ValuationEquation (2) is further specified as Vt 1{τ t} Vt TVt εt (t, T ) E EtTt Et D(t, u; r λ)(ru cu )αu εu (u, T )du FtTD(t, u; r λ)λCu (1 αu )LGDC (εu (u, T )) du FtD(t, u; r λ)λIu (1 αu )LGDI (εu (u, T )) du Ftwhere we introduced the pre-default intensity λIt of the investor and the pre-defaultintensity λCt of the counterparty as1{τI t} λIt dt : Q { τI dt τI t, Ft } , 1{τC t} λCt dt : Q { τC dt τC t, Ft }along with their sum λt and the discount factor for any rate xu , namely D(t, T , x) : Texp{ t xu du}.3 Valuing Collateralized Interest-Rate DerivativesAs we mentioned in the introduction, we will base our analysis on real marketprocesses. All liquid market quotes on the money market (MM) correspond to instruments with daily collateralization at overnight rate (et ), both for the investor and the.counterparty, namely ct et .2 Werefer to [3] and [6] for a precise derivation of the proposition.
256G. Bormetti et al.Notice that the collateral accrual rate is symmetric, so that we no longer have adependency of the accrual rates on the collateral price, as opposed to the general.master equation case. Moreover, we further assume rt et .This makes sense because et being an overnight rate, it embeds a low counterpartyrisk and can be considered a good proxy for the risk-free rate rt . We will describesome of these MM instruments, such as OIS and Interest Rate Swaps (IRS), alongwith their underlying market rates, in the following sections. For the remaining ofthis section we adopt the perfect collateralization approximation of Eq. (1) to derivethe valuation equations for OIS and IRS products, hence assuming no gap-risk,while in the numeric experiments of Sect. 4 we will consider also uncollateralizeddeals. Furthermore, we assume that daily collateralization can be considered as acontinuous-dividend perfect collateralization. See [4] for a discussion on the impactof discrete-time collateralization on interest-rate derivatives.3.1 Overnight Rates and OISAmong other instruments, the MM usually quotes the prices of overnight indexedswaps (OIS). Such contracts exchange a fix-payment leg with a floating leg paying a discretely compounded rate based on the same overnight rate used for theircollateralization. Since we are going to price OIS under the assumption of perfectcollateralization, namely we are assuming that daily collateralization may be viewedas done on a continuous basis, we approximate also daily compounding in OIS floating leg with continuous compounding, which is reasonable when there is no gaprisk. Hence the discounted payoff of a one-period OIS with tenor x and maturity Tis given by Teu duD(t, T , e) 1 xK expT xwhere K is the fixed rate payed by the OIS. Furthermore, we can introduce the (par)fix rates K Et (T , x; e) that make the one-period OIS contract fair, namely priced0 at time t. They are implicitly defined via VtOIS (K) : E 1 xK expT x Teu duD(t, T ; e) Ftwith VtOIS (Et (T , x; e)) 0 leading to1Et (T , x; e) : x Pt (T x; e) 1Pt (T ; e)(3)
Impact of Multiple-Curve Dynamics 257where we define collateralized zero-coupon bonds3 asPt (T ; e) : E [ D(t, T ; e) Ft ] .(4)One-period OIS rates Et (T , x; e), along with multi-period ones, are actively tradedon the market. Notice that we can bootstrap collateralized zero-coupon bond pricesfrom OIS quotes.3.2 LIBOR Rates, IRS and Basis SwapsLIBOR rates (Lt (T )) used to be linked to the term structure of default-free interlinkinterest rates in a fundamental way. In the classical term structure theory, LIBORrates would satisfy fundamental no-arbitrage conditions with respect to zero-couponbonds that we no longer consider to hold, as we pointed out earlier in (1). Wenow deal with a new definition of forward LIBOR rates that may take into accountcollateralization. LIBOR rates are still the indices used as reference rate for manycollateralized interest-rate derivatives (IRS, basis swaps, ). IRS contracts swap afix-payment leg with a floating leg paying simply compounded LIBOR rates. IRScontracts are collateralized at overnight rate et . Thus, a discounted one-period IRSpayoff with maturity T and tenor x is given byD(t, T , e)x(K LT x (T ))where K is the fix rate payed by the IRS. Furthermore, we can introduce the (par) fixrates K Ft (T , x; e) that render the one-period IRS contract fair, i.e. priced at zero.They are implicitly defined viaVtIRS (K) : E (xK xLT x (T )) D(t, T ; e) Ft with VtIRS (Ft (T , x; e)) 0, leading to the following definition of forward LIBOR rate E LT x (T )D(t, T ; e) FtE LT x (T )D(t, T ; e) Ft Ft (T , x; e) : E [ D(t, T ; e) Ft ]Pt (T ; e)The above definition may be simplified by a suitable choice of the measureunder which we take the expectation. In particular, we can consider the followingRadon–Nikodym derivative, defining the collateralized T -forward measure QT ;e ,3 Notice that we are only defining a price process for hypothetical collateralized zero-couponWe are not assuming that collateralized bonds are assets traded on the market.bond.
258G. Bormetti et al. dQT ;e E [ D(0, T ; e) Ft ]D(0, t; e)Pt (T ; e)Zt (T ; e) : : dQ FtP0 (T ; e)P0 (T ; e)which is a positive Q-martingale, normalized so that Z0 (T ; e) 1.Thus, for any payoff φT , perfectly collateralized at overnight rate et , we canexpress prices as expectations under the collateralized T -forward measure and inparticular, we can write LIBOR forward rates as E LT x (T )D(t, T ; e) Ft ET ;e LT x (T ) Ft .Ft (T , x; e) : E [ D(t, T ; e) Ft ](5)One-period forward rates Ft (T , x; e), along with multi-period ones (swap rates),are actively traded on the market. Once collateralized zero-coupon bonds are derived,we can bootstrap forward rate curves from such quotes. See, for instance, [1] or [27]for a discussion on bootstrapping algorithms.Basis swaps are an interesting product that became more popular after the marketswitched to a multi-curve structure. In fact, in a basis swap there are two floatinglegs, one pays a LIBOR rate with a certain tenor and the other pays the LIBOR ratewith a shorter tenor plus a spread that makes the contract fair at inception. Moreprecisely, the payoff of a basis swap whose legs pay respectively a LIBOR rate withtenors x y with maturity T nx my is given byn D(t, T (n i)x, e)x(LT (n i 1)x (T (n i)x) K)i 1 m D(t, T (m j)y, e)yLT (m j 1)y (T (m j)y).j 1It is clear that apart from being traded per se, this instrument is naturally present inthe banks portfolios as result of the netting of opposite swap positions with differenttenors.3.3 Modeling ConstraintsOur aim is to set up a multiple-curve dynamical model starting from collateralizedzero-coupon bonds Pt (T ; e), and LIBOR forward rates Ft (T , x; e). As we have seenwe can bootstrap the initial curves for such quantities from directly observed quotesin the market. Now, we wish to propose a dynamics that preserves the martingaleproperties satisfied by such quantities. Thus, without loss of generality, we can definecollateralized zero-coupon bonds under the Q measure as dPt (T ; e) Pt (T ; e) et dt σtP (T ; e) dWte
Impact of Multiple-Curve Dynamics 259and LIBOR forward rates under the QT ;e measure asdFt (T , x; e) σtF (T , x; e) dZtT ;ewhere W e and Z T ;e are correlated standard (column) vector4 Brownian motions withcorrelation matrix ρ, and the volatility vector processes σ P and σ F may depend onbonds and forward LIBOR rates themselves.The following definition of ft (T , e) is not strictly necessary, and we could keepworking with bonds Pt (T ; e), using their dynamics. However, as it is customaryin interest rate theory to model rates rather than bonds, we may try to formulatequantities that are closer to the standard HJM framework. In this sense we can defineinstantaneous forward rates ft (T ; e), by starting from (collateralized) zero-couponbonds, as given byft (T ; e) : T log Pt (T ; e)We can derive instantaneous forward-rate dynamics by Itô lemma, and we obtain thefollowing dynamics under the QT ;e measuredft (T ; e) σt (T ; e) dWtT ;e , σt (T ; e) : T σtP (T ; e)where the W T ;e s are Brownian motions and partial differentiation is meant to beapplied component-wise.Hence, we can summarize our modeling assumptions in the following way. Sincelinear products (OIS, IRS, basis swaps ) can be expressed in terms of simpler quan
credit and debit valuation adjustments (CVA/DVA) terms, that may also embed collateral and gap risk due to the jump at default of the value of the considered deal (e.g. in a credit derivative), see for example [5]. Notice that the above valuation
behringer ultra-curve pro dsp 24 a/d- d/a dsp ultra-curve pro ultra- curve pro 1.1 behringer ultra-curve pro 24 ad/da 24 dsp ultra-curve pro dsp8024 smd (surface mounted device) iso9000 ultra-curve pro 1.2 ultra-curve pro ultra-curve pro 19 2u 10 ultra-curve pro ultra-curve pro iec . 7 ultra-curve pro dsp8024 .
Business Ready Enhancement Plan for Microsoft Dynamics Customer FAQ Updated January 2011 The Business Ready Enhancement Plan for Microsoft Dynamics is a maintenance plan available to customers of Microsoft Dynamics AX, Microsoft C5, Microsoft Dynamics CRM, Microsoft Dynamics GP, Microsoft Dynamics NAV, Microsoft Dynamics SL, Microsoft Dynamics POS, and Microsoft Dynamics RMS, and
Microsoft Dynamics 365 for Operations on-premises, Microsoft Dynamics NAV, Microsoft Dynamics GP, Microsoft Dynamics SL, Microsoft Dynamics AX 2012 or prior versions, or Microsoft Dynamics CRM 2016 or prior versions. This guide is not intended to influence the choice of Microsoft Dynamics products and services or provide technical specification.
This guide is designed to improve your understanding of how to license Microsoft Dynamics 365, Business edition. This document does not apply to Dynamics 365, Enterprise edition, Microsoft Dynamics NAV, Microsoft Dynamics GP, Microsoft Dynamics SL, Microsoft Dynamics AX 2012, or Microsoft Dynamics CRM 2016 or any other prior version.
Microsoft Dynamics Guide Dynamics GP vs. Dynamics 365 (NAV) vs. Dynamics SL . Dynamics 365 BC -1 Premium User 100/month/user (Subscription) 2000 (On-Premise) . Solomon's application became Dynamics SL. Dynamics SL is geared first and foremost for project-based businesses. This makes SL the
Microsoft Dynamics 365 for Operations on-premises, Microsoft Dynamics NAV, Microsoft Dynamics GP, Microsoft Dynamics SL, Microsoft Dynamics AX 2012 or prior versions, or Microsoft Dynamics CRM 2016 or prior versions. This guide is not intended to influence the choice of Microsoft Dynamics products and services or provide technical specification.
Figure 5 (below) shows a B Trip Curve overlaid onto the chart. The three major components of the Trip Curve are: 1. Thermal Trip Curve. This is the trip curve for the bi-metallic strip, which is designed for slower overcurrents to allow for in rush/startup, as described above. 2. Magnetic Trip Curve. This is the trip curve for the coil or solenoid.
Koch middle-one fifth curve (Fig. 8a) and Koch middle one-sixth curve (Fig. 9a) has dimensions 1.113 and 1.086 respectively that are smaller than the conventional Koch curve. (a) Koch left one-third curve with Generator (b) Koch left one-third Snowflake Fig. 5: Koch Left One-Third Curve and Koch Snowflake for (r1, r2, r3) (1/3,1/3,1/3)
