Search

In this note, we introduce a simple approach for building volatility cubes of an interest-rate index based on the existing volatility cube of another index. Our approach can be formulated as a specific linear factor model, but it is dynamical in nature, and has the advantage of simple, explicit formulas for the ATM implied volatility and skew. As an example, we will construct SOFR volatility smiles from LIBOR-based ones.

In this paper, we define and model forward risk-free term rates, which appear in the payoff definition of derivatives, and possibly cash instruments, based on the new interest-rate benchmarks that will be replacing IBORs globally. We show that the classical interest rate modeling framework can be naturally extended to describe the evolution of both the forward-looking (IBOR-like) and backward-looking (setting-in-arrears) term rates using the same stochastic process. In particular, we show that the extension of the popular LIBOR Market Model (LMM) to the backward-looking rates completes the model by providing additional information about the rate dynamics not accessible in the LMM.

In this note we propose a simple two-factor multi-curve model where Fed-fund, SOFR and LIBOR rates are modeled jointly. The model is used to price the newly quoted SOFR futures as well as Eurodollar futures. We then derive pricing formulas for SOFR-based swaps, and show how the valuations of LIBOR-based swaps as well as LIBOR-SOFR basis swaps are impacted by the introduction of a new LIBOR fallback.

We derive an efficient closed-form approximation for the moment generating function of the integral of a mean-reverting stochastic process, following a linear SDE, which we call GARCH. We then consider a financial application, namely the pricing of a quanto CDS under stochastic intensity of default and an FX devaluation model. Numerical results are finally showcased.

Futures convexity adjustments in the multi-curve world depend on: i) the distribution of forward LIBORs, ii) the distribution of OIS rates, and iii) the correlation between LIBORs and OIS rates. In this article, we introduce a new multi-curve framework for pricing futures convexity adjustments. We assume that forward LIBORs follow a one-factor shifted-lognormal LMM, and that OIS rates evolve according to a general one-factor Cheyette (1992) model. We will consider two particular cases: 1) OIS rates follow a one-factor Hull-White (1990) model; 2) the LIBOR-OIS basis volatility is minimal, which includes the sub-case of deterministic basis when LIBOR and OIS curves are perfectly correlated. We will derive explicit convexity adjustments in both cases, and analyze the impact of the cap skew in their valuation. We will show that the extent a cap smile affects futures pricing depends on the chosen OIS dynamics, or equivalently on the assumed behavior of the LIBOR-OIS basis.

We develop an approximation technique for pricing finite-maturity timer options under Heston-like stochastic volatility models. By approximating the distributions of the accumulated variance and the random variance budget exceeding time, we obtain analytic expressions for timer option prices under zero correlation. For nonzero correlation, we use a simple linear combination approximation which matches the asymptotic correlation behavior. Numerical analysis using the Heston model shows that the method is fairly accurate, especially when the volatility of variance is small or the maximum maturity is large.

There are several pricing and risk model applications where the assumption of a deterministic LIBOR-OIS basis can lead to severe mispricing. By modeling such a basis using a jump-diffusion process, we show how stochastic basis can impact the valuation of specific deals such as zero-coupon swaps, as well as credit valuation adjustments and gap risk in particular. Explicit formulas are provided and numerical examples showcased.

Background: For the expression of late viral genes, HIV-1 efficiently exploits the nuclear export by using Rev viral protein, which specifically binds the RNA Rev Responsive Element (RRE). This region is contained within the gp120-gp41 encoding sequence. Enfuvirtide is the first approved HIV-1 fusion-inhibitor, and gp41 codons associated with primary enfuvirtide-resistance (amino-acids 36–45) are localized within the RRE structure. We previously found the co-presence of V38A+T18A resistance mutations in patients failing enfuvirtide. Methods: Collecting 476 and 135 HIV-1 B-subtype gp41 sequences from enfuvirtide-naïve and enfuvirtide-treated patients, respectively, two mutations previously found associated with enfuvirtide treatment, T18A and V38A, were analyzed. Moreover, the RNA secondary structure was displayed by CONTRAfold-software and the gp41 evolutionary pathways by a mutagenetic tree. Results: By modeling the RRE structure, we show that the T18 and V38 codons are base pairing within the RRE-stem-IIA, an important domain involved in Rev binding. While a structural RRE impairment in the presence of V38A alone was found, a restoration of the original RRE structure occurred in co-presence of V38A+T18A. By mutagenetic tree analysis, a compensatory evolution confirming our hypothesis on the structural modification mechanism was observed. Conclusion: We show that enfuvirtide pressure may also affect specific RRE domains involved in Rev binding, thus requiring a compensatory evolution able to preserve the secondary structure of the RRE.

We introduce an extended LIBOR market model that is compatible with the current market practice of building different yield curves for different tenors and for discounting. The new paradigm is based on modeling the joint evolution of FRA rates and forward rates belonging to the discount curve. We will start by analyzing the basic lognormal case, then we will add stochastic volatility. The dynamics of FRA rates under different measures will be obtained and closed form formulas for caplets and swaptions derived in the lognormal and Heston (1993) cases.

Finance as a whole is linked to correlations among the prices of assets (stocks, bonds, . . . ) or more generally of basic financial quantities (including interest rates, credit spreads, . . . ). In fact, the concept of diversification itself has to do with sensibly exploiting the correlations among such quantities. It is therefore fundamental to be able to safely build and manipulate correlation structures, with which to model the joint behaviour of financial variables, both for pricing purposes and for risk management. We hint at the fact that linear correlation is a satisfactory measure of dependence only for jointly Gaussian variables, finding its correct generalization in the notion of copula function. However, when considering instantaneous returns, the multi-dimensional Brownian motion dynamics typically adopted in finance is Gaussian and therefore correlation parameterization maintains its interest. In this paper, whenever correlation is associated to a dynamics, it is meant as instantaneous correlation. Correlations (often meant as “correlations between returns of financial variables”– be they stock prices, credit spreads, interest rates or else) are usually expressed in terms of matrices, i.e. tables of numbers. Matrices representing correlations between variables must satisfy four basic properties in order to really bear statistical and financial significance

We extend Piterbarg’s result (Piterbarg, Risk, 2:97–102, 2010) on European-style derivative pricing under collateralization by relaxing the assumption of a single unsecured funding rate. Introducing different lending and borrowing rates has the effect of producing nonlinear price functionals for general claims. Buyer and seller prices diverge, and values of derivative portfolios are not the sum of the individual deal values. Conditions under which no-arbitrage price bounds can be derived explicitly are given and numerical examples showcased.

We investigate credit value adjustments (CVAs) in the presence of wrong-way risk (WWR) by introducing jumps at default to model correlation between counterparty's default and relevant risk factors. We focus on the foreign-exchange and interest-rate cases, presenting efficient CVA approximations based on CVA computation under independence assumption. Numerical examples of the CVAs of a cross-currency swap and a vanilla interest-rate swap are showcased.

In this article, we start by briefly reviewing the approach proposed by Jarrow and Yildirim for modelling inflation and nominal rates in a consistent way. Their methodology is applied to the pricing of general inflation-indexed swaps and options. We then introduce two different market model approaches to price inflation swaps, caps and floors. Analytical formulae are explicitly derived. Finally, an example of calibration to swap market data is considered.

In this paper we consider several parametric assumptions for the instantaneous covariance structure of the LIBOR market model, whose role in the modern interest-rate derivatives theory is becoming more and more central. We examine the impact of each different parameterization on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance parameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating.

The aim of this paper is to present two different approximated dynamics for basket-options pricing. The approximations are based on a moment-matching procedure. Single names in the basket are modelled according to geometric Brownian motions (GBMs), as in the classical Black and Scholes framework, driven by instantaneously correlated Brownian motions. The basket distribution resulting from such GBMs for the single assets is unknown. By resorting to Monte Carlo simulation, we compare this distribution to the known approximated distribution coming from the moment-matching dynamics. The comparison is carried out through distances on the space of probability densities, namely the Kullback–Leibler information (KLI) and the Hellinger distance (HD). We are interested in measuring the KLI and the HD between the ‘real’ simulated basket distribution at terminal time and the distributions used for the approximation, both in the log-normal and shifted log-normal families. We isolate influences of instantaneou...

We review the general class of analytically tractable asset-price models that was introduced by Brigo and Mercurio (2001a Mathematical Finance—Bachelier Congr. 2000 (Springer Finance) ed H Geman, D B Madan, S R Pliska and A C F Vorst (Berlin: Springer) pp 151–74), where the considered asset can be an exchange rate, a stock index or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure and includes models featuring (i) explicit asset-price dynamics, (ii) a virtually unlimited number of parameters and (iii) analytical formulae for European options. We also review the fundamental case where the asset-price density is given, at every time, by a mixture of log-normal densities with equal means. We then introduce two other cases: the first is still based on log-normal densities, but it allows for different means in the distributions; the second is based on processes of hyperbolic-sine type. Finally, we test the goodness of calibration to real market data of the ...

We introduce a general class of analytically tractable diffusions for modelling forward LIBOR rates under their canonical measure. The class, which is based on assuming a smooth functional dependence at expiry between a forward rate and an associated Brownian motion, is highly tractable. It implies explicit dynamics, known marginal and transition densities and explicit caplet prices at any time. As an example, we analyse the dynamics given by a linear combination of geometric Brownian motions with perfectly correlated (decorrelated) returns. We finally construct a specific model in the class that reproduces exactly the market caplet volatilities given in input. Examples of the implied-volatility curves produced by the considered models are also shown.

This paper derives bid and ask prices of a general contingent claim in an incomplete market for an investor who can also trade in a given asset and in bonds. The investor's preferences are modelled by a quadratic derived utility function. The ask (resp. bid) price is determined in such a way that the investor is indifferent between the two following policies: (i) selling (resp. buying) the claim at this price and investing optimally in the asset and in bonds; (ii) investing optimally in the asset and in bonds, without trading the contingent claim. The ask (resp. bid) price thus derived is the expectation of the final payoff of the claim under the signed martingale measure that M. Schweizer [Mathematics of Operation Research 20 (1995) 1–32] called “minimal martingale measure”, plus (resp. minus) a quantity that is directly proportional to: (i) the number of claims being traded; (ii) a coefficient of risk-aversion; (iii) the smallest variance of the error of replication due to hedging one single claim. The results we derive apply to very general claims, hence to portfolios of claims.

In the present paper we show how to extend any time-homogeneous short-rate model to a model that can reproduce any observed yield curve, through a procedure that preserves the possible analytical tractability of the original model. In the case of the Vasicek (1977) model, our extension is equivalent to that of Hull and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985) (CIR) model, our extension is more analytically tractable and avoids problems concerning the use of numerical solutions. We also consider the extension of time-homogeneous models without analytical formulas. We then explain why the CIR model is the most interesting model to be extended through our procedure, analyzing it in detail. We also consider an example of calibration to the cap market for two of the presented models. We finally hint at the same extension for multifactor models and explain its advantages for applications.

Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Several researchers have modelled interest rate dynamics in such a way that humped volatility structures are possible and yet analytical formulas for European options on discount bonds are derived. However, these models are Gaussian, and hence interest rates may become negative. Here, a family of interest rate models is proposed where (i) humped volatility structures are possible; (ii) the interest rate volatility may depend on the level of the interest rates themselves; and (iii) the valuation of interest rate derivative securities can be accomplished through recombining lattices. The second item implies that a number of probability distributions are possible for the yield curve dynamics, and some of them ensure that interest rates remain positive. Proportional models of the Ritchken and Sankarasubramanian type and the Black and Karasinski model are proposed. To ensure co...

In the present paper we construct stock-price processes with the same marginal lognormal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can assume any value in-between the no-arbitrage lower and upper bounds. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants that are relevant for trading. The findings of the paper are inspired by a theoretical result, linking density-evolution of diffusion processes to exponential families. Such result is briefly reviewed in an appendix.

Some of the most recent empirical studies on interest rate derivatives have found humped shapes in the volatility structure of interest rates. However, this realistic feature has been hardly addressed in theoretical works. In this paper, we propose an interest rate model which allows for a humped volatility structure and which is analytically tractable, in that it leads to explicit formulae for European options on discount bonds. Empirical evidence supporting the model is also provided.

We develop an asymptotic expansion technique for pricing timer options under general stochastic volatility models around small volatility of variance. Closed-form approximation formulas have been obtained for the Heston model and the 3/2-model. The approximation has an easy-to-understand Black-Scholes-like form and many other attractive properties. Numerical analysis shows that the approximation formulas are very fast and accurate.

We show how to price interest rate swaps under the new market practice of using different curves for generating future Libor rates and for discounting cash flows. Straightforward modifications of the market formulas for caps and swaptions will also be derived.

We introduce trading restrictions in the well known Black-Scholes model and Cox-Ross-Rubinstein model, in the sense that hedging is only allowed at some fixed trading dates. As a consequence, the financial market is incomplete in both modified models. Applying Schweizer's (and Schal's) variance-optimal criterion for pricing and hedging general claims, we first analyse the dynamic consistency of the strategies which minimize the variance of the total loss due to hedging a given claim. Then we establish some convergence results, when the number of trading dates is either kept fixed or increases to infinity.

Some of the key areas of innovation in quantitative finance include new approaches to modeling stochastic processes, development of methods for valuing risk factors and expanding historic models to accommodate new and complex market dynamics. In this interview with Institutional Investor Journals, Fabio Mercurio of Bloomberg shares his observations on volatility, wrong-way risk (WWR) and interest rate models, including the LIBOR market model. He will be speaking at Global Derivatives in Budapest.

The majority of studies have characterized the tropism of HIV-1 subtype-B isolates, but little is known about the determinants of tropism in other subtypes. So, the goal of the present study was to genetically characterize the envelope of viral proteins in terms of co-receptor usage by analyzing 356 full-length env sequences derived from HIV-1 subtype-C infected individuals. The co-receptor usage of V3 sequences was inferred by using the Geno2Pheno and PSSM algorithms, and also analyzed to the "11/25 rule". All reported env sequences were also analyzed with regard to N-linked glycosylation sites, net charge and hydrophilicity, as well as the binomial correlation phi coefficient to assess covariation among gp120(V3) and gp41 signatures and the average linkage hierarchical agglomerative clustering were also performed. Among env sequences present in Los Alamos Database, 255 and 101 sequences predicted as CCR5 and CXCR4 were selected, respectively. The classical V3 signatures at positions 11 and 25, and other specific V3 and gp41 amino acid changes were found statistically associated with different co-receptor usage. Furthermore, several statistically significant associations between V3 and gp41 signatures were also observed. The dendrogram topology showed a cluster associated with CCR5-usage composed by five gp41 mutated positions, A22V, R133M, E136G, N140L, and N166Q that clustered with T2V(V3) and G24T(V3) (bootstrap=1). Conversely, a heterogeneous cluster with CXCR4-usage, involving S11GR(V3), 13-14insIG/LG(V3), P16RQ(V3), Q18KR(V3), F20ILV(V3), D25KRQ(V3), Q32KR(V3) along with A30T(gp41), S107N(gp41), D148E(gp41), A189S(gp41) was identified (bootstrap=0.86). Our results show that as observed for HIV-1 subtype-B, also in subtype-C specific and different gp41 and gp120V3 amino acid changes are associated individually or together with CXCR4 and/or CCR5 usage. These findings strengthen previous observations that determinants of tropism may also reside in the gp41 protein.

We try and apply the single-scenario version of the general model in Castagna, Mercurio and Mosconi (2010) to the pricing of CDOs. We are able to establish a unified approach to both evaluate the Credit VaR and the risk of structured products, and thus evaluate on a consistent and uniform basis the Economic Capital required to face unexpected credit losses, and the risk transferred out of the balance sheet via the securitisation activity. The approach avoids to resort to cumbersome numerical procedure, by retaining a closed-form feature that allows a quick and accurate pricing of CDO structures.

We derive a computable approximation for the value of a European call option when prices satisfy a jump-diffusion model with the coefficients depending explicitly on time. This is achieved by approximating the original coefficients with functions that are piecewise constant in time. We give an interpretation of the approximating option values, in particular in the context of a discrete-time model associated with the approximating continuous-time model.

The lognormal mixture diffusion model is a diffusion model in which the asset price has a marginal density given by a known mixture of lognormal densities. Such a model is analytically tractable and can be easily used to price exotic derivatives under smile effects. Keywords: local-volatility model; lognormal-mixture density; diffusion model; tractability

Caps and floors are call/put options on LIBOR rates. We describe the Black formulas used for their valuation in the market and provide a formal justification for such a use. We also show the corresponding market volatility smiles and describe an extension of Black's formula that accommodates them. Keywords: caps; floors; LIBOR rate; volatility; Black's formula

The combination of pegylated interferons (PEG-IFNs) and ribavirin represents the standard of care for the treatment of chronic HCV-infected patients, yet with a success rate around 50% in genotypes 1 and 4, high costs and side effects. Therefore, early prediction of sustained virological response (SVR) is a relevant issue for HCV-patients. We evaluated the association between SVR and decline of HCV-RNA at 48h in a prospective cohort of 145 HCV-patients treated with PEG-IFNs and ribavirin (males=69.1%; genotypes 1/4=51.0%; HIV-1 coinfected=6.7%). SVR was obtained in 65.5% of patients, while 16.6% experienced relapse and 17.9% no response. The first-phase of HCV-RNA decline clearly differentiated patients with SVR from relapsers and non-responders, independently of genotype (P0.001). In univariate and multivariate analyses, different infralogaritmic thresholds of HCV-RNA decay at 48h were tested, observing the highest predictive potential at 0.5log: decays above this threshold showed a 76.2% negative predictive value for SVR, whereas decays0.5log indicated a 6.8 odds ratio (95% C.I.: 2.0-23.2) for SVR after controlling for genotype, baseline viremia, adherence to therapy and HIV coinfection. Decays beyond the 0.5log threshold were also strongly associated with and highly predictive of early virological response (95.0% positive predictive value, P0.001).

We extend the LIBOR market model to accommodate the new market practice of using different forward and discount curves in the pricing of interest-rate derivatives. Our extension is based on modeling the joint evolution of forward rates belonging to the discount curve and corresponding spreads with FRA rates. We start by considering general stochastic-volatility dynamics and show how to address both the caplet and swaption pricing problems in general. We then consider specific examples, including a model for the simultaneous evolution of different rate and spread tenors. We conclude the article with an example of calibration to real market data. A reduced version of this article can be downloaded at: http://ssrn.com/abstract=1583081

We introduce a new forward CPI model that is based on a multi-factor volatility structure and leads to SABR-like dynamics for forward inflation rates. Our approach is the first in the financial literature to reconcile zero-coupon and year-on-year quotes, granting, at the same time, a both fast and accurate calibration to market data. Explicit formulas for year-on-year caps/floors as well as for zero-coupon options are then derived in terms of the SABR volatility form. An example of calibration to market data is finally provided.

This paper deals with the effects of concentration (single name and sectoral) and contagion risk on credit portfolios. Results are obtained for the value at risk of the portfolio loss distribution, in the analytical framework originally developed by Vasicek in 1991 [1]. V aR is expressed as a sum of terms: the first contribution represents the value at risk of a hypothetical single-factor homogeneous portfolio, the remaining terms are corrections due to contagion, imperfect granularity and multiple industry-geographic sectors. A detailed numerical analysis is also presented.

We extend the model presented in Bonollo et al. by introducing a multiscenario framework that allows for a richer and more realistic specification, including non-static (stochastic) probabilities of default and losses given default. Though more complex from a computational point of view, the model with scenarios is still tractable analytically, yielding results in closed form expressions. The approximated value at risk has been calculated by generalizing the procedure exposed in Bonollo et al. for the single scenario case, in the presence of granularity in the exposures, sector concentration and contagion. The outcome is not simply a weighted sum of the VaRs in the individual scenarios, but results in a more involved function of the single scenarios’ parameters. The theoretical model description is complemented with an in-depth numerical analysis.

The behaviour of a smile model when applied to hedging should be consistent with market evidence that asset prices and market smiles move in the same direction (Hagan et al. 2002). Local volatility models are criticized because not consistent with this desired behaviour, and this has been an important driver towards the use of stochastic volatility models. In this work we perform a simple analysis showing that, if we take into account explicitly the correlation between stochastic volatility and underlying asset which is typical of the most common stochastic volatility models, the hedging behaviour of stochastic volatility models does not always conform with the desired behaviour of a smile model in hedging. With further simple tests we show that the behaviour of local volatility and stochastic volatility models calibrated to market skew is less different than assumed in current market wisdom. Both approaches, when used consistently with model assumptions, do not show the desired behaviour in hedging, while for both models the desired behaviour is obtained in market practice by hedging techniques which are not fully consistent with rigorous model assumptions.

The SABR closed-form formula (Hagan et. al 2002) is the standard for smile-consistent pricing in the swaption market. Here we address the issue of turning SABR assumptions into a consistent and arbitrage-free term structure model in the BGM/Libor Market Model framework. We compute the joint dynamics followed by Libor rates and stochastic volatility of SABR kind under the general pricing measures used for interest rate derivatives, and we observe that the volatility dynamics is non-standard. Based on the analysis of the equation found, we develop and justify theoretically a few approximations aimed at making these no-arbitrage dynamics compatible with the use of the SABR closed-form formula. Then the formulas developed above are confronted both with alternative numerical implementations and with market data. We verify that the formulas for no-arbitrage corrections are acceptably precise, maintain good fitting, and produce regular Libor parameters. Finally we verify that the no-arbitrage corrections to the volatility dynamics make the out-of-calibration-sample prices implied by the model closer to market quotations, compared to prices implied by a trivial multivariate SABR neglecting such corrections.

We present a simple methodology to guarantee that the total correlation structure in a Term Structure Model with one stochastic volatility factor remains positive semidefinite. We design the parameterization with the purpose of keeping as much freedom as possible for the correlation of interest rates and stochastic volatility, while letting the correlation among forward rates reproduce approximately the tendencies usually considered desirable in the market.

In the current markets, options with different strikes or maturities are usually priced with different implied volatilities. This stylized fact, which is commonly referred to asfsmile effect, can be accommodated by resorting to specific models, either for pricing exotic derivatives or for inferring implied volatilities for non quoted strikes or maturities. The former task is typically achieved by introducing alternative dynamics for the underlying asset price, whereas the latter is often tackled by means of statical adjustments or interpolations. In this article, we deal with this latter issue and analyze a possible solution in a foreign exchange (FX) option market. In such a market, in fact, there are only three active quotes for each market maturity (the 0Delta straddle, the risk reversal and the vega-weighted butterfly), thus presenting us with the problem of a consistent determination of the other implied volatilities. FX brokers and market makers typically address this issue by using an empirical procedure to construct the whole smile for a given maturity. Volatility quotes are then provided in terms of the option's Delta, for ranges from the 5Delta put to the 5Delta call. In the following, we will review this market procedure for a given currency. In particular, we will derive closed-form formulas so as to render its construction more explicit. We will then test the robustness (in a static sense) of the resulting smile, in that changing consistently the three initial pairs of strike and volatility produces eventually the same implied volatility curve. We will also show that the same procedure applied to Europeanstyle claims is consistent with static-replication results and consider, as an example, the practical case of a quanto European option. We will finally prove that the market procedure can also be justified in dynamical terms, by defining a hedging strategy that is locally replicating and self-financing.

The price of a CMS based derivative is largely affected by the value of swaption volatilities at extreme strikes. In this article, we propose a very simple procedure for stripping consistently implied volatilities and CMS adjustments from the market quotes of swaption smiles and CMS swap spreads.

We test both the SABR model and the shifted-lognormal mixture model as far as the joint calibration to swaption smiles and CMS swap spreads is concerned. Such a joint calibration is essential to consistently recover implied volatilities for non-quoted strikes and CMS adjustments for any expiry-tenor pair.

In this article, we propose a simple interest rate model, which can well accom-modate swaption smiles, while recovering market prices of CMS swap spreads. Themodel is based on a (possibly multi-factor) Gaussian short rate model coupled withparameter uncertainty. Examples of calibration to real market data will be presentedas well as the pricing of some typical CMS-based derivatives. 1 Introduction Swap rate dependent derivatives have become increasingly popular in the interest ratemarket. Typical examples are the CMS spreads and CMS spread options, whose payofisare based on the difierence between long and short maturities swap rates. Such payofisare usually embedded in swaps, which may be cancelled due to Bermudan-style callabilityfeatures or to TARN-style conditions.To be correctly priced, such derivatives need a model that incorporates at once themarket swaption volatilities of all the relevant swap rates (both at-the-money and away-from-the-money) consistently with a realistic form for their correlation term structure.Swaption volatilities are usually quoted in the market by means of the SABR functionalform, see Hagan et al. (2002), implicitly assuming a difierent (and independent) stochasticvolatility model for each difierent swap rate. Motivating this practice on the basis of aproper market model is still an issue to be addressed. On the other hand, one can simplyregard the swaption smile as a market input to which to calibrate one’s favorite interestrate model. In this case, an efiective and rather straightforward solution is obtained bycoupling a short-rate model with parameter uncertainty in the spirit of Brigo, Mercurio1